on the computational complexity and generalization properties of multi stage and stage wise coupled scenario programs abstract we discuss the computational complexity and feasibility properties of scenario sampling techniques for uncertain optimization programs we propose an alternative way of dealing with a special class of stage wise coupled programs and compare it with existing methods in the literature in terms of feasibility and computational complexity we identify trade offs between different methods depending on the problem structure and the desired probability of constraint satisfaction to illustrate our results an example from the area of approximate dynamic programming is considered

high order numerical methods to a type of delta function integrals we study second to fourth order numerical methods to a type of delta function integrals in one to three dimensions these delta function integrals arise from recent efficient level set methods for computing the multivalued solutions of nonlinear pdes we show that the natural quadrature approach with usual discrete delta functions and support size formulas to the two dimensional delta function integrals suffer from nonconvergence we then design high order numerical methods to this type of delta function integrals based on interpolation approach numerical examples are presented to verify the efficiency and accuracy of our methods

polynomial interpolation results in sobolev spaces abstract in this paper we derive new results related to the lagrange polynomial interpolation on the nodes of gauss type quadrature formulas in particular we obtain asymptotic error estimates of the same order as the distance to the best fit in the polynomial space

a note on the design of hp adaptive finite element methods for elliptic partial differential equations we introduce an hp adaptive finite element algorithm based on a combination of reliable and efficient residual error indicators and a new hp extension control technique which assesses the local regularity of the underlying analytical solution on the basis of its local legendre series expansion numerical experiments confirm the robustness and reliability of the proposed algorithm

sequential randomized algorithms for robust convex optimization sequential randomized algorithms are considered for robust convex optimization which minimizes a linear objective function subject to a parameter dependent convex constraint employing convex optimization and random sampling of parameter these algorithms enable us to obtain a suboptimal solution within reasonable computational time the suboptimal solution is feasible in a probabilistic sense and the suboptimal value belongs to an interval which contains the optimal value the maximum of the interval is the optimal value of the robust convex optimization plus a specified tolerance on the other hand its minimum is the optimal value of the chance constrained optimization which is a probabilistic relaxation of the robust convex optimization with high probability

computing multivalued physical observables for the semiclassical limit of the schrodinger equation we develop a level set method for the computation of multivalued physical observables density velocity etc for the semiclassical limit of the schrodinger equation this method uses an eulerian formulation and applies directly to arbitrary number of space dimensions the main idea is to evolve the density near an n dimensional manifold that is identified as the common zeros of n level set functions in phase space these level set functions are generated from solving the liouville equation with initial data chosen to embed the phase gradient simultaneously we track a new quantity f by solving again the liouville equation near the obtained zero level set but with initial density as initial data the multivalued density and higher moments are thus resolved by integrating f along the n dimensional manifold in the phase directions we show that this is equivalent to using the wigner approach but decomposing the velocity from the density each of which evolves by the same liouville equation the main advantages of this approach in contrast to the standard kinetic equation approach using the liouville equation with a dirac measure initial data include 1 the liouville equations are solved with l initial data and a singular integral involving the email protected function is evaluated only in the post processing step thus avoiding oscillations and excessive numerical smearing 2 a local level set method can be utilized to significantly reduce the computation in the phase space these advantages allow us to compute for the first time all physical observables for multidimensional problems in an eulerian framework

discretization of dirac delta functions in level set methods discretization of singular functions is an important component in many problems to which level set methods have been applied we present two methods for constructing consistent approximations to dirac delta measures concentrated on piecewise smooth curves or surfaces both methods are designed to be convenient for level set simulations and are introduced to replace the commonly used but inconsistent regularization technique that is solely based on a regularization parameter proportional to the mesh size the first algorithm is based on a tensor product of regularized one dimensional delta functions it is independent of the irregularity relative to the grid in the second method the regularization is constructed from a one dimensional regularization that is extended to multi dimensions with a variable support depending on the orientation of the singularity relative to the computational grid convergence analysis and numerical results are given

identifying effective scenarios in distributionally robust stochastic programs with total variation distance traditional stochastic programs assume that the probability distribution of uncertainty is known however in practice the probability distribution oftentimes is not known or cannot be accurately approximated one way to address such distributional ambiguity is to work with distributionally robust convex stochastic programs drsps which minimize the worst case expected cost with respect to a set of probability distributions in this paper we analyze the case where there is a finite number of possible scenarios and study the question of how to identify the critical scenarios resulting from solving a drsp we illustrate that not all but only some scenarios might have effect on the optimal value and we formally define this notion for our general class of problems in particular we examine problems where the distributional ambiguity is modeled by the so called total variation distance we propose easy to check conditions to identify effective and ineffective scenarios for that class of problems computational results show that identifying effective scenarios provides useful insight on the underlying uncertainties of the problem

the scenario approach for stochastic model predictive control with bounds on closed loop constraint violations many practical applications in control require that constraints on the inputs and states of the system are respected while some performance criterion is optimized in the presence of model uncertainties or disturbances it is often sufficient to satisfy the state constraints for at least a prescribed share of the time such as in building climate control or load mitigation for wind turbines for such systems this paper presents a new method of scenario based model predictive control scmpc the basic idea is to optimize the control inputs over a finite horizon subject to robust constraint satisfaction under a finite number of random scenarios of the uncertainty and or disturbances previous scmpc approaches have suffered from a substantial gap between the rate of constraint violations specified in the optimal control problem and that actually observed in closed loop operation of the controlled system this paper identifies the two theoretical explanations for this gap first accounting for the special structure of the optimal control problem leads to a substantial reduction of the problem dimension second the probabilistic constraints have to be interpreted as average in time rather than pointwise in time based on these insights a novel scmpc method can be devised for general linear systems with additive and multiplicative disturbances for which the number of scenarios is significantly reduced the presented method retains the essential advantages of the general scmpc approach namely a low computational complexity and the ability to handle arbitrary probability distributions moreover the computational complexity can be adjusted by a sample and remove strategy

mixed finite element methods of higher order for model contact problems this paper presents mixed finite element methods of higher order for a simplified signorini problem and an idealized frictional problem the discretization is based on a mixed variational formulation proposed by haslinger et al which is extended to higher order finite elements to guarantee the unique existence of the solution of the mixed method a discrete inf sup condition is proven approximation results of the p method of finite elements and some inverse estimates for higher order polynomials are applied numerical results confirm the theoretical findings

a probabilistic framework for problems with real structured uncertainty in systems and control the objective of this paper is twofold first the problem of generation of real random matrix samples with uniform distribution in structured spectral norm bounded sets is studied this includes an analysis of the distribution of the singular values of uniformly distributed real matrices and an efficient i e polynomial time algorithm for their generation second it is shown how the developed techniques may be used to solve in a probabilistic setting several hard problems involving systems subject to real structured uncertainty

