metric completion versus ideal completion complete partial orders have been used for a long time for defining semantics of programming languages in the context of concurrency de bakker and zucker 4 proposed a metric setting for handling concurrency recursion and nontermination which has proved to be very successful in many applications starting with a semantic domain d for finite behaviour we investigate the relation between the ideal completion idl d and the metric completion which are both suitable to model recursion and infinite behaviour we also consider the properties of semantic operators

an optimal one way multigrid algorithm for discrete time stochastic control the numerical solution of discrete time stationary infinite horizon discounted stochastic control problems is considered for the case where the state space is continuous and the problem is to be solved approximately within a desired accuracy after a discussion of problem discretization the authors introduce a multigrid version of the successive approximation algorithm that proceeds one way from coarse to fine grids and analyze its computational requirements as a function of the desired accuracy and of the discount factor they also study the effects of a certain mixing ergodicity condition on the algorithm s performance it is shown that the one way multigrid algorithm improves upon the complexity of its single grid variant and is in a certain sense optimal

pod based feedback control of the burgers equation by solving the evolutionary hjb equation a new approach for solving finite time horizon feedback control problems for distributed parameter systems is proposed it is based on model reduction by proper orthogonal decomposition combined with efficient numerical methods for solving the resulting low order evolutionary hamilton jacobi bellman hjb equation the feasibility of the proposed methodology is demonstrated by means of optimal feedback control for the burgers equation the method for solving the hjb equation is first tested on several 1 d problems and then successfully applied to the control of the reduced order burgers equation the effect of noise is investigated and parallelism is used for computational speedup

an approximation scheme for the minimum time function this paper presents an approximation scheme for the nonlinear minimum time problem with compact target the scheme is derived from a discrete dynamic programming principle and the main convergence result is obtained by applying techniques related to discontinuous viscosity solutions for hamilton jacobi equations the convergence is proved under general controllability assumptions on both the continuous time and the discrete time systems an explicit sufficient condition on the system and the target ensuring the desired controllability is given this condition is shown to be necessary and sufficient for the lipschitz continuity of the minimum time function if the target is smooth an extension to the case of a point shaped target is given

some convergence results for howard s algorithm this paper deals with convergence results of howard s algorithm for the resolution of min a in mathcal a b a x c a 0 where b a is a matrix c a is a vector and mathcal a is a compact set we show a global superlinear convergence result under a monotonicity assumption on the matrices b a extensions of howard s algorithm for a max min problem of the form max b in mathcal b min a in mathcal a b a b x c a b 0 are also proposed in the particular case of an obstacle problem of the form min ax b x g 0 where a is an n times n matrix satisfying a monotonicity assumption we show the convergence of howard s algorithm in no more than n iterations instead of the usual 2 n bound still in the case of an obstacle problem we establish the equivalence between howard s algorithm and a primal dual active set algorithm m hintermuller k ito and k kunisch siam j optim 13 2002 pp 865 888 the algorithms are illustrated on the discretization of nonlinear pdes arising in the context of mathematical finance american option and merton s portfolio problem of front propagation problems and for the double obstacle problem

the barrier attribute of filled functions a number of filled functions was proposed recently this paper presents a new perspective of the filled functions their barrier attribute more filled functions can be constructed according to this new interpretation two of them are with finite barriers while the remains are with infinite barriers

a filled function method for finding a global minimizer of a function of several variables the concept of a filled function is introduced we construct a particular filled function and analyze its properties an algorithm for global minimization is generated based on the concept and properties of the filled function some typical examples with 1 to 10 variables are tested and computational results show that in most cases this algorithm works better than the tunneling algorithm the advantages and disadvantages are analyzed and further research directions are discussed

a computable filled function used for global minimization the filled function method is an approach to find the global minimum of multimodal and multidimensional functions this paper proposes a new filled function this function needs only one parameter and includes neither exponential terms nor logarithmic terms furthermore the lower bound of weight factor a is usually smaller than that of one previous formulation therefore it is reasonable to expect that the proposed function has better computability than the previously reported ones

a patchy dynamic programming scheme for a class of hamilton jacobi bellman equations in this paper we present a new algorithm for the solution of hamilton jacobi bellman equations related to optimal control problems the key idea is to divide the domain of computation into subdomains which are shaped by the optimal dynamics of the underlying control problem this can result in a rather complex geometrical subdivision but it has the advantage that every subdomain is invariant with respect to the optimal dynamics and then the solution can be computed independently in each subdomain the features of this dynamics dependent domain decomposition can be exploited to speed up the computation and for an efficient parallelization since the classical transmission conditions at the boundaries of the subdomains can be avoided for their properties the subdomains are patches in the sense introduced by ancona and bressan esaim control optim calc var 4 1999 pp 445 471 several examples in two and three dimensions illustrate the properties of the new method

