numerical solutions for stochastic differential games with regime switching this paper is concerned with numerical methods for stochastic differential games of regime switching diffusions numerical methods using markov chain approximation techniques are developed a new proof of the existence of a saddle point for the stochastic differential game is provided this new proof enables us to treat certain systems with nonseparable in controls structure convergence of the algorithms is derived by means of weak convergence methods in addition examples are also provided for demonstration purposes

risk sensitive and robust escape criteria the problem of controlling a noisy process so as to prevent it from leaving a prescribed set has a number of interesting applications in this paper new approaches to this problem are considered first a risk sensitive criterion for a stochastic diffusion process model is examined and it is shown that the value is a classical solution of a related pde the qualitative properties of this criterion are favorably contrasted with those of existing criteria in the risk averse limit it is proved that in the risk averse limit the value of the risk sensitive criterion converges to a viscosity solution of a first order pde it is then demonstrated that the value function of a deterministic differential game is also a viscosity solution to the pde this game gives a robust control formulation of the escape time problem and is analogous to h infty control in particular the opposing player attempts to push the process out of the prescribed set and suffers an l2 cost for his efforts lower bounds on the escape time as a function of this cost are obtained

stable particle like solutions to the nonlinear wave equations of three dimensional excitable media so called excitable media support nonlinear waves that propagate as pulses e g the action potential in nerve fibers and heart muscle and the oxidation pulse in certain chemical reactions in two dimensional media the right stimulus will also create rotating vortices which radiate such pulses periodically in every direction their investigation in diverse excitable media has dramatically expanded in the usa and ussr since their discovery in both countries in 1970 in three dimensional media the pivot of the rotating structure is not a point but a curved line generically a closed ring rings linked together and knotted constitute topologically distinctive organizing centers for periodic activity throughout the medium it is of interest to understand how stimuli create organizing centers and how organizing centers persist or decay the numerically discovered stable organizing centers exhibit enough symmetries to encourage mathematical attention this paper illustrates organizing centers by computer g

a general mass conservative numerical solution for the unsaturated flow equation numerical approximations based on different forms of the governing partial differential equation can lead to significantly different results for unsaturated flow problems numerical solution based on the standard h based form of richards equation generally yields poor results characterized by large mass balance errors and erroneous estimates of infiltration depth conversely numerical solutions based on the mixed form of richards equation can be shown to possess the conservative property so that mass is perfectly conserved this leads to significant improvement in numerical solution performance while requiring no additional computational effort however use of the mass conservative method does not guarantee good solutions accurate solution of the unsaturated flow equation also requires use of a diagonal time or mass matrix only when diagonal time matrices are used can the solution be shown to obey a maximum principle which guarantees smooth nonoscillatory infiltration profiles this highlights the fact that proper treatment of the time derivative is critical in the numerical solution of unsaturated flow

mesh refinement and numerical sensitivity analysis for parameter calibration of partial differential equations we consider the calibration of parameters in physical models described by partial differential equations this task is formulated as a constrained optimization problem with a cost functional of least squares type using information obtained from measurements an important issue in the numerical solution of this type of problem is the control of the errors introduced first by discretization of the equations describing the physical model and second by measurement errors or other perturbations our strategy is as follows we suppose that the user defines an interest functional i which might depend on both the state variable and the parameters and which represents the goal of the computation first we propose an a posteriori error estimator which measures the error with respect to this functional this error estimator is used in an adaptive algorithm to construct economic meshes by local mesh refinement the proposed estimator requires the solution of an auxiliary linear equation second we address the question of sensitivity applying similar techniques as before we derive quantities which describe the influence of small changes in the measurements on the value of the interest functional these numbers which we call relative condition numbers give additional information on the problem under consideration they can be computed by means of the solution of the auxiliary problem determined before finally we demonstrate our approach at hand of a parameter calibration problem for a model flow problem

robust l2 gain control for nonlinear systems with projection dynamics and input constraints an example from traffic control we formulate the l 2 gain control problem for a general nonlinear state space system with projection dynamics in the state evolution and hard constraints on the set of admissible inputs we develop specific results for an example motivated by a traffic signal control problem a state feedback control with the desired properties is found in terms of the solution of an associated hamilton jacobi isaacs equation the storage function or value function of the associated game and the critical point of the associated hamiltonian function discontinuities in the resulting control as a function of the state and due to the boundary projection in the system dynamics lead to hybrid features of the closed loop system specifically jumps of the system description between two or more continuous time models trajectories for the closed loop dynamics must be interpreted as a differential set inclusion in the sense of filippov construction of the storage function is via a generalized stable invariant manifold for the flow of a discontinuous hamiltonian vector field which again must be interpreted in the sense of filippov for the traffic control model example the storage function is constructed explicitly the control resulting from this analysis for the traffic control example is a mathematically idealized averaged control which is not immediately implementable implementation issues for traffic problems will be discussed elsewhere

a conceptual model of the soil water retention curve a conceptual model based on the assumption that soil structure evolves from a uniform random fragmentation process is proposed to define the water retention function the fragmentation process determines the particle size distribution of the soil the transformation of particles volumes into pore volumes via a power function and the adoption of the capillarity equation lead to an expression for the water retention curve this expression presents two fitting parameters only the proposed model is tested on water retention data sets of 12 soils representing a wide range of soil textures from sand to clay the agreement between the fitted curves and the measured data is very good the performances of the model are also compared with those of the two parameter models of van genuchten 1980 and russo 1988 for the water retention function in general the proposed model exhibits increased flexibility and improves the fit at both the high and the low water contents range

