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The exponential integrator scheme for stochastic partial differential equations: Pathwise error bounds
Abstract
We present an error analysis for the pathwise approximation of a general semilinear
stochastic evolution equation in d dimensions. We discretise in space by a
Galerkin method and in time by a stochastic exponential integrator. We show that
for spatially regular (smooth) noise the number of nodes needed for the noise can
be reduced and that the rate of convergence degrades as the regularity of the noise
reduces (and the noise is rougher)
... There are numerous numerical methods designed to approximate (1) with linear self-adjoint operator, see e.g. 6,7,8,9 . To classify a numerical method, one also takes in consideration the rate of convergence. ...
... Proof. We only prove that Hence, for ∈ , from the definition of the norm ‖.‖ 1 (Λ,ℝ) , using (9), Hölder inequality and the fact that Λ is bounded, it follows that ...
... One can easily check that (9) holds. In fact, simple estimates yields ...
... Suppose that X is the mild solution of (1) and that X K,M,N is defined by scheme (9). Under Assumptions 1-3 with γ > max{3 − 2H, 3 2 }, it holds that ...
... For equations with additive noise which is fractional in space and white in time, we refer to [1,2] and references therein for optimal error analysis on numerical approximations. As H tends to 1 2 , the parameter γ goes to 2 which coincides with the assumption on the SHEs driven by infinite dimensional standard Brownian motions for the strong order one of accuracy of the exponential integrator [8,9,11]. ...
... Theorem 6. Suppose that X K,M is the mild solution of (8) and that X K,M,N is defined by scheme (9). Under Assumptions 1-3 with γ > max{3 − 2H, 3 2 }, it holds that ...
In this paper, we consider the strong convergence order of the exponential integrator for the stochastic heat equation driven by an additive fractional Brownian motion with Hurst parameter . By showing the strong order one of accuracy of the exponential integrator under appropriote assumptions, we present the first super-convergence result in temporal direction on full discretizations for stochastic partial differential equations driven by infinite dimensional fractional Brownian motions with Hurst parameter . The proof is a combination of Malliavin calculus, the -estimate of the Skorohod integral and the smoothing effect of the Laplacian operator.
... For stochastic systems (SDEs/SPDEs), additional issues arise in handling the contributions of the fluctuations [3][4][5]49]. Work on stochastic exponential integrators for stationary operators L(t) = L 0 has been done in [1,5,21,28]. In these works, stochastic and conventional integral expressions are derived with terms exponential in the evolution operator. ...
... In these works, stochastic and conventional integral expressions are derived with terms exponential in the evolution operator. The integrals and exponentials are either analytically computed, such as using diaognalization in an eigenbasis [5], or approximated using Krylov subspaces [21], finite elements [28,36], or other methods [1,21]. Exponential integrators have also been developed for non-autonomous stochastic systems permitting non-commuting evolution operators in [2,13,26,36,67,69]. ...
We introduce exponential numerical integration methods for stiff stochastic dynamical systems of the form . We consider the setting of time-varying operators where they may not commute , raising challenges for exponentiation. We develop stochastic numerical integration methods using Mangus expansions for preserving statistical structures and for maintaining fluctuation-dissipation balance for physical systems. For computing the contributions of the fluctuation terms, our methods provide alternative approaches without needing directly to evaluate stochastic integrals. We present results for our methods for a class of SDEs arising in particle simulations and for SPDEs for fluctuations of concentration fields in spatially-extended systems. For time-varying stochastic dynamical systems, our introduced discretization approaches provide general exponential numerical integrators for preserving statistical structures while handling stiffness.
... Unlike weak convergence error, the strong convergence error measures the deviation from the trajectory of an exact solution. It has been extensively investigated in various types of SPDEs, see, e.g., [1,11,17,55,50,28,23,31,8,51,52,35,27,36,46,22,19,5,29,39,40] and references therein. We mention here some works on strong convergence of the numerical schemes for linear SEEs with additive or multiplicative noise. ...
... The case of strong convergence of nonlinear SEEs is generally more subtle and challenging, and has received widely attention in the research community in recent years. For instance, Kloeden et al. [28,31] proposed a discretization based on the Galerkin method in space and exponential integrator in time for the nonlinear SEEs with cylindrical additive noise. Kruse [36] analysed the strong convergence error for a finite element method/linear implicit Euler spatiotemporal discretization of semilinear SEEs with multiplicative noise and Lipschitz continuous nonlinearities, and deduced the optimal error estimates. ...
