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Finite element approximation of Lyapunov equations for the computation of quadratic functionals of SPDEs
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Abstract
The computation of quadratic functionals of the solution to a linear stochastic partial differential equation with multiplicative noise is considered. An operator valued Lyapunov equation, whose solution admits a deterministic representation of the functional, is used for this purpose and error estimates are shown in suitable operator norms for a fully discrete approximation of this equation. Weak error rates are also derived for a fully discrete approximation of the stochastic partial differential equation, using the results obtained from the approximation of the Lyapunov equation. In the setting of finite element approximations, a computational complexity comparison reveals that approximating the Lyapunov equation allows for cheaper computation of quadratic functionals compared to applying Monte Carlo or covariance-based methods directly to the discretized stochastic partial differential equation. Numerical simulations illustrates the theoretical results.
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The numerical approximation of solutions to stochastic partial differential equations with additive spatial white noise on bounded domains in is considered. The differential operator is given by the fractional power , , of an integer order elliptic differential operator L and is therefore non-local. Its inverse is represented by a Bochner integral from the Dunford-Taylor functional calculus. By applying a quadrature formula to this integral representation, the inverse fractional power operator is approximated by a weighted sum of non-fractional resolvents at certain quadrature nodes . The resolvents are then discretized in space by a standard finite element method. This approach is combined with an approximation of the white noise, which is based only on the mass matrix of the finite element discretization. In this way, an efficient numerical algorithm for computing samples of the approximate solution is obtained. For the resulting approximation, the strong mean-square error is analyzed and an explicit rate of convergence is derived. Numerical experiments for , , with homogeneous Dirichlet boundary conditions on the unit cube in d=1,2,3 spatial dimensions for varying attest the theoretical results.
We provide new regularity results for the solutions of the Kolmogorov equation associated to a SPDE with nonlinear diffusion coefficients and a Burgers type nonlinearity. This generalizes previous results in the simpler cases of additive or affine noise. The basic tool is a discrete version of a two sided stochastic integral which allows a new formulation for the derivatives of these solutions. We show that this can be used to generalize the weak order analysis performed in [16]. The tools we develop are very general and can be used to study many other examples of applications.
Strong convergence rates for numerical approximations of semilinear stochastic partial differential equations (SPDEs) with smooth and regular nonlinearities are well understood in the literature. Weak convergence rates for numerical approximations of such SPDEs have been investigated for about two decades and are still not yet fully understood. In particular, no essentially sharp weak convergence rates are known for temporal or spatial numerical approximations of space-time white noise driven SPDEs with nonlinear multiplication operators in the diffusion coefficients. In this article we overcome this problem by establishing essentially sharp weak convergence rates for exponential Euler approximations of semilinear SPDEs with nonlinear multiplication operators in the diffusion coefficients. Key ingredients of our approach are applications of the mild It\^{o} type formula in UMD Banach spaces with type 2.
In this work we investigate the interplay of almost sure and mean-square stability for linear SDEs and the Monte Carlo method for estimating the second moment of the solution process. In the situation where the zero solution of the SDE is asymptotically stable in the almost sure sense but asymptotically mean-square unstable, the latter property is determined by rarely occurring trajectories that are sufficiently far away from the origin. The standard Monte Carlo approach for estimating higher moments essentially computes a finite number of trajectories and is bound to miss those rare events. It thus fails to reproduce the correct mean-square dynamics (under reasonable cost). A straightforward application of variance reduction techniques will typically not resolve the situation unless these methods force the rare, exploding trajectories to happen more frequently. Here we propose an appropriately tuned importance sampling technique based on Girsanov’s theorem to deal with the rare event simulation. In addition further variance reduction techniques, such as multilevel Monte Carlo, can be applied to control the variance of the modified Monte Carlo estimators. As an illustrative example we discuss the numerical treatment of the stochastic heat equation with multiplicative noise and present simulation results.
