Meshfree Approximation for Stochastic Optimal Control Problems

In this work,we study the gradient projection method for solving a class of stochastic control problems by using a mesh free approximation ap-proach to implement spatial dimension approximation.Our main contribu-tion is to extend the existing gradient projection method to moderate high-dimensional space.The moving least square method and the general radial basis function interpolation method are introduced as showcase methods to demonstrate our computational framework,and rigorous numerical analysis is provided to prove the convergence of our meshfree approximation approach.We also present several numerical experiments to validate the theoretical re-sults of our approach and demonstrate the performance meshfree approxima-tion in solving stochastic optimal control problems.     

A sequential data assimilation method based on the properties of a diffusion-type process

Data assimilation method, as commonly used in numerical ocean and atmospheric circulation models, produces an estimation of state variables in terms of stochastic processes. This estimation is based on limit properties of a diffusion-type process which follows from the convergence of a sequence of Markov chains with jumps. The conditions for this convergence are investigated. The optimisation problem and the optimal filtering problem associated with the search of the best possible approximation of the true state variable are posed and solved. The results of a simple numerical experiment are discussed. It is shown that the proposed data assimilation method works properly and can be used in practical applications, particularly in meteorology and oceanography.     

Implicit Euler method for numerical solution of nonlinear stochastic partial differential equations with multiplicative trace class noise

In this paper, we consider the numerical approximation of stochastic partial differential equations with nonlinear multiplicative trace class noise. Discretization is obtained by spectral collocation method in space, and semi‐implicit Euler method is used for the temporal approximation. Our purpose is to investigate the convergence of the proposed method. The rate of convergence is obtained, and some numerical examples are included to illustrate the estimated convergence rate.     

An efficient cubic B-spline and bicubic B-spline collocation method for numerical solutions of multidimensional nonlinear stochastic quadratic integral equations

This paper aims to develop a novel numerical approach on the basis of B-spline collocation method to approximate the solution of one-dimensional and two-dimensional nonlinear stochastic quadratic integral equations. The proposed approach is based on the hybrid of collocation method, cubic B-spline, and bi-cubic B-spline interpolation and Itô approximation. Using this method, the problem solving turns into a nonlinear system solution of equations that is solved by a suitable numerical method. Also, the convergence analysis of this numerical approach has been discussed. In the end, examples are given to test the accuracy and the implementation of the method. The results are compared with the results obtained by other methods to verify that this method is accurate and efficient.     

Convergence analysis of a minimax method for finding multiple solutions of hemivariational inequality in Hilbert space

In 2013, a minimax method for finding saddle points of locally Lipschitz continuous functional was designed (Yao Math. Comp. 82 2087–2136 2013). The method can be applied to numerically solve hemivariational inequality for multiple solutions. Its subsequence and sequence convergence results in functional analysis were established in the same paper. But, since these convergence results do not consider discretization, they are not convergence results in numerical analysis. In this paper, we point out what approximation problem is, when this minimax method is used to solve hemivariational inequality and the finite element method is used in discretization. Computation of the approximation problem is discussed, numerical experiment is carried out and its global convergence is verified. Finally, as element size goes to zero, convergence of solutions of the approximation problem to solutions of hemivariational inequality is proved.     

求解log—最优组合投资问题的一个自适应算法及其理论分析

本文给出了一个求解log－最优组合投资问题的自适应算法,它是一个变型的随机逼近方法。该问题是一个约束优化问题,因此,采用基于约束流形的梯度上升方向替代常规梯度上升方向,在一些合理的假设下证明了算法的收敛性并进行了渐近稳定性分析。最后,本文将该算法应用于上海证券交易所提供的实际数据的log-最优组合投资问题求解,获得了理想的数值模拟结果。     

A Fourier method and an extrapolation technique for Stokes’ first problem for a heated generalized second grade fluid with fractional derivative

In this article, we consider Stokes’ first problem for a heated generalized second grade fluid with fractional derivative (SFP-HGSGF). Implicit and explicit numerical approximation schemes for the SFP-HGSGF are presented. The stability and convergence of the numerical schemes are discussed using a Fourier method. In addition, the solvability of the implicit numerical approximation scheme is also analyzed. A Richardson extrapolation technique for improving the order of convergence of the implicit scheme is proposed. Finally, a numerical test is given. The numerical results demonstrate the good performance of our theoretical analysis.     