In this paper, we investigate the stochastic evolution equations (SEEs) driven by -Whittle-Matrn (W-M) random diffusion coefficient field and Q-Wiener multiplicative force noise. First, the well-posedness of the underlying equations is established by proving the existence, uniqueness, and stability of the mild solution. A sampling approach called approximation circulant embedding with padding is proposed to sample the random coefficient field. Then a spatio-temporal discretization method based on semi-implicit Euler-Maruyama scheme and finite element method is constructed and analyzed. An estimate for the strong convergence rate is derived. Numerical experiments are finally reported to confirm the theoretical result.
... By Hölder's inequality and Itô's isometry, it then follows from (18), (20) with θ = 1, contractive property of e tA and semigroup property of P 0 t that ...
... Whereas, our goal in this section is to make an attempt to discuss strong convergence of an exponential integrator (EI) scheme, coupled with a Galerkin scheme for the spatial discretization (see (35) and (36) below), for a class of semi-linear SPDEs with Hölder continuous drifts. With regard to convergence of EI scheme for SPDEs with smooth drift coefficients, we refer to [18,22,23] for further details, to name a few. Also, there is a number of literature on approximation of SPDEs with non-globally Lipschitz continuous nonlinearities; see, for instance, [15]. ...
In this paper, some new results on the the regularity of Kolmogorov equations associated to the infinite dimensional OU-process are obtained. As an application, the average L²-error on [0, T] of exponential integrator scheme for a range of semi-linear stochastic partial differential equations is derived, where the drift term is assumed to be Hölder continuous with respect to the Sobolev norm · β for some appropriate β > 0. In addition, under a stronger condition on the drift, the strong convergence estimate is obtained, which covers the result of the SDEs with Hölder continuous drift. © 2019 American Institute of Mathematical Sciences. All rights reserved.
... Since explicit solutions of (1) do not exist, numerical algorithms are good tools to provide realistic approximations. Numerical methods for SPDE driven only by Gaussian noise are widely investigated in the scientific literature (see e.g., [21,22,23,24,26,30,31,32,48,50,53] and references therein). The case of time-fractional SPDEs driven by Gaussian noise have recently received some attentions (see e.g. ...
... Moreover, (23) and (24) hold if A and X are replaced respectively by their discrete versions A h and X h defined in Section 4. ...
This paper deals with the numerical approximation of semilinear parabolic stochastic partial differential equation (SPDE) driven simultaneously by Gaussian noise and Poisson random measure, more realistic in modeling real world phenomena.
The SPDE is discretized in space with the standard finite element method and in time with the linear implicit Euler method or an exponential integrator, more efficient and stable for stiff problems.
We prove the strong convergence of the fully discrete schemes toward the mild solution. The results reveal how convergence orders depend on the regularity of the noise and the initial data.
In addition, we exceed the classical orders 1/2 in time and 1 in space achieved in the literature when dealing with SPDE driven by Poisson measure with less regularity assumptions on the nonlinear drift function.
In particular, for trace class multiplicative Gaussian noise we achieve convergence order .
For additive trace class Gaussian noise and an appropriate jump function, we achieve convergence order .
Numerical experiments to sustain the theoretical results are provided.
... Nevertheless, it is limited on domains, where the eigenfunctions of the dominant linear operator are not explicitly known. In recent years there has also been a significant interest in analytic results for the rate of approximation using a spectral Galerkin method as a numerical method for SPDEs; see, for example, [18,28] for SPDEs with one-dimensional possibly non-additive noise and globally Lipschitz continuous nonlinearities, [31,32,35,36,25,26] for SPDEs with possibly infinite dimensional additive noise and globally Lipschitz continuous nonlinearities, [30,23] for SPDEs with possibly infinite dimensional additive noise and non-globally Lipschitz continuous nonlinearities, and [20,21,34,33] for SPDEs with possibly infinite dimensional non-additive noise and globally Lipschitz continuous nonlinearities. In most of the above named references also the full discretization is treated including the time discretization. ...