We present an approximation framework for computing the solution of the stochastic linear quadratic control problem on Hilbert spaces. We focus on the finite horizon case and the related differential Riccati equations (DREs). Our approximation framework is concerned with the so-called “singular estimate control systems” (Lasiecka in Optimal control problems and Riccati equations for systems with unbounded controls and partially analytic generators: applications to boundary and point control problems, 2004) which model certain coupled systems of parabolic/hyperbolic mixed partial differential equations with boundary or point control. We prove that the solutions of the approximate finite-dimensional DREs converge to the solution of the infinite-dimensional DRE. In addition, we prove that the optimal state and control of the approximate finite-dimensional problem converge to the optimal state and control of the corresponding infinite-dimensional problem.
In this article we develop a framework for studying parabolic semilinear
stochastic evolution equations (SEEs) with singularities in the initial
condition and singularities at the initial time of the time-dependent
coefficients of the considered SEE. We use this framework to establish
existence, uniqueness, and regularity results for mild solutions of parabolic
semilinear SEEs with singularities at the initial time. We also provide several
counterexample SEEs that illustrate the optimality of our results.
Strong convergence rates for (temporal, spatial, and noise) numerical
approximations of semilinear stochastic evolution equations (SEEs) with smooth
and regular nonlinearities are well understood in the scientific literature.
Weak convergence rates for numerical approximations of such SEEs have been
investigated since about 11 years and are far away from being well understood:
roughly speaking, no essentially sharp weak convergence rates are known for
parabolic SEEs with nonlinear diffusion coefficient functions; see Remark 2.3
in [A. Debussche, Weak approximation of stochastic partial differential
equations: the nonlinear case, Math. Comp. 80 (2011), no. 273, 89-117] for
details. In this article we solve the weak convergence problem emerged from
Debussche's article in the case of spectral Galerkin approximations and
establish essentially sharp weak convergence rates for spatial spectral
Galerkin approximations of semilinear SEEs with nonlinear diffusion coefficient
functions. Our solution to the weak convergence problem does not use Malliavin
calculus. Rather, key ingredients in our solution to the weak convergence
problem emerged from Debussche's article are the use of appropriately modified
versions of the spatial Galerkin approximation processes and applications of a
mild It\^{o} type formula for solutions and numerical approximations of
semilinear SEEs. This article solves the weak convergence problem emerged from
Debussche's article merely in the case of spatial spectral Galerkin
approximations instead of other more complicated numerical approximations. Our
method of proof extends, however, to a number of other kind of spatial,
temporal, and noise numerical approximations for semilinear SEEs.
In this paper we introduce a new family of refined Sobolev-Malliavin spaces
that capture in a fine way integrability in time of the Malliavin derivative.
We consider duality in these spaces and derive a Burkholder type inequality in
a dual norm. The theory we develop allows us to prove weak convergence with
essentially optimal rate for numerical approximations in space and time of
semilinear parabolic stochastic evolution equations driven by Gaussian additive
noise. In particular, we combine Galerkin finite element methods with a
backward Euler scheme in time. The method of proof does not rely on the use of
the Kolmogorov equation or the It\={o} formula and is therefore in nature
non-Markovian. With this method polynomial growth test functions with mild
smoothness assumptions are allowed, meaning in particular that we prove
convergence of arbitrary moments with essentially optimal rate. Our Gronwall
argument also yields weak error estimates which are uniform in time without any
additional effort.
We find the weak rate of convergence of the spatially semidiscrete finite
element approximation of the nonlinear stochastic heat equation. Both
multiplicative and additive noise is considered under different assumptions.
This extends an earlier result of Debussche in which time discretization is
considered for the stochastic heat equation perturbed by white noise. It is
known that this equation has a solution only in one space dimension. In order
to obtain results for higher dimensions, colored noise is considered here,
besides white noise in one dimension. Integration by parts in the Malliavin
sense is used in the proof. The rate of weak convergence is, as expected,
essentially twice the rate of strong convergence.