estimating unsaturated soil hydraulic properties from tension disc infiltrometer data by numerical inversion tension disc infiltrometers are becoming increasingly popular devices for in situ measurement of the unsaturated hydraulic properties of soil tension infiltration data are generally used to evaluate the parametersks and ain gardner s exponential model of the unsaturated hydraulic conductivity either two measurements using different disc diameters or measurements with a single disc but using multiple pressure heads are then used in this paper we describe a parameter estimation procedure which combines the levenberg marquardt nonlinear parameter optimization method involving weighted least squares with a quasi three dimensional numerical model which solves the variably saturatedflow equation by numerical inversion of richards equation the unknown parameters in van genuchten s model of the unsaturated soil hydraulic properties are estimated from observed cumulative infiltration data during transient waterflow additional measurements of the pressure head or water content as well as a penalty function for constraining the unknown parameters to remain in some feasible region bayesian estimation can be optionally included into the parameter estimation procedure the problem of optimal sampling design that is selecting the best points in space and time for making measurements is addressed by studying the sensitivity of the objective function to changes in the optimized hydraulic parameters we calculate objective functions based on available cumulative infiltration pressure head and water content measurements and also on several combinations of these data the behavior of the objective function in three dimensional parameter space is evaluated by means of a series of two dimensional response surfaces the utility of the parameter estimation procedure is demonstrated using numerically generated data the sensitivity of the procedure to different initial estimates of the model parameters is also discussed

on level set regularization for highly ill posed distributed parameter estimation problems the recovery of a distributed parameter function with discontinuities from inverse problems with elliptic forward pdes is fraught with theoretical and practical difficulties better results are obtained for problems where the solution may take on at each point only one of two values thus yielding a shape recovery problem this article considers level set regularization for such problems however rather than explicitly integrating a time embedded pde to steady state which typically requires thousands of iterations methods based on gauss newton are applied more directly one of these can be viewed as damped gauss newton utilized to approximate the steady state equations which in turn are viewed as the necessary conditions of a tikhonov type regularization with a sharpening sub step at each iteration in practice this method is eclipsed however by a special finite time or levenberg marquardt type method which we call dynamic regularization applied to the output least squares formulation our stopping criterion for the iteration does not involve knowledge of the true solution the regularization functional is applied to the smooth level set function rather than to the discontinuous function to be recovered and the second focus of this article is on selecting this functional typical choices may lead to flat level sets that in turn cause ill conditioning we propose a new quartic non local regularization term that penalizes flatness and produces a smooth level set evolution and compare its performance to more usual choices two numerical test cases are considered a potential problem and the classical eit dc resistivity problem

a geometrical theory for spiral waves in excitable media in this paper we develop a geometrical theory for waves in excitable reacting media using singular perturbation arguments and dispersion of traveling plane wave trains we derive an approximate theory of wave front propagation which has strong resemblance to the geometrical diffraction theory of high frequency waves in hyperbolic systems governed by the eikonal equation using this theory we study the effect of curvature on waves in excitable media specifically rotating spiral patterns in planar regions from this theory we are able to determine the frequency and wavelength for spiral patterns in excitable nonoscillatory media

numerical approximations for stochastic differential games the markov chain approximation method is a widely used robust relatively easy to use and efficient family of methods for the bulk of stochastic control problems in continuous time for reflected jump diffusion type models it has been shown to converge under broad conditions and there are good algorithms for solving the numerical problems if the dimension is not too high versions of these methods have been used in applications to various two player differential and stochastic dynamic games for a long time and proofs of convergence are available for some cases mainly using pde type techniques in this paper purely probabilistic proofs of convergence are given for a broad class of such problems where the controls for the two players are separated in the dynamics and cost function and which cover a substantial class not dealt with in previous works discounted and stopping time cost functions are considered finite horizon problems and problems where the process is stopped on first hitting an a priori given boundary can be dealt with by adapting the methods of h j kushner and p dupuis numerical methods for stochastic control problems in continuous time 2nd ed springer verlag berlin new york 2001 as done in this paper for the treated problems the essential conditions are the weak sense existence and uniqueness of solutions an almost everywhere continuity condition and that a weak local consistency condition holds almost everywhere for the numerical approximations just as for the control problem there are extensions to problems with controlled variance and jumps

computational experience with confidence regions and confidence intervals for nonlinear least squares we present the results of a monte carlo study of the leading methods for constructing approximate confidence regions and confidence intervals for parameters estimated by nonlinear least squares we examine three variants of the linearization method the likelihood method and the lack of fit method the linearization method is computationally inexpensive produces easily understandable results and is widely used in practice the likelihood and lack of fit methods are much more expensive and more difficult to report in our tests both the likelihood and lack of fit methods perform very reliably all three variants of the linearization method however often grossly underestimate confidence regions and sometimes significantly underestimate confidence intervals the linearization method variant based solely on the jacobian matrix appears preferable to the two variants that use the full hessian matrix because it is less expensive more numerically stable and at least as accurate the bates and watts curvat

cal h _ infty control of nonlinear systems differential games and viscosity solutions dealing with disturbances is one of the most important questions for controlled systems cal h infty optimal control theory is a deterministic way to tackle the problem in the presence of unfavorable disturbances the theory of differential games and the study of the associated hamilton jacobi isaacs equation appear to be basic tools of the theory we consider a general nonlinear system and prove that the existence of a continuous local viscosity supersolution of the isaacs equation corresponding to the cal h infty control problem is sufficient for its solvability we also show that the existence of a lower semicontinuous viscosity supersolution is necessary

on the recovery of multiple flow parameters from transient head data the problem of estimating groundwater flow parameters from head measurements and other ancillary data is fundamental to the process of modelling a groundwater system we consider here a new method that allows for the simultaneous computation of multiple parameters as the unique minimum of a convex functional

on the recovery of transport parameters in groundwater modelling the problem of recovering the coefficient functions in the groundwater transport equation from piezomet tic head and contaminant concentration measurements is solved by minimization of an associated convex functional

numerical approximations for stochastic differential games the ergodic case the markov chain approximation method is a widely used relatively easy to use and efficient family of methods for the bulk of stochastic control problems in continuous time for reflected jump diffusion type models it has been shown to converge under broad conditions and there are good algorithms for solving the numerical problems if the dimension is not too high we consider a class of stochastic differential games with a reflected diffusion system model and ergodic cost criterion and where the controls for the two players are separated in the dynamics and cost function it is shown that the value of the game exists and that the numerical method converges to this value as the discretization parameter goes to zero the actual numerical method solves a stochastic game for a finite state markov chain and ergodic cost criterion the essential conditions are nondegeneracy and that a weak local consistency condition hold almost everywhere for the numerical approximations just as for the control problem

