Large deviations for stochastic partial differential equations driven by a Poisson random measure

Stochastic partial differential equations driven by Poisson random measures (PRMs) have been proposed as models for many different physical systems, where they are viewed as a refinement of a corresponding noiseless partial differential equation (PDE). A systematic framework for the study of probabilities of deviations of the stochastic PDE from the deterministic PDE is through the theory of large deviations. The goal of this work is to develop the large deviation theory for small Poisson noise perturbations of a general class of deterministic infinite dimensional models. Although the analogous questions for finite dimensional systems have been well studied, there are currently no general results in the infinite dimensional setting. This is in part due to the fact that in this setting solutions may have little spatial regularity, and thus classical approximation methods for large deviation analysis become intractable. The approach taken here, which is based on a variational representation for nonnegative functionals of general PRMs, reduces the proof of the large deviation principle to establishing basic qualitative properties for controlled analogues of the underlying stochastic system. As an illustration of the general theory, we consider a particular system that models the spread of a pollutant in a waterway.     

An optimal switching approach toward cost‐effective control of a stand‐alone photovoltaic panel system under stochastic environment

The operation of a stand‐alone photovoltaic (PV) system ultimately aims for the optimization of its energy storage. We present a mathematical model for cost‐effective control of a stand‐alone system based on a PV panel equipped with an angle adjustment device. The model is based on viscosity solutions to partial differential equations, which serve as a new and mathematically rigorous tool for modeling, analyzing, and controlling PV systems. We formulate a stochastic optimal switching problem of the panel angle, which is here a binary variable to be dynamically controlled under stochastic weather condition. The stochasticity comes from cloud cover dynamics, which is modeled with a nonlinear stochastic differential equation. In finding the optimal control policy of the panel angle, switching the angle is subject to impulsive cost and reduces to solving a system of Hamilton‐Jacobi‐Bellman quasi‐variational inequalities (HJBQVIs). We show that the stochastic differential equation is well posed and that the HJBQVIs admit a unique viscosity solution. In addition, a finite‐difference scheme is proposed for the numerical discretization of HJBQVIs. A demonstrative computational example of the HJBQVIs, with emphasis on a stand‐alone experimental system, is finally presented with practical implications for its cost‐effective operation.     

Solving hyperbolic conservation laws using multiquadric quasi‐interpolation

In this article, we apply the univariate multiquadric (MQ) quasi‐interpolation to solve the hyperbolic conservation laws. At first we construct the MQ quasi‐interpolation corresponding to periodic and inflow‐outflow boundary conditions respectively. Next we obtain the numerical schemes to solve the partial differential equations, by using the derivative of the quasi‐interpolation to approximate the spatial derivative of the differential equation and a low‐order explicit difference to approximate the temporal derivative of the differential equation. Then we verify our scheme for the one‐dimensional Burgers' equation (without viscosity). We can see that the numerical results are very close to the exact solution and the computational accuracy of the scheme is ??(τ), where τ is the temporal step. We can improve the accuracy by using the high‐order quasi‐interpolation. Moreover the methods can be generalized to the other equations. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A multigrid compact finite difference method for solving the one‐dimensional nonlinear sine‐Gordon equation

The aim of this paper is to propose a multigrid method to obtain the numerical solution of the one‐dimensional nonlinear sine‐Gordon equation. The finite difference equations at all interior grid points form a large sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a compact finite difference scheme of fourth‐order for discretizing the spatial derivative and the standard second‐order central finite difference method for the time derivative. The proposed method uses the Richardson extrapolation method in time variable. The obtained system has been solved by V‐cycle multigrid (VMG) method, where the VMG method is used for solving the large sparse linear systems. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional sine‐Gordon equation. Copyright © 2014 John Wiley & Sons, Ltd.     

Minimum action method for the study of rare events

The least action principle from the Wentzell‐Freidlin theory of large deviations is exploited as a numerical tool for finding the optimal dynamical paths in spatially extended systems driven by a small noise. The action is discretized and a preconditioned BFGS method is used to optimize the discrete action. Applications are presented for thermally activated reversal in the Ginzburg‐Landau model in one and two dimensions, and for noise induced excursion events in the Brusselator taken as an example of non‐gradient system arising in chemistry. In the Ginzburg‐Landau model, the reversal proceeds via interesting nucleation events, followed by propagation of domain walls. The issue of nucleation versus propagation is discussed and the scaling for the number of nucleation events as a function of the reversal time and other material parameters is computed. Good agreement is found with the numerical results. In the Brusselator, whose deterministic dynamics has a single stable equilibrium state, the presence of noise is shown to induce large excursions by which the system cycles out of this equilibrium state. © 2004 Wiley Periodicals, Inc.     

