On the numerical structure preservation of nonlinear damped stochastic oscillators

The paper is focused on analyzing the conservation issues of stochastic ??-methods when applied to nonlinear damped stochastic oscillators. In particular, we are interested in reproducing the long-term properties of the continuous problem over its discretization through stochastic ??-methods, by preserving the correlation matrix. This evidence is equivalent to accurately maintaining the stationary density of the position and the velocity of a particle driven by a nonlinear deterministic forcing term and an additive noise as a stochastic forcing term. The provided analysis relies on a linearization of the nonlinear problem, whose effectiveness is proved theoretically and numerically confirmed.

     

Numerical resolution of stochastic focusing NLS equations

In this note, we numerically investigate a stochastic nonlinear Schrödinger equation derived as a perturbation of the deterministic NLS equation. The classical NLS equation with focusing nonlinearity of power law type is perturbed by a random term; it is a strong perturbation since we consider a space-time white noise. It acts either as a forcing term (additive noise) or as a potential (multiplicative noise). For simulations made on a uniform grid, we see that all trajectories blow-up in finite time, no matter how the initial data are chosen. Such a grid cannot represent a noise with zero correlation lengths, so that in these experiments, the noise is, in fact, spatially smooth. On the contrary, we simulate a noise with arbitrarily small scales using local refinement and show that in the multiplicative case, blow-up is prevented by a space-time white noise. We also present results on noise induced soliton diffusion.     

Inverse problem of groundwater modeling by iteratively regularized Gauss-Newton method with a nonlinear regularization term

A nonlinear minimization problem ‖F(d)−u‖?min, ‖u−u_δ‖≤δ, is a typical mathematical model of various applied inverse problems. In order to solve this problem numerically in the lack of regularity, we introduce iteratively regularized Gauss-Newton procedure with a nonlinear regularization term (IRGN-NRT). The new algorithm combines two very powerful features: iterative regularization and the most general stabilizing term that can be updated at every step of the iterative process. The convergence analysis is carried out in the presence of noise in the data and in the modified source condition. Numerical simulations for a parameter identification ill-posed problem arising in groundwater modeling demonstrate the efficiency of the proposed method.     

Order results for Rosenbrock type methods on classes of stiff equations

Summary In this paper we give conditions for theB-convergence of Rosenbrock type methods when applied to stiff semi-linear systems. The convergence results are extended to stiff nonlinear systems in singular perturbation form. As a special case partitioned methods are considered. A third order method is constructed.Dedicated to the memory of Professor Lothar Collatz     

Well-Posedness and Finite Element Approximations for Elliptic SPDEs with Gaussian Noises

The paper studies the well-posedness and optimal error estimates of spectral finite element approximations for the boundary value problems of semi-linear elliptic SPDEs driven by white or colored Gaussian noises. The noise term is approximated through the spectral projection of the covariance operator, which is not required to be commutative with the Laplacian operator.Through the convergence analysis of SPDEs with the noise terms replaced by the projected noises, the well-posedness of the SPDE is established under certain covariance operator-dependent conditions. These SPDEs with projected noises are then numerically approximated with the finite element method. A general error estimate framework is established for the finite element approximations. Based on this framework, optimal error estimates of finite element approximations for elliptic SPDEs driven by power-law noises are obtained. It is shown that with the proposed approach, convergence order of white noise driven SPDEs is improved by half for one-dimensional problems, and by an infinitesimal factor for higher-dimensional problems.     

Overlapping Schwarz preconditioners for spectral element methods in nonstandard domains and heterogeneous media

Overlapping Schwarz preconditioners are constructed and numerically studied for Gauss-Lobatto-Legendre (GLL) spectral element discretizations of heterogeneous elliptic problems on nonstandard domains defined by Gordon-Hall transfinite mappings. The results of several test problems in the plane show that the proposed preconditioners retain the good convergence properties of overlapping Schwarz preconditioners for standard affine GLL spectral elements, i.e. their convergence rate is independent of the number of subdomains, of the spectral degree in the case of generous overlap and of the discontinuity jumps in the coefficients of the elliptic operator, while in the case of small overlap, the convergence rate depends on the inverse of the overlap size.     