Numerical methods for controls for nonlinear stochastic systems with delays and jumps: applications to admission control

We consider numerical methods of the Markov chain approximation type for computing optimal controls and value functions for systems governed by nonlinear stochastic delay equations. Earlier work did not allow Poisson random measure driving processes or delays that are concentrated on points with positive probability. In addition, the Poisson measures can be controlled. Previous proofs are not adequate for the present case. The algorithms are developed and convergence proved as the approximating parameters go to their limits. One motivating example concerns admissions control to a network, where the file arrival process is governed by a Poisson process, and arrivals might be admitted or not, according to the control, which leads to a controlled Poisson process. Numerical data for such an example are presented. The original problem is recast in terms of a transportation equation, which allows the development of practical algorithms. For the problems of interest, alternative methods can entail prohibitive memory and computational requirements.     

A numerical method of structure-preserving model updating problem and its perturbation theory

A method is presented to update a special finite element (FE) analytical model, based on matrix approximation theory with spectral constraint. At first, the model updating problem is treated as a matrix approximation problem dependent on the spectrum data from vibration test and modal parameter identification. The optimal approximation is the first modified solution of FE model. An algorithm is given to preserve the sparsity of the model by multiple correction. The convergence of the algorithm is investigated and perturbation of the modified solution is analyzed. Finally, a numerical example is provided to confirm the convergence of the algorithm and perturbation theory.     

Single and multi-period optimal inventory control models with risk-averse constraints

This paper presents some convex stochastic programming models for single and multi-period inventory control problems where the market demand is random and order quantities need to be decided before demand is realized. Both models minimize the expected losses subject to risk aversion constraints expressed through Value at Risk (VaR) and Conditional Value at Risk (CVaR) as risk measures. A sample average approximation method is proposed for solving the models and convergence analysis of optimal solutions of the sample average approximation problem is presented. Finally, some numerical examples are given to illustrate the convergence of the algorithm.     

非线性l1问题的光滑近似算法

为非线性l1问题的求解构造了光滑逼近函数.首先将非线性l1问题转化为等价的不可微优化问题;其次通过两步提出光滑逼近函数的一般性构造方法;最后进行了数值仿真.文中介绍了光滑逼近函数的有关性质,指出相关文献已有的光滑函数方法是本文的特例,并证明了方法的收敛性及有效性.     

A dynamic contact problem in thermoviscoelasticity with two temperatures

This work is concerned with the study of a one-dimensional dynamic contact problem arising in thermoviscoelasticity with two temperatures. The existence and uniqueness of a solution to the continuous problem is established using the Faedo–Galerkin method. A finite element approximation is proposed, a convergence result given and some numerical simulations described.     

Numerical solution of a long-term average control problem for singular stochastic processes

This paper analyzes numerically a long-term average stochastic control problem involving a controlled diffusion on a bounded region. The solution technique takes advantage of an infinite-dimensional linear programming formulation for the problem which relates the stationary measures to the generators of the diffusion. The restriction of the diffusion to an interval is accomplished through reflection at one end point and a jump operator acting singularly in time at the other end point. Different approximations of the linear program are obtained using finite differences for the differential operators (a Markov chain approximation to the diffusion) and using a finite element method to approximate the stationary density. The numerical results are compared with each other and with dynamic programming. This research has been supported in part by the U.S. National Security Agency under Grant Agreement Number H98230-05-1-0062. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein.     

A new approximation for the generalized fractional derivative and its application to generalized fractional diffusion equation

In this paper, we derive a fourth order approximation for the generalized fractional derivative that is characterized by a scale function z(t) and a weight function w(t) . Combining the new approximation with compact finite difference method, we develop a numerical scheme for a generalized fractional diffusion problem. The stability and convergence of the numerical scheme are proved by the energy method, and it is shown that the temporal and spatial convergence orders are both 4. Several numerical experiments are provided to illustrate the efficiency of our scheme.     

