Numerical convergence for the Bellman equation of stochastic optimal control with quadratic costs and constraints

A predictor-corrector, Crank-Nicolson computer algorithm is examined for the Bellman equation of stochastic optimal control with quadratic costs and constrained control. A linearized comparison equation is heuristically derived for the nonlinear and discontinuous Bellman equation. Convergence of the method is studied using von Neumann's Fourier stability method. A mesh-ratio-type condition for the convergence is derived for the comparison equation. This condition is uniform for both parabolic and hyperbolic versions of the nonlinear equation. The results are valid for Gaussian stochastic noise and Poisson noise.     

On the small time behavior of the nonlinear estimation problem for finite bandwidth signals

In this paper we investigate the small time behavior of solutions of the Zakai equation. We derive a wave equation-like stochastic partial differential equation which is related to the Zakai equation. We are able to solve this equation for sufficiently smooth signals, and (approximately) transform these into solutions of the Zakai equation. We construct a Hadamardtype expansion for solutions of this partial differential equation and show how this expansion is related to a small time expansion of solutions of the Zakai equation.     

Fokker–Planck approximation of the master equation in molecular biology

The master equation of chemical reactions is solved by first approximating it by the Fokker–Planck equation. Then this equation is discretized in the state space and time by a finite volume method. The difference between the solution of the master equation and the discretized Fokker–Planck equation is analyzed. The solution of the Fokker–Planck equation is compared to the solution of the master equation obtained with Gillespie’s Stochastic Simulation Algorithm (SSA) for problems of interest in the regulation of cell processes. The time dependent and steady state solutions are computed and for equal accuracy in the solutions, the Fokker–Planck approach is more efficient than SSA for low dimensional problems and high accuracy.     

A functional approach to the stability of non-linear systems by use of the comparison theorem†

The paper presents some stability conditions for non-linear systems based on the application of the comparison theorem. The non-linear integral equation which gives the output of the system is dominated by another non-linear integral equation of simpler structure. The study of the boundedness of this equation is carried out by considering an algebraic equation; the stability conditions for the non-linear system corresponds to the existence of real positive roots of the algebraic equation.     

Finite element of the heat conduction equation with temperature dependent coefficients

A finite element formulation is derived for the equation of heat conduction with temperature dependent conductivity and heat capacity. The derivation of the finite element model is based on a variational formulation of the heat conduction equation which, together with generalized coordinates, yields an equation of similar form to Lagrange's equation in mechanics. The obtained equation is of special interest for applying the finite element method to solve problems with temperature-dependent properties.     

广义系统具有正定解的Lyapunov方程

本文研究线性广义系统有正定解的Lyapunov方程，给出广义系统稳定等价于Lyapunov方程有正定解，进一步研究了广义系统R－能观，稳定和Lyapunov方程存在正定解三者之间的关系。基于该Lyapunov方程，给出广义系统允许（正则，稳定，无脉冲）的等价条件。     

On the Lyapunov matrix differential equation

A lower bound for the determinant of the solution to the Lyapunov matrix differential equation is derived. It is shown that this bound is obtained as a solution to a simple scalar differential equation. In the limiting case where the solution to the Lyapunov differential equation becomes stationary, the result reduces to one of the existing bounds for the algebraic equation.     

A risk hypothesis and risk measures for throughput capacity in systems

A basic risk hypothesis for system throughput capacity in the presence of risk is proposed. It is expressed as a basic risk equation , derived in the paper, and governs all nongrowth, nonevolving, agent-directed systems. The basic risk equation shows how expected throughput capacity increases linearly with positive risk of loss of throughput capacity. The conventional standard deviation risk measure, from financial systems, may be used. A proposed new measure, the mean-expected loss risk measure with respect to the hazard-free case, is shown to be more appropriate for systems in general. The concept of an efficient system environment is also proposed. The well-known financial risk equation, hitherto deduced empirically, may be derived from the basic risk equation. When there is both risk exposure and resource sharing, the basic risk equation may be combined with a resource-sharing equation that governs how throughput capacity changes with the resource-sharing level. The basic risk equation also allows for risk elimination and reduction. All quantities in the equation are precisely defined, and their units are specified. The risk equation reduces to a useful numerical expression in practice.     

The connection between initial and unique solutions of domain equations in the partial order and metric approach

The purpose of this paper is twofold: First, we show in which way the initial solution of a domain equation for cpo's and the unique solution of a corresponding domain equation for metric spaces are related. Second, we present a technique to lift a given domain equation for cpo's to a corresponding domain equation for metric spaces.     

Correlated noise filtering and invariant directions for the Riccati equation

The Riccati equation associated with a class of discrete-time correlated noise problems is examined, and the concept of invariant directions for this equation is introduced. For single-output systems the set of such directions is completely characterized. Deletion of these directions by an appropriate transformation of the Riccati equation results in a minimal order equation for computation. This transformation also reveals the underlying structure of the optimal filter for the correlated noise problem.     

Development of a sixth-order two-dimensional convection–diffusion scheme via Cole–Hopf transformation

In this paper, we develop a two-dimensional finite-difference scheme for solving the time-dependent convection–diffusion equation. The numerical method exploits Cole–Hopf equation to transform the nonlinear scalar transport equation into the linear heat conduction equation. Within the semi-discretization context, the time derivative term in the transformed parabolic equation is approximated by a second-order accurate time-stepping scheme, resulting in an inhomogeneous Helmholtz equation. We apply the alternating direction implicit scheme of Polezhaev to solve the Helmholtz equation. As the key to success in the present simulation, we develop a Helmholtz scheme with sixth-order spatial accuracy. As is standard practice, we validated the code against test problems which were amenable to exact solutions. Results show excellent agreement for the one-dimensional test problems and good agreement with the analytical solution for the two-dimensional problem.     