an error estimate for the signorini problem with coulomb friction approximated by finite elements the present paper is concerned with the unilateral contact model and the coulomb friction law in linear elastostatics we consider a mixed formulation in which the unknowns are the displacement field and the normal and tangential constraints on the contact area the chosen finite element method involves continuous elements of degree one and continuous piecewise affine multipliers on the contact zone a convenient discrete contact and friction condition is introduced in order to perform a convergence study we finally obtain a first a priori error estimate under the assumptions ensuring the uniqueness of the solution to the continuous problem

equilibration techniques for solving contact problems with coulomb friction abstract in this paper we consider residual and equilibrated error indicators for contact problems with coulomb friction the contact problem is handled within the abstract framework of saddle point problems more precisely the non penetration constraint and the friction law is realized as a variationally consistent weak formulation in terms of a localized dual lagrange multiplier space thus from the displacement we can easily compute in a local post process the lagrange multiplier which acts as a neumann condition on the possible contact zone having computed the discrete lagrange multiplier we can apply standard error estimators by replacing the unknown neumann data by its approximation as it is shown in 1 this results in an error estimator for a one sided contact problem without friction here we consider more general situations and discuss two additional contact terms which measure the non conformity of the discrete lagrange multiplier numerical results in two and three dimensions illustrate the flexibility of the approach and show the influence of the material parameters on the adaptive refinement process

computational high frequency waves through curved interfaces via the liouville equation and geometric theory of diffraction we construct a class of numerical schemes for the liouville equation of geometric optics coupled with the geometric theory of diffractions to simulate the high frequency linear waves with a discontinuous index of refraction in this work s jin x wen a hamiltonian preserving scheme for the liouville equation of geometric optics with partial transmissions and reflections siam j numer anal 44 2006 1801 1828 a hamiltonian preserving scheme for the liouville equation was constructed to capture partial transmissions and reflections at the interfaces this scheme is extended by incorporating diffraction terms derived from geometric theory of diffraction into the numerical flux in order to capture diffraction at the interface we give such a scheme for curved interfaces this scheme is proved to be positive under a suitable time step constraint numerical experiments show that it can capture diffraction phenomena without fully resolving the wave length of the original wave equation

the exact feasibility of randomized solutions of uncertain convex programs many optimization problems are naturally delivered in an uncertain framework and one would like to exercise prudence against the uncertainty elements present in the problem in previous contributions it has been shown that solutions to uncertain convex programs that bear a high probability to satisfy uncertain constraints can be obtained at low computational cost through constraint randomization in this paper we establish new feasibility results for randomized algorithms specifically the exact feasibility for the class of the so called fully supported problems is obtained it turns out that all fully supported problems share the same feasibility properties revealing a deep kinship among problems of this class it is further proven that the feasibility of the randomized solutions for all other convex programs can be bounded based on the feasibility for the prototype class of fully supported problems the feasibility result of this paper outperforms previous bounds and is not improvable because it is exact for fully supported problems

ambiguous chance constrained problems and robust optimization in this paper we study ambiguous chance constrained problems where the distributions of the random parameters in the problem are themselves uncertain we focus primarily on the special case where the uncertainty set of the distributions is of the form where p denotes the prohorov metric the ambiguous chance constrained problem is approximated by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the central measure the main contribution of this paper is to show that the robust sampled problem is a good approximation for the ambiguous chance constrained problem with a high probability this result is established using the strassen dudley representation theorem that states that when the distributions of two random variables are close in the prohorov metric one can construct a coupling of the random variables such that the samples are close with a high probability we also show that the robust sampled problem can be solved efficiently both in theory and in practice

robust filtering for discrete time systems with bounded noise and parametric uncertainty this note presents a new approach to finite horizon guaranteed state prediction for discrete time systems affected by bounded noise and unknown but bounded parameter uncertainty our framework handles possibly nonlinear dependence of the state space matrices on the uncertain parameters the main result is that a minimal confidence ellipsoid for the state consistent with the measured output and the uncertainty description may be recursively computed in polynomial time using interior point methods for convex optimization with n states l uncertain parameters appearing linearly in the state space matrices with rank one matrix coefficients the worst case complexity grows as o l n l sup 3 5 with unstructured uncertainty in all system matrices the worst case complexity reduces to o n sup 3 5

multi phase computations in geometrical optics in this work we propose a new set of partial differential equations pdes which can be seen as a generalization of the classical eikonal and transport equations to allow for solutions with multiple phases the traditional geometrical optics pair of equations suffer from the fact that the class of physically relevant solutions is limited in particular it does not include solutions with multiple phases corresponding to crossing waves our objective has been to generalize these equations to accommodate solutions containing more than one phase the new equations are based on the same high frequency approximation of the scalar wave equation as the eikonal and the transport equations however they also incorporate a finite superposition principle the maximum allowed number of intersecting waves in the solution can be chosen arbitrarily but a higher number means that a larger system of pdes must be solved the pdes form a hyperbolic system of conservation laws with source terms although the equations are only weakly hyperbolic and thus not well posed in the strong sense several examples show the viability of solving the equations numerically the technique we use to capture multivalued solutions is based on a closure assumption for a system of equations representing the moments

scenario min max optimization and the risk of empirical costs we consider convex optimization problems in the presence of stochastic uncertainty the min max sample based solution is the solution obtained by minimizing the max of the cost functions corresponding to a finite sample of the uncertainty parameter the empirical costs are instead the cost values that the solution incurs for the various parameter realizations that have been sampled our goal is to evaluate the risks associated with the empirical costs where the risk associated with a cost is the probability that the cost is exceeded when a new realization of the uncertainty parameter is seen this task is accomplished without resorting to uncertainty realizations other than those used in optimization the theoretical result proved in this paper is that these risks form a random vector whose probability distribution is an ordered dirichlet distribution irrespective of the probability measure of the stochastic uncertainty parameter this result provides a distribution free characterization of the risks as

regularized dirac delta functions for phase field models summary the phase field model is a highly successful computational technique for capturing the evolution and topological change of complex interfaces the main computational advantage of phase field models is that an explicit tracking of the interface is unnecessary the regularized dirac delta function is an important ingredient in many interfacial problems that phase field models have been applied the delta function can be used to postprocess the phase field solution and represent the surface tension force in this paper we present and compare various types of delta functions for phase field models in particular we analytically show which type of delta function works relatively well regardless of whether an interfacial phase transition is compressed or stretched numerical experiments are presented to show the performance of each delta function numerical results indicate that 1 all of the considered delta functions have good performances when the phase field is locally equilibrated and 2 a delta function which is the absolute value of the gradient of the phase field is the best in most of the numerical experiments copyright 2012 john wiley sons ltd