Existence and uniqueness for semilinear stochastic evolution equations with additive noise by means of finite dimensional Galerkin approximations is established and the convergence rate of the Galerkin approximations to the solution of the stochastic evolution equation is estimated. These abstract results are applied to several examples of stochastic partial differential equations (SPDEs) of evolutionary type including a stochastic heat equation, a stochastic reaction diffusion equation and a stochastic Burgers equation. The estimated convergence rates are illustrated by numerical simulations. The main novelty in this article is to estimate the difference of the finite dimensional Galerkin approximations and of the solution of the infinite dimensional SPDE uniformly in space, i.e., in the L^{\infty}-topology, instead of the usual Hilbert space estimates in the L^2-topology, that were shown before.
... The numerical analysis of the stochastic heat equation (1) is an active research area. Without being too exhaustive, beside the above mentioned papers [20] and [21], we mention the following works regarding numerical discretizations of stochastic parabolic partial differential equations: [20,55,5,47] (spatial approximations); [18,22,23,1,48,45,15,17,26,44,52,39,40,31,28,11,30,29,34,38,54,6,12,33,7,53] (temporal and full discretizations); [49,36] (stability). Observe that most of these references are concerned with an interpretation of stochastic partial differential equations in Hilbert spaces and thus error estimates are provided in the L 2 ([0, 1]) norm (or similar norms). ...
A fully discrete approximation of the one-dimensional stochastic heat equation driven by multiplicative space-time white noise is presented. The standard finite difference approximation is used in space and a stochastic exponential method is used for the temporal approximation. Observe that the proposed exponential scheme does not suffer from any kind of CFL-type step size restriction. When the drift term and the diffusion coefficient are assumed to be globally Lipschitz, this explicit time integrator allows for error bounds in , for all , improving some existing results in the literature. On top of this, we also prove almost sure convergence of the numerical scheme. In the case of non-globally Lipschitz coefficients, we provide sufficient conditions under which the numerical solution converges in probability to the exact solution. Numerical experiments are presented to illustrate the theoretical results.
... For additive noise, various pathwise convergence results have been obtained. For example, in [18] the authors consider Galerkin approximations in the same setting as we do but with additive noise and slightly stronger conditions on F . They obtain the same convergence rates as we do. ...
We provide convergence rates for space approximations of semi-linear stochastic differential equations with multiplicative noise in a Hilbert space. The space approximations we consider are spectral Galerkin and finite elements, and the type of convergence we consider is strong and almost sure uniform convergence, i.e., pathwise convergence. The proofs are based on a previously published perturbation result for such equations.
... [37,38]) and stochastic partial differential equations (SPDEs) (e.g. [39,40]), where both strong and weak convergence of exponential integrators were well established at finite time. ...
... Strong convergence rates for spatial and temporal discretization schemes applied to stochastic wave equations have been obtained for instance in the articles [42,43,3,30,13] for semi-discrete and fully-discrete methods using the interpretation of SPDEs of stochastic evolution equations from [18], which is considered in this manuscript. The works mentioned above employ exponential or trigonometric integrator for the temporal discretization, see also [27,28] for a presentation of exponential integrators in the context of SPDEs. The articles [36,14] provide strong convergence analysis using the point of view of random field solutions for SPDEs, developed by [41], see also [19] for a focus on the stochastic wave equation. ...
We study a class of stochastic semilinear damped wave equations driven by additive Wiener noise. Owing to the damping term, under appropriate conditions on the nonlinearity, the solution admits a unique invariant distribution. We apply semi-discrete and fully-discrete methods in order to approximate this invariant distribution, using a spectral Galerkin method and an exponential Euler integrator for spatial and temporal discretization respectively. We prove that the considered numerical schemes also admit unique invariant distributions, and we prove error estimates between the approximate and exact invariant distributions, with identification of the orders of convergence. To the best of our knowledge this is the first result in the literature concerning numerical approximation of invariant distributions for stochastic damped wave equations.
... The strong pairwise coupling approach was introduced and studied experimentally in the work on finite-dimensional Langevin SDE by Müller et al. [38] and extended to filtering methods for (infinite-dimensional) SPDE by Chernov et al. [8]. The coupling idea is based on the exponential Euler integrator [11,23,33,36] for time-discretization of reaction-diffusion type SDE/SPDE. For the finite-dimensional SDE in [38], strong coupling is shown to produce constant-factor efficiency gains in numerical experiments, whereas for the herein considered class of SPDE, we show that strong coupling reduces the asymptotic rate of growth in the computational cost. ...