Modification of upwind finite difference fractional step methods by the transient state of the semiconductor device

The mathematical model of the three‐dimensional semiconductor devices of heat conduction is described by a system of four quasi‐linear partial differential equations for initial boundary value problem. One equation of elliptic form is for the electric potential; two equations of convection‐dominated diffusion type are for the electron and hole concentration; and one heat conduction equation is for temperature. Upwind finite difference fractional step methods are put forward. Some techniques, such as calculus of variations, energy method multiplicative commutation rule of difference operators, decomposition of high order difference operators, and the theory of prior estimates and techniques are adopted. Optimal order estimates in L² norm are derived to determine the error in the approximate solution.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Determination of minimum required damping in stochastic following seas modeled by using Gaussian white noise

The aim of this study is to present an analytical method to determine the minimum required damping moment for a stable ship in stochastic following seas modeled by using Gaussian white noise. Stochastic differential equation is used as a mathematical model to represent rolling motion of a ship. First, the minimum required damping is obtained analytically by using Lyapunov function. Second, analytically obtained damping values are verified by integrating the nonlinear stochastic rolling motion equation by stochastic Euler method (Euler–Maruyama Schema) to deduce whether rolling motion is stable or not. It can be seen from the results of numerical computation that the ship is sufficiently stable for the minimum required damping value obtained by the use of Lyapunov function and the minimum required damping is highly dependent on natural frequency of roll, diffusion constant and maximum variation of initial metacentric height.     

Mixed two‐grid finite difference methods for solving one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations

The aim of this paper is to propose mixed two‐grid finite difference methods to obtain the numerical solution of the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. The finite difference equations at all interior grid points form a large‐sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a family of finite difference methods for discretizing the spatial and time derivatives. The obtained system has been solved by two‐grid method, where the two‐grid method is used for solving the large‐sparse linear systems. Also, in the proposed method, the spectral radius with local Fourier analysis is calculated for different values of h and Δt. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. Copyright © 2016 John Wiley & Sons, Ltd.     

Reconstruction of random-disturbance amplitude in linear stochastic equations from measurements of some of the coordinates

The problem of reconstructing the unknown amplitude of a random disturbance in a linear stochastic differential equation is studied in a fairly general formulation by applying dynamic inversion theory. The amplitude is reconstructed using discrete information on several realizations of some of the coordinates of the stochastic process. The problem is reduced to an inverse one for a system of ordinary differential equations satisfied by the elements of the covariance matrix of the original process. Constructive solvability conditions in the form of relations on the parameters of the system are discussed. A finite-step software implementable solving algorithm based on the method of auxiliary controlled models is tested using a numerical example. The accuracy of the algorithm is estimated with respect to the number of measured realizations.     

Computational Krylov‐based methods for large‐scale differential Sylvester matrix problems

In the present paper, we propose Krylov‐based methods for solving large‐scale differential Sylvester matrix equations having a low‐rank constant term. We present two new approaches for solving such differential matrix equations. The first approach is based on the integral expression of the exact solution and a Krylov method for the computation of the exponential of a matrix times a block of vectors. In the second approach, we first project the initial problem onto a block (or extended block) Krylov subspace and get a low‐dimensional differential Sylvester matrix equation. The latter problem is then solved by some integration numerical methods such as the backward differentiation formula or Rosenbrock method, and the obtained solution is used to build the low‐rank approximate solution of the original problem. We give some new theoretical results such as a simple expression of the residual norm and upper bounds for the norm of the error. Some numerical experiments are given in order to compare the two approaches.     

The quasi‐reversibility method for a final value problem of the time‐fractional diffusion equation with inhomogeneous source

This paper is devoted to discuss a multidimensional backward heat conduction problem for time‐fractional diffusion equation with inhomogeneous source. This problem is ill‐posed. We use quasi‐reversibility regularization method to solve this inverse problem. Moreover, the convergence estimates between regularization solution and the exact solution are obtained under the a priori and the a posteriori choice rules. Finally, the numerical examples for one‐dimensional and two‐dimensional cases are presented to show that our method is feasible and effective.     

Approximate solutions to large nonsymmetric differential Riccati problems with applications to transport theory

In this paper, we consider large‐scale nonsymmetric differential matrix Riccati equations with low‐rank right‐hand sides. These matrix equations appear in many applications such as control theory, transport theory, applied probability, and others. We show how to apply Krylov‐type methods such as the extended block Arnoldi algorithm to get low‐rank approximate solutions. The initial problem is projected onto small subspaces to get low dimensional nonsymmetric differential equations that are solved using the exponential approximation or via other integration schemes such as backward differentiation formula (BDF) or Rosenbrock method. We also show how these techniques can be easily used to solve some problems from the well‐known transport equation. Some numerical examples are given to illustrate the application of the proposed methods to large‐scale problems.     

