stochastic uncertain systems subject to relative entropy constraints induced norms and monotonicity properties of minimax games entropy and relative entropy are fundamental concepts on which information theory is founded on and in general telecommunication systems design on the other hand dissipation inequalities minimax strategies and induced norms are the basic concepts on which robustness of uncertain control and estimation of systems are founded on in this paper the precise relation between these notions is investigated in particular it will be shown that the higher the dissipation the higher the entropy of the system which has implications in computing the induced norm associated with robustness these connections are obtained by considering stochastic optimal uncertain control systems in which uncertainty is described by a relative entropy constraint between the nominal and uncertain measures while the pay off is a linear functional of the uncertain measure this is a minimax game in which the controller measure seeks to minimize the pay off while the disturbance measure aims at maximizing the pay off salient properties of the minimax solution are derived including a characterization of the optimal sensitivity reduction computation of the induced norm monotonicity properties of minimax solution and relations between dissipation and relative entropy of the system the theory is developed in an abstract setting and then applied to nonlinear partially observable continuous time uncertain controlled systems in which the nominal and uncertain systems are described by conditional distributions in addition existence of the optimal control policy among the class of policies known as wide sense control laws is shown and an explicit formulae for the worst case conditional measure is derived the results are applied to linear quadratic gaussian problems

the riemann problem for the two dimensional zero pressure euler equations abstract the riemann problem for the two dimensional zero pressure euler equations is considered the initial data are constant values in each quadrant which satisfy an assumption that each initial discontinuity projects only one two dimensional wave the phenomenon of two dimensional delta shock wave with a dirac delta function in both density and internal energy is identified both generalized rankine hugoniot relation and entropy condition for this type of two dimensional delta shock wave are proposed the qualitative behavior of entropy solutions to this relation with certain special initial data is established based on these preparations we obtain twenty three explicit solutions and their corresponding criteria in particular the mach reflection like patterns arise in the exact solutions

convergence of relaxed multisplitting usaor methods for h matrices linear systems relaxed technique is one of the techniques for improving convergence rate of splitting iterative methods in this paper based on the methods in frommer and mayer a frommer f mayer convergene of relaxed parallel multisplitting methods linear algebra and its applications 119 1989 141 152 and zhang et al l t zhang t z huang t x gu global relaxed non stationary multisplitting multi parameters methods international journal of computer mathematics 85 2 2008 211 224 we present local relaxed parallel multisplitting method global relaxed parallel multisplitting method local relaxed non stationary parallel multisplitting multi parameters method and global relaxed non stationary parallel multisplitting multi parameters method and study the convergence of our methods associated with usaor multisplitting for solving a large sparse linear system whose coefficient matrix is an h matrix when choosing the approximately optimal relaxed parameters our methods have faster convergence rate which is showed through numerical examples c 2008 elsevier inc all rights reserved

a pregenerator for burgers equation forced by conservative noise we consider the stationarity of a burgers equation with an external random force of gradient type in one space dimension the expected stationary measure is the white noise measure on the space of tempered distributions as a consequence the nonlinearity of the formal equation u t u u x u xx x is ill defined introducing a pregenerator we can formulate a generalized martingale problem leading to a meaningful version of the formal equation which was an open problem

the stochastic heat equation feynman kac formula and intermittence we study in one space dimension the heat equation with a random potential that is a white noise in space and time this equation is a linearized model for the evolution of a scalar field in a space time dependent random medium it has also been related to the distribution of two dimensional directed polymers in a random environment to the kpz model of growing interfaces and to the burgers equation with conservative noise we show how the solution can be expressed via a generalized feynman kac formula we then investigate the statistical properties the two point correlation function is explicitly computed and the intermittence of the solution is proven this analysis is carried out showing how the statistical moments can be expressed through local times of independent brownian motions

exponential linear quadratic optimal control with discounting a direct derivation of the solution of the discounted exponential linear quadratic gaussian optimal control problem is presented some extensions to partial observations problems are also given the solution involves time varying riccati equations providing an example of an infinite horizon constant coefficient regulator problem with a time varying optimal control

convergence of parallel multisplitting iterative methods for m matrices abstract despite the major recent activity in parallel processing few effective new algorithms designed exclusively for multiprocessors have been put forth one such is the multisplitting iterative algorithm suggested by o leary and white although they have given some sufficient conditions for convergence a general convergence theory has not been developed even for the classical situation where the coefficient matrix a is an m matrix or is symmetric positive definite in this paper we study the m matrix case in detail the multisplitting process for a rn n is recast as an ordinary iterative process for a certain block matrix a r kn kn where k is the number of processors and standard convergence results are used to develop a convergence theory for multisplitting iterative methods where a is an m matrix comparison results between multisplitting methods are established in terms of monotonic norms and for the case where a is irreducible in terms of the asymptotic convergence rate a key observation here is that in certain cases the rate of global convergence of these parallel iterative methods is inherent in the splitting of a and is independent of the manner in which the work is distributed among the processors thus in general one can distribute the work for load balancing purposes without affecting the convergence rate

