George Em Karniadakis

Numerical Methods for Stochastic Partial Differential Equations with White Noise

Preface -- Prologue -- Brownian Motion and Stochastic Calculus -- Numerical Methods for Stochastic Differential Equations -- Part I Stochastic Ordinary Differential Equations -- Numerical Schemes for SDEs with Time Delay Using the Wong-Zakai Approximation -- Balanced Numerical Schemes for SDEs with non-Lipschitz Coefficients -- Part II Temporal White Noise -- Wiener Chaos Methods for Linear Stochastic Advection-Diffusion-Reaction Equations -- Stochastic Collocation Methods for Differential Equations with White Noise -- Comparison Between Wiener Chaos Methods and Stochastic Collocation Methods -- Application of Collocation Method to Stochastic Conservation Laws -- Part III Spatial White Noise -- Semilinear Elliptic Equations with Additive Noise -- Multiplicative White Noise: The Wick-Malliavin Approximation -- Epilogue -- Appendices -- A. Basics of Probability -- B. Semi-analytical Methods for SPDEs -- C. Gauss Quadrature -- D. Some Useful Inequalities and Lemmas -- E. Computation of Convergence Rate.

A Petrov–Galerkin spectral element method for fractional elliptic problems

Abstract We develop a new C 0 -continuous Petrov–Galerkin spectral element method for one-dimensional fractional elliptic problems of the form 0 D x α u ( x ) − λ u ( x ) = f ( x ) , α ∈ ( 1 , 2 ] , subject to homogeneous boundary conditions. We employ the standard (modal) spectral element basis and the Jacobi poly-fractonomials as the test functions (Zayernouri and Karniadakis (2013)). We formulate a new procedure for assembling the global linear system from elemental (local) mass and stiffness matrices. The Petrov–Galerkin formulation requires performing elemental (local) construction of mass and stiffness matrices in the standard domain only once. Moreover, we efficiently obtain the non-local (history) stiffness matrices, in which the non-locality is presented analytically for uniform grids. We also investigate two distinct choices of basis/test functions: (i) local basis/test functions, and (ii) local basis with global test functions. We show that the former choice leads to a better-conditioned system and accuracy, while the latter results in ill-conditioned linear systems, and therefore, that is not an efficient and a proper choice of test function. We consider smooth and singular solutions, where the singularity can occur at boundary points as well as in the interior domain. We also construct two non-uniform grids over the whole computational domain in order to capture singular solutions. Finally, we perform a systematic numerical study of non-local effects via full and partial history fading in order to further enhance the efficiency of the scheme.

Semilinear elliptic equations with additive noise

With temporal white noise, the solutions of stochastic parabolic equations have low regularity in time (Holder continuous with exponent 1∕2 −e and e > 0 is arbitrarily small) and thus the spectral approximation of Brownian motion leads to only half order convergence in its truncation mode n. With spatial white noise, however, the solution can be smoother and we can expect higher-order convergence with spectral approximation of Brownian motion.

Numerical schemes for SDEs with time delay using the Wong-Zakai approximation

Will a spectral approximation of Brownian motion lead to higher-order numerical methods for stochastic differential equations as the Karhunen-Loeve truncation of a smooth stochastic process does?

Stochastic collocation methods for differential equations with white noise

Stochastic collocation methods can lead to a fully decoupled system of PDEs, which can be readily implemented on parallel computers. However, stochastic collocation methods do not work when longer time integration is required. Though these methods are also cursed by the dimensionality, we apply the recursive strategy for longer time integration discussed in Chapter 6 to investigate the error behavior of stochastic collocation methods in one-time step approximation or short time integration.

Brownian motion and stochastic calculus

In this chapter, we review some basic concepts for stochastic processes and stochastic calculus as well as numerical integration methods in random space for obtaining statistics of stochastic processes.

Numerical methods for stochastic differential equations

In this chapter, we discuss some basic aspects of stochastic differential equations (SDEs) including stochastic ordinary (SODEs) and partial differential equations (SPDEs).

Balanced numerical schemes for SDEs with non-Lipschitz coefficients

In this chapter, we discuss numerical methods for SDEs with coefficients of polynomial growth. The nonlinear growth of the coefficients induces instabilities, especially when the nonlinear growth is polynomial or even exponential. For stochastic differential equations (SDEs) with coefficients of polynomial growth at infinity and satisfying a one-sided Lipschitz condition, we prove a fundamental mean-square convergence theorem on the strong convergence order of a stable numerical scheme in Chapter 5.2. We apply the theorem to a number of existing numerical schemes. We present in Chapter 5.3 a special balanced scheme, which is explicit and of half-order mean-square convergence. Some numerical results are presented in Chapter 5.4. We summarize the chapter in Chapter 5.5 and present some bibliographic notes on numerical schemes for nonlinear SODEs. Three exercises are presented for interested readers.

Wiener chaos methods for linear stochastic advection-diffusion-reaction equations

In this chapter, we discuss numerical algorithms using Wiener chaos expansion (WCE) for solving second-order linear parabolic stochastic partial differential equations (SPDEs). The algorithm for computing moments of the SPDE solutions is deterministic, i.e., it does not involve any statistical errors from generating random numbers.

Multiplicative white noise: The Wick-Malliavin approximation

In this chapter, we consider Wiener chaos expansion (WCE) for elliptic equations with multiplicative noise. Unlike the stochastic collocation methods (SCM), a direct application of WCE will lead to a fully coupled linear system. To sparsify the resulting linear system, we present WCE with the use of Ito-Wick product and an approximation/reduction technique called Wick-Malliavin approximation. Specifically, we consider Wick-Malliavin approximation for elliptic equations with lognormal coefficients and use the Wick product for elliptic equations with spatial white noise as coefficients. Numerical results demonstrate that high-order Wick-Malliavin approximation is efficient even when the noise intensity is relatively large.