techniques for making finite elements competitve in modeling flow in variably saturated porous media a galerkin finite element formulation is developed for the numerical simulation of water flow in variably saturated soil systems included in this formulation is a solution strategy based on picard and newton raphson algorithms both algorithms are designed especially to cope with severely nonlinear field problems the two algorithms are formulated for both rectangular and triangular elements the element matrices are evaluated in a simple and efficient manner using a technique referred to as the influence coefficient technique this technique avoids numerical integration and leads to a substantial saving of computational cost four examples are presented to demonstrate the effectiveness of the present finite element approach these examples show that the nonlinear solution schemes are capable of accomodating cases involving large variations in the saturated hydraulic conductivity as well as highly nonlinear soil moisture characteristics a comparative study of the picard and the newton raphson algorithms is also provided the study indicates that despite the higher cost per iteration of the newton raphson scheme it usually requires a substantially smaller number of iterations than the picard scheme in some instances where convergence difficulties are experienced with the latter scheme it is desirable to use the newton raphson scheme in order to obtain a cost effective solution to the problem

spiral waves in the belousov zhabotinskii reaction abstract the beautiful spiral waves of oxidation in the belousov zhabotinskii reaction are the source of many interesting and important questions about periods structures in excitable media it has long been known that these spirals are similar to involutes of circles at least some distance from the center but until now no way has been known to determine the correct wavelength and frequency in this paper we show that the parameters of a spiral wave can be viwed s eigenvalues of a problem with unique solution the critical ingredients of the theory are the effects of curvature on the propagation of wavefronts in two dimensional media and the dispersion of plane waves our analytical results are shown to be in good agreement with experimental data for the belousov zhabotinskii reagent

the eikonal approximation to excitable reaction diffusion systems travelling non planar wave fronts on the plane abstract exact non planar travelling solutions of the eikonal equation on an infinite plane are presented for the first time these solutions are matched to produce corrugated wave fronts and patterns such as spot solutions as well as extended parabolic type wave fronts the stability of these solutions is also analysed the variational equation which belongs to a generalised wangerin class of differential equations is solved first with the aid of the liouville green approximation for the estimated eigenvalues characterising stability and then by a more elaborate shooting matching method all of the three types of travelling solutions are found to be geometrically stable it is suggested that some of these predictions are experimentally testable

comparison of parametric and nonparametric methods for the analysis and inversion of immittance data recently two methods for the estimation of discrete and or continuous distributions of relaxation times from small signal electrical frequency response data have been compared for discrete line distributions the parametric method used was found to be inferior in some ways to the nonparametric one which involved tikhonov regularization and it was concluded that the parametric one could not be employed to estimate continuous distributions at all here it is shown by monte carlo simulation that both conclusions are incorrect the same data situations analyzed in the earlier work were reanalyzed using a complex nonlinear least squares parametric method that has been employed to estimate discrete line distributions since 1982 and continuous ones since 1993 quite different results from those presented earlier were obtained and the original parametric method was shown to be far superior to the nonparametric one for the estimation of discrete line distributions since inversion is unnecessary and resolution is far greater for continuous or mixed distribution inversions the parametric method was again superior and it allows unambiguous distinction between discrete line points and those associated with a continuous distribution while the nonparametric inversion method does not allow such distinction and approximates all distributional points as continuous distribution ones the parametric method used and described here is also valuable for other data analysis tasks other than those involving inversion some of its error characteristics are investigated herein and the importance of matching the weighting error model to the form of the errors in the data is illustrated it was found that with normally distributed random errors added to exact data the distributions of estimated parameters were not normal but were closer to normal for proportional errors than for additive ones

parameter estimation in flow through partially saturated porous materials a class of numerical simulators were developed and critically evaluated to be incorporated as the solver of a forward problem in the framework of an inverse modeling strategy the strategy couples a mass lumped galerkin linear finite element solution of the mixed form richards equation with an experimental time space series and the osborne more revised form of the levenberg marquardt algorithm to retrieve hydraulic parameters of a partially saturated porous medium the numerical simulator shows excellent agreement with a reference solution obtained on a dense grid and infinitesimal time step in terms of fluid pressure head fluid content and fluid volumetric flux density and perfectly conserves the global mass an adaptive algorithm was implemented to estimate sensitivity matrix in the inverse algorithm a multi criterion stopping rule was developed and successfully implemented to end the inverse code at the solution the result of the optimization was compared with a large scale in situ soil moisture space time series measured during the course of a drainage experiment and excellent agreements were found analysis of the parameter response surfaces and hyper space plots closeness of the gradient of the penalty function at minimum to zero and positive definiteness of the approximation for the hessian at the solution eigs h 0 indicate that the obtained solution is a strong local minimum a state of the art sensitivity analysis carried out to quantify sensitivity of the state variable with respect to uncertainty and changes in different model parameters

a transformed pressure head based approach to solve richards equation for variably saturated soils this paper presents a new approach to solve richards equation it introduces a nonlinear transformed pressure pt as the dependent variable with the modified picard method the new approach was compared to and contrasted with two efficient existing methods the based transformation method kirkland et al 1992 and the h based modified picard method celia et al 1990 a total of 12 different one dimensional cases were considered saturated unsaturated layered and uniform soil profiles with pressure and flux type boundary conditions the results show that the new method offers excellent cpu efficiency and unlike the h based method is numerically robust for all cases of variably saturated heterogeneous media and first or second type boundary conditions the method does not require difficult numerical coding and its cpu efficiency is not affected by complicated heterogeneous and hysteretic media the pt transformation is easy to incorporate into existing h based codes

maximum likelihood adjoint state finite element estimation of groundwater parameters under steady and nonsteady state conditions a method is presented for estimating the hydraulic parameters of groundwater flow models under steady and nonsteady state conditions the estimation problem is posed in the framework of maximum likelihood theory by means of a log likelihood criterion that includes prior estimates of the parameters to allow for an incomplete knowledge of the covariances of the prior head and parameter errors these covariances are expressed in terms of a few unknown statistical parameters that may be estimated jointly with the hydraulic parameters computational efficiency is achieved by evaluating the gradient of the estimation criterion with an adjoint state finite element scheme and using a combination of conjugate gradient algorithms coupled with newton s method for determining the step size to be taken at each iteration model structure identification criteria developed in the time series literature all of which utilize the maximum likelihood concept are shown to be useful for selecting the best way to parametrize a groundwater flow region when a number of alternative schemes of parametrization are given the paper also demonstrates the potential utility of the proposed estimation method for the optimum design of space time measurement networks a case study dealing with three dimensional flow in a multiaquifer system is briefly discussed

