a radial basis collocation method for hamilton jacobi bellman equations in this paper we propose a semi meshless discretization method for the approximation of viscosity solutions to a first order hamilton jacobi bellman hjb equation governing a class of nonlinear optimal feedback control problems in this method the spatial discretization is based on a collocation scheme using the global radial basis functions rbfs and the time variable is discretized by a standard two level time stepping scheme with a splitting parameter q a stability analysis is performed showing that even for the explicit scheme that q 0 the method is stable in time since the time discretization is consistent the method is also convergent in time numerical results performed to verify the usefulness of the method demonstrate that the method gives accurate approximations to both of the control and state variables

optimal control of diffusions a verification theorem for viscosity solutions we consider an optimal control problem for a diffusion process with finite time horizon a verification theorem for viscosity solutions is proved by using recent results on the connection between maximum principle and dynamic programming 8 and a uniqueness result for viscosity solution by lions

maximal abelian subalgebras of e p q algebras abstract maximal abclian subalgebras masas of one of the classical real inhomogencous lie algebras are constructed namely those of the pseudoeuclidean lie algebra e p q use is made of the semidireet sum structure of e p q with the translations t p q as an abclian ideal we first construct splitting masas that arc themselves direct sums of abelian subalgebras of o p q and of subalgebras of t p q the splitting subalgebras are used to construct the complementary nonsplitting ones here the results are less complete than in the splitting case we present general decomposition theorems and construct indecomposable masas for all algebras e p q p q 0 the case of q 0 and 1 were treated earlier in a physical context the case q 2 is analyzed here in detail as an illustration of the general results

a probabilistic collocation eulerian lagrangian localized adjoint method on sparse grids for assessing co2 leakage through wells in randomly heterogeneous porous media abstract we develop a probabilistic collocation eulerian lagrangian localized adjoint method on sparse grids for assessing co2 leakage through wells in randomly heterogeneous porous media by utilizing the intrinsic mathematical numerical and physical properties of the mathematical model we model the process in which co2 is injected into a deep aquifer spreads within the aquifer and upon reaching a leaky well rises up to a shallower aquifer to quantify the leakage rate which depends on the pressure build up in the aquifer due to injection and the buoyancy of co2 the underlying eulerian lagrangian framework has high potential to improve the efficiency and accuracy for the numerical simulation of complex flow and transport processes in co2 sequestration the sparse grid probabilistic collocation framework adds computationally efficient uncertainty quantification functionality onto pre existing eulerian lagrangian methods in a nonintrusive manner it also provides a scalable framework to consider uncertainty in a straightforward parallel manner preliminary numerical experiments show the feasibility and potential of the method

deterministic near optimal controls part ii dynamic programming and viscosity solution approach near optimization is as sensible and important as optimization for both theory and applications this paper concerns dynamic near optimization or near optimal controls for systems governed by deterministic ordinary differential equations and uses dynamic programming to study the near optimality since nonsmoothness is inherent in this subject the viscosity solution approach is employed to investigate the problem the dynamic programming equation is derived in terms of e superdifferential subdifferential the relationships among the adjoint functions the value functions and the hamiltonian along near optimal trajectories are revealed verification theorems with which near optimal feedback controls can be constructed are obtained

characterization of optimality for controlled diffusion processes this paper studies controlled systems governed by ito s stochastic differential equations where control variables are allowed to enter both drift and diffusion terms optimal controls are characterized by necessary and sufficient conditions which involve super differentials and sub differentials rather than derivatives of the value functions

fixed final time constrained optimal control of nonlinear systems using neural network hjb approach in this paper fixed final time constrained optimal control laws using neural networks nns to solve hamilton jacobi bellman hjb equations for general affine in the constrained nonlinear systems are proposed an nn is used to approximate the time varying cost function using the method of least squares on a predefined region the result is an nn nearly constrained feedback controller that has time varying coefficients found by a priori offline tuning convergence results are shown the results of this paper are demonstrated in two examples including a nonholonomic system