a level set method for the semiclassical limit of the schrodinger equation with discontinuous potentials we propose a level set method for the semiclassical limit of the schrodinger equation with discontinuous potentials the discontinuities in the potential corresponds to potential barriers at which incoming waves can be partially transmitted and reflected previously such a problem was handled by jin and wen using the liouville equation which arises as the semiclassical limit of the schrodinger equation with an interface condition to account for partial transmissions and reflections s jin x wen siam j num anal 44 2006 1801 1828 however the initial data are dirac delta functions which are difficult to approximate numerically with a high accuracy in this paper we extend the level set method introduced in s jin h liu s osher r tsai j comp phys 210 2005 497 518 for this problem instead of directly discretizing the delta functions our proposed method decomposes the initial data into finite sums of smooth functions that remain smooth in finite time along the phase flow and hence can be solved much more easily using conventional high order discretization schemes two ideas are introduced here 1 the solutions of the problems involving partial transmissions and partial reflections are decomposed into a finite sum of solutions solving problems involving only complete transmissions and those involving only complete reflections for problems involving only complete transmission or complete reflection the method of jlot applies and is used in our simulations 2 a reinitialization technique is introduced so that waves coming from multiple transmissions and reflections can be combined seamlessly as new initial value problems this is implemented by rewriting the sum of several delta functions as one delta function with a suitable weight which can be easily implemented numerically we carry out numerical experiments in both one and two space dimensions to verify this new algorithm

a direct boundary element method for signorini problems in this paper a signorini problem is reduced to a variational inequality on the boundary and a direct boundary element method is presented for its solution furthermore error estimates for the approximate solutions of signorini problems are given in addition we show that the signorini problem may be formulated as a saddle point problem on the boundary

high frequency wave propagation by the segment projection method geometrical optics is a standard technique used for the approximation of high frequency wave propagation computational methods based on partial differential equations instead of the traditional ray tracing have recently been applied to geometrical optics these new methods have a number of advantages but typically exhibit difficulties with linear superposition of waves in this paper we introduce a new partial differential technique based on the segment projection method in phase space the superposition problem is perfectly resolved and so is the problem of computing amplitudes in the neighborhood of caustics the computational complexity is of the same order as that of ray tracing the new algorithm is described and a number of computational examples are given including a simulation of waveguides

a p version finite element method for nonlinear elliptic variational inequalities in 2d this article introduces and analyzes a p version fem for variational inequalities resulting from obstacle problems for some quasi linear elliptic partial differential operators we approximate the solution by controlling the obstacle condition in images of the gauss lobatto points we show existence and uniqueness for the discrete solution up from the p version for the obstacle problem we prove the convergence of up towards the solution with respect to the energy norm and assuming some additional regularity for the solution we derive an a priori error estimate in numerical experiments the p version turns out to be superior to the h version concerning the convergence rate and the number of unknowns needed to achieve a certain exactness of the approximation

stochastic control with uncertain parameters via chance constrained control chance constrained methods handle problem uncertainty by formulating the constraints probabilistically and only guaranteeing their satisfaction up to a given violation level this paper extends existing chance constrained methods to handle both uncertainty in the system state as well as in the constraint parameters due to the imperfect knowledge of the system state caused by motion sensor and environment uncertainty the system constraints cannot be guaranteed to be satisfied and consequently must be considered probabilistically prior work has primarily only dealt with uncertainty in the system state or constraint parameters but not both simultaneously however in many motivating applications both frequently occur to handle this case previous approaches either approximate the distribution leading to a nonconvex optimization program or use sampling alone to represent the uncertainty which requires a large number of samples to accurately represent the distribution to address these limitations a novel hybrid method is proposed that uses both analytical functions and sampling to represent the uncertainty it is shown that under certain conditions the resulting optimization program using this hybrid representation is convex to check the convexity an efficient a priori sufficient condition is developed furthermore by using this hybrid representation this method drastically reduces the computational complexity over previous methods which is demonstrated through two examples consequently this method has the ability to enable real time stochastic control for the motivating applications

randomized sampling for large zero sum games this paper addresses the solution of large zero sum matrix games using randomized methods we formalize a procedure termed as the sampled security policy ssp algorithm by which a player can compute policies that with a high confidence are security policies against an adversary using randomized methods to explore the possible outcomes of the game the ssp algorithm essentially consists of solving a stochastically sampled subgame that is much smaller than the original game we also propose a randomized algorithm termed as the sampled security value ssv algorithm which computes a high confidence security level i e worst case outcome for a given policy which may or may not have been obtained using the ssp algorithm for both the ssp and the ssv algorithms we provide results to determine how many samples are needed to guarantee a desired level of confidence we start by providing results when the two players sample policies with the same distribution and subsequently extend these results to the case of mismatched distributions we demonstrate the usefulness of these results in a hide and seek game that exhibits exponential complexity

homogenization limits and wigner transforms we present a theory for carrying out homogenization limits for quadratic functions called energy densities of solutions of initial value problems ivps with anti self adjoint spatial pseudo differential operators pdos the approach is based on the introduction of phase space wigner matrix measures that are calculated by solving kinetic equations involving the spectral properties of the pdo the weak limits of the energy densities are then obtained by taking moments of the wigner measure the very general theory is illustrated by typical examples like semi classical limits of schrodinger equations with or without a periodic potential the homogenization limit of the acoustic equation in a periodic medium and the classical limit of the dirac equation 1997 john wiley sons inc

on h p adaptive bem for frictional contact problems in linear elasticity a mixed formulation for a tresca frictional contact problem in linear elasticity is considered in the context of boundary integral equations which is later extended to coulomb friction the discrete lagrange multiplier an approximation of the surface traction on the contact boundary part is a linear combination of biorthogonal basis functions in case of curved isoparametric elements these are the solutions of local problems in particular the biorthogonality allows to rewrite the variational inequality constraints as a simple set of complementarity problems thus enabling an efficient application of a semi smooth newton solver for the discrete mixed problem converging locally super linearly in the frictional case and quadratically in the frictionless case typically the solution of frictional contact problems is of reduced regularity at the interface between contact to non contact and from stick to slip to identify the a priori unknown locations of these interfaces two a posteriori error estimations are introduced in a first step the error is split into specific error contributions resulting from the contact and friction conditions and from the discretization error of a variational equation for the latter a residual and a bubble error estimation are considered explicitly the numerical experiments show the applicability of the derived error estimations and the superiority of h p adaptivity compared to low order uniform and adaptive approaches