Multilevel Monte Carlo (MLMC) has become an important methodology in applied mathematics for reducing the computational cost of weak approximations. For many problems, it is well-known that strong pairwise coupling of numerical solutions in the multilevel hierarchy is needed to obtain efficiency gains. In this work, we show that strong pairwise coupling indeed is also important when MLMC is applied to stochastic partial differential equations (SPDE) of reaction-diffusion type, as it can improve the rate of convergence and thus improve tractability. For the MLMC method with strong pairwise coupling that was developed and studied numerically on filtering problems in (Chernov in Num Math 147:71-125, 2021), we prove that the rate of computational efficiency is higher than for existing methods. We also provide numerical comparisons with alternative coupling ideas on linear and nonlinear SPDE to illustrate the importance of this feature.
... The timestep is chosen between t = 10 −2 and t = 10 −3 . In order to include stochasticity, we generalize first-order ETD to include the stochastic increment, similar to [31,32]. ...
The subcritical transition to turbulence, as occurs in pipe flow, is believed to generically be a phase transition in the directed percolation universality class. At its heart is a balance between the decay rate and proliferation rate of localized turbulent structures, called puffs in pipe flow. Here, we propose the first-ever dynamical mechanism for puff proliferation—the process by which a puff splits into two. In the first stage of our mechanism, a puff expands into a slug. In the second stage, a laminar gap is formed within the turbulent core. The notion of a split-edge state, mediating the transition from a single puff to a two-puff state, is introduced and its form is predicted. The role of fluctuations in the two stages of the transition, and how splits could be suppressed with increasing Reynolds number, are discussed. Using numerical simulations, the mechanism is validated within the stochastic Barkley model. Concrete predictions to test the proposed mechanism in pipe and other wall-bounded flows, and implications for the universality of the directed percolation picture, are discussed.
... The case of strong convergence of nonlinear SEEs is generally more subtle and challenging, and has received widely attention in the research community in recent years. For instance, Kloeden et al. [35,39] proposed a discretization based on the Galerkin method in space and exponential integrator in time for the nonlinear SEEs with cylindrical additive noise. Kruse [44] analyzed the strong convergence error for a finite element method/linear implicit Euler spatio-temporal discretization of semilinear SEEs with multiplicative noise and Lipschitz continuous nonlinearities, and deduced the optimal error estimates. ...
abstract>
In this paper, we investigate the stochastic evolution equations (SEEs) driven by a bounded -Whittle-Matrn (W-M) random diffusion coefficient field and Q -Wiener multiplicative force noise. First, the well-posedness of the underlying equations is established by proving the existence, uniqueness, and stability of the mild solution. A sampling approach called approximation circulant embedding with padding is proposed to sample the random coefficient field. Then a spatio-temporal discretization method based on semi-implicit Euler-Maruyama scheme and finite element method is constructed and analyzed. An estimate for the strong convergence rate is derived. Numerical experiments are finally reported to confirm the theoretical result.
</abstract
... The time-step is chosen between ∆t = 10 −2 and ∆t = 10 −3 . In order to include stochasticity, we generalize first-order ETD to include the stochastic increment, similar to [28,29]. ...
The subcritical transition to turbulence, as occurs in pipe flow, is believed to generically be a phase transition in the directed percolation universality class. At its heart is a balance between the decay rate and proliferation rate of localized turbulent structures, called puffs in pipe flow. Here we propose the first-ever dynamical mechanism for puff proliferation -- the process by which a puff splits into two. In the first stage of our mechanism, a puff expands into a slug. In the second stage, a laminar gap is formed within the turbulent core. The notion of a split-edge state, mediating the transition from a single puff to a two puff state, is introduced and its form is predicted. The role of fluctuations in the two stages of the transition, and how splits could be suppressed with increasing Reynolds number, are discussed. Using numerical simulations, the mechanism is validated within the stochastic Barkley model. Concrete predictions to test the proposed mechanism in pipe and other wall bounded flows, and implications for the universality of the directed percolation picture, are discussed.
... For γ ∈ [2,4], the order of the convergence in time is higher than the Hölder regularity in time of the mild solution, this is due to the fact that that noise is additive noise. This standard phenomenon is known for SPDEs, see [33,48]. ...
