Robust Modified Policy Iteration Robust dynamic programming (robust DP) mitigates the effects of ambiguity in transition probabilities on the solutions of Markov decision problems. We consider the computation of robust DP solutions for discrete-stage, infinite-horizon, discounted problems with finite state and action spaces. We present robust modified policy iteration (RMPI) and demonstrate its convergence. RMPI encompasses both of the previously known algorithms, robust value iteration and robust policy iteration. In addition to proposing exact RMPI, in which the "inner problem" is solved precisely, we propose inexact RMPI, in which the inner problem is solved to within a specified tolerance. We also introduce new stopping criteria based on the span seminorm. Finally, we demonstrate through some numerical studies that RMPI can significantly reduce computation time. Markov decision models typically assume that state transition probabilities are known with certainty. In practice, however, these probabilities are estimated from data. It may be the case that data are scarce for some state-action pairs; that is, the true transition probability measures may not be known. Unfortunately, dynamic programming (DP) solutions may be sensitive to errors in the estimation of these probabilities. Robust dynamic programming mitigates the effects of ambiguity in transition probabilities on resulting decisions. In this paper we consider the computation of robust DP solutions in the context of discrete-stage, The study of Markov decision processes (MDPs) with uncertain transition probabilities goes back at least to the work of Silver (1963) , which was expanded upon by Martin (1967) .. These works use a Bayesian framework for updating information about unknown transition probabilities. At the start of the decision process, some prior distribution is assumed. A chosen conjugate prior may actually assume very little initial information about the transition probabilities. The decision-maker's problem is to maximize reward while at the same time update relevant information. In the decision process then, there is a tradeoff between "exploitation" of taking an action that presumably has the higher payoff and "exploration" to get more information about the transition probabilities. The framework we are interested in is different. We are not concerned with exploration in the decision process, and the decision process does not incorporate Bayesian updating. The max-min setting that we consider only uses statistical information available a priori.. MDPs with ambiguous transition probabilities in a max-min framework were previously studied by Satia and Lave (1973) , White and Eldeib (1994) , and Bagnell et al. (2001) . Bagnell et al. (2001) present a robust value iteration algorithm for solving robust DP problems when uncertainty sets are convex and satisfy some conditions. Robust value iteration is similar to standard value iteration, but it includes both a maximization and a minimization in each step. The minimization, which is due to nature's objective, is referred to as the "inner problem.". More recently, the theory of robust DP was advanced in two concurrent papers by Iyengar (2005) and Nilim and El Ghaoui (2005) . In addition to establishing theoretical grounds for robust value iteration, these papers propose methods for efficient computation when uncertainty sets are based on relative entropy bounds or on likelihood regions. These advances open the door for practical applications of robust DP, which include path planning for robots moving around dynamic objects (Bagnell et al. 2001 ) and for aircraft moving around storms (Nilim and El Ghaoui 2005) . Our work is an extension of the work of Iyengar (2005) and Nilim and El Ghaoui (2005) since robust value iteration is a special case of our RMPI algorithm. Other related works include the papers of Harmanec (2002) , Mannor et al. (2007) , Li and Si (2008) , and Delage and Mannor (2010) .. Both Iyengar (2005) and Nilim and El Ghaoui (2005) provide conditions for and proof of convergence of robust value iteration. As for linear programming, there is no efficient robust counterpart; the analogous problem is a non-convex optimization problem (Iyengar 2005, p. 269) . A robust policy iteration algorithm is presented by Satia and Lave (1973) .. As Iyengar (2005, p. 269 ) points out, however, "it is not clear, and [Satia and Lave (1973)] do not show, that this iterative procedure converges." The issue, it seems, is not whether their procedure converges, but rather it is whether it converges to the right value function.. Proposition 4 of Satia and Lave (1973) claims that their policy evaluation procedure results in an -optimal policy for nature, but it has not been proven. Iyengar proposes an alternative robust policy iteration algorithm. For policy evaluation, he presents a robust optimization problem (Iyengar 2005, Lemma 3.2) . According to Iyengar (2005, p. 268) though, "for most practical applications, the policy evaluation step is computationally expensive and is usually replaced by an m-step look-ahead value iteration." He does not provide such an algorithm nor prove convergence. Prior to RMPI, the most viable option for solving robust DPs has been robust value iteration. Therefore, our main interest is in comparing RMPI to robust value iteration. White and Eldeib (1994) present the only other known modified policy iteration algorithm for a problem with ambiguity. They combine modified policy iteration with a reward revision algorithm, and present an algorithm in Proposition 7 of White and Eldeib (1994) . However, they are not able to provide conditions that guarantee convergence of this algorithm. (Note that Assumption (i) of White and Eldeib (1994, Proposition 7) needs to hold at every step, but there is no guarantee that it will.) According to White and Eldeib (1994) , "Unfortunately, the proof of Lemma 5a in [White et al. (1985) ] does not generalize when the [P] are sets under the max-min strategy.". We provide conditions that guarantee convergence of RMPI. Our results do not depend on a specific form for P. Depending on how the uncertainty sets are constructed, it might not be efficient to solve the inner problem precisely. Similar to analysis in Nilim and El Ghaoui (2005) , we distinguish between two cases: exact RMPI in which the inner problem is solved precisely, and inexact RMPI in which the inner problem is solved only to within a specified tolerance. Our main contributions are as follows:. • We are the first to present a robust modified policy iteration algorithm together with conditions for and a proof of convergence.. • For inexact RMPI, we provide conditions on the tolerance that guarantee convergence to a decision rule that is -optimal : its total reward is within a pre-specified value of the optimal reward. Moreover, convergence is guaranteed within a finite number of iterations.. • We provide new stopping criteria based on the span seminorm that guarantee that, after a finite number of iterations, the decision rule is -optimal. In addition to applying to RMPI, these conditions apply to robust value iteration and are in many cases an improvement over previously known stopping criteria based on the supremum norm.. • We consider two numerical studies that demonstrate that, as compared to robust value iteration, the RMPI algorithm can substantially reduce the time it takes to compute solutions.. The rest of this paper is outlined as follows: In Section 2 we discuss preliminaries and introduce some notation. In Section 3 we present the RMPI algorithm along with proof of its convergence. We analyze the exact case first, in Section 3.1. We then analyze the inexact case, in Section 3.2. We present new stopping criteria based on the span seminorm in Section 4. Numerical results are presented in Section 5. We conclude in Section 6. The solutions to dynamic programs are often sensitive to errors in the transition matrices.. To alleviate this, we have introduced exact and inexact robust modified policy iteration and demonstrated the convergence of both. Additionally, we have presented new stopping criteria based on the span seminorm that will allow these algorithms to stop sooner than under the previously known stopping criteria.. Except for perhaps the inexact RMPI results, our results might seem natural as they are analogous to results in the standard DP setting. While standard DP has additional algorithms available to it, robust DP has only RMPI, robust value iteration, and robust policy iteration. Recall that linear programming can be used to solve standard DPs, but the analogous robust problem is a non-convex optimization problem Iyengar (2005, p. 269) .. Even policy iteration does not extend most naturally to the robust DP setting since policies cannot be evaluated by solving a system of linear equations. Prior to RMPI, robust value iteration was perhaps the natural choice for solving infinite horizon robust DPs.. For the standard DP setting, Puterman (1994, p. 193) states that, "in practice, value iteration should never be used. Implementation of modified policy iteration requires little additional programming effort yet attains superior convergence." We echo this statement for the robust DP setting. We conjecture that, for many applications, RMPI will be significantly more efficient than robust value iteration. We have demonstrated the benefits of RMPI for some inventory control problems. These benefits are reaped with very little additional implementation effort.. The added complexity of robust DP over standard DP is due to the solutions of the "inner problem." The complexities of the algorithms are heavily dependent on the choice of uncertainty sets, and our results are independent of the choice of construction for P(s, a). We refer to the discussions of Iyengar (2005) and Nilim and El Ghaoui (2005) on the complexity of robust value iteration under various choices. Outside of the fact that robust value iteration is a special case of RMPI, we cannot guarantee that RMPI will necessarily be more efficient.. What we have proved, in relationship (8), is that the iterates of RMPI, {v n } where v n+1 = Ψ mn v n , converge at least as fast as the iterates of robust value iteration, {u n } where u n+1 = Υu n . The operator Ψ mn includes an additional m n − 1 successive approximations of the value function. That is, Ψ mn is more computationally expensive than Υ, and in particular it solves an additional (m n − 1)|S| inner problems. Hence, there is a tradeoff between faster convergence and additional complexity for each iteration.. The intuition as to why RMPI can be more efficient than robust value iteration and robust policy iteration carries over from the standard DP setting. RMPI might outperform robust value iteration by avoiding the need to solve inner problems for all feasible actions each time it updates the value function. RMPI might outperform robust policy iteration by performing only a partial policy evaluation prior to a policy update. Because the solution of the inner problem might be substantially more expensive to compute than a simple expectation operator, the benefits of RMPI in the robust DP setting might be significantly more pronounced than the benefits of modified policy iteration in the standard DP setting.. For standard DP, Puterman and Shin (1978) prove that modified policy iteration converges at least linearly, and they provide a bound on its rate of convergence. Moreover, they. show that, "Whenever the transition matrices of successive decision rules approach that of an optimal decision rule, the convergence rate of the algorithm is bounded by that of [m n ] steps of value iteration" (Puterman 1994, p. 193) . This is the "appeal" of modified policy iteration. The validity of similar claims for RMPI remain an open question. The analysis of Puterman and Shin (1978) makes use of the fact that the transition probabilities are fixed for each decision rule. The robust DP setting is more complicated because nature's choices for p ∈ P(s, a) may change each time the value function changes, irrespective of whether the decision rule remains fixed.. We have not analyzed the optimal choice for {m n }. For standard DP, the choice of {m n } is not straightforward. There are rules of thumb, such as the algorithm may perform better if m n gets larger as n gets larger, but there is no clear best choice.. Finally, we point out that for standard DP, there exist extensions of modified policy iteration to algorithms that are asynchronous. Examples include the algorithms of Williams and Baird III (1993) and Singh and Gullapalli (1993) . Similar extensions might apply to RMPI in the robust DP setting. This remains a topic for future research.