a bloch band based level set method for computing the semiclassical limit of schrodinger equations a novel bloch band based level set method is proposed for computing the semiclassical limit of schrodinger equations in periodic media for the underlying equation subject to a highly oscillatory initial data a hybrid of the wkb approximation and homogenization leads to the bloch eigenvalue problem and an associated hamilton jacobi system for the phase in each bloch band with the bloch eigenvalue be part of the hamiltonian we formulate a level set description to capture multi valued solutions to the band wkb system and then evaluate total homogenized density over a sample set of bands a superposition of band densities is established over all bands and solution branches when away from caustic points the numerical approach splits the solution process into several parts i initialize the level set function from the band decomposition of the initial data ii solve the bloch eigenvalue problem to compute bloch waves iii evolve the band level set equation to compute multi valued velocity and density on each bloch band iv evaluate the total position density over a sample set of bands using bloch waves and band densities obtained in steps ii and iii respectively numerical examples with different number of bands are shown to demonstrate the capacity of the method

an eulerian method for capturing caustics a robust numerical method for the localization of caustics is proposed for general hamiltonians it is based on the direct resolution of a system of partial differential equations obtained through a local change of the time variable in the hamilton jacobi equation and complemented by a set of transport equations numerical results 1 to 3 d are presented

coupling regularization and adaptive hp bem for the solution of a delamination problem in this paper we couple regularization techniques with the adaptive hp version of the boundary element method hp bem for the efficient numerical solution of linear elastic problems with nonmonotone contact boundary conditions as a model example we treat the delamination of composite structures with a contaminated interface layer this problem has a weak formulation in terms of a nonsmooth variational inequality the resulting hemivariational inequality is first regularized and then discretized by an adaptive hp bem we give conditions for the uniqueness of the solution and provide an a priori error estimate furthermore we prove the very first a posteriori error estimate for the nonsmooth variational problem utilizing a novel mixed regularized formulation thus enabling hp adaptivity various numerical experiments illustrate the behavior strengths and limitations of the proposed high order approximation scheme

the scenario approach to robust control design this paper proposes a new probabilistic solution framework for robust control analysis and synthesis problems that can be expressed in the form of minimization of a linear objective subject to convex constraints parameterized by uncertainty terms this includes the wide class of np hard control problems representable by means of parameter dependent linear matrix inequalities lmis it is shown in this paper that by appropriate sampling of the constraints one obtains a standard convex optimization problem the scenario problem whose solution is approximately feasible for the original usually infinite set of constraints i e the measure of the set of original constraints that are violated by the scenario solution rapidly decreases to zero as the number of samples is increased we provide an explicit and efficient bound on the number of samples required to attain a priori specified levels of probabilistic guarantee of robustness a rich family of control problems which are in general hard to solve in a deterministically robust sense is therefore amenable to polynomial time solution if robustness is intended in the proposed risk adjusted sense

a statistical learning theory approach for uncertain linear and bilinear matrix inequalities abstract in this paper we consider the problem of minimizing a linear functional subject to uncertain linear and bilinear matrix inequalities which depend in a possibly nonlinear way on a vector of uncertain parameters motivated by recent results in statistical learning theory we show that probabilistic guaranteed solutions can be obtained by means of randomized algorithms in particular we show that the vapnik chervonenkis dimension vc dimension of the two problems is finite and we compute upper bounds on it in turn these bounds allow us to derive explicitly the sample complexity of these problems using these bounds in the second part of the paper we derive a sequential scheme based on a sequence of optimization and validation steps the algorithm is on the same lines of recent schemes proposed for similar problems but improves both in terms of complexity and generality the effectiveness of this approach is shown using a linear model of a robot manipulator subject to uncertain parameters

a quadrature based third order moment method for dilute gas particle flows dilute gas particle flows can be described by a kinetic equation containing terms for spatial transport gravity fluid drag and particle particle collisions however the direct numerical solution of the kinetic equation is intractable for most applications due to the large number of independent variables a useful alternative is to reformulate the problem in terms of the moments of the velocity distribution function closure of the moment equations is challenging for flows away from the equilibrium maxwellian limit in this work a quadrature based third order moment closure is derived that can be applied to gas particle flows at any knudsen number a key component of quadrature based closures is the moment inversion algorithm used to find the weights and abscissas a robust inversion procedure is proposed for moments up to third order and tested for three example applications riemann shock problem impinging jets and vertical channel flow extension of the moment inversion algorithm to fifth or higher order is possible but left to future work the spatial fluxes in the moment equations are treated using a kinetic description and hence a gradient diffusion model is not used to close the fluxes because the quadrature based moment method employs the moment transport equations directly instead of a discretized form of the boltzmann equation the mass momentum and energy are conserved for arbitrary knudsen number including the euler limit while developed here for dilute gas particle flows quadrature based moment methods can in principle be applied to any application that can be modeled by a kinetic equation e g thermal and non isothermal flows currently treated using lattice boltzmann methods and examples are given from the literature

consistency of the scenario approach this paper is meant to prove the consistency of the scenario approach a la calafiore campi and garatti with convex constraints scenario convex problems are usually stated in two equivalent forms

reliable approximations of probability constrained stochastic linear quadratic control here we consider a state constrained stochastic linear quadratic control problem this problem has linear dynamics and a quadratic cost and states are required to satisfy a probabilistic constraint in this paper the joint probabilistic constraint in the model is converted to a conservative deterministic constraint using a multi dimensional chebyshev bound a maximum volume inscribed ellipsoid problem is solved to obtain this probability bound using the probability bound we develop a recursive state feedback control algorithm for a special class of state constrained stochastic linear quadratic regulator lqr the performance of this approach is explored in a numerical example

using k branch entropy solutions for multivalued geometric optics computations this paper is devoted to a numerical simulation of the classical wkb system arising in geometric optics expansions it contains the nonlinear eikonal equation and a linear conservation law whose coefficient can be discontinuous we address the problem of treating it in such a way that superimposed signals can be reproduced by means of the kinetic formulation of multibranch solutions originally due to brenier and corrias some existence and uniqueness results are given together with computational test cases of increasing difficulty displaying up to five multivaluations

high order numerical methods to two dimensional delta function integrals in level set methods in this paper we design and analyze a class of high order numerical methods to delta function integrals appearing in level set methods in two dimensional case the methods comprise approximating the mesh cell restrictions of the delta function integral in each mesh cell the two dimensional delta function integral can be rewritten as a one dimensional ordinary integral with the smooth integrand being a one dimensional delta function integral and thus is approximated by applying standard one dimensional high order numerical quadratures and high order numerical methods to one dimensional delta function integrals proposed in x wen high order numerical methods to a type of delta function integrals j comput phys 226 2007 1952 1967 we establish error estimates for the method which show that the method can achieve any desired accuracy by assigning the corresponding accuracy to the sub algorithms and has better accuracy under an assumption on the zero level set of the level set function which holds generally numerical examples are presented showing that the second to fourth order methods implemented in this paper achieve or exceed the expected accuracy and demonstrating the advantage of using our high order numerical methods

direct computation of multivalued phase space solutions for hamilton jacobi equations a method is presented for the exact and complete resolution of hamiltonian systems only using the corresponding hamilton jacobi equation we show that it possible to get the ode solution from the corresponding pde