stochastic nonlinear minimax dynamic games with noisy measurements this note is concerned with nonlinear stochastic minimax dynamic games which are subject to noisy measurements the minimizing players are control inputs while the maximizing players are square integrable stochastic processes the minimax dynamic game is formulated using an information state which depends on the paths of the observed processes the information state satisfies a partial differential equation of the hamilton jacobi bellman hjb type the hjb equation is employed to characterize the dissipation properties of the system to derive a separation theorem between the design of the estimator and the controller and to introduce a certainty equivalence principle along the lines of whittle finally the separation theorem and the certainty equi valence principle are applied to solve the linear quadratic gaussian minimax game the results of this note generalize the l sup 2 gain of deterministic systems to stochastic analogs they are related to the controller design of stochastic systems which employ risk sensitive performance criteria and to the controller design of deterministic systems which employ minimax performance criteria

a musta scheme for a nonconservative two fluid model we present a multistage centered scheme of the kind proposed by toro appl numer math 56 2006 pp 1464 1479 for numerically resolving the simultaneous flow of two fluids through a transport pipeline this model contains nonconservative terms in both the temporal and spatial derivatives and an extension of the standard numerical framework for conservation laws is needed in this paper we rewrite the model in an equivalent mathematical form eliminating the nonconservative time derivatives this allows us to use the framework described by pares siam j numer anal 44 2006 pp 300 321 we develop force and musta type schemes which are consistent with pares formalism numerical simulations demonstrate a high degree of stability of our proposed schemes comparisons with the roe and rusanov schemes indicate that convergence to near identical solutions is obtained when the nonconservative terms are discretized with respect to the same evaluation of the path dependent integrals however if the schemes are not mutually formally path consistent in the sense of pares different converged solutions are obtained

risk sensitivity large deviations and stochastic control abstract an exponential function of cost is adopted as a risk sensitive criterion and reasons given that this choice should be a natural one it is shown that the analysis leads to risk sensitive versions of the certainty equivalence principle separation principle and of the maximum principle and that these have a validity even outside the usual linear quadratic gaussian framework the methods are applied to some simple examples of economic interest

the equivalence between infinite horizon optimal control of stochastic systems with exponential of integral performance index and stochastic differential games a new method based on the theory of large deviations from the invariant measure is introduced for the analysis of stochastic systems with an infinite horizon exponential of integral performance index it is shown that the infinite horizon optimal exponential of integral stochastic control problem is equivalent to a stationary stochastic differential game for an auxiliary system as an application of the developed technique the infinite horizon risk sensitive lqg problem is analyzed for both the completely observed and partially observed case

dynamic programming subject to total variation distance ambiguity the aim of this paper is to address optimality of stochastic control strategies via dynamic programming subject to total variation distance ambiguity on the conditional distribution of the controlled process we formulate the stochastic control problem using minimax theory in which the control minimizes the payoff while the conditional distribution from the total variation distance set maximizes it first we investigate the maximization of a linear functional on the space of probability measures on abstract spaces among those probability measures which are within a total variation distance from a nominal probability measure and then we give the maximizing probability measure in closed form second we utilize the solution of the maximization to solve minimax stochastic control with deterministic control strategies under a markovian and a non markovian assumption on the conditional distributions of the controlled process the results of this part include 1 minimax optimization subject to total va

solving the kpz equation we introduce a new concept of solution to the kpz equation which is shown to extend the classical cole hopf solution this notion provides a factorisation of the cole hopf solution map into a universal measurable map from the probability space into an explicitly described auxiliary metric space composed with a new solution map that has very good continuity properties the advantage of such a formulation is that it essentially provides a pathwise notion of a solution together with a very detailed approximation theory in particular our construction completely bypasses the cole hopf transform thus laying the groundwork for proving that the kpz equation describes the uctuations of systems in the kpz universality class as a corollary of our construction we obtain very detailed new regularity results about the solution as well as its derivative with respect to the initial condition other byproducts of the proof include an explicit approximation to the stationary solution of the kpz equation a well posedness result for the fokker planck equation associated to a particle diusing in a rough space time dependent potential and a new periodic homogenisation result for the heat equation with a space time periodic potential one ingredient in our construction is an example of a non gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge

convergence improvement of relaxed multisplitting usaor methods for h matrices linear systems in this paper based on local relaxed parallel multisplitting usaor lusaor method presented by zhang et al 2008 and similar ideas used by zhang and li 2014 we further analyze local relaxed parallel multisplitting usaor lusaor method and obtain the better convergence results compared to zhang et al s when the system matrix is an h matrix moreover convergence graph and numerical examples clearly show that our new convergence domain is wider