The first aim of this paper is to examine existence, uniqueness and regularity for the Cahn-Hilliard-Cook (CHC) equation in space dimension . By applying a spectral Galerkin method to the infinite dimensional equation, we elaborate the well-posedness and regularity of the finite dimensional approximate problem. The key idea lies in transforming the stochastic problem {\color{black}{with additive noise}} into an equivalent random equation. The regularity of the solution to the equivalent random equation is obtained, in one dimension, with the aid of the Gagliardo-Nirenberg inequality and done in two and three dimensions, by the energy argument. Further, the approximate solution is shown to be strongly convergent to the unique mild solution of the original CHC equation, whose spatio-temporal regularity can be attained by similar arguments. In addition, a fully discrete approximation of such problem is investigated, performed by the spectral Galerkin method in space and the backward Euler method in time. The previously obtained regularity results of the problem help us to identify strong convergence rates of the fully discrete scheme.
... The time-step is chosen between ∆t = 10 −2 and ∆t = 10 −3 . In order to include stochasticity, we generalize first-order ETD to include the stochastic increment, similar to [22,23]. ...
The transition to turbulence in pipes is characterized by a coexistence of laminar and turbulent states. At the lower end of the transition, localized turbulent pulses, called puffs, can be excited. Puffs can decay when rare fluctuations drive them close to an edge state lying at the phase-space boundary with laminar flow. At higher Reynolds numbers, homogeneous turbulence can be sustained, and dominates over laminar flow. Here we expand this landscape of states, placing it within a unified bifurcation picture, and reveal the role it plays in transitions. We demonstrate our claims within the Barkley model, and motivate them generally. We first suggest the existence of an anti-puff and a gap-edge -- states which mirror the puff and related edge state. Previously observed laminar gaps forming within homogeneous turbulence are then naturally identified as anti-puffs nucleating and decaying through the gap edge. We further propose a novel mechanism for puff splits, possible in a coexistence region between puffs, homogeneous turbulence and the gap edge: (i) a puff expands into a slug, which has a homogeneous turbulent core, and (ii) a laminar gap is formed within the core, mediated through the gap edge. We present the corresponding split-edge state, discuss the effect of the Reynolds number on the two transition mechanisms, and confirm our picture for stochastic splits within the Barkley model.
... The strong pairwise coupling approach was introduced and studied experimentally in the work on finite-dimensional Langevin SDE by Müller et al. [38] and extended to filtering methods for (infinite-dimensional) SPDE by Chernov et al. [8]. The coupling idea is based on the exponential Euler integrator [11,23,33,36] for time-discretization of reaction-diffusion type SDE/SPDE. For the finite-dimensional SDE in [38], strong coupling is shown to produce constant-factor efficiency gains in numerical experiments, whereas for the herein considered class of SPDE, we show that strong coupling reduces the asymptotic rate of growth in the computational cost. ...
Multilevel Monte Carlo (MLMC) has become an important methodology in applied mathematics for reducing the computational cost of weak approximations. For many problems, it is well-known that strong pairwise coupling of numerical solutions in the multilevel hierarchy is needed to obtain efficiency gains. In this work, we show that strong pairwise coupling indeed is also important when (MLMC) is applied to stochastic partial differential equations (SPDE) of reaction-diffusion type, as it can improve the rate of convergence and thus improve tractability. For the (MLMC) method with strong pairwise coupling that was developed and studied numerically on filtering problems in [{\it Chernov et al., Numer. Math., 147 (2021), 71-125}], we prove that the rate of computational efficiency is higher than for existing methods. We also provide numerical comparisons with alternative coupling ideas illustrate the importance of this feature. The comparisons are conducted on a range of SPDE, which include a linear SPDE, a stochastic reaction-diffusion equation, and stochastic Allen--Cahn equation.
... There are numerous numerical methods designed to approximate (1) with linear self-adjoint operator, see e.g. 6,7,8,9 . To classify a numerical method, one also takes in consideration the rate of convergence. ...
In this paper, we investigate the numerical approximation of stochastic convection-reaction-diffusion equations using two explicit exponential integrators. The stochas-tic partial differential equation (SPDE) is driven by additive Wiener process. The approximation in space is done via a combination of the standard finite element method and the Galerkin projection method. Using the linear functional of the noise, we construct two accelerated numerical methods, which achieve higher convergence orders. In particular, we achieve convergence rates approximately 1 for trace class noise and 1 2 for space-time white noise. These convergences orders are obtained under less regularities assumptions on the nonlinear drift function than those used in the literature for stochastic reaction-diffusion equations. Numerical experiments to illustrate our theoretical results are provided.