a stabilized lagrange multiplier method for the finite element approximation of contact problems in elastostatics in this work we consider a stabilized lagrange or kuhn tucker multiplier method in order to approximate the unilateral contact model in linear elastostatics the particularity of the method is that no discrete inf sup condition is needed in the convergence analysis we propose three approximations of the contact conditions well adapted to this method and we study the convergence of the discrete solutions several numerical examples in two and three space dimensions illustrate the theoretical results and show the capabilities of the method

a posteriori error estimates of higher order finite elements for frictional contact problems abstract in this paper a posteriori estimates are derived for higher order finite element methods and frictional contact problems the discretization is based on a mixed approach where the geometrical and frictional constraints are captured by lagrange multipliers the use of higher order polynomials leads to a certain non conformity in the discretization which requires special attention in the error analysis as a main result an error estimation is proposed which consists of the dual norm of a residual plus some computable remainder terms the residual is estimated by well known a posteriori error estimates for variational equations the remainder terms represent typical sources resulting from the non conforming mixed discretization numerical experiments confirm the applicability of the a posteriori estimates to adaptive mesh refinements

on the road between robust optimization and the scenario approach for chance constrained optimization problems we propose a new method for solving chance constrained optimization problems that lies between robust optimization and scenario based methods our method does not require prior knowledge of the underlying probability distribution as in robust optimization methods nor is it based entirely on randomization as in the scenario approach it instead involves solving a robust optimization problem with bounded uncertainty where the uncertainty bounds are randomized and are computed using the scenario approach to guarantee that the resulting robust problem is solvable we impose certain assumptions on the dependency of the constraint functions with respect to the uncertainty and show that tractability is ensured for a wide class of systems our results lead immediately to guidelines under which the proposed methodology or the scenario approach is preferable in terms of providing less conservative guarantees or reducing the computational cost

a field space based level set method for computing multi valued solutions to 1d euler poisson equations we present a field space based level set method for computing multi valued solutions to one dimensional euler poisson equations the system of these equations has many applications and in particular arises in semiclassical approximations of the schrodinger poisson equation the proposed approach involves an implicit eulerian formulation in an augmented space called field space which incorporates both velocity and electric fields into the configuration both velocity and electric fields are captured through common zeros of two level set functions which are governed by a field transport equation simultaneously we obtain a weighted density f by solving again the field transport equation but with initial density as starting data the averaged density is then resolved by the integration of the obtained f against the dirac delta function of two level set functions in the field space moreover we prove that such obtained averaged density is simply a linear superposition of all multi valued densities and the averaged field quantities are weighted superposition of corresponding multi valued ones computational results are presented and compared with some exact solutions which demonstrate the effectiveness of the proposed method

wait and judge scenario optimization we consider convex optimization problems with uncertain probabilistically described constraints in this context scenario optimization is a well recognized methodology where a sample of the constraints is used to describe uncertainty one says that the scenario solution generalizes well or has a high robustness level if it satisfies most of the other constraints besides those in the sample over the past 10 years the main theoretical investigations on the scenario approach have related the robustness level of the scenario solution to the number of optimization variables this paper breaks into the new paradigm that the robustness level is a posteriori evaluated after the solution is computed and the actual number of the so called support constraints is assessed wait and judge a new theory is presented which shows that a posteriori observing k support constraints in dimension d k allows one to draw conclusions close to those obtainable when the problem is from the outset in dimension k this new theory provides evaluations of the robustness that largely outperform those carried out based on the number of optimization variables

the sample average approximation method for stochastic discrete optimization in this paper we study a monte carlo simulation based approach to stochastic discrete optimization problems the basic idea of such methods is that a random sample is generated and the expected value function is approximated by the corresponding sample average function the obtained sample average optimization problem is solved and the procedure is repeated several times until a stopping criterion is satisfied we discuss convergence rates stopping rules and computational complexity of this procedure and present a numerical example for the stochastic knapsack problem

comparison of mixed h p bem stabilized and non stabilized for frictional contact problems we present boundary integral equation procedures for contact problems with tresca or coulomb friction which are based on mixed formulations where besides the displacement also the traction on the contact boundary part appears as a lagrange multiplier this approach allows for an easy and efficient way to perform an h p be method by the use of biorthogonal basis functions this is especially suited for applying the semi smooth newton ssn method which is a very efficient solver superior to standard algorithm like uzawa with an adaptive algorithm we perform locally mesh refinements and increase of polynomial degrees for the be solution thus correctly representing the contact phenomena we also present as stabilized version of our mixed h p bem scheme with gauss lobatto lagrange basis which circumvents the discrete inf sup condition numerical results supporting our theory are reported

numerical solution of the high frequency asymptotic expansion for the scalar wave equation new numerical methods are derived for calculation of high frequency asymptotic expansion of the scalar wave equation the nonlinear partial differential equations defining the terms in the expansio

multi phase computations of the semiclassical limit of the schrodinger equation and related problems whitham vs wigner abstract we present and compare two different techniques to obtain the multi phase solutions for the schrodinger equation in the semiclassical limit the first is whitham s averaging method which gives the modulation equations governing the evolution of multi phase solutions the second is the wigner transform a convenient tool to derive the semiclassical limit equation in the phase space the vlasov equation for the linear schrodinger equation motivated by the linear superposition principle we derive and prove the multi phase ansatz for the wigner function by the stationary phase method and then use the ansatz to close the moment equations of the vlasov equation and obtain the multi phase equations in the physical space we show that the multi phase equations so derived agree with those derived by whitham s averaging method which can be proved using different arguments generic way of obtaining and computing the multi phase equations by the wigner function is given and kinetic schemes are introduced to solve the multi phase equations the numerical schemes based on multiphase closure of the liouville equation are purely eulerian and only operate in the physical space several numerical examples are given to explore the validity of this approach similar studies are conducted for the linearized korteweg de vries equation and the linear wave equation

randomized algorithms for probabilistic robustness with real and complex structured uncertainty there has been a growing interest in developing randomized algorithms for probabilistic robustness of uncertain control systems unlike classical worst case methods these algorithms provide probabilistic estimates assessing for instance if a certain design specification is met with a given probability one of the advantages of this approach is that the robustness margins can be often increased by a considerable amount at the expense of a small risk in this sense randomized algorithms may be used by the control engineer together with standard worst case methods to obtain additional useful information the applicability of these probabilistic methods to robust control is presently limited by the fact that the sample generation is feasible only in very special cases which include systems affected by real parametric uncertainty bounded in rectangles or spheres sampling in more general uncertainty sets is generally performed through overbounding at the expense of an exponential rejection rate in the paper randomized algorithms for stability and performance of linear time invariant uncertain systems described by a general m spl delta configuration are studied in particular efficient polynomial time algorithms for uncertainty structures spl delta consisting of an arbitrary number of full complex blocks and uncertain parameters are developed