diversity of three dimensional chemical waves abstract the propagation of spiral waves in the belousov zhabotinsky bz reaction has been extensively investigated the aim of this paper is to show the diversity of phenomenon in the three dimensional bz reagent with the aid of photographs of the evolving structures not all of which have been studied mathematically the most frequent observed wave forms are simple scrolls and toroidal scrolls a brief discussion of the existence of toroidal scroll solution to the eikonal equation in three dimensions is presented

jump diffusions with controlled jumps existence and numerical methods a comprehensive development of effective numerical methods for stochastic control problems in continuous time for reflected jump diffusion models is given in earlier work by the author while these methods cover the bulk of models which have been of interest to date they do not explicitly deal with the case where the jump itself is controlled in the sense that the value of the control just before the jump affects the distribution of the jump we do not deal explicitly with the numerical algorithms but develop some of the concepts which are needed to provide the background which is necessary to extend the proofs of earlier work to this case a critical issue is that of closure i e defining the model such that any sequence of systems controls has a convergent subsequence of the same type one needs to introduce an extension of the poisson measure which serves a purpose analogous to that served by relaxed controls which we call the relaxed poisson measure analogously to the use of the martingale measure concept given earlier to deal with controlled variance the existence of an optimal control is a consequence of the development

risk sensitive zero sum differential games we consider two player risk sensitive zero sum differential games rszsdgs in our problem setup both the drift term and the diffusion term in the controlled stochastic differential equation are dependent on the state and controls of both players and the objective functional is of the risk sensitive type first a stochastic maximum principle type necessary condition for an open loop saddle point of the rszsdg is established via nonlinear transformations of the adjoint processes of the equivalent risk neutral stochastic zero sum differential game in particular we obtain two variational inequalities namely the pair of saddle point inequalities of the rszsdg next we obtain the hamilton jacobi isaacs partial differential equation for the rszsdg which provides a sufficient condition for a feedback saddle point of the rszsdg using a logarithmic transformation of the associated value function finally we study the extended linear quadratic rszsdg lq rszsdg we show intractability of the extended lq rszsdg with the state and or controls of both players appearing in the diffusion term this unexpected intractability could lead to nonlinear open loop and feedback saddle points even if the problem itself is essentially lq and the isaacs condition holds

technical communique policy set iteration for markov decision processes this communique presents an algorithm called policy set iteration psi for solving infinite horizon discounted markov decision processes with finite state and action spaces as a simple generalization of policy iteration pi psi generates a monotonically improving sequence of stationary markovian policies p k based on a set manipulation as opposed to pi s single policy manipulation at each iteration k when the set involved with psi at k contains n independently generated sample policies from a given distribution d the probability that the expected value of any sampled policy from d with respect to an initial state distribution is greater than that of p k converges to zero with o n k rate moreover psi converges to an optimal policy no slower than pi in terms of the number of iterations for any d

parameter identification in stationary groundwater flow problems in drainage basins we discuss the recovery of some parameters in an elliptic boundary value problem which models a specific stationary flow problem in drainage basins by using the measured values of the hydraulic head in discrete points through the physical domain the underlying direct problem is the one considered by toth j geophys res 67 1962 4375 67 1963 4795 among others the inverse problem is solved by means of the levenberg marquardt method in g a watson ed numerical analysis lecture notes in mathematics vol 630 springer berlin new york 1977 pp 105 116 we also show how the infinite element method allows the identification of the far field value of the hydraulic head

a mass conservative switching algorithm for modeling fluid flow in variably saturated porous media numerical solutions of flow equation in fluid content based form or in fluid pressure head based form are often tradeoffs between speed accuracy and convenience the fluid content based form can be solved quite rapidly with low cpu time and perfect mass balance however it cannot be used in saturated regions as diffusivity function becomes infinite and strictly becomes invalid in composite layered and real heterogeneous porous materials due to singularity and discontinuity in fluid content profile this formulation also gives misleading impression that gradient in fluid content causes the flow of fluid in porous materials where in reality gravity and fluid pressure potential gradient produce the motion the pressure head based form on the other hand is more flexible but due to its highly nonlinear nature is much more time consuming and produces poor global mass balance for dry initial conditions very fine spatial and temporal discretizations are needed to maintain mass balance property for these scenarios the mixed form of the flow equation partially solves these issues as it maintains acceptable mass balance and is applicable to layered heterogeneous and composite fractured foundations however it is only applicable in unsaturated zones in this study a switching algorithm was proposed and implemented in which the mass conservative mixed form and the pressure head based form were respectively used in the unsaturated and saturated zones of an initial boundary value flow problem involving a variably saturated porous medium the algorithm showed excellent agreement with a reference solution obtained on a very fine spatiotemporal mesh the simulator was then calibrated with several real world large scale experimental datasets in all cases the proposed algorithm exhibited close agreements with the experimental time space series the algorithm poses excellent mass balance property and can easily be used in both saturated and unsaturated regions without special treatment of fluid content discontinuities in heterogeneous and layered porous media the proposed algorithm can also be extended to simulate multiphase and multidimensional flow problems

risk sensitive control on an infinite time horizon stochastic control problems on an infinite time horizon with exponential cost criteria are considered the donsker varadhan large deviation rate is used as a criterion to be optimized the optimum rate is characterized as the value of an associated stochastic differential game with an ergodic expected average cost per unit time cost criterion if we take a small noise limit a deterministic differential game with average cost per unit time cost criterion is obtained this differential game is related to robust control of nonlinear systems

open queueing networks in heavy traffic this paper presents heavy traffic limit theorems for the queue length and sojourn time processes associated with open queueing networks these limit theorems state that properly normalized sequences of queue length and sojourn time processes converge weakly to a certain diffusion as the network traffic intensity converges to unity the limit diffusion is reflected brownian motion on the nonnegative orthant this process behaves like brownian motion on the interior of its state space and reflects instantaneously on the boundaries the reflection direction is a constant for each boundary hyperplane