a two space dimensional semilinear heat equation perturbed by gaussian white noise a two space dimensional heat equation perturbed by a white noise in a bounded volume is considered the equation is perturbed by a non linearity of the type f au where means wick re ordering with respect to the free solution a are small parameters u denotes a solution f is the fourier transform of a complex measure with compact support

interactions of delta shock waves for the transport equations with split delta functions this paper is concerned with the interactions of shock waves and the vacuum states between the two contact discontinuities for the transport equations the solutions are obtained constructively when the initial data are three piecewise constant states the global structure and large time asymptotic behaviors of the solutions are analyzed case by case moreover it can be found that the riemann solutions are stable for such small perturbations with initial data by studying the limits of the solutions when the perturbed parameter e tends to zero

partially observable nonlinear risk sensitive control problems dynamic programming and verification theorems in this paper we consider continuous time partially observable optimal control problems with exponential of integral cost criteria we derive a rigorous verification theorem when the state and control enter nonlinear in the dynamics in addition we show that the quadratic sensor problem is estimation solvable with respect to a certain cost criterion the framework relies on dynamic programming and the hamilton jacobi theory

the stochastic burgers equation we study burgers equation perturbed by a white noise in space and time we prove the existence of solutions by showing that the cole hopf transformation is meaningful also in the stochastic case the problem is thus reduced to the anaylsis of a linear equation with multiplicative half white noise an explicit solution of the latter is constructed through a generalized feynman kac formula typical properties of the trajectories are then discussed a technical result concerning the regularizing effect of the convolution with the heat kernel is proved for stochastic integrals

hybrid flux splitting schemes for a two phase flow model in this paper we deal with the construction of hybrid flux vector splitting fvs schemes and flux difference splitting fds schemes for a two phase model for one dimensional flow the model consists of two mass conservation equations one for each phase and a common momentum equation the complexity of this model as far as numerical computation is concerned is related to the fact that the flux cannot be expressed in terms of its conservative variables this is the motivation for studying numerical schemes which are not based on approximate riemann solvers and or calculations of jacobian matrix this work concerns the extension of an fvs type scheme a van leer type scheme and an advection upstream splitting method ausm type scheme to the current two phase model our schemes are obtained through natural extensions of corresponding schemes studied by y wada and m s liou 1997 siam j sci comput 18 633 657 for euler equations we explore the various schemes for flow cases which involve both fast and slow transients in particular we demonstrate that the fvs scheme is able to capture fast propagating acoustic waves in a monotone way while it introduces an excessive numerical dissipation at volume fraction contact steady and moving discontinuities on the other hand the ausm scheme gives accurate resolution of contact discontinuities but produces oscillatory approximations of acoustic waves this motivates us to propose other hybrid fvs fds schemes obtained by removing numerical dissipation at contact discontinuities in the fvs and van leer schemes

optimal control of uncertain stochastic systems subject to total variation distance uncertainty this paper is concerned with optimization of uncertain stochastic systems in which uncertainty is described by a total variation distance constraint between the measures induced by the uncertain systems and the measure induced by the nominal system while the payoff is a linear functional of the uncertain measure robustness at the abstract setting is formulated as a minimax game in which the control seeks to minimize the payoff over the admissible controls while the uncertainty aims at maximizing it over the total variation distance constraint it is shown that the maximizing measure in the total variation distance constraint exists while the resulting payoff is a linear combination of l 1 and l infty norms further the maximizing measure is characterized by a linear combination of a tilted measure and the nominal measure giving rise to a payoff which is a nonlinear functional on the space of measures to be minimized over the admissible controls the abstract formulation and results are subsequ

microscopic profiles of shock waves and ambiguities in multiplications of distributions a rigorous mathematical method is introduced for studying discontinuous solutions for systems in nonconservative form the evidence of the agreement between the theoretical and numerical results is shown on simplified models of physics

distributions as initial values in a triangular hyperbolic system of conservation laws the present paper concerns the system ut u x 0 vt u v x 0 having distributions as initial conditions under certain conditions and supposing functions we explicitly solve this cauchy problem within a convenient space of distributions u v for this purpose a consistent extension of the classical solution concept defined in the setting of a distributional product not constructed by approximation processes is used shock waves shock waves and also waves defined by distributions that are not measures are presented explicitly as examples this study is carried out without assuming classical results about conservation laws for reader s convenience a brief survey of the distributional product is also included

generalized solution to a system of conservation laws which is not strictly hyperbolic abstract in this paper we study a non strictly hyperbolic system of conservation laws when viscosity is present and when viscosity is zero which has been studied in 12 we show the existence and uniqueness of the solution in the space of generalized functions of colombeau for the viscous problem and construct a solution to the inviscid system in the sense of association also we construct a solution using shadow wave approach 17 and volpert product which was partially determined as vanishing viscosity limit in 12