Numerical methods for controlled and uncontrolled multiplexing and queueing systems

We deal with a very useful numerical method for both controlled and uncontrolled queuing and multiplexing type systems. The basic idea starts with a heavy traffic approximation, but it is shown that the results are very good even when working far from the heavy traffic regime. The underlying numerical method is a version of what is known as the Markov chain approximation method. It is a powerful methodology for controlled and uncontrolled stochastic systems, which can be approximated by diffusion or reflected diffusion type systems, and has been used with success on many other problems in stochastic control. We give a complete development of the relevant details, with an emphasis on multiplexing and particular queueing systems. The approximating process is a controlled or uncontrolled Markov chain which retains certain essential features of the original problem. This problem is generally substantially simpler than the original physical problem, and there are associated convergence theorems. The non-classical associated ergodic cost problem is derived, and put into a form such that reliable and good numerical algorithms, based on multigrid type ideas, can be used. Data for both controlled and uncontrolled problems shows the value of the method.Supported by ARO contract DAAL-03-92-G-0115, AFOSR contract F49620-92-J-0081, and DARPA contract AFOSR-91-0375.Formerly at Brown University. Supported by DARPA contract AFOSR-91-0375.     

A penalty function-based random search algorithm for optimal control of switched systems with stochastic constraints and its application in automobile test-driving with gear shifts

Practical industrial process is usually a dynamic process including uncertainty. Stochastic constraints can be used for industrial process modeling, when system sate and/or control input constraints cannot be strictly satisfied. Thus, optimal control of switched systems with stochastic constraints can be available to address practical industrial process problems with different modes. In general, obtaining an analytical solution of the optimal control problem is usually very difficult due to the discrete nature of the switching law and the complexity of stochastic constraints. To obtain a numerical solution, this problem is formulated as a constrained nonlinear parameter selection problem (CNPSP) based on a relaxation transformation (RT) technique, an adaptive sample approximation (ASA) method, a smooth approximation (SA) technique, and a control parameterization (CP) method. Following that, a penalty function-based random search (PFRS) algorithm is designed for solving the CNPSP based on a novel search rule-based penalty function (NSRPF) method and a novel random search (NRS) algorithm. The convergence results show that the proposed method is globally convergent. Finally, an optimal control problem in automobile test-driving with gear shifts (ATGS) is further extended to illustrate the effectiveness of the proposed method by taking into account some stochastic constraints. Numerical results show that compared with other typical methods, the proposed method is less conservative and can obtain a stable and robust performance when considering the small perturbations in initial system state. In addition, to balance the computation amount and the numerical solution accuracy, a tolerance setting method is also provided by the numerical analysis technique.     

Numerical solution of a nonlinear reaction diffusion equation

In this paper, the authors propose a numerical method to compute the solution of a Cauchy problem with blow-up of the solution. The problem is split in two parts: a hyperbolic problem which is solved by using Hopf and Lax formula and a parabolic problem solved by a backward linearized Euler method in time and a finite element method in space. It is proved that the numerical solution blows up in a finite time as the exact solution and the support of the approximation of a self-similar solution remains bounded. The convergence of the scheme is obtained.     

Highly Accurate Numerical Schemes for Stochastic Optimal Control via FBSDEs

This work is concerned with numerical schemes for stochastic optimal control problems (SOCPs) by means of forward backward stochastic differential equations (FBSDEs). We first convert the stochastic optimal control problem into an equivalent stochastic optimality system of FBSDEs. Then we design an efficient second order FBSDE solver and an quasi-Newton type optimization solver for the resulting system. It is noticed that our approach admits the second order rate of convergence even when the state equation is approximated by the Euler scheme. Several numerical examples are presented to illustrate the effectiveness and the accuracy of the proposed numerical schemes.     

Spectral discretization of a mixed Schrödinger boundary-value problem and application

Paraxial approximation to the Helmholtz equation in ocean acoustic leads to solve a mixed Schrödinger boundary-value problem. The numerical analysis is performed combining a spectral method in the depth direction and a leap-frog scheme in the propagation direction. A convergence analysis for the approximation is developed, providing error estimates.