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.     

Criterion for the convergence of the solution of the Riccati differential equation

The optimal control problem for a linear system with a quadratic cost function leads to the matrix Riccati differential equation. The convergence of the solution of this equation for increasing time interval is investigated as a function of the final state penalty matrix. A necessary and sufficient condition for convergence is derived for stabilizable systems, even if the output in the cost function is not detectable. An algorithm is developed to determine the limiting value of the solution, which is one of the symmetric positive semidefinite solutions of the algebraic Riccati equation. Examples for convergence and nonconvergence are given. A discussion is also included of the convergence properties of the solution of the Riccati differential equation to any real symmetric (not necessarily positive semidefinite) solution of the algebraic Riccati equation.     

Parametrizing and graphing nonsingular cubic curves

Elliptic functions are used to parametrize and graph nonsingular cubic curves. First, a sequence of transformations is derived that reduces a third degree equation to a standard equation. A parametrization in terms of elliptic functions is given for the graph of the standard equation. The transformations convert this parametrization to a parametrization of the graph of the given equation. The details of using this parametrization for computer graphing are presented.     

Methods of solving a polynomial equation for anH∞ optimal control problem for a single-inputsingle-output discrete-time system

A polynomial equation for the H_∞ optimal control problem is reduced to a nonlinear algebraic equation. Two methods are proposed for solving the algebraic equation. One method uses the singularity of a linear algebraic equation as the optimality index. The other gives an approximate solution by solving an eigenvalue problem. A numerical example is presented     

Hybrid Hamilton–Jacobi–Poisson wall distance function model

Expensive to compute wall distances are used in key turbulence models and also for the modeling of peripheral physics. A potentially economical, robust, readily parallel processed, accuracy improving, differential equation based distance algorithm is described. It is hybrid, partly utilising an approximate Poisson equation. This also allows auxiliary front propagation direction/velocity information to be estimated, effectively giving wall normals. The Poisson normal can be used fully, in an approximate solution of the eikonal equation (the exact differential equation for wall distance). Alternatively, a weighted fraction of this Poisson front direction (effectively, front velocity, in terms of the eikonal equation input) information and that implied by the eikonal equation can be used. Either results in a hybrid Poisson–eikonal wall distance algorithm. To improve compatibility of wall distance functions with turbulence physics a Laplacian is added to the eikonal equation. This gives what is termed a Hamilton–Jacobi equation. This hybrid Poisson–Hamilton–Jacobi approach is found to be robust on poor quality grids. The robustness largely results from the elliptic background presence of the Poisson equation. This elliptic component prevents fronts propagated from solid surfaces, by the hyperbolic eikonal equation element, reflecting off zones of rapidly changing grid density. Where this reflection (due to poor grid quality) is extreme, the transition of front velocity information from the Poisson to Hamilton–Jacobi equation can be done more gradually. Consistent with turbulence modeling physics, under user control, the hybrid equation can overestimate the distance function strongly around convex surfaces and underestimate it around concave. If the former trait is not desired the current approach is amenable to zonalisation. With this, the Poisson element is automatically removed around convex geometry zones.     

A Fitted Finite Volume Method for the Valuation of Options on Assets with Stochastic Volatilities

In this paper, we present a finite volume method for a two-dimensional Black-Scholes equation with stochastic volatility governing European option pricing. In this work, we first formulate the Black-Scholes equation with a tensor (or matrix) diffusion coefficient into a conservative form. We then present a finite volume method for the resulting equation, based on a fitting technique proposed for a one-dimensional Black-Scholes equation. We show that the method is monotone by proving that the system matrix of the discretized equation is an M-matrix. Numerical experiments, performed to demonstrate the usefulness of the method, will be presented.     

Limit Theorem for Multidimensional Renewal Equation

The multidimensional renewal equation in matrix form is considered. The renewal equation for the process with independent increments and states of the Markov process is found. The renewal function is investigated. The limit theorem for the renewal equation is proved.

     

PDE工具箱实现偏微分方程的有限元求解

偏微分方程在科学和工程上有着广泛的应用。有限元法是一种重要的偏微分方程数值解法。编程实现从偏微分方程到有限元求解全过程需要很好的理论基础和编程技巧,难度较高。该文介绍了偏微分方程有限元求解的基本理论和一般Neumann条件下椭圆型方程的有限元求解具体过程,并通过两个实例,电机磁场问题和热传导问题,介绍了使用PDE工具箱实现偏微分方程的有限元解法。实验结果表明这一方法具有操作简单明了,运算速度快,计算误差可控制等优点。     

Commutativity of source generation procedure and Bäcklund transformation: A BKP equation

The source generation procedure is applied to a BKP-type equation and a modified BKP-type (m-BKP) equation, respectively. As a result, a BKP equation with self-consistent sources (ESCS) and a m-BKP ESCS are derived. It is also proved that the bilinear m-BKP ESCS constitutes a bilinear Bäcklund transformation for the BKP ESCS. This means that the commutativity between the source generation procedure and Bäcklund transformation is valid for the BKP equation.