a posteriori error estimation and basis adaptivity for reduced basis approximation of nonaffine parametrized linear elliptic partial differential equations in this paper we extend the earlier work m barrault y maday n c nguyen a t patera an empirical interpolation method application to efficient reduced basis discretization of partial differential equations c r acad sci paris serie i 339 2004 667 672 m a grepl y maday n c nguyen a t patera efficient reduced basis treatment of nonaffine and nonlinear partial differential equations m2an math model numer anal 41 3 2007 575 605 to provide a posteriori error estimation and basis adaptivity for reduced basis approximation of linear elliptic partial differential equations with nonaffine parameter dependence the essential components are i rapidly convergent reduced basis approximations galerkin projection onto a space w n u spanned by n global hierarchical basis functions which are constructed from solutions of the governing partial differential equation at judiciously selected points in parameter space ii stable and inexpensive interpolation procedures methods which allow us to replace nonaffine parameter functions with a coefficient function expansion as a sum of m products of parameter dependent coefficients and parameter independent functions iii a posteriori error estimation relaxations of the error residual equation that provide inexpensive yet sharp error bounds for the error in the outputs of interest iv optimal basis construction processes which make use of the error bounds as an inexpensive surrogate for the expensive true error to explore the parameter space in the quest for an optimal sampling set and v offline online computational procedures methods which decouple the generation and projection stages of the approximation process the operation count for the online stage in which given a new parameter value we calculate the output of interest and associated error bounds depends only on n m and the affine parametric complexity of the problem the method is thus ideally suited for repeated and reliable evaluation of input output relationships in the many query or real time contexts

a hybrid global optimization method for inverse estimation of hydraulic parameters annealing simplex method inverse estimation of unsaturated hydraulic parameters is often a highly nonlinear optimization problem with multiple parameters the objective functions involved are often topographically complex and may contain many local minima because of these reasons the inverse solutions are commonly very sensitive to the initial guess of the parameters when conventional optimizers are used this paper presents an annealing simplex method that incorporates simulated annealing strategies into a classical downhill simplex method an upward infiltration experiment was used as an example of inverse estimation to test the method numerical experiments of both minimizing an algebraic function and inversion of upward infiltration data showed that the new method successfully converged to the global minimum in all cases irrespective of the initial hydraulic parameter estimates while the classical downhill method often converged to unfavorable local minima the cpu times needed for the annealing simplex method to estimate 5 and 7 hydraulic parameters simultaneously are about a half hour and 1 hour on a pc respectively additionally no special requirements are imposed on the objective function and the method is independent of the details of the simulation submodel therefore the proposed method should be applicable to other optimization problems in water resources when it is important to have a robust global search capability

algorithms for solving richards equation for variably saturated soils previous work has shown that a one dimensional water content based finite difference algorithm can offer improved cpu efficiency in modeling infiltration into very dry heterogeneous soil when compared to similar pressure based algorithms the usefulness of the water content based method is limited however since it cannot be applied to saturated soils in this paper we develop two new methods which retain the advantages of water content based methods while still permitting fully saturated conditions in the first method we define a new variable for the transformed richards equation which has the characteristics of water content when soil is unsaturated and of pressure when soil is at or near saturation in addition to the transformed richards equation method an improved pressure based method which uses flux updating is presented both methods are implemented in algorithms using a preconditioned conjugate gradient method equation solver the performance of the algorithms is compared to that of an algorithm based on a mixed form of richards equation using modified picard iteration two test cases are examined the first test case examines two dimensional infiltration into very dry heterogeneous soil which remains unsaturated throughout the simulation period the second test case examines a two dimensional developing perched water table results indicate that the new methods retain the advantages of the water content based method for fully unsaturated heterogeneous problems while allowing for fully saturated conditions for the unsaturated problem the new algorithms were from 16 to 59 times faster than the algorithm based on the mixed form of richards equation using modified picard iteration for the saturated problem the new algorithms were from 6 to 25 times faster than the mixed base algorithm results also indicate that the performance of the mixed base algorithm could be significantly improved if a more effective adaptive time step were developed

survey paper a partial history of the early development of continuous time nonlinear stochastic systems theory this article is a survey of the early development of selected areas in nonlinear continuous time stochastic control key developments in optimal control and the dynamic programming principle existence of optimal controls under complete and partial observations nonlinear filtering stochastic stability the stochastic maximum principle and ergodic control are discussed issues concerning wide bandwidth noise for stability modeling filtering and ergodic control are dealt with the focus is on the earlier work but many important topics are omitted for lack of space

spirals in excitable media the free boundary limit with diffusion abstract we solve numerically for the steady state spiral in the thin interface limit including the effects of diffusion of the slow field the calculation is performed using a generalization of the hybrid scheme of keener in this method the diffusion equation is solved on a suitable mapped lattice while the eikonal equation relating the field on the interface to the interfacial velocity and curvature is solved independently we present results for the selected frequency and tip radius as a function of the various parameters we note that a stability analysis based on these results may be performed

brief paper convergence of markov chain approximation on generalized hjb equation and its applications this work is concerned with numerical methods for a class of stochastic control optimizations and stochastic differential games numerical procedures based on markov chain approximation techniques are developed in a framework of generalized hamilton jacobi bellman equations convergence of the algorithms is derived by means of viscosity solution methods

shaping wave patterns in reaction diffusion systems we present a method to control the two dimensional shape of traveling wave solutions to reaction diffusion systems such as interfaces and excitation pulses control signals that realize a pregiven wave shape are determined analytically from nonlinear evolution equation for isoconcentration lines as the perturbed nonlinear phase diffusion equation or the perturbed linear eikonal equation while the control enforces a desired wave shape perpendicular to the local propagation direction the wave profile along the propagation direction itself remains almost unaffected provided that the one dimensional wave profile of all state variables and its propagation velocity can be measured experimentally and the diffusion coefficients of the reacting species are given the new approach can be applied even if the underlying nonlinear reaction kinetics are unknown

