Full-Discrete Finite Element Method for Stochastic Hyperbolic Equation

This paper is concerned with the finite element method for the stochastic wave equation and the stochastic elastic equation driven by space-time white noise. For simplicity, we rewrite the two types of stochastic hyperbolic equations into a unified form. We convert the stochastic hyperbolic equation into a regularized equation by discretizing the white noise and then consider the full-discrete finite element method for the regularized equation. We derive the modeling error by using "Green's method" and the finite element approximation error by using the error estimates of the deterministic equation. Some numerical examples are presented to verify the theoretical results.     

Identification for Linear Stochastic Systems Driven by Fractional Brownian Motion

Abstract

We apply Grenander's method of sieves to the problem of identification or estimation of the “drift” function for linear stochastic systems driven by a fractional Brownian motion (fBm). We use an increasing sequence of finite dimensional subspaces of the parameter space as the natural sieves on which we maximise the likelihood function.     

Full‐discrete finite element method for the stochastic elastic equation driven by additive noise

We study the full‐discrete finite element method for the stochastic elastic equation driven by additive noise. To analyze the error estimates, we write the stochastic elastic equation as an abstract stochastic equation. Strong convergence estimates in the root mean square L₂ ‐norm are obtained by using the error estimates for the deterministic problem and the semigroup theory. Numerical experiments are carried out to verify the theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

The Minimal Entropy Martingale Measure and Numerical Option Pricing for the Barndorff–Nielsen–Shephard Stochastic Volatility Model

Abstract

We develop and apply a numerical scheme for pricing options in the stochastic volatility model proposed by Barndorff–Nielsen and Shephard. This non-Gaussian Ornstein–Uhlenbeck type of volatility model gives rise to an incomplete market, and we consider the option prices under the minimal entropy martingale measure. To numerically price options with respect to this risk neutral measure, one needs to consider a Black and Scholes type of partial differential equation, with an integro-term arising from the volatility process. We suggest finite difference schemes to solve this parabolic integro-partial differential equation, and derive appropriate boundary conditions for the finite difference method. As an application of our algorithm, we consider price deviations from the Black and Scholes formula for call options, and the implications of the stochastic volatility on the shape of the volatility smile.     

Uncertainty quantification for linear elastic bodies with two fluctuating input parameters

In this work, we present a numerical method for solving partial differential equations (PDEs) with stochastic coefficients for a linear elastic body. To this end, a stochastic finite element method is applied. We distinguish two different cases for an isotropic material with two fluctuating input parameters in order to analyse the optimal choice of input parameters. Using the GALERKIN projection, the final stochastic equation system is reduced to a system of deterministic PDEs. Subsequently, the solution is determined iteratively. Finally, a numerical example for a plate with a ring hole is presented. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A posteriori analysis of finite element discretizations of stochastic partial differential delay equations

In this paper, we study a posteriori error estimates for finite element approximation of stochastic partial differential delay equations containing a noise. We derive an energy norm a posteriori bounds for an Euler time-stepping method combined with a standard Galerkin schemes for the problems. For accessibility, we first address the spatially semidiscrete case and then move to the fully discrete scheme.     

Crank–Nicolson Finite Element Scheme and Modified Reduced-Order Scheme for Fractional Sobolev Equation

Abstract

In this paper, a Crank–Nicolson finite difference/finite element method is considered to obtain the numerical solution for a time fractional Sobolev equation. Firstly, the classical finite element method is presented. Stability and error estimation for the fully discrete scheme are rigorously established. However, the amount of calculation and computing time are too large due to many degrees of freedom of classical finite element scheme and nonlocality of fractional differential operator. And then the modified reduced-order finite element scheme with low dimensions and sufficiently high accuracy, which is based on proper orthogonal decomposition technique, is provided. Stability and convergence for the reduced-order scheme are also studied. At last, numerical examples show that the results of numerical computation are consistent with previous theoretical conclusions.     

Stochastic Differential Equations with Non-Lipschitz Coefficients in Hilbert Spaces

Abstract

In this article, we discuss the successive approximations problem for the solutions of the semilinear stochastic differential equations in Hilbert spaces with cylindrical Wiener processes under some conditions which are weaker than the Lipschitz one. We establish the existence and the uniqueness of the solution and additionally, in our framework we consider a limiting problem for the mild solution. It is shown that the mild solution tends to the solution of the stochastic differential equation of Itô type in finite dimensional space.     

Mean-field stochastic control with elephant memory in finite and infinite time horizon

ABSTRACT

Our purpose of this paper is to study stochastic control problems for systems driven by mean-field stochastic differential equations with elephant memory, in the sense that the system (like the elephants) never forgets its history. We study both the finite horizon case and the infinite time horizon case.

• In the finite horizon case, results about existence and uniqueness of solutions of such a system are given. Moreover, we prove sufficient as well as necessary stochastic maximum principles for the optimal control of such systems. We apply our results to solve a mean-field linear quadratic control problem.

• For infinite horizon, we derive sufficient and necessary maximum principles.

  As an illustration, we solve an optimal consumption problem from a cash flow modelled by an elephant memory mean-field system.

     

Implementation of an adaptive finite-element approximation of the Mumford-Shah functional

Summary. We present and detail a method for the numerical solving of the Mumford-Shah problem, based on a finite element method and on adaptive meshes. We start with the formulation introduced in [13], detail its numerical implementation and then propose a variant which is proved to converge to the Mumford-Shah problem. A few experiments are illustrated. Received October 8, 1998 / Published online: April 20, 2000     

A wavelet-based stochastic finite element method of thin plate bending

A wavelet-based stochastic finite element method is presented for the bending analysis of thin plates. The wavelet scaling functions of spline wavelets are selected to construct the displacement interpolation functions of a rectangular thin plate element and the displacement shape functions are expressed by the spline wavelets. A new wavelet-based finite element formulation of thin plate bending is developed by using the virtual work principle. A wavelet-based stochastic finite element method that combines the proposed wavelet-based finite element method with Monte Carlo method is further formulated. With the aid of the wavelet-based stochastic finite element method, the present paper can deal with the problem of thin plate response variability resulting from the spatial variability of the material properties when it is subjected to static loads of uncertain nature. Numerical examples of thin plate bending have demonstrated that the proposed wavelet-based stochastic finite element method can achieve a high numerical accuracy and converges fast.     