On the convergence of Runge–Kutta methods for stiff non linear differential equations

Summary. This paper studies the convergence properties of general Runge–Kutta methods when applied to the numerical solution of a special class of stiff non linear initial value problems. It is proved that under weaker assumptions on the coefficients of a Runge–Kutta method than in the standard theory of B-convergence, it is possible to ensure the convergence of the method for stiff non linear systems belonging to the above mentioned class. Thus, it is shown that some methods which are not algebraically stable, like the Lobatto IIIA or A-stable SIRK methods, are convergent for the class of stiff problems under consideration. Finally, some results on the existence and uniqueness of the Runge–Kutta solution are also presented. Received November 18, 1996 / Revised version received October 6, 1997     

The composite Milstein methods for the numerical solution of Ito stochastic differential equations

In this paper, we present the composite Milstein methods for the strong solution of Ito stochastic differential equations. These methods are a combination of semi-implicit and implicit Milstein methods. We give a criterion for choosing either the implicit or the semi-implicit scheme at each step of our numerical solution. The stability and convergence properties are investigated and discussed for the linear test equation. The convergence properties for the nonlinear case are shown numerically to be the same as the linear case. The stability properties of the composite Milstein methods are found to be more superior compared to those of the Milstein, the Euler and even better than the composite Euler method. This superiority in stability makes the methods a better candidate for the solution of stiff SDEs.     

Convergence of a numerical scheme for SPDEs with correlated noise and global Lipschitz coefficients

The aim of this paper is to investigate the pathwise numerical solution of semilinear parabolic stochastic partial differential equations (SPDEs) with colored noise instead of the usual space–time white noise. We estimate the numerical solution in the L^∞ topology by a method that takes advantages of the smoothing effect of the dominant linear operator. We consider the case the covariance operator of the forcing does not necessarily commute with the linear operator of the SPDE because of the fact that the Brownian motions are not necessarily independent. We show convergence of this method, and numerical examples give insight into the reliability of the theoretical study. Copyright © 2015 John Wiley & Sons, Ltd.     

A-stability of Runge-Kutta methods for systems with additive noise

Numerical stability of both explicit and implicit Runge-Kutta methods for solving ordinary differential equations with an additive noise term is studied. The concept of numerical stability of deterministic schemes is extended to the stochastic case, and a stochastic analogue of Dahlquist'sA-stability is proposed. It is shown that the discretization of the drift term alone controls theA-stability of the whole scheme. The quantitative effect of implicitness uponA-stability is also investigated, and stability regions are given for a family of implicit Runge-Kutta methods with optimal order of convergence.This author was partially supported by the Italian Consiglio Nazionale delle Ricerche.     

Finite volume schemes for nonhomogeneous scalar conservation laws: error estimate

Summary. In this paper, we study finite volume schemes for the nonhomogeneous scalar conservation law with initial condition . The source term may be either stiff or nonstiff. In both cases, we prove error estimates between the approximate solution given by a finite volume scheme (the scheme is totally explicit in the nonstiff case, semi-implicit in the stiff case) and the entropy solution. The order of these estimates is in space-time -norm (h denotes the size of the mesh). Furthermore, the error estimate does not depend on the stiffness of the source term in the stiff case. Received October 21, 1999 / Published online February 5, 2001     

B-convergence properties of defect correction methods. I

Summary B-convergence properties of defect correction methods based on the implicit Euler and midpoint schemes are discussed. The property ofB-convergence means that there exist global error bounds for nonlinear stiff problems independent of their stiffness. It turns out that the orders ofB-convergence of these methods coincide with the conventional orders of convergence of these methods derived under the assumption that.hL is small (whereL is a Lipschitz constant of the right-hand side). In Part I these assertions are reduced to the validity of the so-called Hypothesis A which is discussed in greater detail in Part II. Numerical experiments confirming the theoretical analysis are also given in Part II.     

A choice of forcing terms in inexact Newton method

Inexact Newton method is one of the effective tools for solving systems of nonlinear equations. In each iteration step of the method, a forcing term, which is used to control the accuracy when solving the Newton equations, is required. The choice of the forcing terms is of great importance due to their strong influence on the behavior of the inexact Newton method, including its convergence, efficiency, and even robustness. To improve the efficiency and robustness of the inexact Newton method, a new strategy to determine the forcing terms is given in this paper. With the new forcing terms, the inexact Newton method is locally Q-superlinearly convergent. Numerical results are presented to support the effectiveness of the new forcing terms.     

Convex mean curvature flow with a forcing term in direction of the position vector

A smooth, compact and strictly convex hypersurface evolving in ℝ ⁿ⁺¹ along its mean curvature vector plus a forcing term in the direction of its position vector is studied in this paper. We show that the convexity is preserving as the case of mean curvature flow, and the evolving convex hypersurfaces may shrink to a point in finite time if the forcing term is small, or exist for all time and expand to infinity if it is large enough. The flow can converge to a round sphere if the forcing term satisfies suitable conditions which will be given in the paper. Long-time existence and convergence of normalization of the flow are also investigated.     