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

inverse type estimates on hp finite element spaces and applications this work is concerned with the development of inverse type inequalities for piecewise polynomial functions and in particular functions belonging to hp finite element spaces the cases of positive and negative sobolev norms are considered for both continuous and discontinuous finite element functions the inequalities are explicit both in the local polynomial degree and the local mesh size the assumptions on the hp finite element spaces are very weak allowing anisotropic shape irregular elements and varying polynomial degree across elements finally the new inverse type inequalities are used to derive bounds for the condition number of symmetric stiffness matrices of hp boundary element method discretisations of integral equations with element wise discontinuous basis functions constructed via scaled tensor products of legendre polynomials

fast phase space computation of multiple arrivals we present a fast general computational technique for computing the phase space solution of static hamilton jacobi equations starting with the liouville formulation of the characteristic equations we derive escape equations which are static time independent eulerian pdes they represent all arrivals to the given boundary from all possible starting configurations the solution is numerically constructed through a one pass formulation building on ideas from semi lagrangian methods dijkstra like methods for the eikonal equation and ordered upwind methods to compute all possible trajectories corresponding to all possible boundary conditions the technique is of computational order o n log n where n is the total number of points in the computational phase space domain any particular set of boundary conditions then is extracted through rapid post processing suggestions are made for speeding up the algorithm in the case when the particular distribution of sources is provided in advance as an application we apply the technique to the problem of computing first multiple and most energetic arrivals to the eikonal equation

hp fem for a stabilized three field formulation of the biharmonic problem abstract in this paper a three field formulation for the biharmonic equation is considered that consists of the potential u the flux and a lagrange multiplier the use of stabilization techniques circumvents the discrete inf sup condition and thus enables the application of arbitrary conforming finite element spaces for the three fields a priori error estimates as well as reliable and efficient a posteriori error estimates are derived the proof of the latter is based on an implicit never to be computed h 2 reconstruction of the discrete potential several numerical experiments underline the theoretical results and show the existence of discretization spaces for which the a posteriori error estimate is p robust with an efficiency index tending towards the optimal value of one the numerical experiments indicate that the proposed method is applicable even if the theoretically required minimal regularity of the lagrange multiplier for the a priori error estimate is not fulfilled in particular h and h p adaptivity recover optimal algebraic and exponential converge rates respectively

big ray tracing different traveltimes are identified as viscosity solutions of the eikonal equation on locally selected domains called big rays big rays are constructed by an ad hoc algorithm based on a coarse ray tracing viscosity solutions in the big rays are computed using an upwind finite difference scheme this hybrid method allows an easy and fast computation of multivalued traveltime fields

efficient monte carlo procedures for generating points uniformly distributed over bounded regions we consider the monte carlo problem of generating points uniformly distributed within an arbitrary bounded measurable region the class of markovian methods considered generate points asymptotically uniformly distributed within the region computational experience suggests the methods are potentially superior to conventional rejection techniques for large dimensional regions

chance constrained h control for a class of time varying systems with stochastic nonlinearities the finite horizon case in this paper a new finite horizon h control problem is considered for a class of time varying systems with stochastic nonlinearities measurements degradation and chance constraints the purpose of the addressed problem is to design the time varying controller such that the closed loop system satisfies the prespecified h disturbance attenuation requirement and certain chance constraints on the controlled output vector zk i e the probability of the controlled output zk belonging to a given set is larger than a prescribed value a modified maximum volume inscribed ellipsoid mvie method is employed to convert the chance constraint into some tractable inequalities which could be conveniently handled by the recursive linear matrix inequality rlmi approach then by using stochastic control analysis method sufficient conditions are derived for the existence of the desired multi objective controller and furthermore the gains of the desired controllers are characterized by means of rlmis finally two illustrative examples and a practical system are proposed to highlight the effectiveness and applicability of the presented controller design technology

risk and complexity in scenario optimization scenario optimization is a broad methodology to perform optimization based on empirical knowledge one collects previous cases called scenarios for the set up in which optimization is being performed and makes a decision that is optimal for the cases that have been collected for convex optimization a solid theory has been developed that provides guarantees of performance and constraint satisfaction of the scenario solution in this paper we open a new direction of investigation the risk that a performance is not achieved or that constraints are violated is studied jointly with the complexity as precisely defined in the paper of the solution it is shown that the joint probability distribution of risk and complexity is concentrated in such a way that the complexity carries fundamental information to tightly judge the risk this result is obtained without requiring extra knowledge on the underlying optimization problem than that carried by the scenarios in particular no extra knowledge on the distribution by which scenarios are generated is assumed so that the result is broadly applicable this deep seated result unveils a fundamental and general structure of data driven optimization and suggests practical approaches for risk assessment

on the connection between compression learning and scenario based single stage and cascading optimization problems we investigate the connections between compression learning and scenario based optimization we first show how to strengthen or relax the consistency assumption at the basis of compression learning and provide novel learnability conditions for the underlying algorithms we then consider different constrained optimization problems affected by uncertainty represented by means of scenarios we show that the compression learning perspective provides a unifying framework for scenario based optimization since the issue of providing guarantees on the probability of constraint violation reduces to a learning problem for an appropriately chosen algorithm that satisfies some consistency assumption to illustrate this we revisit the scenario approach within the developed context moreover using the compression learning machinery we provide novel results on the probability of constraint violation for the class of cascading optimization problems

a level set approach for computing discontinuous solutions of hamilton jacobi equations we introduce two types of finite difference methods to compute the l solution and the proper viscosity solution recently proposed by the second author for semi discontinuous solutions to a class of hamilton jacobi equations by regarding the graph of the solution as the zero level curve of a continuous function in one dimension higher we can treat the corresponding level set equation using the viscosity theory introduced by crandall and lions however we need to pay special attention both analytically and numerically to prevent the zero level curve from overturning so that it can be interpreted as the graph of a function we demonstrate our lax friedrichs type numerical methods for computing the l solution using its original level set formulation in addition we couple our numerical methods with a singular diffusive term which is essential to computing solutions to a more general class of hj equations that includes conservation laws with this singular viscosity our numerical methods do not require the divergence structure of equations and do apply to more general equations developing shocks other than conservation laws these numerical methods are generalized to higher order accuracy using weighted eno local lax friedrichs methods as developed recently by jiang and peng we verify that our numerical solutions approximate the proper viscosity solutions obtained by the second author in a recent hokkaido university preprint finally since the solution of scalar conservation law equations can be constructed using existing numerical techniques we use it to verify that our numerical solution approximates the entropy solution