a boundary property of semimartingale reflecting brownian motions we consider a class of reflecting brownian motions on the non negative orthant inr k in the interior of the orthant such a process behaves like brownian motion with a constant covariance matrix and drift vector at each of the k 1 dimensional faces that form the boundary of the orthant the process reflects instantaneously in a direction that is constant over the face we give a necessary condition for the process to have a certain semimartingale decomposition and then show that the boundary processes appearing in this decomposition do not charge the set of times that the process is at the intersection of two or more faces this boundary property plays an essential role in the derivation performed in a separate work of an analytical characterization of the stationary distributions of such semimartingale reflecting brownian motions

saddle points of discrete markov zero sum game with stopping we study the sufficient conditions for the existence of a saddle point of a time dependent discrete markov zero sum game up to a given stopping time the stopping time is allowed to take either a finite or an infinite non negative random variable with its associated objective function being well defined the result enables us to show the existence of the saddle points of discrete games constructed by markov chain approximation of a class of stochastic differential games

zero sum ergodic stochastic games with feller transition probabilities we consider zero sum stochastic games with borel state spaces satisfying a generalized geometric ergodicity condition the main objective of this paper is to establish a minimax theorem for a class of ergodic stochastic games with the feller transition probability function all previous results on ergodic stochastic games are based on a strong continuity of the transition probabilities in the actions of the players

symmetric spirals in media with relaxation kinetics and two diffusing species abstract we find an approximate solution for the shape and rotation frequency of spirals in excitable media with anti symmetric kinetics and diffusion in both species using singular perturbation theory we approximate the governing diffusion reaction system with a free boundary problem and then map this free boundary problem onto a fixed domain problem the resulting fixed domain problem is viewed as a nonlinear eigenvalue problem which can be solved iteratively using a numerical shooting algorithm solutions of the approximate free boundary problem are compared with numerical solutions of the full system of diffusion reaction equations and shown to be in good agreement

three parameter lognormal distribution model for soil water retention many models for soil water retention have been proposed however most of these models are curve fitting equations and do not emphasize the physical significance of their empirical parameters a new retention model that exhibits increased flexibility was developed by applying three parameter lognormal distribution laws to the pore radius distribution function r and to the water capacity function which was taken to be the pore capillary pressure distribution function this model contains three parameters that are closely related to the statistics of the bubbling pressure c the mode 0 of and the standard deviation of transformed by comparison of this model with three existing models the van genuchten model the brooks corey model and the modified tani model it was shown that c 0 and are all essential for a general retention model

review of parameter identification procedures in groundwater hydrology the inverse problem the purpose of this survey is to review parameter identification procedures in groundwater hydrology and to examine computational techniques which have been developed to solve the inverse problem parameter identification methods are classified under the error criterion used in the formulation of the inverse problem the problem of ill posedness in connection with the inverse problem is addressed typical inverse solution techniques are highlighted the review also includes the evaluation of methods used for computing the sensitivity matrix statistics which can be used to estimate the parameter uncertainty are outlined attempts have been made to compare and contrast representative inverse procedures and direction for future research is suggested

lognormal distribution model for unsaturated soil hydraulic properties the soil water retention model developed by kosugi was modified to be compatible with mualem s model in order to derive an analytical expression for the relative hydraulic conductivity kr the modified water retention model is to be derived by applying a lognormal distribution law to the soil pore radius distribution function parameters of this retention model have physical significance on the water content capillary pressure curve and are related directly to the statistics of the pore radius distribution the accuracy of the resulting combined water retention hydraulic conductivity model is verified for observed data sets for six soils results showed that the proposed model produces acceptable matches with observed water retention curves and adequate predictions of hydraulic conductivities in five out of six cases the and kr or kr curves generated by this model are generally similar to those generated by van genuchten s model

analyses of infiltration events in relation to determining soil hydraulic properties by inverse problem methodology at the field scale because of the relatively large number of observation points required to estimate the spatial distributions of hydraulic properties application of inversion procedures must be based on a characteristic of the flow that for a given point in the field can be measured with relative ease and can be used as input for the inversion procedure suggesting the use of infiltration data for this purpose and recognizing the fact that a basic requirement for the design of a transient flow experiment is that the resulting inverse problem be sufficiently well posed to allow solution this paper addresses the question under which circumstances the resulting inverse problem is well posed utilizing the concepts of identifiability uniqueness and stability for this purpose three different models of the soil hydraulic properties were analyzed the main conclusion of this study is that when infiltration data measured at the darcy scale are used as input for the inversion procedure the inclusion of prior information on a single measurable parameter the saturated conductivity ks in the estimation criterion will enhance the likelihood of uniqueness and stability of the inverse solution provided that the structure of the hydraulic model is sufficiently simple however since for a particular situation it is impossible to determine a priori whether the resultant inverse problem is well posed or not this must be carried out only a posteriori by solving the problem several times with different initial parameter estimates accompanied by an analysis of the associated estimation errors

the gauss seidel numerical procedure for markov stochastic games consider the problem of value iteration for solving markov stochastic games one simply iterates backward via a jacobi like procedure the convergence of the gauss seidel form of this procedure is shown for both the discounted and ergodic cost problems under appropriate conditions with extensions to problems where one stops when a boundary is hit or if any one of the players chooses to stop with associated costs generally the gauss seidel procedure accelerates convergence

travelling waves and static structures in a two dimensional exactly solvable reaction diffusion system exact travelling wave solutions of a fisher type reaction diffusion equation in two spatial dimensions are obtained the waves have nontrivial geometries and possess velocities smaller and larger than that of a plane wave the dispersion relationship and velocity curvature dependence generated by these solutions are characterized static structures with and without circular symmetry are constructed