The entities A,
B,
X,
Y in the title are
operators, by which we mean either linear
transformations on a finite-dimensional vector space (matrices) or
bounded (= continuous) linear transformations on a Banach space.
(All scalars will be complex
numbers.) The definitions and statements below are valid in both
the finite-dimensional and the
infinite-dimensional cases, unless the contrary is stated.
We study the error of the Euler scheme applied to a stochastic partial differential equation. We prove that, as is often the case, the weak order of convergence is twice the strong order. A key ingredient in our proof is Malliavin calculus which enables us to get rid of the irregular terms of the error. We apply our method to the case of a semilinear stochastic heat equation driven by a space-time white noise.
We study backward stochastic Riccati equations (BSREs) arising in quadratic optimal control problems with infinite dimensional stochastic differential state equations. We allow the coefficients, both in the state equation and in the cost, to be random. In such a context BSREs are backward stochastic differential equations existing in a non-Hilbert space and involving quadratic nonlinearities. We propose two different notions of solutions to BSREs and prove, for both of them, existence and uniqueness results. We also show that such solutions allow us to perform the synthesis of the optimal control. Finally we apply our results to the optimal control of a delay equation and of a wave equation with random damping.
. A finite element method for the Cahn-Hilliard equation (a semilinear parabolic equation of fourth order) is analyzed, both in a spatially semidiscrete case and in a completely discrete case based on the backward Euler method. Error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data. A detailed study of the regularity of the exact solution is included. The analysis is based on local Lipschitz conditions for the nonlinearity with respect to Sobolev norms, and the existence of a Ljapunov functional for the exact and the discretized equations is essential. A result concerning the convergence of the attractor of the corresponding approximate nonlinear semigroup (upper semicontinuity with respect to the discretization parameters) is obtained as a simple application of the nonsmooth data error estimate. 1. Introduction. The Cahn-Hilliard equation (1.1) u t + Delta 2 u Gamma DeltaOE(u) = 0; x 2Omega ae R 3 ; t ? 0; w...
The efficient simulation of the mean value of a non-linear functional of the solution to a linear stochastic partial differential equation (SPDE) with additive Gaussian noise is considered. A Galerkin finite element method is employed along with an implicit Euler scheme to arrive at a fully discrete approximation of the mild solution to the equation. A scheme is presented to compute the covariance of this approximation, which allows for rapid sampling in a Monte Carlo method. This is then extended to a multilevel Monte Carlo method, for which a scheme to compute the cross-covariance between the approximations at different levels is presented. In contrast to traditional path-based methods it is not assumed that the Galerkin subspaces at these levels are nested. The computational complexities of the presented schemes are compared to traditional methods and simulations confirm that, under suitable assumptions, the costs of the new schemes are significantly lower.
Given the square matrices A, B, D, E and the matrix C of conforming dimensions, we consider the linear matrix equation AXE + DXB = C in the unknown matrix X. Our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems, which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing.
This chapter deals with semilinear stochastic evolution equations in Hilbert spaces. The first two sections are more introductory and review some important properties of Wiener processes and the stochastic Itô integral in Hilbert spaces. The main references for the presented material are Da Prato and Zabczyk (Stochastic Equations in Infinite Dimensions, vol. 44, 1992) and Prévôt and Röckner (A Concise Course on Stochastic Partial Differential Equations, vol. 1905, 2007).
Introduction Part I. Foundations: 1. Why equations with Levy noise? 2. Analytic preliminaries 3. Probabilistic preliminaries 4. Levy processes 5. Levy semigroups 6. Poisson random measures 7. Cylindrical processes and reproducing kernels 8. Stochastic integration Part II. Existence and Regularity: 9. General existence and uniqueness results 10. Equations with non-Lipschitz coefficients 11. Factorization and regularity 12. Stochastic parabolic problems 13. Wave and delay equations 14. Equations driven by a spatially homogeneous noise 15. Equations with noise on the boundary Part III. Applications: 16. Invariant measures 17. Lattice systems 18. Stochastic Burgers equation 19. Environmental pollution model 20. Bond market models Appendix 1. Operators on Hilbert spaces Appendix 2. C0-semigroups Appendix 3. Regularization of Markov processes Appendix 4. Ito formulae Appendix 5. Levy-Khinchin on [0,+ ) Appendix 6. Proof of Lemma List of symbols Bibliography Index.