risk sensitive control for a class of nonlinear systems with multiplicative noise abstract in this paper we consider the problem of optimal control for a class of nonlinear stochastic systems with multiplicative noise the nonlinearity consists of quadratic terms in the state and control variables the optimality criteria are of a risk sensitive and generalised risk sensitive type the optimal control is found in an explicit closed form by the completion of squares and the change of measure methods as applications we outline two special cases of our results we show that a subset of the class of models which we consider leads to a generalised quadratic affine term structure model qatsm for interest rates we also demonstrate how our results lead to generalisation of exponential utility as a criterion in optimal investment

applicable stochastic control from theory to practice stochastic control has come of age it is currently applied in a wide variety of fields spanning cancer research and chemotherapy economics and finance engineering management and many other areas of scientific and social investigation both theoretical models formulated to elucidate some characteristics of a problem under study and practical applications which have sought to provide specific answers to well stated problems have been the subjects of intense research for example economists have simultaneously groped with the underlying theoretical constructs justifying the adoption of particular econometric models while at the same time attempting to devise algorithms and optimization approaches to provide economic policy guidelines based on these models in the same spirit t remendous interest in options management which are based on stochastic dynamic models has stimulated much research and numerical analyses not only in universities but also in the back rooms of trading firms the usefulness of stochastic control was recognized once it became clear that it can be used as a unifying framework for treating multi stage or multi period optimization problems subject to stochastic disturbances however just as it is convenient as a formalism to formulate such problems it is becoming increasingly clear that it is practically difficult to solve them there are in fact only few problems one can solve completely and obtain optimal solutions for this reason there is today an increased awareness that a broad range of techniques including stochastic approximations perturbation techniques markov decision processes etc are an essential part of stochastic control kits it is in such a spirit that computer software such as pandore of inria the adaptive control algorithms for econometric models developed by kendrick and many others has been devised providing thereby computer aided expertise for managing stochastic control problems this software is essential even after the great effort applied to solving such problems over the last three decades ever since bellman s principle of optimality there are very few non trivial problems we can solve theoretically these include the well known linear quadratic stochastic control scp problem some version of the multi armed bandit problem and some specific linear optimization problems for example even the adaptive control problems studied over the last few years by kendrick and his co workers and presumed to be convex are believed today to include some unspecified nonconvexities private communication by d kendrick and hans amman in this spirit the intent in this paper is to introduce salient aspects of stochastic control which are useful for applications in management a review of applications can be found in neck 1984 for example as well as in books such as tapiero 1977 1988 bensoussan hurst and naslund 1974 bensoussan kleindorfer and tapiero 1978 1980 sethi and thompson 1981 kamien and schwarz 1981 malliaris and brock 1982 and whittle 1982 further due to an awesome amount of research published in this area appearing under various connotations such a task has become unwieldy rather

preconditioned parallel multisplitting usaor method for h matrices linear systems in this paper the preconditioned multisplitting usaor method is established for solving the system of linear equations the convergence and comparison results of the method are given when the coefficient matrices of the linear systems are h matrices the method for h matrices is proved to be more efficient than the multisplitting usaor method for m matrices finally a numerical example is given to illustrate the efficiency of our method

convergence of relaxed parallel multisplitting methods with different weighting schemes1supported by the state major key project for basic researches and the doctorial point foundation of china 1 multisplitting methods are parallel methods for the solution of a linear system ax b it has been observed that the convergence of multisplitting methods is often improved if some kind of relaxation and some different weighting other than the traditional postweighting are used in this paper by using a result on extrapolation method cf lemma 6 below and the comparison theorem of elsner l elsner numer math 56 1989 283 289 we study the convergence of two different variants of relaxed multisplitting methods with different weighting schemes the results obtained considerably improve and generalize the previous ones moreover some comparison theorems are also presented when a is a monotone matrix or an m matrix

δ shock waves as a new type of solutions to systems of conservation laws abstract a concept of a new type of singular solutions to systems of conservation laws is introduced it is so called n shock wave where n is n th derivative of the dirac delta function n 1 2 in this paper the case n 1 is studied in details we introduce a definition of shock wave type solution for the system u t f u x 0 v t f u v x 0 w t f u v 2 f u w x 0 within the framework of this definition the rankine hugoniot conditions for shock are derived and analyzed from geometrical point of view we prove shock balance relations connected with area transportation finally a solitary shock wave type solution to the cauchy problem of the system of conservation laws u t u 2 x 0 v t 2 u v x 0 w t 2 v 2 u w x 0 with piecewise continuous initial data is constructed these results first show that solutions of systems of conservation laws can develop not only dirac measures as in the case of shocks but their derivatives as well