... It is known that exponential integrators, as explicit time-stepping schemes, are successfully used to solve deterministic stiff problems such as parabolic partial differential equations and their spatial discretizations (see the survey article [24] and references therein). The extension to SPDEs has been extensively studied in [4,5,16,17,27,28,30,35,43,44,45], where both strong and weak convergence of exponential integrators were well established for SPDEs over finite time intervals. However, the weak error analysis in infinite horizon for exponential integrators is missing, which partly motivates this work. ...
We discrete the ergodic semilinear stochastic partial differential equations in space dimension d≤3 with additive noise, spatially by a spectral Galerkin method and temporally by an exponential Euler scheme. It is shown that both the spatial semi-discretization and the spatio-temporal full discretization are ergodic. Further, convergence orders of the numerical invariant measures, depending on the regularity of noise, are recovered based on an easy time-independent weak error analysis without relying on Malliavin calculus. To be precise, the convergence order is 1−ϵ in space and 12−ϵ in time for the space-time white noise case and 2−ϵ in space and 1−ϵ in time for the trace class noise case in space dimension d=1, with arbitrarily small ϵ>0. Numerical results are finally reported to confirm these theoretical findings.
... Both tasks rely heavily upon the ability to efficiently sample different realizations from these SDEs and typically use Monte Carlo approaches (e.g. [8], [9], [14]). ...
We consider linearizations of stochastic differential equations with additive noise using the Karhunen-Lo\`eve expansion. We obtain our linearizations by truncating the expansion and writing the solution as a series of matrix-vector products using the theory of matrix functions. Moreover, we restate the solution as the solution of a system of linear differential equations. We obtain strong and weak error bounds for the truncation procedure and show that, under suitable conditions, the mean square error has order of convergence and the second moment has a weak order of convergence , where m denotes the size of the expansion. We also discuss efficient numerical linear algebraic techniques to approximate the series of matrix functions and the linearized system of differential equations. These theoretical results are supported by experiments showing the effectiveness of our algorithms when compared to standard methods such as the Euler-Maruyama scheme.
... For semilinear SPDEs with super-linearly growing drift driven by Wiener or Gaussian noise, numerical schemes are considered in [7], [11], [27], [41], [65], [68], [70], [74], [75], [77], [78], [80], [81], [98], [112], [113], [131], [134]. ...
Stochastic evolution equations with compensated Poisson noise are considered in the variational approach with monotone and coercive coefficients. Here the Poisson noise is assumed to be time homogeneous with -finite intensity measure on a metric space. By using finite element methods and Galerkin approximations, some explicit and implicit discretizations for this equation are presented and their convergence is proved. Polynomial growth condition and linear growth condition are assumed on the drift operator, respectively for the implicit and explicit schemes.
... Thus, we need to rely on numerical discretization schemes to estimate moments or statistics of the solution. The numerical approximation of SPDEs has been an active field of research in the last decade, see for instance [7,8,16,21,22,25,27,28,31] and the references therein. Lévy fields as driving noise of the SPDE have been investigated, among others, in [6,10,13,18,29,34]. ...
Semilinear hyperbolic stochastic partial differential equations have various applications in the natural and engineering sciences. From a modelling point of view the Gaussian setting can be too restrictive, since phenomena as porous media, pollution models or applications in mathamtical finance indicate an influence of noise of a different nature. In order to capture temporal discontinuities and allow for heavy-tailed distributions, Hilbert space valued-L\'evy processes (or L\'evy fields) as driving noise terms are considered. The numerical discretization of the corresponding SPDE involves several difficulties: Low spatial and temporal regularity of the solution to the problem entails slow convergence rates and instabilities for space/time-discretization schemes. Furthermore, the L\'evy process admits values in a possibly infinite-dimensional Hilbert space, hence projections into a finite-dimensional subspace for each discrete point in time are necessary. Finally, unbiased sampling from the resulting L\'evy field may not be possible. We introduce a fully discrete approximation scheme that addresses these issues. A discontinuous Galerkin approach for the spatial approximation is coupled with a suitable time stepping scheme to avoid numerical oscillations. Moreover, we approximate the driving noise process by truncated Karhunen-Lo\'eve expansions. The latter essentially yields a sum of scaled and uncorrelated one-dimensional L\'evy processes, which may be simulated with controlled bias by Fourier inversion techniques.