a comparison of picard and newton iteration in the numerical solution of multidimensional variably saturated flow problems picard iteration is a widely used procedure for solving the nonlinear equation governing flow in variably saturated porous media the method is simple to code and computationally cheap but has been known to fail or converge slowly the newton method is more complex and expensive on a per iteration basis than picard and as such has not received very much attention its robustness and higher rate of convergence however make it an attractive alternative to the picard method particularly for strongly nonlinear problems in this paper the picard and newton schemes are implemented and compared in one two and three dimensional finite element simulations involving both steady state and transient flow the eight test cases presented highlight different aspects of the performance of the two iterative methods and the different factors that can affect their convergence and efficiency including problem size spatial and temporal discretization initial solution estimates convergence error norm mass lumping time weighting conductivity and moisture content characteristics boundary conditions seepage faces and the extent of fully saturated zones in the soil previous strategies for enhancing the performance of the picard and newton schemes are revisited and new ones are suggested the strategies include chord slope approximations for the derivatives of the characteristic equations relaxing convergence requirements along seepage faces dynamic time step control nonlinear relaxation and a mixed picard newton approach the tests show that the picard or relaxed picard schemes are often adequate for solving richards equation but that in cases where these fail to converge or converge slowly the newton method should be used the mixed picard newton approach can effectively overcome the newton scheme s sensitivity to initial solution estimates while comparatively poor performance is reported for the various chord slope approximations finally given the reliability and efficiency of current conjugate gradient like methods for solving linear nonsymmetric systems the only real drawback of using newton rather than picard iteration is the algebraic complexity and computational cost of assembling the derivative terms of the jacobian matrix and it is suggested that both methods can be effectively implemented and used in numerical models of richards equation

value set iteration for two person zero sum markov games we present a novel exact algorithm called value set iteration vsi for solving two person zero sum markov games mgs as a generalization of value iteration vi and as a general framework of combining multiple solution methods we introduce a novel operator in the value function space and iteratively apply the operator with any sequence of the set of policies extending chang s vsi for mdps into the mg setting we show that vsi for mgs converges to the equilibrium value function with at least linear convergence rate and establish that vsi can potentially improve the convergence speed in terms of the number of iterations by proper setting of the sequence of the set of policies

reflected brownian motion on an orthant we consider a k dimensional diffusion process z whose state space is the nonnegative orthant on the interior of the orthant z behaves like a kdimensional brownian motion with arbitrary covariance matrix and drift vector at each of the k 1 dimensional hyperplanes that form the boundary of the orthant z reflects instantaneously in a direction that is constant over that hyperplane there is no extant theory of multidimensional diffusion that applies to this process because the boundary of its state space is not smooth we adopt an approach that requires a restriction on the directions of reflection but reiman has shown that this restriction is met by all diffusions z arising as heavy traffic limits in open k station queuing networks our process z is defined as a path to path mapping of k dimensional brownian motion from this construction it follows that z is a continuous markov process and a semimartingale using the latter property we obtain a change of variable formula from which one can develop a complete analytical theory for the process z

efficient stochastic sensitivity analysis of discrete event systems sensitivity analysis quantifies the dependence of a system s behavior on the parameters that could possibly affect the dynamics calculation of sensitivities of stochastic chemical systems using kinetic monte carlo and finite difference based methods is not only computationally intensive but direct calculation of sensitivities by finite difference based methods of parameter perturbations converges very poorly in this paper we develop an approach to this issue using a method based on the girsanov measure transformation for jump processes to smooth the estimate of the sensitivity coefficients and make this estimation more accurate we demonstrate the method with simple examples and discuss its appropriate use

multi objective optimization in variably saturated fluid flow a general purpose multi objective inverse modeling strategy was developed and implemented to quantify fluid flow parameters in variably saturated porous materials the strategy combines a robust and mass conservative numerical simulator of the flow equation with an optimization algorithm and an experimental data set to estimate the parameters the numerical simulator of the direct problem shows excellent agreement with a reference solution and conserves global mass with near perfection an adaptive method was proposed in which the sensitivity matrix was calculated by one sided finite difference approximation at the early stages of the optimization and the more accurate two sided differentiation as the search approaches the minimum a combined termination criterium was developed to stop the inverse code at the solution the results of the multi objective optimization were compared with those of single objective minimization while single objective optimization generates reasonable results for either the fluid pressure head profile or the fluid content data the proposed multi objective optimization shows excellent agreement with both profiles simultaneously

numerical evaluation of iterative and noniterative methods for the solution of the nonlinear richards equation several noniterative procedures for solving the nonlinear richards equation are introduced and compared to the conventional newton and picard iteration methods noniterative strategies for the numerical solution of transient nonlinear equations arise from explicit or linear time discretizations or they can be obtained by linearizing an implicit differencing scheme we present two first order accurate linearization methods a second order accurate two level implicit factored scheme and a second order accurate three level lees method the accuracy and computational efficiency of these four schemes and of the newton and picard methods are evaluated for a series of test problems simulating one dimensional flow processes in unsaturated porous media the results indicate that first order accurate schemes are inefficient compared to second order accurate methods that second order accurate noniterative schemes can be quite competitive with the iterative newton and picard methods and that the newton scheme is no less efficient than the picard method and for strongly nonlinear problems can outperform the picard scheme the two second order accurate noniterative schemes appear to be attractive alternatives to the iterative methods although there are concerns regarding the stability behavior of the three level scheme which need to be resolved we conclude that of the four noniterative strategies presented the implicit factored scheme is the most promising and we suggest improved formulations of the method

admission control for combined guaranteed performance and best effort communications systems under heavy traffic communications systems often have many types of users since the users share the same resource there is a conflict in their needs this conflict leads to the imposition of controls on admission or elsewhere in this paper there are two types of customers gp guaranteed performance and be best effort we consider an admission control of gp customer which has two roles first to guarantee the performance of the existing gp customers and second to regulate the congestion for the be users the optimal control problem for the actual physical system is difficult a heavy traffic approximation is used with optimal or nearly optimal controls it is shown that the optimal values for the physical system converge to that for the limit system and that good controls for the limit system are also good for the physical system this is done for both the discounted and average cost per unit time cost criteria additionally asymptotically the pathwise average not mean costs for the physical system are nearly minimal when good nearly optimal controls for the limit system are used numerical data show that the heavy traffic optimal control approach can lead to substantial reductions in waiting time for be with only quite moderate rejections of gp under heavy traffic it also shows that the controls are often linear in the state variables the approach has many advantages it is robust simplifies the analysis both analytical and numerical and allows a more convenient study of the parametric dependencies even if optimal control is not wanted the approach is very convenient for a systematic exploration of the possible tradeoffs among the various cost components this is done by numerically solving a series of problems with different weights on the costs we can then get the best tradeoffs and the control policies which give them