O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Finite element computation of absorbing boundary conditions for time-harmonic wave problems This paper proposes a new method, in the frequency domain, to define absorbing boundary conditions for general two-dimensional problems. The main feature of the method is that it can obtain boundary conditions from the discretized equations without much knowledge of the analytical behavior of the solutions and is thus very general. It is based on the computation of waves in periodic structures and needs the dynamic stiffness matrix of only one period in the medium which can be obtained by standard finite element software. Boundary conditions at various orders of accuracy can be obtained in a simple way. This is then applied to study some examples for which analytical or numerical results are available. Good agreements between the present results and analytical solutions allow to check the efficiency and the accuracy of the proposed method. Wave problems in unbounded media can occur in many applications in mechanics and engineering such as in acoustics, solid mechanics, electromagnetics, etc. It is well known that analytical solutions for such problems are available only for some special cases. On the contrary, numerical methods can be applied to many complex problems. Physically, for problems in infinite domains, the energy is produced by sources in the region to be analyzed and must escape to infinity. For methods solving the problem on a bounded domain like the finite element method, it introduces the difficulty of an artificial boundary to get a bounded domain. This boundary must be such that the energy crosses it without reflection and special conditions must be specified at the artificial boundary to reproduce this phenomena. Generally, these can be classified into local or global boundary conditions. With a global condition all degrees of freedom (dofs) on the boundary are coupled while a local condition connects only neighboring dofs.. The first global method which has been used for solving such problems was the boundary element method. This method is well adapted for infinite domains and is described in numerous classical textbooks like [1, 2, 3, 4, 5] . It consists in solving an equation on the boundary of the domain only and the radiation conditions are taken into account analytically. It also reduces the dimension of the problem to a surface in 3D and to a curve in 2D decreasing thus the size of the linear problem to solve. However, the final problem involves full matrices which are also generally non symmetrical. It is also mainly limited to linear problems and to homogeneous domains or otherwise one has to introduce special and complex techniques to deal with non linear or non homogeneous situations. There are also singularities in the integrals which need special attention for the numerical integrations. So this method is interesting and has been extensively used but it can lead to heavy computations when the number of degrees of freedom increases. More information on such techniques can be found in the historical and review papers [6, 7] .. In the other approaches, the computational domain is truncated at some distance and boundary conditions are imposed at this artificial boundary. These conditions at finite distance must simulate as closely as possible the exact radiation condition at infinity. An approach leading to a global boundary condition is the Dirichlet to Neumann (DtN) mapping proposed by [8, 9] and in an earlier version by [10, 11] . It consists in dividing the domain into a finite part containing the sources and an infinite domain of simple shape. The solution in the infinite domain is solved analytically, for example by series expansions, and an exact impedance relation is obtained on the boundary between the finite and infinite domains. This relation links the variable and its normal derivative on the whole boundary. The DtN mapping is thus non local and every node on the boundary is connected to all other nodes. This gives a full matrix for the nodes of the boundary which partially destroys the sparse matrix of the FEM and increases substantially the computing resources needed to get the solution. The solution has to be found in the exterior domain by analytical or numerical methods. When the analytical solution can be found, it is generally under the form of a series expansion. The number of terms in the expansion must be sufficient for an accurate solution which can lead to heavy computations. Developments of the method can be found in [12, 13] . An application to the case of wave scattering in plates is also found in [14] .. The other methods are local and the condition at a node involves only neighboring nodes which make them less demanding in computing resources and much easier to implement in a finite element code but also less accurate. A first possibility of such approaches is the use of infinite elements as proposed by [15, 16, 17, 18, 19] . It consists in developing special elements with a behavior at infinity reflecting that of analytical solutions obtained for the same type of problems. For wave problems, it involves complex-valued basis functions with outwarding propagation wave-like behavior in the radial direction. The elements were further developed by [20] to considered other coordinate systems such as expansions in prolate coordinates. This method is interesting but the inclusion of infinite elements requires the development of special elements and these elements can depend on decay parameters which have to be accurately chosen. A review of these methods has been proposed by [21] .. In the perfectly matched layer proposed by [22, 23] , originally for electromagnetic waves, an exterior layer of finite thickness is introduced around the bounded domain. The absorption in this domain is increasing as we move towards the exterior such that outgoing waves are absorbed before reaching the exterior truncation boundary. The number of elements in the layer, its thickness, the variation of the absorption properties have to be carefully chosen to optimize the efficiency of the method. This efficiency is better for a layer with a large thickness but this can lead to a significant increase in the number of elements in the finite element model. Various developments of the method can be found in [24] and its optimization in [25] . Otherwise, various classes of absorbing boundary conditions were also developed by [26] . They consist in the numerical approximation of differential operators on the boundary. For instance, examples of the application of Bayliss-Turkel conditions are presented by [27] . However, the more accurate boundary conditions involve high order derivatives on the boundary which are difficult to implement in the finite element method [28] . Finally, it was proved by [25, 29] that, in fact, the perfectly matched layer and the absorbing boundary conditions were closely connected. The Helmholtz equation was also solved with these two boundary conditions by [30] and these conditions were compared and optimized to minimize the reflection. Other boundary conditions involving only second order derivatives have also been proposed. They introduce auxiliary variables and systems of equations on the boundary which lead to high order boundary conditions, see [31] for a review of such non-reflecting boundary (NRBC) methods. They were mainly developed for acoustic problems but in [32] a local boundary condition for elastic waves has been proposed. In [33] an impedance boundary condition in new coordinates was developed for the convected Helmholtz equation. For fluid dynamic problems, [34] developed Lagrange multipliers for imposing various absorbing boundary conditions for cases where the type and the number of boundary conditions can change, for instance as the flow changes from subsonic to supersonic regimes and its direction varies with time. A general review of the methods described in the precedent paragraphs for various dynamic, acoustic and wave propagation problems can be found in [35, 36, 37, 38] . Comparisons are also made between the different methods.. In the present study, another local method is proposed. This method works on discrete systems directly, in contrast with many existing absorbing boundary conditions which are written on the continuous differential equations and discretized after. The principle of the method is to compute wave propagations in groups of elements near the boundary from the dynamic stiffness matrix of these elements. Then, a boundary condition is obtained for cancelling the reflected waves. This condition is finally written as an impedance boundary condition relating the force and displacement degrees of freedom on the boundary. The approach is based on the waveguide theory proposed by [39, 40, 41] and is used to determine absorbing boundary conditions at the truncation boundary of 2D periodic media. Only information related to one period, obtained from any standard FE software (the discrete stiffness and mass matrices and the nodal coordinates) are required to formulate the method. The advantage is that it can be applied to media with various complex behaviors. This paper is outlined as follows. In section 2, the methodology for determining absorbing boundary conditions for periodic media is described. Then, a discussion for the application of the method to general media is proposed. In section 3, a simple application is proposed to show the results of the method in a case where detailed computations can be done. In section 4, two examples of finite element computations in acoustics and elastodynamics are presented. They allow to check the efficiency and accuracy of the proposed method for more complex cases. Finally, the paper is closed with some conclusions. In this paper, a method to determine absorbing boundary conditions for two dimensional periodic media has been presented. It works directly on the discretized equations. The boundary condition is first obtained as a global impedance relation and is then localized into boundary conditions of various orders. In two examples, good agreements were observed when compared with analytical solutions.. In any case, the proposed method is efficient because it requires only the discrete dynamic matrices which can be obtained by any standard finite element software. This method could be used for media with more complex behaviors than those presented in the precedent examples.