... The numerical analysis of the stochastic heat equation (1.1) is an active research area. Without being too exhaustive, beside the above mentioned papers Gyöngy (1998b) and Gyöngy (1999), we men- tion the following works regarding numerical discretizations of stochastic parabolic partial differential equations: Gyöngy (1998b); Yan (2005); Barth & Lang (2012); Sauer & Stannat (2015) (spatial ap- proximations); Gaines (1995); Gyöngy & Nualart (1995, 1997; Allen et al. (1998); Shardlow (1999); Printems (2001); Davie & Gaines (2001); Du & Zhang (2002); Hausenblas (2003); Pettersson & Signahl (2005); Walsh (2005); Millet & Morien (2005); Müller-Gronbach & Ritter (2007); Jentzen & Kloeden (2009); ; Cox & van Neerven (2010);; Jentzen (2011); Kloeden et al. (2011); Lord & Tambue (2013); Wang & Gan (2013); Barth & Lang (2013); Cox & van Neerven (2013); Jentzen & Röckner (2015); ; Wang (2017) (temporal and full discretizations); Ta et al. (2015); Lang et al. (2017) (stability); Shardlow (2003); Jentzen & Kurniawan (2015); Debussche & Printems (2009); Debussche (2011); Andersson & Larsson (2016) (weak approximations). Observe that most of these references are concerned with an interpretation of stochastic partial differential equa- tions in Hilbert spaces and thus error estimates are provided in the L 2 ( [0,1]) norm (or similar norms). ...
A fully discrete approximation of the one-dimensional stochastic heat equation driven by multiplicative space-time white noise is presented. The standard finite difference approximation is used in space and a stochastic exponential method is used for the temporal approximation. Observe that the proposed exponential scheme does not suffer from any kind of CFL-type step size restriction. When the drift term and the diffusion coefficient are assumed to be globally Lipschitz, this explicit time integrator allows for error bounds in L q (Ω), for all q ⩾ 2, improving some existing results in the literature. On top of this, we also prove almost sure convergence of the numerical scheme. In the case of non-globally Lipschitz coefficients, under a strong assumption about pathwise uniqueness of the exact solution, convergence in probability of the numerical solution to the exact solution is proved. Numerical experiments are presented to illustrate the theoretical results.
... It is known that exponential integrators, as explicit time-stepping schemes, are successfully used to solve deterministic stiff problems such as parabolic partial differential equations and their spatial discretizations (see the survey article [24] and references therein). The extension to SPDEs has been extensively studied in [4,5,16,17,27,28,30,35,43,44,45], where both strong and weak convergence of exponential integrators were well established for SPDEs over finite time intervals. However, the weak error analysis in infinite horizon for exponential integrators is missing, which partly motivates this work. ...
We discrete the ergodic semilinear stochastic partial differential equations in space dimension with additive noise, spatially by a spectral Galerkin method and temporally by an exponential Euler scheme. It is shown that both the spatial semi-discretization and the spatio-temporal full discretization are ergodic. Further, we recover the convergence orders of the numerical invariant measures, depending on the regularity of noise, based on an easy time-independent weak error analysis without relying on Malliavin calculus. To be precise, the convergence order is in space and in time for the space-time white noise case and in space and in time for the trace class noise case in space dimension , with arbitrarily small . Numerical results are finally reported to confirm these theoretical findings.
... An important ingredient to achieve that optimal convergence order is the application of Taylor's formula in Banach space to the drift function, see Section 2.2.2. It is worth to mention that such approach and assumptions on the nonlinear drift function F were also used in [17,26,30,44] for exponential integrators and semi-implicit Euler method for autonomous SPDEs driven by additive noise to achieve optimal convergence order 1 in time. Due to the complexity of the linear operator and the corresponding semi discrete linear operator after space dis- cretisation, novel additional technical estimates are provided on the terms involving the noise to achieve higher convergence order, see e.g., Lemma 2.15 and Section 2.2.3. ...
In this paper, we investigate a numerical approximation of a general second order
semilinear parabolic non-autonomous stochastic partial differential equation (SPDE)
driven by additive noise. Numerical approximations for autonomous SPDEs are thoroughly
investigated in the literature while the non-autonomous case is not yet well understood.