An efficient method for fractional nonlinear differential equations by quasi‐Newton's method and simplified reproducing kernel method

An efficient method for nonlinear fractional differential equations is proposed in this paper. This method consists of 2 steps. First, we linearize the nonlinear operator equation by quasi‐Newton's method, which is based on Fréchet derivative. Then we solve the linear fractional differential equations by the simplified reproducing kernel method. The convergence of the quasi‐Newton's method is discussed for the general nonlinear case as well. Finally, some numerical examples are presented to illustrate accuracy, efficiency, and simplicity of the method.     

Numerical method for coupling the macro and meso scales in stochastic chemical kinetics

A numerical method is developed for simulation of stochastic chemical reactions. The system is modeled by the Fokker–Planck equation for the probability density of the molecular state. The dimension of the domain of the equation is reduced by assuming that most of the molecular species have a normal distribution with a small variance. The numerical approximation preserves properties of the analytical solution such as non-negativity and constant total probability. The method is applied to a nine dimensional problem modelling an oscillating molecular clock. The oscillations stop at a fixed point with a macroscopic model but they continue with our two dimensional, mixed macroscopic and mesoscopic model. Dedicated to the memory of Germund Dahlquist (1925–2005). AMS subject classification (2000)  65M20, 65M60     

A computational method for large‐scale differential symmetric Stein equation

We propose a numerical method for solving large‐scale differential symmetric Stein equations having low‐rank right constant term. Our approach is based on projection the given problem onto a Krylov subspace then solving the low dimensional matrix problem by using an integration method, and the original problem solution is built by using obtained low‐rank approximate solution. Using the extended block Arnoldi process and backward differentiation formula (BDF), we give statements of the approximate solution and corresponding residual. Some numerical results are given to show the efficiency of the proposed method.     

Semidiscretisation Methods for Quasi‐periodic Solutions

The aim of our approach is a reliable numerical approximation of quasi‐periodic solutions of periodically forced ODEs without using a‐priori transformations into new coordinates [1]. The invariant torus is computed as a solution of a special invariance equation. In the case of two basic frequencies this system can be solved by semidiscretisation, which transforms the system into a higher dimensional autonomous ODE system with periodic solutions.     

Numerical analysis nonlinear multi‐term time fractional differential equation with collocation method via fractional B‐spline

In this work, we present numerical analysis for nonlinear multi‐term time fractional differential equation which involve Caputo‐type fractional derivatives for . The proposed method is based on utilization of fractional B‐spline basics in collocation method. The scheme can be readily obtained efficient and quite accurate with less computational work numerical result. The proposal approach transform nonlinear multi‐term time fractional differential equation into a suitable linear system of algebraic equations which can be solved by a suitable numerical method. The numerical experiments will be verify to demonstrate the effectiveness of our method for solving one‐ and two‐dimensional multi‐term time fractional differential equation.     

Balanced POD for linear PDE robust control computations

A mathematical model of a physical system is never perfect; therefore, robust control laws are necessary for guaranteed stabilization of the nominal model and also ??nearby?? systems, including hopefully the actual physical system. We consider the computation of a robust control law for large-scale finite dimensional linear systems and a class of linear distributed parameter systems. The controller is robust with respect to left coprime factor perturbations of the nominal system. We present an algorithm based on balanced proper orthogonal decomposition to compute the nonstandard features of this robust control law. Convergence theory is given, and numerical results are presented for two partial differential equation systems.     

Fractional quasi AKNS‐technique for nonlinear space–time fractional evolution equations

This paper aims to formulate the fractional quasi‐inverse scattering method. Also, we give a positive answer to the following question: can the Ablowitz‐Kaup‐Newell‐Segur (AKNS) method be applied to the space–time fractional nonlinear differential equations? Besides, we derive the Bäcklund transformations for the fractional systems under study. Also, we construct the fractional quasi‐conservation laws for the considered fractional equations from the defined fractional quasi AKNS‐like system. The nonlinear fractional differential equations to be studied are the space–time fractional versions of the Kortweg‐de Vries equation, modified Kortweg‐de Vries equation, the sine‐Gordon equation, the sinh‐Gordon equation, the Liouville equation, the cosh‐Gordon equation, the short pulse equation, and the nonlinear Schrödinger equation.     

A maximum principle for relaxed stochastic control of linear SDEs with application to bond portfolio optimization

We study relaxed stochastic control problems where the state equation is a one dimensional linear stochastic differential equation with random and unbounded coefficients. The two main results are existence of an optimal relaxed control and necessary conditions for optimality in the form of a relaxed maximum principle. The main motivation is an optimal bond portfolio problem in a market where there exists a continuum of bonds and the portfolio weights are modeled as measure-valued processes on the set of times to maturity.