the comparison of a cpo based semantics with cms based semantics for csp we study two semantics for csp both semantics are compositional and interleaving the first uses the concept of complete partial orders cpo for the handling of repetitive constructs and for the creation of suitable semantic domains the second uses the concept of complete metric spaces cms for the same purposes our results show that the cms semantics gives more detailed information about a program and that we can obtain the cpo semantics from it from this investigation we can draw conclusions concerning the influence of the mathematical discipline on the kind of semantics obtained

local minimization algorithms for dynamic programming equations the numerical realization of the dynamic programming principle for continuous time optimal control leads to nonlinear hamilton jacobi bellman equations which require the minimization of a nonlinear mapping over the set of admissible controls this minimization is often performed by comparison over a finite number of elements of the control set in this paper we demonstrate the importance of an accurate realization of these minimization problems and propose algorithms by which this can be achieved effectively the considered class of equations includes nonsmooth control problems with ell 1 penalization which lead to sparse controls

recursively defined metric spaces without contraction in this paper we use the theory of accessible categories to find fixed points of endofunctors on the category of 1 bounded complete metric spaces and nonexpansive functions in contrast to previous approaches we do not assume that the endofunctors are locally contractive and our results do not depend on banach s fixed point theorem our approach is particularly suitable for constructing models of systems that feature quantitative data for instance using the kantorovich metric on probability measures we construct a quantitative model for probabilistic transition systems the metric in our model can reasonably be seen as measuring the behavioural distance between states of the system it depends exclusively on the transition probabilities and not on an arbitrary discount factor

the category theoretic solution of recursive domain equations recursive specifications of domains plays a crucial role in denotational semantics as developed by scott and strachey and their followers the purpose of the present paper is to set up a categorical framework in which the known techniques for solving these equations find a natural place the idea is to follow the well known analogy between partial orders and categories generalizing from least fixed points of continuous functions over cpos to initial ones of continuous functors over omega categories to apply these general ideas we introduce wand s bf o categories where the morphism sets have a partial order structure and which include almost all the categories occurring in semantics the idea is to find solutions in a derived category of embeddings and we give order theoretic conditions which are easy to verify and which imply the needed categorical ones the main tool is a very general form of the limit colimit coincidence remarked by scott in the concluding section we outline how compatibilit

ordered upwind methods for static hamilton jacobi equations theory and algorithms we develop a family of fast methods for approximating the solutions to a wide class of static hamilton jacobi pdes these fast methods include both semi lagrangian and fully eulerian versions numerical solutions to these problems are typically obtained by solving large systems of coupled nonlinear discretized equations our techniques which we refer to as ordered upwind methods oums use partial information about the characteristic directions to decouple these nonlinear systems greatly reducing the computational labor our techniques are considered in the context of control theoretic and front propagation problems we begin by discussing existing oums focusing on those designed for isotropic problems we then introduce a new class of oums which decouple systems for general anisotropic problems we prove convergence of one such scheme to the viscosity solution of the corresponding hamilton jacobi pde next we introduce a set of finite differences methods based on an analysis of the role played by anisotropy in the context of front propagation and optimal trajectory problems the performance of the methods is analyzed and computational experiments are performed using test problems from computational geometry and seismology

numerical stabilization of bilinear control systems extremal lyapunov exponents for bilinear control systems with constrained control values are computed numerically by solving discounted optimal control problems based on this computation a numerical algorithm to calculate stabilizing control functions is developed

error analysis for pod approximations of infinite horizon problems via the dynamic programming approach in this paper infinite horizon optimal control problems for nonlinear high dimensional dynamical systems are studied nonlinear feedback laws can be computed via the value function characterized as the unique viscosity solution to the corresponding hamilton jacobi bellman hjb equation which stems from the dynamic programming approach however the bottleneck is mainly due to the curse of dimensionality and hjb equations are solvable only in a relatively small dimension therefore a reduced order model is derived for the dynamical system using the method of proper orthogonal decomposition pod the resulting errors in the hjb equations are estimated by an a priori error analysis which is utilized in the numerical approximation to ensure a desired accuracy for the pod method numerical experiments illustrates the theoretical findings

trees and semantics abstract several types of trees are proposed in the literature to model the behaviour of nondeterministic processes the purpose of this paper is to clarify the distinction between the various types of trees and to discuss their use as semantic domains trees in the sense of milner s derivation trees and bisimulation equivalence classes of trees are considered in three different settings in the category of sets in the metric setting and in the partial order setting each type of tree is characterized by a domain equation special attention is given to the use of trees as models for processes which allow for unbounded nondeterminism as for instance milners ccs processes in this case one has to deal with infinitely branching trees which cause problems to give denotational semantics that are fully abstract w r t a transition system based operational semantics

