A class of split-step balanced methods for stiff stochastic differential equations

In this paper we design a class of general split-step balanced methods for solving It? stochastic differential systems with m-dimensional multiplicative noise, in which the drift or deterministic increment function can be taken from any chosen one-step ODE solver. We then give an analysis of their order of strong convergence in a general setting, but for the mean-square stability analysis, we confine our investigation to a special case in which the drift increment function of the methods is replaced by the one from the well known Rosenbrock method. The resulting class of stochastic differential equation (SDE) solvers will have more appropriate and useful mean-square stability properties for SDEs with stiffness in their drift and diffusion parts, compared to some other already reported split-step balanced methods. Finally, numerical results show the effectiveness of these methods.     

Heterogeneous multiscale methods for stiff ordinary differential equations

The heterogeneous multiscale methods (HMM) is a general framework for the numerical approximation of multiscale problems. It is here developed for ordinary differential equations containing different time scales. Stability and convergence results for the proposed HMM methods are presented together with numerical tests. The analysis covers some existing methods and the new algorithms that are based on higher-order estimates of the effective force by kernels satisfying certain moment conditions and regularity properties. These new methods have superior computational complexity compared to traditional methods for stiff problems with oscillatory solutions.

     

Contractive methods for stiff differential equations part I

An integration method for ordinary differential equations is said to be contractive if all numerical solutions of the test equationx=x generated by that method are not only bounded (as required for stability) but non-increasing. We develop a theory of contractivity for methods applied to stiff and non-stiff, linear and nonlinear problems. This theory leads to the design of a collection of specific contractive Adams-type methods of different orders of accuracy which are optimal with respect to certain measures of accuracy and/or contractivity. Theoretical and numerical results indicate that some of these novel methods are more efficient for solving problems with a lack of smoothness than are the familiar backward differentiation methods. This lack of smoothness may be either inherent in the problem itself, or due to the use of strongly varying integration steps. In solving smooth problems, the efficiency of the low-order contractive methods we propose is approximately the same as that of the corresponding backward differentiation methods.This work was done during the first author's stay at the IBM Thomas J. Watson Research Center under his appointment as Senior Researcher of the Academy of Finland. It was sponsored in addition by the IBM Corporation and by the AirForce Office of Scientific Research (AFSC), United States Air Force, under contracts No. F44620-75-C-0058 and F49620-77-C-0088. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon.     

A family of fully implicit Milstein methods for stiff stochastic differential equations with multiplicative noise

In this paper a family of fully implicit Milstein methods are introduced for solving stiff stochastic differential equations (SDEs). It is proved that the methods are convergent with strong order 1.0 for a class of SDEs. For a linear scalar test equation with multiplicative noise terms, mean-square and almost sure asymptotic stability of the methods are also investigated. We combine analytical and numerical techniques to get insights into the stability properties. The fully implicit methods are shown to be superior to those of the corresponding semi-implicit methods in term of stability property. Finally, numerical results are reported to illustrate the convergence and stability results.     

Exponentially-fitted Obrechkoff methods for second-order differential equations

We develop two-step exponentially-fitted Obrechkoff methods. The combination of exponential fitting and methods of Obrechkoff type is discussed here for the first time. We shall construct such methods of different orders and study their linear stability along the line of the Coleman/Ixaru definition. Some numerical results are introduced to show the applicability of such methods.     

Energy methods for stochastic differential equations

This article is devoted to the existence of strong solutions to stochastic differential equations (SDEs). Compared with Ito's theory, we relax the assumptions on the volatility term and replace the global Lipschitz continuity condition with a local Lipschitz continuity condition and a Hoelder continuity condition. In particular, our general SDE covers the Cox–Ingersoll–Ross SDE as a special case. We note that the general weak existence theory presumably extends to our general SDE (although the explicit time dependence of the drift term and the volatility term might require some extra considerations). However, avoiding weak existence theory we prove the existence of a strong solution directly using a priori estimates (the so-called energy estimates) derived from the SDE. The benefit of this approach is that the argument only requires some basic knowledge about stochastic and functional analysis. Moreover, the underlying principle has developed to become one of the cornerstones of the modern theory of partial differential equations (PDEs). In this sense, the general goal of this article is not just to establish the existence of a strong solution to the SDE under consideration but rather to introduce a new principle in the context of SDEs that has already proven to be successful in the context of PDEs.     

Rational-fractional methods for solving stiff systems of differential equations

This paper proposes new numerical methods for solving stiff systems of first-order differential equations not resolved with respect to the derivative. These methods are based on rational-fractional approximations of the vector-valued function of solution of the system considered. The authors study the stability of the constructed methods of arbitrary finite order of accuracy. Analysis of the results of experimental studies of these methods by test examples of various types confirms their efficiency. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 203–208, 2006.     