the iterative methods for centrosymmetric matrices in this paper we give some new centrosymmetric splittings of a centrosymmetric matrix a and obtain corresponding iterative methods for a linear system ax b these iterative schemes are particularly based on the centrosymmetric property of a and aimed at reduction of the cost of computation and storage

shadow waves entropies and interactions for delta and singular shocks this paper deals with two key problems for delta or singular shock solutions to systems of conservation laws that of their entropy admissibility conditions which is connected to the notorious uniqueness problem and that of their interaction we choose to represent systems of conservation laws by nets of functions which are piecewise constant or constant with respect only to the space variable here called shadow waves all the calculations can then be done on each element of such nets using only the usual rankine hugoniot conditions a 3 3 pressureless gas dynamics model is the main example in the paper

on the optimal control of stochastic systems with an exponential of integral performance index abstract in this paper a theory of optimal control is developed for stochastic systems whose performance is measured by the exponential of an integral form such a formulation of the cost function is shown to be not only general and useful but also analytically tractable starting with very general classes of stochastic systems optimality conditions are obtained which exploit the multiplicative decomposability of the exponential of integral form specializing to partially observed systems of stochastic differential equations with brownian motion disturbances optimality conditions are obtained which parallel those for systems with integral costs also treated are the special cases of linear systems with exponential of quadratic costs for which explicit optimal controls are obtainable in addition several general results of independent interest are obtained which concern optimality of stochastic systems

weak asymptotic methods for scalar equations and systems abstract in this paper we show how one can construct families of continuous functions which satisfy asymptotically scalar equations with discontinuous nonlinearity and systems having irregular solutions this construction produces weak asymptotic methods which are issued from maslov asymptotic analysis we obtain a sequence of functions which tend to satisfy the equation s in the weak sense in the space variable and in the strong sense in the time variable to this end we reduce the partial differential equations to a family of ordinary differential equations in a classical banach space for scalar equations we prove that the initial value problem is well posed in the l 1 sense for the approximate solutions we construct then we prove that this method gives back the widely accepted solutions when they are known for systems we obtain existence in the general case and uniqueness in the analytic case using an abstract cauchy kovalevska theorem

grpm style methods and comparisons of convergent and divergent rates relaxed technique is one of techniques for improving convergence rate of splitting iterative methods based on local relaxed method and system relaxed method of parallel multisplitting frommer and mayer a frommer g mayer convergence of relaxed parallel multisplitting methods linear algebra appl 119 1989 141 152 we give the global relaxed parallel multisplitting grpm method by introducing some relaxed parameters and study the convergence of our methods grpm style when the coefficient matrices are h matrices numerical experiments show that when choosing the approximately optimal relaxed parameters our methods have faster convergent rate than the methods in chang d w chang convergence analysis of the parallel multisplitting tor methods j comput appl math 72 1996 169 177 frommer and mayer a frommer g mayer convergence of relaxed parallel multisplitting methods linear algebra appl 119 1989 141 152 furthermore the convergent and divergent rates of local relaxed parallel multisplitting lrpm style methods about multislitting tor aor sor g s extraolated jacobi methods as well as jacobi iterative method are compared in detail

centralized and decentralized solutions of the linear exponential gaussian problem a particular class of stochastic control problems constrained to different information patterns is considered this class consists of minimizing the expectation of an exponential cost criterion with quadratic argument subject to a discrete time gauss markov dynamic system i e the linear exponential gaussian leg control problem besides the one step delayed information pattern previously considered the classical and the one step delayed information sharing osdis patterns are assumed after determining the centralized controller based upon the classical information pattern the optimal decentralized controller based upon the osdis pattern and the solution to a static team problem is found to be affine a unifying approach to determine controllers based upon these three information patterns is obtained by noting that the value of a quadratic exponent of an exponential function is independent of the information structure both necessary and sufficient conditions for the controllers to be minimizing are obtained regardless of the exponential form the negative exponential form is included which is unimodal but not convex

the elastoplastic shock problem as an example of the resolution of ambiguities in the multiplication of distributions one dimensional collisions between two isotropic solids in which the equations of physics lead to multiplications of distributions are considered based on this example a general physicomathematical method to be adapted to each particular case is proposed to resolve the ambiguity inherent in such products this can be achieved with the aid of a new mathematical theory of generalized functions which permits dealing with mathematical phenomena of a microscopic nature that govern products of distributions having singularities at the same point this tool has recently been applied in various situations in continuum mechanics in which the equations of physics lead to heuristic products of distributions one obtains new algebraic formulas in the simplest cases and new numerical schemes in more general cases the key to the resolution of ambiguities lies in more precise statements of the laws of physics than are permitted within distribution theory and have no analog in classical analysis so that in general a resolution cannot be obtained from formal calculations