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

maximal abelian subgroups of the isometry and conformal groups of euclidean and minkowski spaces the maximal abelian subalgebras masas of the euclidean and pseudo euclidean lie algebras are classified into conjugacy classes under the action of the corresponding lie groups and and also under the conformal groups and respectively the results are presented in terms of decomposition theorems for orthogonally indecomposable masas exist only for p 1 and p 2 for on the other hand orthogonally indecomposable masas exist for all values of p the results are used to construct new coordinate systems in which wave equations and hamilton jacobi equations allow the separation of variables

risk sensitive production planning of a stochastic manufacturing system this paper is concerned with long run average risk sensitive control of production planning in a manufacturing system with machines that are subject to breakdown and repair by using a logarithmic transformation it is shown that the associated hamilton jacobi bellman equation has a viscosity solution the risk sensitive control problem has a dynamic stochastic game interpretation finally a limiting problem is obtained when the rates of machine breakdown and repair go to infinity

on the transport diffusion algorithm and its applications to the navier stokes equations this paper deals with an algorithm for the solution of diffusion and or convection equations where we mixed the method of characteristics and the finite element method globally it looks like one does one step of transport plus one step of diffusion or projection but the mathematics show that it is also an implicit time discretization of thepde in lagrangian form we give an error bound h t h h t in the interesting case that holds also for the navier stokes equations even when the reynolds number is infinite euler equation

a hybrid mixed finite element method for miscible displacement problem with mcc procedure abstract a new combined method is developed for solving incompressible miscible displacement in porous media in this method a hybrid mixed element method is constructed for the pressure and velocity equation while a mass conservative characteristic mcc finite element method is presented for concentration equation the introduction of these two numerical algorithms not only makes the coefficient matrixes symmetric positive definite but also keeps the mass balance the stability and consistence of this method are analyzed and the optimal error estimates in l2 norm for velocity concentration and pressure are derived finally some numerical results are presented

nearly optimal control laws for nonlinear systems with saturating actuators using a neural network hjb approach the hamilton jacobi bellman hjb equation corresponding to constrained control is formulated using a suitable nonquadratic functional it is shown that the constrained optimal control law has the largest region of asymptotic stability ras the value function of this hjb equation is solved for by solving for a sequence of cost functions satisfying a sequence of lyapunov equations le a neural network is used to approximate the cost function associated with each le using the method of least squares on a well defined region of attraction of an initial stabilizing controller as the order of the neural network is increased the least squares solution of the hjb equation converges uniformly to the exact solution of the inherently nonlinear hjb equation associated with the saturating control inputs the result is a nearly optimal constrained state feedback controller that has been tuned a priori off line

statistics of multiple avoided crossings the author considers the energy levels of a generic quantum mechanical system with hamiltonian h depending on a parameter x if the energy levels en are plotted as a function of x the curves do not cross the points of closest approach are called avoided crossings these have a distinctive geometry and are important because they determine the limits of applicability of the adiabatic theorem he describes some theoretical results on the parameter space density of avoided crossings for systems with the spectral statistics of the gaussian orthogonal ensemble goe these results are in good agreement with numerical experiments

the minimal dimension of maximal commutative subalgebras of full matrix algebras abstract let f be a field and a a maximal commutative subalgebra of the full matrix algebra mn f it is shown that dim a 2n 2 3 1 it is also shown that if the radical of a has cube zero then dim a 3n 2 3 4 and that this result is best possible for infinitely many natural numbers n

stochastic verification theorem of forward backward controlled systems for viscosity solutions abstract in this paper we investigate the controlled systems described by forward backward stochastic differential equations with the control contained in drift diffusion and generator of bsdes a new verification theorem is derived within the framework of viscosity solutions without involving any derivatives of the value functions it is worth pointing out that this theorem has wider applicability than the restrictive classical verification theorems as a relevant problem the optimal stochastic feedback controls for forward backward systems are discussed as well

separation of variables and subgroup bases on n dimensional hyperboloids a graphical formalism is introduced for describing subgroup type coordinates on n dimensional lorentzian hyperboloids imbedded into n 1 dimensional minkowski spaces the o n 1 group element is parametrized according to different subgroup chains involving lorentz rotation and euclidean subgroups the coordinates are then induced by the corresponding group action eigenfunctions of the laplace beltrami operator are obtained as products of jacobi functions associated legendre functions and modified bessel functions

statistical properties of eigenvectors in non hermitian gaussian random matrix ensembles statistical properties of eigenvectors in non hermitian random matrix ensembles are discussed with an emphasis on correlations between left and right eigenvectors two approaches are described one is an exact calculation for ginibre s ensemble in which each matrix element is an independent identically distributed gaussian complex random variable the other is a simpler calculation using n 1 as an expansion parameter where n is the rank of the random matrix this is applied to girko s ensemble consequences of eigenvector correlations which may be of physical importance in applications are also discussed it is shown that eigenvalues are much more sensitive to perturbations than in the corresponding hermitian random matrix ensembles it is also shown that in problems with time evolution governed by a non hermitian random matrix transients are controlled by eigenvector correlations

verification theorems within the framework of viscosity solutions abstract in optimal control theory the verification technique plays an important role in testing for the optimality of a given control and in constructing optimal feedback controls however the existing classical verification theorem is restrictive in that it requires the associated dynamic programming equation to have smooth solutions in this paper some verification theorems are presented within the framework of viscosity solutions under mild assumptions these theorems are shown to have wider applicability than the classical verification theorem as a relevant problem some differences and relationships between the viscosity solution and clarke s generalized gradient are also discussed