Weak martingale solutions to the stochastic Landau–Lifshitz–Gilbert equation with multi-dimensional noise via a convergent finite-element scheme

We propose an unconditionally convergent linear finite element scheme for the stochastic Landau–Lifshitz–Gilbert (LLG) equation with multi-dimensional noise. By using the Doss–Sussmann technique, we first transform the stochastic LLG equation into a partial differential equation that depends on the solution of the auxiliary equation for the diffusion part. The resulting equation has solutions absolutely continuous with respect to time. We then propose a convergent $\theta$-linear scheme for the numerical solution of the reformulated equation. As a consequence, we are able to show the existence of weak martingale solutions to the stochastic LLG equation.     

Space-time finite element methods stabilized using bubble function spaces

ABSTRACT

In this paper, a stabilized space-time finite element method for solving linear parabolic evolution problems is analyzed. The proposed method is developed on a base of a space-time variational setting, that helps on the simultaneous and unified discretization in space and in time by finite element techniques. Stabilization terms are constructed by means of classical bubble spaces. Stability of the discrete problem with respect to an associated mesh dependent norm is proved, and a priori discretization error estimates are presented. Numerical examples confirm the theoretical estimates.     

A GLOBAL-LOCAL FINITE ELEMENT METHOD IN SPACE-TIME FOR A HYPERBOLIC PROBLEM

ABSTRACT

A finite element method for solving the wave equation with couples boundary conditions is presented. In this approach finite elements are applied globally with respect to space and simultaneously but locally with respect to time. This gives rise to a single-step method in time. The method is a practical and economic one and the numerical results obtained compare favourably with the available analytic solution.     

Mixed finite element methods for a dynamical Ginzburg-Landau model in superconductivity

Summary. In this paper we consider the numerical solutions of the nonlinear time-dependent Ginzburg-Landau model which describes the phase transitions taking place in superconducting films. We propose a semi-implicit finite element scheme which is based on a linear finite element approximation of the order parameter and a mixed finite element discretization for the equation involving the magnetic potential A. The error estimates of the scheme are derived. Received September 5, 1994 / Revised version received April 23, 1995     

A note on conjugate gradient convergence – Part III

Summary. In this paper we again consider the rate of convergence of the conjugate gradient method. We start with a general analysis of the conjugate gradient method for uniformly bounded solutions vectors and matrices whose eigenvalues are uniformly bounded and positive. We show that in such cases a fixed finite number of iterations of the method gives some fixed amount of improvement as the the size of the matrix tends to infinity. Then we specialize to the finite element (or finite difference) scheme for the problem . We show that for some classes of function we see this same effect. For other functions we show that the gain made by performing a fixed number of iterations of the method tends to zero as the size of the matrix tends to infinity. Received July 9, 1998 / Published online March 16, 2000     

Decomposition of test sets in stochastic integer programming

 We study Graver test sets for linear two-stage stochastic integer programs and show that test sets can be decomposed into finitely many building blocks whose number is independent on the number of scenarios of the stochastic program. We present a finite algorithm to compute the building blocks directly, without prior knowledge of test set vectors. Once computed, building blocks can be employed to solve the stochastic program by a simple augmentation scheme, again without explicit knowledge of test set vectors. Finally, we report preliminary computational experience. Received: March 14, 2002 / Accepted: March 27, 2002 Published online: September 27, 2002 Key words. test sets – stochastic integer programming – decomposition methods Mathematics Subject Classification (2000): 90C15, 90C10, 13P10     

Applied topics of control theory,mathematical statistics,and mathematical cybernetics

A stochastic discrepancy method is proposed for the construction of a solving system of equations of the projective grid method in Bubnov-Galerkin form. The interpolation polynomial on a finite element is viewed as the result of weighted averaging of the nodal parameters of the element. A simple posterior bound on discretization error is considered.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 100–103, 1988.     

Conforming and nonconforming harmonic finite elements

ABSTRACT

Instead of using the full polynomial space, a conforming and a nonconforming finite element methods are designed where only harmonic polynomials (a much smaller space) are employed in the computation. The conforming quadratic harmonic polynomial finite element is defined only on a special triangular grid. The nonconforming quadratic harmonic finite element is defined on general triangular grids. The optimal order of convergence is proved for both finite element methods, and confirmed by numerical computations. In addition, numerical comparisons with the standard conforming and nonconforming finite elements are presented.     

A two-grid finite element method for the Allen-Cahn equation with the logarithmic potential

In this paper, we present a two-grid finite element method for the Allen-Cahn equation with the logarithmic potential. This method consists of two steps. In the first step, based on a fully implicit finite element method, the Allen-Cahn equation is solved on a coarse grid with mesh size H. In the second step, a linearized system whose nonlinear term is replaced by the value of the first step is solved on a fine grid with mesh size h. We give the energy stabilities of the traditional finite element method and the two-grid finite element method. The optimal convergence order of the two-grid finite element method in H¹ norm is achieved when the mesh sizes satisfy h = O(H²). Numerical examples are given to demonstrate the validity of the proposed scheme. The results show that the two-grid method can save the CPU time while keeping the same convergence rate.