B I O O B I I O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O


Learning similarity matching in multimedia content-based retrieval AbstractÐMany multimedia content-based retrieval systems allow query formulation with user setting of relative importance of features (e.g., color, texture, shape, etc) to mimic the user's perception of similarity. However, the systems do not modify their similarity matching functions, which are defined during the system development. In this paper, we present a neural network-based learning algorithm for adapting similarity matching function toward the user's query preference based on his/her relevance feedback. The relevance feedback is given as ranking errors (misranks) between the retrieved and desired lists of multimedia objects. The algorithm is demonstrated for facial image retrieval using the NIST Mugshot Identification Database with encouraging results. SIMILARITY matching is crucial in our daily life. We rely on it to categorize the physical and information worlds and to understand new concepts and situations by comparison with existing ones. For multimedia databases wherein exact matching is not appropriate, similarity matching plays a very important role. Since humans originate the queries and have expectations of the retrieval results, it is necessary that the similarity measure implemented in a multimedia content-based retrieval system approximates the users' perception of similarity matching closely [6] . Many multimedia content-based retrieval systems support query by example, with specification of relative weights of features (e.g., color, texture, shape, etc.) to mimic a user's perception of similarity (e.g., [3] , [7] , [8] ). However, these systems do not modify their similarity matching functions, which are determined during system development [5] .. In this paper, we present a neural network-based learning algorithm for adapting similarity matching function toward the user's query preference based on his/her relevance feedback. The relevance feedback is given as ranking errors (misranks) between the retrieved and desired lists of multimedia objects. More specifically, a user presents a sample query object to the system. The system responds with an ordered sequence of retrieved objects. The user reorders the sequence according to his/her perception of similarity. The misrank information derived from the difference between the retrieved sequence and the desired sequence is fed back as an error signal to the system for learning.. There are two possible types of learning problems. The first type involves short-term learning using only a single misrank sequence. The aim here is to learn a user's perception of similarity from this sequence. The intention is that the system will be able to produce a newly retrieved sequence that is closer to the user's perception of similarity for query refinement. The probable consequence is that objects considered similar to the query by the user but not retrieved previously are now retrieved or ordered more to the front in the retrieved sequence. Objects considered dissimilar to the query by the user but retrieved previously are now not retrieved or ordered more to the rear of the retrieved sequence.. The second type of learning (dubbed long-term learning) is more ambitious. Its objective is to capture a user's perceptual consistency in similarity matching. Learning requires the user's misrank feedback on a number of retrieved sequences corresponding to the sample query objects. The intention is that the system will be able to learn the regularities in a user's perception of similarity for the multimedia objects in the target application domain. After learning, a new query sample will produce a retrieved sequence with less misrank compared to the one without learning.. In any case, learning is user-specific and a user is only required to provide desired ranking on a retrieved list of objects (not the whole database). The similarity matching function is either tuned toward a user's perceived similarity in a user query session or over a number of query sessions. As the learning is based on ranking errors, it is most effective when the domain objects can be ranked easily by the user. The learning algorithm is demonstrated for facial image retrieval using the NIST Mugshot Identification Database with encouraging results. Although we are using error-correcting supervised learning for both types of learning mentioned above, it is neither similar to conventional supervised learning for pattern classification [2] nor adaptive Information Retrieval (IR) [1] .. To train a pattern classifier, x training samples of the form i Y d i , where i is a feature vector and d i is the desired class label for i , are presented to the classifier. For each i , the classifier will output a target class label t i . The difference between d i and t i is treated as an error i i d i À t i which can be used to modify the parameters in the classifier toward a less erroneous classification performance.. In our proposed approach, the training patterns and objectives are quite different. First, the input vectors for learning are not feature vectors, but similarities (or distances) between the feature vectors of two multimedia objects. Second, the output vectors for learning are not desired and target class labels, but misrank information. Last, but not least, the learning process modifies the representation of the similar matching function, which is fundamental in comparing two multimedia objects for retrieval purpose, instead of some discrimination function used in defining the class or decision boundaries.. While learning in multimedia content-based retrieval is new [5] , adaptive IR has been an active research area [1] . In fact, Crestani and van Rijsbergen [1] have given a comprehensive and up-to-date survey of past works in adaptive IR in their paper. We shall not repeat the review here, but would like to point out some important observations. First, in IR, the feature vectors are usually binary vector representation of query and document descriptors (i.e., index terms). But, in multimedia content-based retrieval, the feature vectors are mostly real vectors. They are much richer to capture color, shape, texture, and even spatial information. In fact, due to their complex nature, they are best described in a multilevel structure. Second, learning in IR typically uses relevance feedback directly in the form of class labels (i.e., relevant or irrelevant classes) as in the case of supervised learning for pattern classification. In this paper, errors are derived from ranking information provided by a user. Last, but not least, adaptive IR, as typified by the approach described in [1] , attempts to acquire application domain knowledge by learning the association between the query descriptors and the document descriptors into a query adaptation or expansion function. The latter can then be used to modify a query for matching (by a ªMatcherº [1] ) with documents in the database during retrieval. However, in our approach, it is the similarity matching function (i.e., theªMatcherº) that gets adapted using misrank feedback. In short, the uniqueness in our approach surrounds a common notion of relativity: Since the similarity matching function is the function that determines the relative closeness between any two multimedia objects, it is the focus of adaptation. By the same token, the input and supervised information for learning are also computed and presented in relative sense, namely, similarities between pairwise feature vector elements of query and retrieved objects and ranking error of retrieved sequence with respect to the desired sequence, respectively.