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

maximal abelian subalgebras of pseudounitary lie algebras abstract the task of classifying and constructing all maximal abelian subalgebras of su p q p q 1 is reduced to that of classifying orthogonally indecomposable masas these are either maximal abelian nilpotent subalgebras represented by nilpotent matrices in any finite dimensional representation or for p q they can be nonorthogonally decomposable and their study can be reduced to a construction of manss of sl p c two types of manss of su p q are shown to exist one rowed and non one rowed numerous classification theorems are proven and applied to obtain all masas of su p 1 su p 2 and su p q with p q 6 physical applications are discussed

conservative multimoment transport along characteristics for discontinuous galerkin methods a set of algorithms based on characteristic discontinuous galerkin methods is presented for tracer transport on the sphere the algorithms are designed to reduce message passing interface communica

a two grid eulerian lagrangian localized adjoint method to miscible displacement problems with dispersion term abstract in this paper we propose a novel two grid method of eulerian lagrangian localized adjoint method ellam mixed finite element method mfem solutions for nonlinear and coupled miscible displacement problems which describe complex fluid flow processes in porous media an ellam is used to solve the nonlinear convection diffusion equation for concentration and we present a mfem to compute the pressure equation for the pressure and darcy velocity a two grid scheme is investigated to linearize and decouple the mixed method equations it is illustrated that the coarse space can be selected as coarse as h o h 1 2 and the asymptotically optimal approximation as the nonlinear schemes can be still achieved as a result a large class of such nonlinear and coupled problems can be solved efficiently with large time steps

risk sensitive control with hara utility a control methodology based on the hyperbolic absolute risk averse hara utility function is presented as an alternative to the exponential of an integral approach to finding robust controllers the work is inspired by the intuition that hara controllers while being robust may give better performance than exponential controllers in normal situations the hara problem is shown to be equivalent to a certain differential game and the asymptotic properties of the hara problem and this differential game are studied as an example a linear quadratic hara problem is studied where the problem of finding a robust hara controller is proved to be equivalent to solving a standard linear quadratic problem for a system with a higher noise intensity this reveals an interesting relationship between robustness and uncertainty

a multivariate adaptive regression b spline algorithm bmars for solving a class of nonlinear optimal feedback control problems in this paper we present a novel method for solving a class of nonlinear optimal feedback control problems with moderately high dimensional state spaces based on an adapted version of the bmars algorithm numerical experiments were performed using problems with up to six state variables the numerical results clearly demonstrate the efficiency and potential of the method for solving high dimensional problems

normal forms of elements of classical real and complex lie and jordan algebras elements of the classical complex and real lie and jordan algebras with involutions are classified into conjugacy classes under the action of the corresponding classical lie group normal forms of representatives of each conjugacy class are chosen so as to resemble the jordan normal forms of n n complex matrices for completeness similar results are given for gl n c gl n r and gl n h

correlation functions of eigenvalues of multi matrix models and the limit of a time dependent matrix the universality of correlation functions of eigenvalues of large random matrices has been observed in various physical systems and proved in some particular cases as the hermitian one matrix model with polynomial potential here we consider the more difficult case of a unidimensional chain of hermitian matrices with first neighbour couplings and polynomial potentials an asymptotic expression of the orthogonal polynomials and a generalization of the darboux christoffel theorem allow us to find new results for the correlations of eigenvalues of different matrices of the chain eventually we consider the limit of the infinite chain of matrices which can be interpreted as a time dependent hermitian one matrix model and give the correlation functions of eigenvalues at different times