convergence properties of policy iteration this paper analyzes asymptotic convergence properties of policy iteration in a class of stationary infinite horizon markovian decision problems that arise in optimal growth theory these problems have continuous state and control variables and must therefore be discretized in order to compute an approximate solution the discretization may render inapplicable known convergence results for policy iteration such as those of puterman and brumelle math oper res 4 1979 pp 60 69 under certain regularity conditions we prove that for piecewise linear interpolation policy iteration converges quadratically also under more general conditions we establish that convergence is superlinear we show how the constants involved in these convergence orders depend on the grid size of the discretization these theoretical results are illustrated with numerical experiments that compare the performance of policy iteration and the method of successive approximations

several filled functions with mitigators several filled functions were reported to seek the global minimum of multimodal functions of multiple variables this paper proposes an alternative formulation that may reduce the negative definite effect of the hessian of a filled function proposed before furthermore a class of mitigators is defined and applied to improve the computational characteristics of filled functions results of numerical experiments on typical testing functions are also reported

can local single pass methods solve any stationary hamilton jacobi bellman equation the use of local single pass methods like e g the fast marching method has become popular in the solution of some hamilton jacobi equations the prototype of these equations is the eikonal equation for which the methods can be applied saving cpu time and possibly memory allocation then some questions naturally arise can local single pass methods solve any hamilton jacobi equation if not where should the limit be set this paper tries to answer these questions in order to give a complete picture we present an overview of some fast methods available in the literature and briefly analyze their main features we also introduce some numerical tools and provide several numerical tests which are intended to exhibit the limitations of the methods we show that the construction of a local single pass method for general hamilton jacobi equations is very hard if not impossible nevertheless some special classes of problems can actually be solved making local single pass methods very useful from a p

denotational semantics in the cpo and metric approach abstract we investigate the implications of choosing a cpo framework resp a complete metric space framework for defining denotational semantics of languages that allow for recursion iteration communication and concurrency we first establish a general framework for the cpo and the metric approach the existence and uniqueness of meaning functions is studied in the metric case the existence and uniqueness of a meaning function can be established under some reasonable assumptions in the cpo case we obtain the existence of a least meaning function from these theorems consistency results can be concluded in the second part we study the impact of the choice between cpo and metric for semantics based on event structures and for semantics based on pomset classes

two new classes of filled functions the filled function method is an effective approach to find the global minimizer two of the recently proposed filled functions are l x and l 2 x having observed that the second term in l x or l 2 x may result in undesirable computational behaviors in this paper we propose a new function l 3 x having also realized that the essence of the mitigation is to eliminate the discontinuity we propose another new function l 4 x both theoretical analysis and numerical testing results are presented

stable numerical schemes for solving hamilton jacobi bellman isaacs equations stable approximation schemes for solving hamilton jacobi bellman isaacs equations in r n are proposed the efficiency of numerical procedures based on such approximations is demonstrated by solving several differential games in two three and four dimensions

connections between two theories of concurrency metric spaces and synchronization trees a connection is established between the semantic theories of concurrency and communication in the works of de bakker and zucker who develop a denotational semantics of concurrency using metric spaces instead of complete partial orders and milner who develops an algebraic semantics of communication based upon observational equivalence between processes his rigid synchronization trees rsts are endowed with a simple pseudometric distance induced by milner s weak equivalence relation and the quotient space is shown to be complete an isometry between this space and the solution to a domain equation of de bakker and zucker is established presenting their solution in a conceptually simpler framework under an additional assumption the equivalence between the weak equivalence relation over rsts and the elementary equivalence relation induced by the sentences of a modal logic due to hennessy and milner is established

an efficient policy iteration algorithm for dynamic programming equations we present an accelerated algorithm for the solution of static hamilton jacobi bellman equations related to optimal control problems our scheme is based on a classic policy iteration procedure which is known to have superlinear convergence in many relevant cases provided the initial guess is sufficiently close to the solution this limitation often degenerates into a behavior similar to a value iteration method with an increased computation time the new scheme circumvents this problem by combining the advantages of both algorithms with an efficient coupling the method starts with a coarse mesh value iteration phase and then switches to a fine mesh policy iteration procedure when a certain error threshold is reached a delicate point is to determine this threshold in order to avoid cumbersome computations with the value iteration and at the same time to ensure the convergence of the policy iteration method to the optimal solution we analyze the methods and efficient coupling in a number of examples

an adaptive grid scheme for the discrete hamilton jacobi bellman equation in this paper an adaptive finite difference scheme for the solution of the discrete first order hamilton jacobi bellman equation is presented local a posteriori error estimates are established and certain properties of these estimates are proved based on these estimates an adapting iteration for the discretization of the state space is developed an implementation of the scheme for two dimensional grids is given and numerical examples are discussed