B B I O O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B O O O B I O O O O O O O O O B I O B I O O O B O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B O O O B I O O O O O O O O O B I O B I O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O B I B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O B O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O B O O O O O B I O O O O O O O O O O O O O O O B O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B I O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O


Rainbow induced subgraphs in proper vertex colorings Abstract. For a given graph H we define ρ(H) to be the minimum order of a graph G such that every proper vertex coloring of G contains a rainbow induced subgraph isomorphic to H. We give upper and lower bounds for ρ(H), compute the exact value for some classes of graphs, and consider an interesting combinatorial problem connected with computation of ρ(H) for paths. This research is motivated by some ideas in on-line graph coloring algorithms.Rainbow induced subgraphs have been considered in many papers in connection with various problems of extremal graph theory. They have been considered both for edge-colorings and vertex-colorings, and both in terms of existence or in terms of avoiding (see [1, 2, 3, 4, 9, 10, 11] ). Our special motivation comes from research in on-line coloring (see [5, 8] ), where the base for some algorithms is the existence of rainbow anticliques to force a player to use a new color. In particular, in [8], a problem has been formulated to estimate the minimal number of moves in the game considered one needs to force the appearance of a rainbow copy of a fixed graph H in a fixed class of graphs C.In this paper we deal with proper vertex colorings of graphs, colorings, in short, and rainbow induced subgraphs, by which we mean induced subgraphs whose all vertices have different colors. We consider a problem of constructing a minimal graph G such that for a given graph H every coloring of G contains a rainbow induced subgraph isomorphic to H. We shall write simply G r → H to denote such a situation. We define ρ(H) to be the least number m such that there exist a graph G with G r → H. A similar definitions are introduced in [3] in connection with anti-Ramsey numbers. There is no requirement for colorings to be proper. Instead, there is a restriction on the number of colors used to be not smaller than the cardinality |H|. As a consequence it may happen there are only finitely many graphs that in every such coloring contain rainbow colored copies of H, and the authors concentrate on such situations. Proper vertex colorings and a problem connected with the existence of rainbow induced stars and paths have been considered in [10] . Interesting results on proper coloring for edges are contained in [9] and [11] .

B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O


Low-Power Memory Mapping Through Reducing Address Bus Activity Abstract-Arrays in behavioral specifications that are too large to fit into on-chip registers are usually mapped to off-chip memories during behavioral synthesis. We address the problem of system power reduction through transition count minimization on the memory address bus when these arrays are accessed from memory. We exploit regularity and spatial locality in the memory accesses and determine the mapping of behavioral array references to physical memory locations to minimize address bus transitions. We describe array mapping strategies for two important memory configurations: all behavioral arrays mapped to a single off-chip memory and arrays mapped into multiple memory modules drawn from a library. For the single memory configuration, we describe a heuristic for selecting a memory mapping scheme to achieve low power for each behavioral array. For mapping into a library of multiple memory modules, we formulate the problem as three logical-to-physical memory mapping subtasks and present experiments demonstrating the transition count reductions based on our approach. Our experiments on several image processing benchmarks show power savings of up to 63% through reduced transition activity on the memory address bus in the single memory case. We also observe a further transition count reduction by a factor of 1.5-6.7 over a straightforward mapping scheme in the multiple memories configuration.