numerical analysis of convection diffusion reaction problems with higher order characteristics finite elements part ii fully discretized scheme and quadrature formulas in this paper a higher order lagrange galerkin discretization method is analyzed when applied to a variable coefficient convection possibly degenerated diffusion reaction equation with mixed dirichlet robin boundary conditions in a previous paper a bermu dez m r nogueiras and c va zquez siam j numer anal to appear the proposed second order time discretization scheme has been rigorously introduced for exact and approximated characteristics moreover the l infty l 2 stability property and l infty l 2 error estimates of order o delta t 2 have been obtained as a continuation of that work consistency error estimates of order o delta t 2 h k are obtained for the fully discretized lagrange galerkin scheme moreover adequate quadrature formulas are proposed for the practical implementation of the method with particular finite element spaces finally some numerical tests illustrate the theoretical results and the performance of the combination of second order lagrange galerkin schemes with quadrature formulas

sensitivity of the eigenfunctions and the level curvature distribution in quantum billiards in searching for the manifestations of the sensitivity of the eigenfunctions in quantum billiards with dirichlet boundary conditions with respect to the boundary data the normal derivative we have performed instead various numerical tests for the robnik billiard a quadratic conformal map of the unit disk for 600 shape parameter values where we have looked at the sensitivity of the energy levels with respect to the shape parameter we show the energy level flow diagrams for three stretches of fifty consecutive odd eigenstates each with index 1000 to 2000 in particular we have calculated the unfolded and normalized level curvature distribution and found that it continuously changes from a delta distribution for the integrable case circle to a broad distribution in the classically ergodic regime for some shape parameters the agreement with the goe von oppen formula is very good whereas we have also cases where the deviation from goe is significant and of physical origin in the intermediate case of mixed classical dynamics we have a semiclassical formula in the spirit of the berry robnik 1984 surmise here the agreement with theory is not good partially due to the localization phenomena which are expected to disappear in the semiclassical limit we stress that even for classically ergodic systems there is no global universality for the curvature distribution not even in the semiclassical limit

maximal abelian subalgebras of pseudoorthogonal lie algebras abstract the classification of maximal abelian subalgebras masas of o p q is reduced to the construction of orthogonally indecomposable oid masas five types of oid masas exist and their classification amounts to an analysis of maximal abelian nilpotent subalgebras manss of o p q su p q sl n r or sl n c theorems are provided making possible the classification of all manss of o p q and low dimensional examples are analyzed in detail

some characterizations of optimal trajectories in control theory several characterizations of optimal trajectories for the classical mayer problem in optimal control are provided for this purpose the regularity of directional derivatives of the value function is studied for instance it is shown that for smooth control systems the value function v is continuously differentiable along an optimal trajectory x t 0 1 to bf r n provided v is differentiable at the initial point t 0 x t 0 then the upper semicontinuity of the optimal feedback map is deduced the problem of optimal design is addressed obtaining sufficient conditions for optimality finally it is shown that the optimal control problem may be reduced to a viability one

brief paper a neural network solution for fixed final time optimal control of nonlinear systems in this paper fixed final time optimal control laws using neural networks and hjb equations for general affine in the input nonlinear systems are proposed the method utilizes kronecker matrix methods along with neural network approximation over a compact set to solve a time varying hjb equation the result is a neural network feedback controller that has time varying coefficients found by a priori offline tuning convergence results are shown the results of this paper are demonstrated on an example

maximal abelian subalgebras of real and complex symplectic lie algebras we provide guidelines for classifying maximal abelian subalgebras masa s of the symplectic lie algebras sp 2n r and sp 2n c into conjugacy classes under the lie groups sp 2n r and sp 2n c respectively the task of classifying all masa s is reduced to the classification of orthogonally indecomposable oid masa s two types of orthogonally indecomposable masa s of sp 2n c exist 1 indecomposable maximal abelian nilpotent subalgebras mans s 2 decomposable masa s their classification reduces to a classification of mans s of sl n c four types of orthogonally indecomposable masa s of sp 2n r exist 1 absolutely indecomposable masa s mans s 2 relatively indecomposable masa s their classification reduces to a classification of mans s of su p q for p q n 3 decomposable absolutely oid masa s involving mans s of sl n r 4 decomposable relatively oid masa s involving mans s of sl n 2 c for n even low dimensional cases of sp 2n f n 1 2 3 f r or c are treated exhau

on zero sum stochastic differential games with jump diffusion driven state a viscosity solution framework a zero sum differential game with controlled jump diffusion driven state is considered and studied by a combination of dynamic programming and viscosity solution techniques we prove under certain conditions that the value of the game exists moreover the value function is shown to be the unique viscosity solution of a fully nonlinear integro partial differential equation in addition we formulate and prove a verification theorem for such games within the viscosity solution framework for nonlocal equations