In this paper, we analyze the weak error of a semi-discretization in time by the linear implicit Euler method for semilinear stochastic partial differential equations (SPDEs) with additive noise. The main result reveals how the weak order depends on the regularity of noise and that the order of weak convergence is twice that of strong convergence. In particular, the linear implicit Euler method for SPDEs driven by trace class noise achieves an almost optimal order 1−ϵ1−ϵ for arbitrarily small ϵ>0ϵ>0.
A unified approach is given for the analysis of the weak error of spatially semidiscrete finite element methods for linear
stochastic partial differential equations driven by additive noise. An error representation formula is found in an abstract
setting based on the semigroup formulation of stochastic evolution equations. This is then applied to the stochastic heat,
linearized Cahn-Hilliard, and wave equations. In all cases it is found that the rate of weak convergence is twice the rate
of strong convergence, sometimes up to a logarithmic factor, under the same or, essentially the same, regularity requirements.
This paper solves a quadratic optimal control problem governed by a linear stochastic evolution equation with unbounded coefficients. It is assumed that the stochastic noise depends both on the state and on the control. The dynamic programming approach is used and attention is focussed on the Riccati equation. Some attractivity and maximality properties of the solutions of the algebraic Riccati equation are proved and it is shown that, in some special cases, there exists a maximal solution.
Stochastic regulator problems and optimal stationary control as well as stability are studied for inifinite dimensional systems with state and control dependent noise. The stochastic model is described by a semigroup and Wiener processes in Hilbert space and W. M. Wonham's approach using differential generators and dynamic programming is extended to infinite dimensions.
The stability of the -projection onto some standard finite element spaces , considered as a map in and , is shown under weaker regularity requirements than quasi-uniformity of the triangulations underlying the definitions of the .
An abstract is not available.
Göttingen, Math.-naturwiss. F., Diss. v. 22. März 1951 (Nicht f. d. Aust.).
We study a regulator problem for a system governed by a linear stochastic differential equation with unbounded coefficients in Hilbert spaces.
The optimal control of a class of stochastic parabolic systems is studied. This class includes systems with noise depending on spatial derivatives of the state, Neumann boundary control, and Dirichlet boundary observation, and extends a class of stochastic systems with distributed control studied by Da Prato [3] and Da Prato and Ichikawa [4]. The work is based on the direct study of the Riccati equation arising in the optimal control problem over finite time horizon. The problem over infinite time horizon and the corresponding algebraic Riccati equation are also considered.
Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Itô formula is proved which is applied to prove the existence of strong solutions for a class of stochastic evolution equations in UMD Banach spaces. The abstract results are applied to prove regularity in space and time of the solutions of the Zakai equation.
Boundary control of a stochastic parabolic equation with nonsmooth final cost
- F Flandoli
F. Flandoli. Boundary control of a stochastic parabolic equation with nonsmooth final cost.
In Analysis and optimization of systems (Antibes, 1990), volume 144 of Lecture Notes in
Control and Inform. Sci., pages 694-703. Springer, Berlin, 1990.
Stochastic LQ and Associated Riccati equation of PDEs Driven by State-and Control-Dependent White Noise
- Y Hu
- S Tang
Y. Hu and S. Tang. Stochastic LQ and Associated Riccati equation of PDEs Driven by
State-and Control-Dependent White Noise. arXiv e-prints, arXiv:1809.05308, Sep 2018.
Computational aspects of Lévy-driven SPDE approximations
- A Petersson
A. Petersson. Computational aspects of Lévy-driven SPDE approximations, December 2017.
Licentiate Thesis, Chalmers University of Technology, Gothenburg, Sweden.