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Fairness in Multiuser Systems with Polymatroid Capacity Region For a wide class of multi-user systems, a subset of capacity region which includes the corner points and the sum-capacity facet has a special structure known as polymatroid. Multiaccess channels with fixed input distributions and multiple-antenna broadcast channels are examples of such systems. Any interior point of the sum-capacity facet can be achieved by time-sharing among corner points or by an alternative method known as rate-splitting. The main purpose of this paper is to find a point on the sum-capacity facet which satisfies a notion of fairness among active users. This problem is addressed in two cases: (i) where the complexity of achieving interior points is not feasible, and (ii) where the complexity of achieving interior points is feasible. For the first case, the corner point for which the minimum rate of the active users is maximized (max-min corner point) is desired for signaling. A simple greedy algorithm is introduced to find the optimum max-min corner point. For the second case, the polymatroid properties are exploited to locate a rate-vector on the sum-capacity facet which is optimally fair in the sense that the minimum rate among all users is maximized (max-min rate). In the case that the rate of some users can not increase further (attain the max-min value), the algorithm recursively maximizes the minimum rate among the rest of the users. It is shown that the problems of deriving the time-sharing coefficients or rate-spitting scheme can be solved by decomposing the problem to some lower-dimensional subproblems. In addition, a fast algorithm to compute the timesharing coefficients to attain a general point on the sum-capacity facet is proposed. In the multi-user scenarios, multiple transmitters/receivers share a common communication medium, and therefore, there is an inherent competition in accessing the channel. Information theoretic results for such systems imply that in order to achieve a high spectral efficiency, the users with stronger channel should have a higher portion of the resources. The drawback to this is the loss of the fairness among the users. Providing fairness, while achieving high-spectral efficiency, is thus a challenging problem.. A lot of research has addressed this problem and suggested different criteria to design a fair system. One of the first criteria is known as max-min measure. In this method, the main effort is to maximize the minimum rate of the users, by giving the highest priority to the user with the worst channel. In other words, this method penalizes the users with better channel and sacrifices overall efficiency.. By relaxing the strict condition on fairness, the spectral efficiency can be increased. By compromising between fairness and throughput, proportional fairness is proposed in [1] . Based on this criterion, the rates of users with a stronger channel can be increased with the cost of decreasing the rates of users with a weaker channel. Any change in the rates is acceptable if the total proportional increase in the rates of some users is larger than the total proportional decrease in the rates of the rest. In fact, by relaxing the strict condition on fairness, the spectral efficiency increases. In [2] , a criterion based on Nash Bargaining solution in the context of Game Theory is proposed. This method generalizes the proportional fairness and increases the efficiency of the system.. All of the aforementioned methods deal with a general multi-user system. However, for a wide class of multi-user systems, the capacity region has a special structure that we can exploit to provide fairness. Particularly in some multiuser systems, the boundary of the capacity region includes a facet on which the sum-rate is maximum (sum-capacity facet). In such systems, one can benefit from the available degrees of freedom, and determine the fairest rate-vector on the sum-capacity facet.. As a special case, we consider a class of multi-user systems, in which the whole or a subset of the capacity region which includes the corner points and the sum-capacity facet forms a structure known as polymatroid. For this class of multi-user systems, the sum-capacity facet has a! corner points, where a is the number of users with non-zero power (active users). The sum-capacity facet is the convex hull of these corner points. This means that the interior points of the sumcapacity facet can be attained by time-sharing among such corner points. As an example of such systems, it is shown that the capacity region of multiaccess channels (MAC) with fixed and independent input distributions forms a polymatroid [3] . In MAC, the sum-capacity is achieved by successive decoding. Applying different orders for the users in successive decoding results in different rate-vectors, all with the sum-rate equal to the sum-capacity. The resulting rate-vectors correspond to the corner points of the sum-capacity facet. Any point in the convex hull of these corner points is on the boundary. In [4] , it is proven that the Marton inner bound (see [5] ) for capacity region of the broadcast channel under fixed joint probability of the auxiliary and input variables, with some conditions, has a polymatroid structure 1 . As another example, we will show that a subset of the capacity region for multiple-input multiple-output (MIMO) broadcast channel which includes the corner points forms a polymatroid.. In [3] , the optimal dynamic power allocation strategy for time-varying single-antenna multipleaccess channel is established. To this end, the polymatroid properties of the capacity region for time-invariant multiple-access channel with fixed input distributions have been exploited. In [6] , the polymatroid properties have been used to find a fair power allocation strategy. This problem is formulated by representing a point on the face of the contra-polymatroid (see [3] , [7] ) as a convex combination of its extreme points.. This article aims at finding a point on the sum-capacity facet which satisfies a notion of fairness among active users by exploiting the properties of polymatroids. In order to provide fairness, the minimum rate among all users is maximized (max-min rate). In the case that the rate of some users can not increase further (attain the max-min value), the algorithm recursively maximizes the minimum rate among the rest of the users. Since this rate-vector is in the face of the polymatroid, it can be achieved by time sharing among the corner points. It is shown that the problem of deriving the time-sharing coefficients to attain this point can be decomposed to some lower-dimensional subproblems. An alternative approach to attain an interior point for multiple access channels is rate splitting [8] , [9] . This method is based on splitting all input sources except one into two parts and treating each spilt input as two virtual inputs (or two virtual users). By splitting the sources appropriately and successive decoding of virtual users in a suitable order, any point on the sum-capacity facet can be attained [8] , [9] . Similar to the time-sharing procedure, we show that the problem of rate-splitting can be decomposed to some lower dimensional subproblems.. There are cases that the complexity of achieving interior points is not feasible. This motivates us to compute the corner point for which the minimum rate of the active users is maximized (max-min corner point). A simple greedy algorithm is introduced to find the max-min corner point.. The rest of the paper is organized as follows. In Section II, the structure of the polymatroid is presented. In addition, the relationship between the capacity region of some channels and the polymatroid structure is described. Section III discusses the case in which the optimal fair corner point is computed. In Section IV, the optimal fair rate-vector on the sum-capacity facet is computed by exploiting polymatroid structures. In addition, it is shown that the problem of deriving the time-sharing coefficients and rate-splitting can be solved by decomposing the problem into some lower-dimensional subproblems.. Notation: All boldface letters indicate vectors (lower case) or matrices (upper case). det(H) and H † denote the determinant and the transpose conjugate of the matrix H, respectively. M 0 represents that the matrix M is positive semi-definite. 1 n represents an n dimensional vector with all entries equal to one. E is a set of integers E = {1, · · · , |E|}, where |E| denotes the cardinality of the set E. The set function f : 2 E −→ R + is a mapping from all subsets of E (there are a total of 2 |E| subsets) to the positive real numbers. A permutation of the set E is denoted by π and π(i), 1 ≤ i ≤ |E|, represents the element of the set E located in the i th position after the permutation. For an a-dimensional vector x = {x 1 , x 2 , . . . , x a } ∈ R a and S ⊂ E, x(S). denotes i∈S x i . Also, for a set of positive semi-definite matrices D i , D(S) represents i∈S D i . We considered the problem of fairness for a class of systems for which a subset of the capacity region forms a polymatroid structure. The main purpose is to find a point on the sum-capacity facet which satisfies a notion of fairness among active users. This problem is addressed in cases where the complexity of achieving interior points is not feasible, and where the complexity of achieving interior points is feasible. For the first case, the corner point for which the minimum rate of the active users is maximized (max-min corner point) is desired for signaling. A simple greedy algorithm is introduced to find the optimum max-min corner point. For the second case, the polymatroid properties are exploited to locate a rate-vector on the sum-capacity facet which is optimally fair in the sense that the minimum rate among all users is maximized (max-min rate).. In the case that the rate of some users can not increase further (attain the max-min value), the algorithm recursively maximizes the minimum rate among the rest of the users. It is shown that the problems of deriving the time-sharing coefficients and rate-spitting scheme can be solved by decomposing the problem to some lower-dimensional subproblems. In addition, a fast algorithm to compute the time-sharing coefficients to attain a general point on the sum-capacity facet is proposed.