Approximation with frequency filter for backward parabolic equations

Summary. We propose a numerical method for the initial (and boundary) value problem for the equation of the form where A is an unbounded, selfadjoint operator with negative spectrum. Roundoff errors in the numerical solution of such problem may generate a parasite term growing very quickly with time. To eliminate this parasite term, we apply a special finite difference equation with r free parameters. Similar ideas may be useful also for another numerically difficult differential problems. Received October 6, 1997 / revised version received November 26, 1998 / Published online October 16, 2000     

Stochastic evolution equations with general white noise disturbance

Existence and uniqueness theorems are proved for a general class of stochastic linear abstract evolution equations, with a general type of stochastic forcing term. The abstract evolution equation is modeled using an evolution operator (or 2-parameter semigroup) approach and this includes linear partial differential equations and linear differential delay equations. The stochastic forcing term is modeled by defining an Itô stochastic integral with respect to a Hilbert space-valued orthogonal increments process, which can be used to model both Gaussian and non-Gaussian white noise processes. The theory is illustrated by examples of stochastic partial differential equations and delay equations, which arise in filtering problems for distributed and delay systems.     

A Levin-type algorithm for accelerating the convergence of Fourier series

A nonlinear sequence transformation is presented which is able to accelerate the convergence of Fourier series. It is tailored to be exact for a certain model sequence. As in the case of the Levin transformation and other transformations of Levin-type, in this model sequence the partial sum of the series is written as the sum of the limit (or antilimit) and a certain remainder, i.e., it is of Levin-type. The remainder is assumed to be the product of a remainder estimate and the sum of the first terms oftwo Poincaré-type expansions which are premultiplied by two different phase factors. This occurrence of two phase factors is the essential difference to the Levin transformation. The model sequence for the new transformation may also be regarded as a special case of a model sequence based on several remainder estimates leading to the generalized Richardson extrapolation process introduced by Sidi. An algorithm for the recursive computation of the new transformation is presented. This algorithm can be implemented using only two one-dimensional arrays. It is proved that the sequence transformation is exact for Fourier series of geometric type which have coefficients proportional to the powers of a numberq, |q|<1. It is shown that under certain conditions the algorithm indeed accelerates convergence, and the order of the convergence is estimated. Finally, numerical test data are presented which show that in many cases the new sequence transformation is more powerful than Wynn's epsilon algorithm if the remainder estimates are properly chosen. However, it should be noted that in the vicinity of singularities of the Fourier series the new sequence transformation shows a larger tendency to numerical instability than the epsilon algorithm.     

Error analysis of finite element approximations of the stochastic Stokes equations

Numerical solutions of the stochastic Stokes equations driven by white noise perturbed forcing terms using finite element methods are considered. The discretization of the white noise and finite element approximation algorithms are studied. The rate of convergence of the finite element approximations is proved to be almost first order (h|ln h|) in two dimensions and one half order ( h^\frac12h^{\frac{1}{2}}) in three dimensions. Numerical results using the algorithms developed are also presented.     

Convergence of a finite volume scheme for nonlinear degenerate parabolic equations

Summary. One approximates the entropy weak solution u of a nonlinear parabolic degenerate equation by a piecewise constant function using a discretization in space and time and a finite volume scheme. The convergence of to u is shown as the size of the space and time steps tend to zero. In a first step, estimates on are used to prove the convergence, up to a subsequence, of to a measure valued entropy solution (called here an entropy process solution). A result of uniqueness of the entropy process solution is proved, yielding the strong convergence of to{\it u}. Some on a model equation are shown. Received September 27, 2000 / Published online October 17, 2001     

Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term

 We study existence and uniqueness of a mild solution in the space of continuous functions and existence of an invariant measure for a class of reaction-diffusion systems on bounded domains of ℝ ^d , perturbed by a multiplicative noise. The reaction term is assumed to have polynomial growth and to be locally Lipschitz-continuous and monotone. The noise is white in space and time if d=1 and coloured in space if d>1; in any case the covariance operator is never assumed to be Hilbert-Schmidt. The multiplication term in front of the noise is assumed to be Lipschitz-continuous and no restrictions are given either on its linear growth or on its degenaracy. Our results apply, in particular, to systems of stochastic Ginzburg-Landau equations with multiplicative noise. Received: 1 November 2001 / Revised version: 17 June 2002 / Published online: 14 November Mathematics Subject Classification (2000): 60H15, 35R60, 47A35