randomized methods for design of uncertain systems in this paper we study randomized methods for feedback design of uncertain systems the first contribution is to derive the sample complexity of various constrained control problems in particular we show the key role played by the binomial distribution and related tail inequalities and compute the sample complexity this contribution significantly improves the existing results by reducing the number of required samples in the randomized algorithm these results are then applied to the analysis of worst case performance and design with robust optimization the second contribution of the paper is to introduce a general class of sequential algorithms denoted as sequential probabilistic validation spv in these sequential algorithms at each iteration a candidate solution is probabilistically validated and corrected if necessary to meet the required specifications the results we derive provide the sample complexity which guarantees that the solutions obtained with spv algorithms meet some pre specified probabilistic accuracy and confidence the performance of these algorithms is illustrated and compared with other existing methods using a numerical example dealing with robust system identification

an accurate spectral discontinuous finite element formulation of a phase space based level set approach to geometrical optics in this paper we introduce a new numerical procedure for simulations in geometrical optics that based on the recent development of eulerian phase space formulations of the model can deliver very accurate uniformly resolved solutions which can be made to converge with arbitrarily high orders in general geometrical configurations following previous treatments the scheme is based on the evolution of a wavefront in phase space defined as the intersection of level sets satisfying the relevant liouville equation in contrast with previous work however our numerical approximation is specifically designed i to take full advantage of the smoothness of solutions ii to facilitate the treatment of scattering obstacles all while retaining high order convergence characteristics indeed to incorporate the full regularity of solutions that results from the unfolding of singularities our method is based on their spectral representation to enable a simple high order treatment of scattering boundaries on the other hand we resort to a discontinuous galerkin finite element method for the solution of the resulting system of equations the procedure is complemented with the use of a recently derived strong stability preserving runge kutta ssp rk scheme for the time integration that as we demonstrate allows for overall approximations that are rapidly convergent

notes on the scenario design approach the scenario optimization method developed in is a theoretically sound and practically effective technique for solving in a probabilistic setting robust convex optimization problems arising in systems and control design that would otherwise be hard to tackle via standard deterministic techniques in this note we explore some further aspects of the scenario methodology and present two results pertaining to the tightness of the sample complexity bounds we also state a new theorem that enables the user to make a priori probabilistic claims on the scenario solution with one level of probability only

residual error estimators for coulomb friction this paper is concerned with residual error estimators for finite element approximations of coulomb frictional contact problems a recent uniqueness result by renard in siam j math anal 38 2006 pp 452 467 for the continuous problem allows us to perform an a posteriori error analysis we propose study and implement numerically two residual error estimators associated with two finite element discretizations in both cases the estimators permit us to obtain upper and lower bounds of the discretization error

stabilized mixed hp bem for frictional contact problems in linear elasticity we analyze stabilized mixed hp boundary element methods for frictional contact problems for the lame equation the stabilization technique circumvents the discrete inf sup condition for the mixed problem and thus allows us to use the same mesh and polynomial degree for the primal and dual variables we prove a priori convergence rates in the case of tresca friction using gauss legendre lagrange polynomials as test and trial functions for the lagrange multiplier additionally a residual based a posteriori error estimate for a more general class of discretizations is derived it in particular applies to discretizations based on bernstein polynomials for the discrete lagrange multiplier which we also analyze the discretization and the a posteriori error estimate are extended to the case of coulomb friction several numerical experiments underline our theoretical results demonstrate the behavior of the method and its insensitivity to the scaling and perturbations of the stabilization term

geometric optics in a phase space based level set and eulerian framework we introduce a level set approach for ray tracing and the construction of wavefronts in geometric optics this is important in a wide variety of applications in wave propagation our approach automatically handles the multivalued solutions that appear and automatically resolves the wavefronts this is achieved through solving for the bicharacteristic strips whose projection to spatial space gives the wavefronts in a reduced phase space under a eulerian and partial differential equation based framework the bicharacteristic strips are represented using a level set approach for handling higher codimensional objects and the partial differential equations responsible for the evolution are reduced forms of the liouville equations results for the two dimensional case for constant and variable indices of refraction are shown and compared to those of other current methods in the field results are also introduced to show the ability to handle reflection and to extend the method to the three dimensional case

stochastic algorithms for exact and approximate feasibility of robust lmis in this note we discuss fast randomized algorithms for determining an admissible solution for robust linear matrix inequalities lmis of the form f x spl delta spl les 0 where x is the optimization variable and spl delta is the uncertainty which belongs to a given set spl delta the proposed algorithms are based on uncertainty randomization the first algorithm finds a robust solution in a finite number of iterations with probability one if a strong feasibility condition holds in case no robust solution exists the second algorithm computes an approximate solution which minimizes the expected value of a suitably selected feasibility indicator function the theory is illustrated by examples of application to uncertain linear inequalities and quadratic stability of interval matrices

time domain boundary elements for dynamic contact problems abstract this article considers a unilateral contact problem for the wave equation the problem is reduced to a variational inequality for the dirichlet to neumann operator for the wave equation on the boundary which is solved in a saddle point formulation using boundary elements in the time domain as a model problem also a variational inequality for the single layer operator is considered a priori estimates are obtained for galerkin approximations both to the variational inequality and the mixed formulation in the case of a flat contact area where the existence of solutions to the continuous problem is known numerical experiments demonstrate the performance of the proposed mixed method they indicate the stability and convergence beyond flat geometries

computing multi valued physical observables for the high frequency limit of symmetric hyperbolic systems we develop a level set method for the computation of multi valued physical observables density velocity energy etc for the high frequency limit of symmetric hyperbolic systems in any number of space dimensions we take two approaches to derive the method the first one starts with a weakly coupled system of an eikonal equation for phase s and a transport equation for density r ts h x s 0 t x r xr n email protected email protected x email protected ph x xs 0 the main idea is to evolve the density near the n dimensional bi characteristic manifold of the eikonal hamiltonian jacobi equation which is identified as the common zeros of n level set functions in phase space x k r 2 n these level set functions are generated from solving the liouville equation with initial data chosen to embed the phase gradient simultaneously we track a new quantity email protected t x k det email protected by solving again the liouville equation near the obtained zero level set f 0 but with initial density as initial data the multi valued density and higher moments are thus resolved by integrating f along the bi characteristic manifold in the phase directions the second one uses the high frequency limit of symmetric hyperbolic systems derived by the wigner transform this gives rise to liouville equations in the phase space with measure valued solution in its initial data due to the linearity of the liouville equation we can decompose the density distribution into products of function each of which solves the liouville equation with l initial data on any bounded domain it yields higher order moments such as energy and energy flux the main advantages of these new approaches in contrast to the standard kinetic equation approach using the liouville equation with a dirac measure initial data include 1 the liouville equations are solved with l initial data and a singular integral involving the email protected function is evaluated only in the post processing step thus avoiding oscillations and excessive numerical smearing 2 a local level set method can be utilized to significantly reduce the computation in the phase space these methods can be used to compute all physical observables for multi dimensional problems our method applies to the wave fields corresponding to simple eigenvalues of the dispersion matrix one such example is the wave equation which will be studied numerically in this paper