B O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O


Abstract interpretation of resolution-based semantics We extend the abstract interpretation point of view on context-free grammars by Cousot and Cousot to resolution-based logic programs and proof systems. Starting from a transition-based small-step operational semantics of Prolog programs (akin to the Warren Machine), we consider maximal finite derivations for the transition system from most general goals. This semantics is abstracted by instantiation to terms and furthermore to ground terms, following the so-called c-and s-semantics approach. Orthogonally, these sets of derivations can be abstracted to SLD-trees, call patterns and models, as well as interpreters providing effective implementations (such as Prolog). These semantics can be presented in bottom-up fixpoint form. This abstract interpretation-based construction leads to classical bottom-up semantics (such as the s-semantics of computed answers, the c-semantics of correct answers of Keith Clark, and the minimal-model semantics of logical consequences of Maarten van Emden and Robert Kowalski). The approach is general and can be applied to infinite and top-down semantics in a straightforward way. The semantics of logic programs is characterised by a variety of forms and methods, ranging from the more traditional operational and denotational semantics towards the more logic-based ones related with the view of the interpreter as a theorem prover. Examples of this variety include the semantics of predicate logic as a programming language in [1, 2] , the operational and denotational semantics of Prolog [3] [4] [5] [6] [7] , the logical models of Prolog control features [8] [9] [10] , the (fixpoint) observational models in [11] [12] [13] [14] [15] [16] [17] [18] [19] 47] , and the so-called or-compositional models in [20] [21] [22] [23] [24] [25] , the last culminating in the so-called s-semantics approach to logic program semantics [26] , which is comprehensive of different observational semantics.. The essence in the study of comparative semantics in logic programming is the attempt to capture the difference between the various aspects of logic as a programming language, ranging from its view in theorem proving to its use in concrete programming. Theorem proving corresponds to restricting programs as theories of definite Horn clauses. In this context both a model-theoretic semantics (unique Herbrand representative model) and a proof procedure (e.g. SLD resolution) are given [1, 27] . Programming instead considers resolution strategies and backtracking control mechanisms such as cut as essential parts of the art of logic programming [28, 29] . For these latter aspects, the model-theoretic semantics is not adequate. This wide range of possible interpretation for a logic program has led researchers to develop a number of different semantics capturing specific aspects of logic as a programming language, from operational (resolution-based) semantics to denotational, model-theoretic, etc. Several attempts have been made in order to construct a comprehensive hierarchy of semantics [18] , some of them using abstract interpretation for specifying semantics at different levels of abstraction [30] [31] [32] [33] .. In this paper we develop a hierarchy of semantics for resolution-based languages by incremental abstractions of a maximal trace semantics. The trace semantics is constructed by generalising transitional semantics of context-free grammars akin push-down automata to resolution-based derivations of Horn-like clauses. The result is a hierarchy of topdown and bottom-up semantics of logic programs including as abstract interpretations most of the well-known semantics: the partial correctness semantics, the success semantics, the ground (Herbrand) models, the SLD-semantics, the breadthfirst semantics and the cut semantics. All semantics are derived as abstract interpretations, where consecutive abstractions specify a Prolog interpreter modelling the different observable properties of the program. We showed how abstract interpretation of the maximal trace semantics of a simple grammar-based language, akin the semantics of context-free grammars and push-down automata [35] , can provide a comprehensive view of most well-known semantics of resolution-based languages such as logic programming and Prolog. Other semantics can be derived similarly, for instance for modelling infinite computations by combining inductive and co-inductive semantics [50] and for modelling different forms of negation [27] . The result is a uniform specification framework for interpreters of logic programs which can be systematically designed by consecutive abstractions of a basic abstract machine. The analogy with the semantics of grammars is, in this context, striking. We believe that both the semantics of grammars and that of resolution-based languages can be specified in a uniform way as instances of a unique transition system semantics involving more expressive grammars inspired by Prolog, rewriting, etc. Having formalized logic program semantics by abstract interpretation, may provide the way to integrate its correctness proofs with that of its decidable abstractions, such as those for static analysis. Because abstraction can be constructed by calculational design [51] , as shown in the formal proofs, a proof assistant or theorem prover can be used to automatically check or perform these calculations. This leads to formally verified implementations and static analyzers.

B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


The State of the Art of Application Restrictions and Sandboxes: A Survey of Application-oriented Access Controls and their Shortfalls Abstract: Under most widely-used security mechanisms the programs users run possess more authority than is strictly necessary, with each process typically capable of utilising all of the user's privileges. Consequently such security mechanisms often fail to protect against contemporary threats, such as previously unknown ('zero-day') malware and software vulnerabilities, as processes can misuse a user's privileges to behave maliciously. Application restrictions and sandboxes can mitigate threats that traditional approaches to access control fail to prevent by limiting the authority granted to each process. This developing field has become an active area of research, and a variety of solutions have been proposed. However, despite the seriousness of the problem and the security advantages these schemes provide, practical obstacles have restricted their adoption. This paper describes the motivation for application restrictions and sandboxes, presenting an indepth review of the literature covering existing systems. This is the most comprehensive review of the field to date. The paper outlines the broad categories of existing application-oriented access control schemes, such as isolation and rule-based schemes, and discusses their limitations. Adoption of these schemes has arguably been impeded by workflow, policy complexity, and usability issues. The paper concludes with a discussion on areas for future work, and points a way forward within this developing field of research with recommendations for usability and abstraction to be considered to a further extent when designing application-oriented access controls. Traditional approaches to access control have typically been user-oriented, that is, they make access decisions based primarily on the identity of users. These methods generally fail to protect users from previously unknown software vulnerabilities and malware, because when executed this malicious code may utilise all of the user's privileges. Application restrictions and sandboxing can mitigate this threat by limiting the authority granted to processes based on the privileges or resource name-space they require in order to carry out their legitimate functions. The field of application-oriented access control, schemes which primarily base access decisions on the programs involved rather than on the identity of users, has therefore become an active area of research, and a number of approaches have been proposed. However, despite the security advantages they provide, adoption of these schemes has been limited. This paper aims to review the various approaches taken and identify their limitations.. This paper is structured as follows. We begin with a discussion of the limitations of the traditional security approaches, introducing the concept of user-oriented access controls and the inability of this method to sufficiently mitigate the threat posed by malware and vulnerabilities. Limitations of trust-based selective execution (such as traditional anti-malware software) and integrity-level schemes are also discussed. This is followed by a comprehensive overview and comparison of application-oriented access control schemes. Isolation-based schemes, including traditional sandboxes, virtual machines and containers, are described and practical drawbacks to these approaches are explored. Rule-based schemes M A N U S C R I P T This paper has presented an overview of the motivation for application-oriented access controls, and described in depth the application-oriented access controls that have been proposed in the literature. Unlike user-oriented access controls, application-oriented access controls specify policy primarily in terms of programs or groups of programs, rather than users or groups of users. By identifying the privileges required by each application and restricting their actions to those privileges, the ability of applications to act beyond their legitimate purpose is limited, and attempts to act maliciously are restricted. Application-oriented schemes include models that isolate programs to only access the set of resources available from within the isolated environment, and rule-based schemes that provide controls over shared resources while allowing applications to access the same resources in a restricted manner.. Both of these main approaches pose a number of challenges. Isolation does not suit typical user workflows, can result in redundancy of resources, and requires frequent circumvention of the security model to be practical. Rule-based schemes typically result in complex policies that are hard to specify, review, and manage.. There are many opportunities for future research, and there is a need for usable application-oriented security solutions. Although application-oriented access controls can provide substantial security benefits by restricting the activities of individual applications, to date adoption remains relatively rare and targeted, arguably due to the inadequate usability of these schemes.