new finite dimensional risk sensitive filters small noise limits the paper is concerned with continuous time nonlinear risk sensitive filters it is shown that for large classes of nonlinearities entering both the dynamics and measurements these filters are finite dimensional generalizations of the benes filters specific examples are discussed the small noise limiting analog is discussed using change of probability measures

nonlinear generalized functions and jump conditions for a standard one pressure liquid gas model abstract the mixture of a liquid and a gas is classically represented by one pressure models these models are a system of pdes in nonconservative form and shock wave solutions do not make sense within the theory of distributions they give rise to products of distributions that are not defined within distribution theory but they make sense by applying a theory of nonlinear generalized functions to these equations in contrast to the familiar case of conservative systems the jump conditions cannot be calculated a priori jump conditions for these nonconservative systems can be obtained using the theory of nonlinear generalized functions by inserting some adequate physical information into the equations the physical information that we propose to insert for the one pressure models of a mixture of a liquid and a gas is a natural mathematical expression in the theory of nonlinear generalized functions of the fact that liquids are practically incompressible while gases are very compressible and so they do not satisfy equally well their respective state laws on the shock waves this modelization gives well defined explicit jump conditions the great numerical difficulty for solving numerically nonconservative systems is due to the fact that slightly different numerical schemes can give significantly different results the jump conditions obtained above permit to select the numerical schemes and validate those that give numerical solutions that satisfy these jump conditions which can be an important piece of information in the absence of other explicit discontinuous solutions and of precise observational results we expose with care the mathematical originality of the theory of nonlinear generalized functions an original abstract analysis issued by the leopoldo nachbin team on infinite dimensional holomorphy that permits to state mathematically physical facts that cannot be formulated within distribution theory and are the key for the removal of ambiguities that classically appear when one tries to calculate on multiplications of distributions that occur in the differential equations of physics

generalized variational principles global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics we study systems of conservation laws arising in two models of adhesion particle dynamics the first is the system of free particles which stick under collision the second is a system of gravitationally interacting particles which also stick under collision in both cases mass and momentum are conserved at the collisions so the dynamics is described by 2 2 systems of conservations laws we show that for these systems global weak solutions can be constructed explicitly using the initial data by a procedure analogous to the lax oleinik variational principle for scalar conservation laws however this weak solution is not unique among weak solutions satisfying the standard entropy condition we also study a modified gravitational model in which instead of momentum some other weighted velocity is conserved at collisions for this model we prove both existence and uniqueness of global weak solutions we then study the qualitative behavior of the solutions with random initial data we show that for continuous but nowhere differentiable random initial velocities all masses immediately concentrate on points even though they were continuously distributed initially and the set of shock locations is dense

limits of nonlinear weak asymptotic methods we propose a mathematical limit of l1 stable weak asymptotic methods a family of l1 stable approximate solutions is transformed into a normal family of holomorphic functions defined in a complex domain having the real space on its boundary this provides a holomorphic function which is the same mathematical object as the solutions from explicit calculations the weak limit of the approximate solutions from weak asymptotic methods in the space of bounded radon measures is recovered as a boundary value of this holomorphic function

comparisons of weak regular splittings and multisplitting methods comparison results for weak regular splittings of monotone matrices are derived as an application we get upper and lower bounds for the convergence rate of iterative procedures based on multisplittings this yields a very simple proof of results of neumann plemmons on upper bounds and establishes lower bounds which has in special cases been conjectured by these authors

spaces of weighted measures for conservation laws with singular shock solutions abstract we study a model system of two strictly hyperbolic conservation laws which is genuinely nonlinear but for which the riemann problem has no global solution singular solutions are defined by means of a generalized rankine hugoniot relation and an overcompressive condition on the discontinuity we show that approximate solutions which can be constructed by several standard methods converge in a weighted measure space and that the error in the approximation converges to zero viscous approximations satisfy approximate entropy inequalities which imply the overcompressive condition

linear exponential quadratic stochastic differential games in this technical note a two person noncooperative stochastic differential game described by a linear stochastic equation with a brownian motion and a payoff that is the exponential of a quadratic functional in the state and the control strategies of the two players is explicitly solved the optimal strategies form a nash equilibrium the value of the game is also determined explicitly the solution of this problem is naturally related to a linear quadratic stochastic differential game and a linear exponential quadratic control problem

on convergence of nested stationary iterative methods abstract we study the convergence of nested stationary iterative methods according to lanzkron rose and szyld in contrast with their assumption that the co efficient matrix is monotone our assumption on the coefficient matrix includes h matrices