Efficient parallel multivalue hybrid methods for stiff differential equations

A class of efficient parallel multivalue hybrid methods for stiff differential equations are presented, which are all extremely stable at infinity,A-stable for orders 1–3 and A(α)-stable for orders 4–8. Each method of the class can be performed parallelly using two processors with each processor having almost the same computational amount per integration step as a backward differentiation formula (BDF) of the same order with the same stepsize performed in serial, whereas the former has not only much better stability properties but also a computational accuracy higher than the corresponding BDF. Theoretical analysis and numerical experiments show that the methods constructed in this paper are superior in many respects not only to BDFs but also to some other commonly used methods.     

On a dangerous property of methods for stiff differential equations

Two mathematically unstable problems are proposed as tests for numerical methods for stiff differential equations. Several methods failed to detect the instability of the problems and produced invalid solutions that for an unsuspecting user could appear to be quite reasonable.     

Split-step forward methods for stochastic differential equations

In this paper we discuss split-step forward methods for solving Itô stochastic differential equations (SDEs). Eight fully explicit methods, the drifting split-step Euler (DRSSE) method, the diffused split-step Euler (DISSE) method and the three-stage Milstein (TSM 1a-TSM 1f) methods, are constructed based on Euler-Maruyama method and Milstein method, respectively, in this paper. Their order of strong convergence is proved. The analysis of stability shows that the mean-square stability properties of the methods derived in this paper are improved on the original methods. The numerical results show the effectiveness of these methods in the pathwise approximation of Itô SDEs.     

Stability of implicit Runge-Kutta methods for nonlinear stiff differential equations

Under the assumption that an implicit Runge-Kutta method satisfies a certain stability estimate for linear systems with constant coefficientsl ₂-stability for nonlinear systems is proved. This assumption is weaker than algebraic stability since it is satisfied for many methods which are not evenA-stable. Some local smoothness in the right hand side of the differential equation is needed, but it may have a Jacobian and higher derivatives with large norms. The result is applied to a system derived from a strongly nonlinear parabolic equation by the method of lines.     

Convergence analysis of a global optimization algorithm using stochastic differential equations

We establish the convergence of a stochastic global optimization algorithm for general non-convex, smooth functions. The algorithm follows the trajectory of an appropriately defined stochastic differential equation (SDE). In order to achieve feasibility of the trajectory we introduce information from the Lagrange multipliers into the SDE. The analysis is performed in two steps. We first give a characterization of a probability measure (Π) that is defined on the set of global minima of the problem. We then study the transition density associated with the augmented diffusion process and show that its weak limit is given by Π.     

The efficiency of Singly-implicit Runge-Kutta methods for stiff differential equations

Singly-implicit Runge-Kutta methods are considered to be good candidates for stiff problems because of their good stability and high accuracy. The existing methods, SIRK (Singly-implicit Runge-Kutta), DESI (Diagonally Extendable Singly-implicit Runge-Kutta), ESIRK (Effective order Singly-implicit Rung-Kutta) and DESIRE (Diagonally Extended Singly-implicit Runge-Kutta Effective order) methods have been shown to be efficient for stiff differential equations, especially for high dimensional stiff problems. In this paper, we measure the efficiency for the family of singly-implicit Runge-Kutta methods using the local truncation error produced within one single step and the count of number of operations. Verification of the error and the computational costs for these methods using variable stepsize scheme are presented. We show how the numerical results are effected by the designed factors: additional diagonal-implicit stages and effective order.     

B-Theory of Runge-Kutta methods for stiff Volterra functional differential equations

B-stability and B-convergence theories of Runge-Kutta methods for nonlinear stiff Volterra func-tional differential equations(VFDEs)are established which provide unified theoretical foundation for the studyof Runge-Kutta methods when applied to nonlinear stiff initial value problems(IVPs)in ordinary differentialequations(ODEs),delay differential equations(DDEs),integro-differential equatioons(IDEs)and VFDEs of     

Hermite interpolation and A-stable methods for stiff ordinary differential equations

Hermite interpolation in the form of Newton's divided difference expressions is employed to give a generating function for A-stable difference methods of order 2n. These methods can be used to solve the initial value ordinary differential equation y′=g(y,t), y(a)=η. The extension to higher dimensions is considered, and practical suggestions are given for step size changes and order changes.     

Analysis of multiscale methods for stochastic differential equations

We analyze a class of numerical schemes proposed [26] for stochastic differential equations with multiple time scales. Both advective and diffusive time scales are considered. Weak as well as strong convergence theorems are proven. Most of our results are optimal. They in turn allow us to provide a thorough discussion on the efficiency as well as optimal strategy for the method. © 2005 Wiley Periodicals, Inc.     

Analytic stochastic process solutions of second-order random differential equations

In this work, trigonometric stochastic processes arise as mean square solutions of random differential equations, using a random Fröbenius method. Important operational properties of the trigonometric stochastic processes are established.