nash equilibrium approximation of some class of stochastic differential games a combined chebyshev spectral collocation method with policy iteration abstract this study relates to nonzero sum stochastic games with perfect information it proposes an efficient combined chebyshev spectral collocation method cscm with the policy iteration pi algorithm for solving nonlinear coupled hamilton jacobi hj equations the proposed approach is comprised of two steps first the pi algorithm is used to reduce the nonlinear coupled hj equations to a sequence of linear uncoupled pdes then these equations are approximated by the cscm the cscm pi is especially useful when the cscm fails due to the increasing number of collocation points for solving the associated system of nonlinear algebraic equations the main advantage of the resulting method is that it converts nonlinear coupled hj equations to the systems of linear algebraic equations which can be solved readily convergence analysis of this method is also provided in detail to confirm the accuracy and validity of the proposed computational algorithm several illustrative examples are presented

estimation of two phase petroleum reservoir properties by regularization abstract an algorithm is developed based on the theory of regularization and on spline approximation to estimate the absolute permeability in two phase petroleum reservoirs from noisy well pressure data the regularization feature of the algorithm converts the ill posed estimation problem to a well posed one the algorithm which employs the partial conjugate gradient method of nazareth as its core minimization technique automatically chooses the regularization parameter based on the non regularized estimation it is shown that regularized estimation is more stable and insensitive to the choice of initial guess as compared to non regularized conventional estimation

the eikonal equation stability of reaction diffusion waves on a sphere abstract the eikonal approach to excitable reaction diffusion systems provides an analytical hold on the phenomenon of wave propagation on curved surfaces and the evolution of toroidal scroll structures in the neighbourhood of organising filaments the extent to which the eikonal equation contains information on the stability of these wave solutions is important from a theoretical and experimental point of view we initiated this study in 1987 by demonstrating the stability of an expanding spherical wave recent experiments on wave propagation on a sphere encouraged us further to investigate stability of waves on surfaces we analyse in this paper the geometrical stability of symmetric counter rotating spiral waves propagating on the unit sphere we demonstrate that these solutions are stable under small perturbations normal to the wave fron lying on the unit sphere

identification of the thermal conductivity and heat capacity in unsteady nonlinear heat conduction problems using the boundary element method in this study the inverse problem of the identification of temperature dependent thermal properties of a heat conducting body is investigated the solution of the corresponding direct problem is obtained using a time marching boundary element method bem which allows without any need of interpolation and solution domain discretisation efficient and accurate evaluation of the temperature everywhere inside the space time dependent domain since the inverse problem which requires the determination of the thermal conductivity and heat capacity from a finite set of temperature measurements taken inside the body possesses poor uniqueness features additional information is achieved by assuming that the thermal properties belong to a set of polynomials thus the inverse problem reduces to a parameter system estimation problem which is solved using the nonlinear least squares method convergent and stable numerical results are obtained for the finite set of parameters which characterise the thermal properties for various test examples once the thermal properties are accurately obtained then the bem determines automatically the temperature inside the solution domain and the remaining unspecified boundary values and the numerically obtained results show good agreement with the corresponding analytical solutions

label setting methods for multimode stochastic shortest path problems on graphs stochastic shortest path ssp problems arise in a variety of discrete stochastic control contexts an optimal solution to such a problem is typically computed using the value function which can be found by solving the corresponding dynamic programming equations in the deterministic case these equations can be often solved by highly efficient label setting methods such as dijkstra s and dial s algorithms in this paper we define and study a class of multimode stochastic shortest path mssp problems and develop sufficient conditions for the applicability of label setting methods we illustrate our approach in a number of discrete stochastic control examples we also discuss the relationship of ssps with discretizations of static hamilton jacobi equations and provide an alternative derivation for several fast noniterative numerical methods for these partial differential equations pdes

zero sum risk sensitive stochastic differential games we study zero sum risk sensitive stochastic differential games on the infinite horizon with discounted and ergodic payoff criteria under certain assumptions we establish the existence of values and saddle point equilibria we obtain our results by studying the corresponding hamilton jacobi isaacs equations finally we show that the value of the ergodic payoff criterion is a constant multiple of the maximal eigenvalue of the generators of the associated nonlinear semigroups

singular perturbation theory of traveling waves in excitable media a review abstract waves of chemical or electrical activity traveling through space have been observed in several contexts chemical reaction mixtures in non convecting liquid phase cell suspensions nerve axons and neuromuscular tissues typically wave supporting preparations are excitable that is they respond and sensitivity to perturbations are rapidly damped out but suprathreshold disturbances trigger an abrupt and substantial response the abruptness of the response can be exploited by the methods of singular perturbation theory to obtain a mathematical description of wave propagation in spatially distributed excitable media singular perturbation analysis of propagating waves in one spatial dimension is straightforward and uncontentious but the analysis of propagating waves in one spatial dimension is straightforward and several fundamentally different ways we compare and contrast the approaches taken by greenberg zykov fife krinskii and his collaborators and ourselves with particular emphasis on the case of rotating spiral waves our intention is to bring some order to the important but difficult theory of propagating waves in two dimensional excitable media in conclusion we discuss briefly some possible extensions of the singular perturbation approach to propagating wave surfaces in three dimensional space

parametrization of flow processes in porous media by multiobjective inverse modeling a multiobjective optimization algorithm was developed and applied to parameterize biofluid flow processes in partially saturated porous media the forward problem was formulated as a nonlinear partial differential equation and solved by an efficient galerkin finite element method the numerical simulator was validated with reference and analytical solutions the inverse problem was formulated in a nonlinear optimization framework in which model parameters were estimated by minimizing a complex penalty function representing the discrepancies between the observed and predicted attributes of the physical system a physical model was designed built and operated to collect the experimental datasets needed for model calibration the proposed strategy was then coupled with the fluid pressure head fluid content and fluid flux density time space series to estimate model parameters several optimization scenarios were investigated and it was concluded that while single objective optimization approach has limited accuracy and results in reasonable outputs for one or some attributes of the physical system the proposed multiobjective optimization shows excellent agreements with the experimental datasets for all state variables