finite element methods for nonlinear flows in porous media abstract a model problem is presented for fluid flow in porous media which has numerical difficulties common with many other areas of nonlinear mechanics special finite element techniques are described which have proved useful in alleviating many of these difficulties a modified method of characteristics mixed finite element methods for fluid velocities singularity removal and self adaptive local grid refinement techniques are presented both theoretical and computational results are described together with an indication of the present and future uses of finite element methods in large scale reservoir simulation

optimal portfolios with regime switching and value at risk constraint we consider the optimal portfolio selection problem subject to a maximum value at risk mvar constraint when the price dynamics of the risky asset are governed by a markov modulated geometric brownian motion gbm here the market parameters including the market interest rate of a bank account the appreciation rate and the volatility of the risky asset switch over time according to a continuous time markov chain whose states are interpreted as the states of an economy the mvar is defined as the maximum value of the vars of the portfolio in a short time duration over different states of the chain we formulate the problem as a constrained utility maximization problem over a finite time horizon by utilizing the dynamic programming principle we shall first derive a regime switching hamilton jacobi bellman hjb equation and then a system of coupled hjb equations we shall employ an efficient numerical method to solve the system of coupled hjb equations for the optimal constrained portfolio we shall provide numerical results for the sensitivity analysis of the optimal portfolio the optimal consumption and the var level with respect to model parameters these results are also used to investigating the effect of the switching regimes

maximal abelian subalgebras of complex orthogonal lie algebras abstract the study of maximal abelian subalgebras masas of o n c is reduced to the study of orthogonally indecomposable masas their classification is in turn reduced to that of maximal abelian nilpotent subalgebras manss of sl n 2 c for n even and of o n c itself general results and guidelines are given making it possible to construct all manss and hence all masas for any given n representatives of all o n c conjugacy classes of masas of o n c are constructed for 2 n 7 as an application of the general techniques

distribution of the spacing between two adjacent avoided crossings we consider the frequency at which avoided crossings appear in an energy level structure when an external field is applied to a quantum chaotic system the distribution of the spacing in the parameter between two adjacent avoided crossings is investigated using a random matrix model we find that the distribution of these spacings is well fitted by a power law distribution for small spacings the powers are 2 and 3 for the gaussian orthogonal ensemble and gaussian unitary ensemble respectively we also find that the distributions decay exponentially for large spacings the distributions in concrete quantum chaotic systems agree with those of the random matrix model

on application of an alternating direction method to hamilton jacobin bellman equations this paper presents a numerical method for the approximation of viscosity solutions to a hamilton jacobi bellman hjb equation governing a class of optimal feedback control problems the first order hjb equation is first perturbed by adding a diffusion term with a singular perturbation parameter the time and spatial variables in the resulting equation are then discretized respectively by an implicit modified method of characteristics and the alternating direction ad scheme we show that the ad procedure s perturbation error is virtually negligible due to the small perturbation parameter and the efficient ad scheme can be applied to our hjb equation without generating splitting error numerical results performed to verify the usefulness of the method show that the method gives accurate approximate solutions to both of the control and the state variables

distribution of multiple avoided crossings numerical evaluation the authors compute numerically the distribution rho delta in of delta in the local minimum distance avoided crossing gap size between neighbouring energy levels of two model systems they find that limit delta in to 0 rho approximately delta in the result differs from that expected for a goe model for which wilkinson 1988 has shown that in the small delta in limit rho delta in is constant they suggest that the discrepancy arises from parametric correlations which are known to exist but are absent from the goe

a semiclassical sum rule for matrix elements of classically chaotic systems in the semiclassical limit the sum s e delta e sigma nm mod anm mod 2 delta e 1 2 en em delta delta e en em of matrix elements of an arbitrary operator a can be related to the classical correlation function of the weyl symbol a q p of a ca e t integral d alpha delta e h alpha a alpha a alpha t s e delta e is proportional to the fourier transform of ca e t over t plus a set of correction terms associated with periodic trajectories in phase space if the system has a chaotic classical limit the matrix elements are independently gaussian distributed with mean value zero and s e delta e gives the variance of this distribution

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

stochastic verification theorems within the framework of viscosity solutions this paper studies controlled systems governed by ito s stochastic differential equations in which control variables are allowed to enter both drift and diffusion terms a new verification theorem is derived within the framework of viscosity solutions without involving any derivatives of the value functions this theorem is shown to have wider applicability than the restrictive classical verification theorems which require the associated dynamic programming equations to have smooth solutions based on the new verification result optimal stochastic feedback controls are obtained by maximizing the generalized hamiltonians over both the control regions and the superdifferentials of the value functions