on constraint sampling in the linear programming approach to approximate dynamic programming in the linear programming approach to approximate dynamic programming one tries to solve a certain linear program the alp that has a relatively small numberk of variables but an intractable numberm of constraints in this paper we study a scheme that samples and imposes a subset ofm

on a class of interval predictor models with universal reliability abstract an interval predictor model ipm is a rule by which some observable variables system inputs are mapped into an interval that is used to predict an inaccessible variable system output ipms have been studied in campi et al 2009 where the problem of fitting an ipm on a set of observations has been considered in the same paper upper bounds on the probability that a future system output will fall outside the predicted interval misprediction have also been derived in a stationary and independent framework while these bounds have the notable property of being valid independently of the unknown mechanism that has generated the data in general the actual probability distribution of the misprediction does depend on the data generation mechanism and hence these bounds may introduce conservatism when applied to a specific case in this paper we study the reliability of an important class of ipms called minimax layers and show that this class exhibits the special property that the probability distribution of the misprediction is known exactly and is universal i e is always the same irrespective of the data generation mechanism this result carries important consequences on the use of minimax layers in practice

hamiltonian preserving schemes for the liouville equation of geometrical optics with discontinuous local wave speeds in this paper we construct two classes of hamiltonian preserving numerical schemes for a liouville equation with discontinuous local wave speed this equation arises in the phase space description of geometrical optics and has been the foundation of the recently developed level set methods for multivalued solution in geometrical optics we extend our previous work in s jin x wen hamiltonian preserving schemes for the liouville equation with discontinuous potentials commun math sci 3 2005 285 315 for the semiclassical limit of the schrodinger equation into this system the designing principle of the hamiltonian preservation by building in the particle behavior at the interface into the numerical flux is used here and as a consequence we obtain two classes of schemes that allow a hyperbolic stability condition when a plane wave hits a flat interface the hamiltonian preservation is shown to be equivalent to snell s law of refraction in the case when the ratio of wave length over the width of the interface goes to zero when both length scales go to zero positivity and stabilities in both l 1 and l norms are established for both schemes the approach also provides a selection criterion for a unique solution of the underlying linear hyperbolic equation with singular discontinuous and measure valued coefficients benchmark numerical examples are given with analytic solution constructed to study the numerical accuracy of these schemes

hp fem convergence for unilateral contact problems with tresca friction in plane linear elastostatics abstract this paper is concerned with the convergence of the h p version of the finite element method h p fem for some nonsmooth unilateral problems in linear elastostatics we consider in particular the deformation of an elastic body unilaterally supported by a rigid foundation admitting tresca friction given friction along the rigid foundation solely subjected to body forces and surface tractions without being fixed along some part of its boundary for the discretization of the unilateral constraint and the nonsmooth friction functional we employ gauss lobatto quadrature we show convergence of the h p fem approximations for mechanically definite problems without imposing any regularity assumption moreover we treat the coercive case when the body is fixed along some part of the boundary based on an abstract cea falk estimate and operator interpolation arguments we establish an a priori error estimate in the energy norm under a reasonable regularity assumption

stochastic model predictive control of lpv systems via scenario optimization a stochastic receding horizon control approach for constrained linear parameter varying discrete time systems is proposed in this paper it is assumed that the time varying parameters have stochastic nature and that the system s matrices are bounded but otherwise arbitrary nonlinear functions of these parameters no specific assumption on the statistics of the parameters is required by using a randomization approach a scenario based finite horizon optimal control problem is formulated where only a finite number m of sampled predicted parameter trajectories scenarios are considered this problem is convex and its solution is a priori guaranteed to be probabilistically robust up to a user defined probability level p the p level is linked to m by an analytic relationship which establishes a tradeoff between computational complexity and robustness of the solution then a receding horizon strategy is presented involving the iterated solution of a scenario based finite horizon control problem at each time step our key result is to show that the state trajectories of the controlled system reach a terminal positively invariant set in finite time either deterministically or with probability no smaller than p the features of the approach are illustrated by a numerical example

chance constrained problems and rare events an importance sampling approach we study chance constrained problems in which the constraints involve the probability of a rare event we discuss the relevance of such problems and show that the existing sampling based algorithms cannot be applied directly in this case since they require an impractical number of samples to yield reasonable solutions we argue that importance sampling is techniques combined with a sample average approximation saa approach can be effectively used in such situations provided that variance can be reduced uniformly with respect to the decision variables we give sufficient conditions to obtain such uniform variance reduction and prove asymptotic convergence of the combined saa is approach as it often happens with is techniques the practical performance of the proposed approach relies on exploiting the structure of the problem under study in our case we work with a telecommunications problem with bernoulli input distributions and show how variance can be reduced uniformly over a suitable approximation of the feasibility set by choosing proper parameters for the is distributions although some of the results are specific to this problem we are able to draw general insights that can be useful for other classes of problems we present numerical results to illustrate our findings

learning noisy functions via interval models this paper considers the problem of identification of an interval model for an unknown static function using a finite batch of stochastic input output data u i y i u i y i i 1 ni 1 n the criterion used for identification is that the width of the interval output of the model should be minimized while containing a given fraction of the observed outputs y i y i we show that for suitable finite nn the resulting model will be reliable that is it will explain any other unseen output up to a given and arbitrary high probability

a level set approach for dilute non collisional fluid particle flows gas particle and other dispersed phase flows can be described by a kinetic equation containing terms for spatial transport acceleration and particle processes such as evaporation or collisions however computing the dispersed velocity is a challenging task due to the large number of independent variables a level set approach for computing dilute non collisional fluid particle flows is presented we will consider the sprays governed by the williams kinetic equation subject to initial distributions away from equilibrium of the form i 1 n i x u i x the dispersed velocity is described as the zero level set of a smooth function which satisfies a transport equation this together with the density weight recovers the particle distribution at any time moments of any desired order can be evaluated by a quadrature formula involving the level set function and the density weight it is shown that the method can successfully handle highly non equilibrium flows e g impinging particle jets jet crossing particle rebound off walls finite stokes number flows

space time adaptive finite elements for nonlocal parabolic variational inequalities abstract this article considers the error analysis of finite element discretizations and adaptive mesh refinement procedures for nonlocal dynamic contact and friction both in the domain and on the boundary for a large class of parabolic variational inequalities associated to the fractional laplacian we obtain a priori and a posteriori error estimates and study the resulting space time adaptive mesh refinement procedures particular emphasis is placed on mixed formulations which include the contact forces as a lagrange multiplier corresponding results are presented for elliptic problems our numerical experiments for 2 dimensional model problems confirm the theoretical results they indicate the efficiency of the a posteriori error estimates and illustrate the convergence properties of space time adaptive as well as uniform and graded discretizations