burgers turbulence and random lagrangian systems we consider a spatially periodic inviscid random forced burgers equation in arbitrary dimension and the random time dependent lagrangian system related to it we construct a unique stationary distribution for viscosity solutions of the burgers equation we also show that with probability 1 there exists a unique minimizing trajectory for the random lagrangian system which generates a non trivial ergodic invariant measure for the non random skew product extension of the lagrangian system

a nearly implicit hydrodynamic numerical scheme for two phase flows this article presents an extension of the semi implicit numerical scheme for two phase flow simulation the extension uses an implicit evaluation of the convective fluxes and thus eliminates the material courant stability restriction the new algorithm involves a two step approach for the mass and energy equations and a single fully implicit step for the momentum evaluations some analysis and accuracy considerations for the scheme are also presented

delta shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws abstract for simple models of hyperbolic systems of conservation laws we study a new type of nonlinear hyperbolic wave a delta shock wave which is a dirac delta function supported on a shock we prove that delta shock waves are w limits in l1 of solutions to some reasonable viscous perturbations as the viscosity vanishes further we solve the multiplication problem of a delta function with a discontinuous function to show that delta shock waves satisfy the equations in the sense of distributions under suitable generalized rankine hugoniot and entropy conditions we establish the existence and uniqueness of solutions involving delta shock waves for the riemann problems the existence of solutions to the cauchy problem is also investigated

risk sensitive estimation and a differential game a large deviation result is employed to solve the state estimation problem of a continuous time gauss markov system with an exponential cost the exponential cost is the expected value of an exponential function of the state estimation error a scalar spl theta appearing in the cost termed as the risk factor determines the penalty on the higher order moments of the error in contrast to the minimum variance estimate penalty on large deviations of the estimation error is possible

stochastic burgers and kpz equations from particle systems we consider two strictly related models a solid on solid interface growth model and the weakly asymmetric exclusion process both on the one dimensional lattice it has been proven that in the diffusive scaling limit the density field of the weakly asymmetric exclusion process evolves according to the burgers equation and the fluctuation field converges to a generalized ornstein uhlenbeck process we analyze instead the density fluctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the non linear burgers equation with a random noise on the density current for the solid on solid model we prove that the fluctuation field of the interface profile if suitably rescaled converges to the kardar parisi zhang equation this provides a microscopic justification of the so called kinetic roughening i e the non gaussian fluctuations in some non equilibrium processes our main tool is the cole hopf transformation and its microscopic version we also develop a mathematical theory for the macroscopic equations

generalized solutions of the multidimensional stochastic burgers equation abstract we introduce a new concept of weak solution for the conservative stochastic burgers equation in any dimension the definition is based on weak solution concepts introduced by various authors in order to make sense of equations which do not have solutions in the sense of distributions in one dimension the solution reduces to the classical distributional solution of the 1 d stochastic burgers equation

a simple numerical scheme for the 3 d system of ideal gases and a study of approximation in the sense of distributions we present a 3 d numerical scheme for the approximation of the system of gas dynamics we prove that as long as the boundedness of the velocity field in the cfl condition and the positiveness of the total energy are numerically verified when the space step tends to 0 the scheme provides a numerical solution which satisfies the conservation laws in the sense of distributions modulo a small remainder of order one in the space step and a slightly weaker order for the state law in the case of irregular approximate solutions in the case of classical tests sod woodward colella toro one provides numerical verifications that these conditions are verified even for very small values of the space step and correspondingly one observes that the scheme produces the known exact solutions with great precision full proofs are given in the pressureless case the scheme produces the numerical results on the six 2 d riemann problems presented by p d lax p d lax mathematics and physics bull ams 45 1 2008 135 152 up to the smallest detail therefore this result contributes to a mathematical justification of the fact that the numerical results presented by p d lax on these 2 d riemann problems are approximations of a solution this simple order one low cost 3 d scheme is obtained from the convection pressure correction method proposed by le roux et al r baraille g bourdin f dubois a y le roux une version a pas fractionnaire du schema de godunov pour l hydrodynamique comptes rendus acad sci paris 314 1992 147 152

an approximate linearized riemann solver for a two fluid model an approximate linearized riemann solver is presented for the numerical simulation of two phase flows this new solver is based on a linearization of nonconservative products and uses an extension of roe s approximate riemann solver the scheme is applied to shock tube problem and to a standard test for two fluid codes

dynamics of propagation and interaction of δ shock waves in conservation law systems abstract we introduce a new definition of a shock wave type solution for a class of systems of conservation laws in the one dimensional case the weak asymptotics method developed by the authors is used to construct formulas describing the propagation and interaction of shock waves the dynamics of merging two shocks is described by explicit formulas continuously in time