O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O B I I O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Predicting linear B-cell epitopes using amino acid anchoring pair composition Background: Accurate identification of linear B-cell epitopes plays an important role in peptide vaccine designs, immunodiagnosis, and antibody productions. Although several prediction methods have been reported, unsatisfied accuracy has limited the broad usages in linear B-cell epitope prediction. Therefore, developing a reliable model with significant improvement on prediction accuracy is highly desirable. Results: In this study, we developed a novel model for prediction of linear B-cell epitopes, APCpred, which was derived from the combination of amino acid anchoring pair composition (APC) and Support Vector Machine (SVM) methods. Systematic comparisons with the existing prediction models demonstrated that APCpred method significantly improved the prediction accuracy both in fivefold cross-validation of training datasets and in independent blind datasets. In the fivefold cross-validation test with Chen872 dataset at window size of 20, APCpred achieved AUC of 0.809 and accuracy of 72.94%, which was much more accurate than the existing models, e.g., Bayesb, Chen's AAP methods and the enhanced combination method of AAP with five AP scales. For the fivefold cross-validation test with ABC16 dataset, APCpred achieved an improved AUC of 0.794 and A CC of 73.00% at window size of 16, and attained an AUC of 0.748 and A CC of 67.96% on Blind387 dataset after being trained with ABC16 dataset. Trained with Lbtope_Confirm dataset, APCpred achieved an increased Acc of 55.09% on FBC934 dataset. Within sequence window sizes from 12 to 20, APCpred final model on homology-reduced dataset achieved an optimal AUC of 0.748 and A CC of 68.43% in fivefold cross-validation at the window size of 20. Conclusion: APCpred model demonstrated a significant improvement in predicting linear B-cell epitopes using the features of amino acid anchoring pair composition (APC). Based on our study, a webserver has been developed for on-line prediction of linear B-cell epitopes, which is a free access at: http:/ccb.bmi.ac.cn/APCpred/.Background B-cell epitope is a part of an antigen recognized region that is bound to immunoglobulin molecules to stimulate B-cell response [1] . Based on structural characteristics, Bcell epitopes can be categorized into two types: linear (continuous) epitopes and conformational (discontinuous) epitopes. Linear epitopes are made up of short contiguous amino acids, whereas, conformational epitopes are composed of amino acids that are not contiguous in primary sequence but are brought together by the folded BioData Mining B-cell linear epitopes are short sequences on the antigenic proteins, which contain structure characters to exposure themselves to antibodies, and easily bind to the antibodies, even if they are disengaged from the source proteins. In order to have antigenic functions, epitope sequences must be different from the random sequences generated from Swissprot database. We believe that B-cell linear epitopes sequences must fold into a stable structure to show the sequences' information for being bound to antibodies. We propose that the amino acid anchoring pairs play important roles in stabilizing folding of epitopes structure by producing the force for folding in three-dimensional spaces. Thus, in this paper, we studied the roles of amino acid pairs in prediction of B-cell linear epitopes. Since it has been reported that 86.7% epitopes' length was at most 20 amino acids in Bcipep database [12] , during dealing with the large variability in the length of the epitopes, we chose to fix length of epitopes with lengths ranging from 12 to 20 peptides in the method of El-Manzalawy [12] and Saha [10] , instead of windows of five or seven amino acids at the center of a linear epitope as Parker [5] and Karplus [6] did. The existing B-cell linearepitope finding methods are far less than optimal or may only find part of epitope sequences, which may indicate that the prediction methods based on composition of The four classifiers were trained using ABC16 dataset and evaluated using the third dataset of Blind287. "*" denotes the information was obtained on online prediction of ABCpred with the third dataset though an automatic program script. The bold denotes the largest A CC value of the prediction. Both models were trained on Lbtope_Confirm dataset. The bold denotes the largest Acc value of the prediction.

O B I I O B I I I I O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O B I I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O B I I I I O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Predictive RANS simulations via Bayesian Model-Scenario Averaging is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. The turbulence closure model is the dominant source of error in most Reynolds Averaged Navier-Stokes simulations, yet no reliable estimators for this error component currently exist. Here we develop a stochastic, a posteriori error estimate, calibrated to specific classes of flow. It is based on variability in model closure coefficients across multiple flow scenarios, for multiple closure models. The variability is estimated using Bayesian calibration against experimental data for each scenario, and Bayesian Model-Scenario Averaging (BMSA) is used to collate the resulting posteriors, to obtain an stochastic estimate of a Quantity of Interest (QoI) in an unmeasured (prediction) scenario. The scenario probabilities in BMSA are chosen using a sensor which automatically weights those scenarios in the calibration set which are similar to the prediction scenario. The methodology is applied to the class of turbulent boundary-layers subject to various pressure gradients. For all considered prediction scenarios the standard-deviation of the stochastic estimate is consistent with the measurement ground truth. Furthermore, the mean of the estimate is more consistently accurate than the individual model predictions. Numerical predictions of continuum mechanics are affected by several sources of error: discretization error, parametric uncertainty, and physical modelling error, are the three most significant. Yet only the latter can not be estimated and controlled with standard techniques. As such it represents the bottleneck to a robust predictive capability in many fields.. In industrial Computational Fluid Dynamics (CFD) simulations, the Reynolds-Averaged Navier-Stokes (RANS) equations are the workhorse tool for modelling turbulent flow fields. The equations require a turbulence closure model, of which there are many [36] ; each of which requires empirical closure coefficients parameterising modelling assumptions. Standard practice is to calibrate these coefficients against simple flow configurations (generally, incompressible homogeneous isotropic turbulence, free shear layers, and zero pressure gradient flat plate boundary layers [25] ) using available experimental data. Even so there is no consensus on the best coefficient values for most models, as evidenced by the wide range of coefficients seen in the literature. As a consequence model predictions can differ significantly [34, 38] .. In the Bayesian framework, model error is typically accounted for with a stochastic term added to (or multiplying) the output of the simulation. For instance Kennedy and O'Hagan [13] define "model inadequacy" as the discrepancy that remains between the true process and the prediction of the simulation code evaluated using the bestfit parameter values. We believe however that modelling this discrepancy directly is inappropriate when the simulation is an approximation of continuum mechanics, which is characterized by perfectly known conservation laws (e.g. conservation of momentum), containing imperfect, subordinate, empirical models, such as turbulence closure models in RANS. In this context, it makes much more sense to include stochastic modelling of inadequacy at the level of the empirical model, not at the level of the simulation code output. This choice has the ancillary benefit that predictions of quantities of other type than the calibration data can be made -and in other geometries -while still incorporating estimates of model inadequacy. This is not possible in the original Kennedy and O'Hagan framework.. There are two natural ways to incorporate stochastic model inadequacy into turbulence closure models: via the Reynolds-stress tensor [9] , and via the closure coefficients [8] . It is the latter approach that we will use; and our goal is to quantify model error in the predictions of RANS codes.. Our novel approach is based on Bayesian statistics [10] , and can be summarized as follows: we first choose a class of flows for which we wish to make reliable predictions with quantified model error, in this work the class is flat-plate boundary-layers at a variety of pressure gradients. We select a number of examples of this class for which we have reliable experimental data; these flows form our calibration scenarios. Finally we select a set of turbulence closure models, that are both expected to perform acceptably on the flow-class, but which are also heterogeneous in their modelling approaches, in this work we use k − ε, k − ω, Spalart-Allmaras, Baldwin-Lomax and stress-ω models [36] . For each model and each scenario we perform Bayesian calibration to obtain posteriors on closure coefficients [3] as well as on model probabilities, i.e. probabilities of recovering the data by using a given model with the calibrated coefficients. For all models a strong dependence of coefficients on scenario is observed. This variation is interpreted as the extent to which the coefficients must be adjusted to match scenarios in the calibration set; informally a kind of coefficient uncertainty. Precisely, to make predictions for some Quantity of Interest (QoI) in some unmeasured scenario, we use Bayesian Model Averaging (BMA) [12] to collate the individual coefficient posteriors. A key step is propagating these posteriors through the simulation code for the prediction scenario. By using the freedom inherent in BMA to define prior scenario probabilities, we automatically assign a higher weight to those calibration scenarios which allow recovering consistent predictions for a new scenario with all alternative models considered; these are often found to be similar to the prediction scenario. The prediction is summarized by the mean and 2. variance of the posterior predictive distribution [6, 27] . Because in our framework the variation between scenarios is at least as important as the variation between models, we speak of Bayesian Model-Scenario Averaging (BMSA), rather than BMA.. There is very little work in the literature on numerical methods for estimating RANS closure model error. The authors of [22] perform calibration with multiple turbulence models and model-inadequacy terms on the simulation output, but they do not combine them to make reliable predictions. Other authors have applied Bayesian approaches to problems in aeroelasticity, but rely on expert judgement [26] . Recently Emory et al. [9] proposed a physically-motivated method for perturbing the Reynolds stress tensor, but again the uncertainty is determined by physically possible states, and not informed by experimental data. Finally in [8] the present authors used Bayesian statistics to estimate the model error of the k − ε turbulence model [15] , via multiple posterior distributions of the closure coefficients. However our use of p-boxes limited the flexibility of the predictions, leading to model error estimates that contained the true solution, but which were impractically large.. This work proposes one of the first error estimates for turbulence closure models. However the computational cost of the method investigated here is extremely high, if it would be applied to complex flow topologies. Our primary goal in this article is to investigate the potential of our framework for estimating model error, in the absence of simplifying approximations. Success at this stage motivates investigation of both numerical and statistical techniques for making the framework computationally feasible for expensive simulation codes. As a proof of concept, one such a technique is already proposed in the current article.. The paper is laid out as follows: the four turbulence models are described in Section 2, and the scenarios together with experimental data in Section 3. Section 4.1 lays out our calibration framework, Section 4.2 prediction with Bayesian Model-Scenario Averaging, and Section 4.3 automatic scenario weighting. Section 5 discusses results. We performed Bayesian calibrations of the closure coefficients of 5 turbulence models: the Launder-Sharma k − ε, Wilcox k − ω, Spalart-Allmaras and Baldwin-Lomax model and Wilcox stress-ω. Each model was calibrated for 14 scenarios consisting of flatplate boundary-layers at varying pressure-gradients, using experimental velocity measurements. Substantial variation in closure coefficients for all models was observed across this large range of simple flow cases.. In order to synthesize these results, and make predictions for unmeasured flows, we utilized Bayesian Model Averaging, with an emphasis on scenarios, i.e. Bayesian Scenario Averaging. The framework requires the computation of the posterior model probabilities, which can be thought of as a measure of consistency that a model has with the experimental data. We found that, like the closure coefficients, the posterior model probabilities vary greatly with the applied pressure-gradient scenario. These results suggest that there is no single best choice of turbulence model or closure coefficients, and no obvious way to choose an appropriate model a priori.. Instead we used BMSA to make stochastic predictions of unmeasured velocity profiles. Closing BMSA with uniform scenario weighting produced predictions which matched reference data, but with excessively large variance. We therefore developed a smart scenario sensor, to automatically preferentially weight those scenarios in the calibration set that are similar to the prediction case. This resulted in substantially improved predictors, both in terms of mean and variance. For almost all of 14 prediction cases, the predictions lie within one standard deviation of the experimental validation data, and the variance was of the same order as the experimental measurement error.. Despite the fact that only experimental u + data was used to inform our posterior distributions, we also used their topology independent nature to make predictions for skin-friction profiles. Although these posterior distributions were not optimal for this particular quantity of interest, we were still able to make consistently more robust predictions than can be expected from a single turbulence model. 36 Finally, an approach was suggested to significantly reduce the computational effort required by the predictive phase of our methodology. By averaging the posterior distributions over different scenarios, the number of pdf propagations drops from I × K to just I. Here, I is the number of turbulence models and K the number of scenarios. Even though expert opinion is now needed to specify the scenario weighting, initial results look promising.. Future work will involve the application of our method to more complex flow topologies. We see the methods developed here as a step towards providing estimates of the error due to turbulence closure models, and thereby lending a robust predictive power to RANS.