riemann problems for a class of coupled hyperbolic systems of conservation laws abstract we study a class of coupled hyperbolic systems of conservation laws which contain the one dimensional zero pressure gas dynamics as a prototypical example the riemann problems are solved constructively the riemann solutions exactly include two kinds delta shock wave solutions and vacuum solutions under the generalized rankine hugoniot relation and entropy condition all of the existence uniqueness and stability of solutions to viscous perturbations are proved two typical examples are presented finally

numerical solutions of one pressure models in multifluid flows flows of a mixture of several fluids can be described by one pressure models their formulation is formally simple and their efficiency has been recognized but from the mathematical viewpoint they present severe peculiarities they are not hyperbolic appear to be ill posed and are in nonconservative form so they show ambiguous products of distributions in case of shock waves a resolution of these ambiguities is proposed it gives jump conditions similar to the classical rankine hugoniot jump conditions of conservative systems this resolution attempts to solve the riemann problem and then the cauchy problem by godunov type schemes following nonconservative techniques initiated by a y le roux and collaborators

risk sensitive filters for recursive estimation of motion from images in this paper an extended risk sensitive filter ersf is used to estimate the motion parameters of an object recursively from a sequence of monocular images the effect of varying the risk factor spl theta on the estimation error is examined the performance of the filter is compared with the extended kalman filter ekf and the theoretical cramer rao lower bound when the risk factor spl theta and the uncertainty in the measurement noise are large the initial estimation error of the ersf is less than that of the corresponding ekf the ersf is also found to converge to the steady state value of the error faster than the ekf in situations when the uncertainty in the initial estimate is large and the ekf diverges the ersf converges with small errors in confirmation with the theory as spl theta tends to zero the behavior of the ersf is the same as that of the ekf

asymptotic study of the initial value problem to a standard one pressure model of multifluid flows in nondivergence form abstract we construct families of approximate solutions to the initial value problem and provide complete mathematical proofs that they tend to satisfy the standard system of isothermal one pressure two fluid flows in 1 d when the data are l 1 in densities and l in velocities to this end we use a method that reduces this system of pdes to a family of systems of four odes in banach spaces whose smooth solutions are these approximate solutions this method is constructive using standard numerical methods for odes one can easily and accurately compute these approximate solutions which therefore from the mathematical proof can serve for comparison with numerical schemes one observes agreement with previously known solutions from scientific computing evje and flatten 2003 16 we show that one recovers the solutions of these authors exactly in one case with a slight difference in another case then we propose an efficient numerical scheme for the original system of two fluid flows and show it gives back exactly the same results as the theoretical solutions obtained above

a variational representation for certain functionals of brownian motion in this paper we show that the variational representation log ee f w inf v e 1 2 int 0 1 parallel v s parallel 2 ds f w int 0 cdot v s ds holds where w is a standard d dimensional brownian motion f is any bounded measurable function that maps c 0 1 mathbb r d into mathbb r and the infimum is over all processes v that are progressively measurable with respect to the augmentation of the filtration generated by w an application is made to a problem concerned with large deviations and an extension to unbounded functions is given

a risk sensitive maximum principle the case of imperfect state observation the risk sensitive maximum principle for optimal stochastic control derived by the author in an earlier work system control letters vol 15 1990 is restated this is an immediate generalization of the classic pontryagin principle to which it reduces in the deterministic case and is expressed immediately in terms of observables it is derived on the assumption that the criterion function is the exponential of an additive cost function and is exact under linear quadratic gaussian assumptions but is otherwise valid as a large deviation approximation the principle is extended to the case of imperfect state observation after preliminary establishment of a certainty equivalence principle the derivation yields as byproduct a large deviation version of the updating equation for nonlinear filtering the development is heuristic it is believed that the mathematical arguments given are the essential ones and provide a self contained treatment at this level

a hamiltonian formulation of risk sensitive linear quadratic gaussian control we consider the optimal control problem with time homogeneous linear and in general non markov dynamics and risk sensitive criterion appeal to the extremal principle prescribed by the risk sensitive certainty equivalence principle rscep yields a symmetric equation system indicating that the extended hamiltonian formulation generalizes naturally to the risk sensitive case the conjugate variable of a hamiltonian formulation now has an interpretation in terms of forecasted process noise and the rscep in fact provides a stochastic maximum principle for which all variables have a clear interpretation and the desired measurability properties in the infinite horizon case meaningful under generalized controllability conditions optimal control is determined explicitly in terms of a canonical factorization for the case of imperfect process observation the rscep leads to coupled equation systems that can again be solved in terms of canonical factorizations in the time invariant stationary case

convergence of relaxed parallel multisplitting methods abstract multisplitting methods are parallel methods for the solution of a linear system ax b it has been observed that the convergence of multisplitting methods is often improved if some kind of relaxation is used we investigate two different variants of relaxed multisplitting methods if a is an h matrix these methods converge if the relaxation parameter is from an interval 0 0 with 0 1