O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Green's conjecture and testing linear-invariant properties Given a set of linear equations M x = b, we say that a set of integers S is (M, b)-free if it contains no solution to this system of equations. Motivated by questions related to testing linear-invariant properties of boolean functions, as well as recent investigations in additive number theory, the following conjecture was raised (implicitly) by Green [27] and by Bhattacharyya, Chen, Sudan and Xie [14]: we say that a set of integers S ⊆ [n], is -far from being (M, b)-free if one needs to remove at least n elements from S in order to make it (M, b)-free. The conjecture of [14, 27] was that for any system of homogenous linear equations M x = 0 and for any > 0 there is a constant time algorithm that can distinguish with high probability between sets of integers that are (M, 0)-free from sets that are -far from being (M, 0)-free. Or in other words, that for any M there is an efficient testing algorithm for the property of being (M, 0)-free. In this paper we confirm the above conjecture by showing that such a testing algorithm exists even for non-homogenous linear equations. As opposed to most results on testing boolean functions, which rely on algebraic and analytic arguments, our proof relies on results from extremal hypergraph theory, such as the recent removal lemmas of Gowers [25] , Rödl et al. [38, 39] and Austin and Tao [10] . The basic question we address here is deciding if a given set of integers S contains a solution to a fixed set of linear equations. More formally, fixing a set of linear equations in p unknowns M x = b, we say that a set of integers S is (M, b)-free if S contains no solution to the set of linear equations M x = b, that is, if there is no vector v ∈ S p for which M v = b. Perhaps the simplest special case of this general family of problems is the so called 3SUM problem, in which we are asked to decide if S contains three numbers a, b, c satisfying a + b = c. As it turns out, even this special case is of particular interest (e.g. in computational geometry, see [23] ) and finding an algorithm with running time O(|S| 2−c ), for some c > 0, is a major open problem (see [11] for a weaker result in this direction). Erickson [19] has shown that in certain models of computation, such an algorithm does not exist, and that more generally, the problem of deciding if a set of integers contains a solution to a single linear equation in p unknowns requires time Ω(|S| p/2 ). Motivated by several recent works in property testing of boolean functions as well as in additive number theory, we consider the following relaxed version of this problem. For a set of integers S ⊆ [n] = {1, . . . , n}, we say that S is -far from being (M, b)-free if one needs to remove at least n elements from S in order to make it (M, b)-free. The relaxed version of the problem we consider here is that of distinguishing, with high probability, between the case that the input set S is (M, b)-free from the case that it is -far from satisfying this property. Such an algorithm will be called an -tester for the property of being (M, b)-free. So when = 0, we have that -testing the property of being (M, b)-free is precisely the decision problem we have discussed in the previous paragraph. Our main result in this paper is that whenever > 0, one can design surprisingly fast algorithms for the relaxed version of the problem. In fact, we show that for any set of linear equations M x = b and for any > 0, there is an -tester for the property of being (M, b)-free whose running time depends only on , and is independent of the size of S. This result confirms conjectures that were raised by Green [27] and Bhattacharyya, Chen, Sudan and Xie [14] . We next discuss the computational as well as the combinatorial background and significance of our results.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O