We discretize the non-autonomous SPDE in space by the finite element method and in time by the Magnus-type integrator. We provide a strong convergence proof of the fully discrete scheme toward the mild solution in the root-mean-square norm. Appropriate assumptions on the drift term and the noise allow to achieve optimal
convergence order in time greater than 1/2, without any logarithmic reduction of convergence order in time.In particular, for trace class noise, we achieve optimal convergence orders , where is a positive number small enough. Numerical simulations are provided to illustrate our theoretical results.
... Whereas, our goal in this paper is to make an attempt to discuss strong convergence of an exponential integrator (EI) scheme, coupled with a Galerkin scheme for the spatial discretization (see (1.8) and (1.9) below), for a class of SPDEs with Hölder continuous drifts. With regard to convergence of EI scheme for SPDEs with smooth drift coefficients, we refer to [13,14,17,18] for further details, to name a few. Also, there is a number of literature on approximation of SPDEs with non-globally Lipschitz continuous nonlinearities; see, for instance, [6,9]. ...
In this paper, exploiting the regularities of the corresponding Kolmogorov equations involved we investigate strong convergence of exponential integrator scheme for a range of stochastic partial differential equations, in which the drift term is H\"older continuous, and reveal the rate of convergence.
In this paper, we consider the numerical approximation of a general second order semilinear stochastic partial differential equation (SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part also called stochastic reactive dominated transport equations. Most numerical techniques, including current stochastic exponential integrators lose their good stability properties on such equations. Using finite element for space discretization, we propose a new scheme appropriated on such equations, called stochastic exponential Rosenbrock scheme (SERS) based on local linearization at every time step of the semi-discrete equation obtained after space discretization. We consider noise that is in a trace class and give a strong convergence proof of the new scheme toward the exact solution in the root-mean-square norm. Numerical experiments to sustain theoretical results are provided.
How predictable are turbulent flows? Here, we use theoretical estimates and shell model simulations to argue that Eulerian spontaneous stochasticity, a manifestation of the nonuniqueness of the solutions to the Euler equation that is conjectured to occur in Navier-Stokes turbulence at high Reynolds numbers, leads to universal statistics at finite times, not just at infinite time as for standard chaos. These universal statistics are predictable, even though individual flow realizations are not. Any small-scale noise vanishing slowly enough with increasing Reynolds number can trigger spontaneous stochasticity, and here we show that thermal noise alone, in the absence of any larger disturbances, would suffice. If confirmed for Navier-Stokes turbulence, our findings would imply that intrinsic stochasticity of turbulent fluid motions at all scales can be triggered even by unavoidable molecular noise, with implications for modeling in engineering, climate, astrophysics, and cosmology.
We revisit the issue of whether thermal fluctuations are relevant for incompressible fluid turbulence and estimate the scale at which they become important. As anticipated by Betchov in a prescient series of works more than six decades ago, this scale is about equal to the Kolmogorov length, even though that is several orders of magnitude above the mean free path. This result implies that the deterministic version of the incompressible Navier-Stokes equation is inadequate to describe the dissipation range of turbulence in molecular fluids. Within this range, the fluctuating hydrodynamics equation of Landau and Lifschitz is more appropriate. In particular, our analysis implies that both the exponentially decaying energy spectrum and the far-dissipation-range intermittency predicted by Kraichnan for deterministic Navier-Stokes will be generally replaced by Gaussian thermal equipartition at scales just below the Kolmogorov length. Stochastic shell model simulations at high Reynolds numbers verify our theoretical predictions and reveal furthermore that inertial-range intermittency can propagate deep into the dissipation range, leading to large fluctuations in the equipartition length scale. We explain the failure of previous scaling arguments for the validity of deterministic Navier-Stokes equations at any Reynolds number and we provide a mathematical interpretation and physical justification of the fluctuating Navier-Stokes equation as an "effective field theory" valid below some high-wave-number cutoff Λ, rather than as a continuum stochastic partial differential equation. At Reynolds number around a million, comparable to that in Earth's atmospheric boundary layer, the strongest turbulent excitations observed in our simulation penetrate down to a length scale of about eight microns, still two orders of magnitude greater than the mean free path of air. However, for longer observation times or for higher Reynolds numbers, more extreme turbulent events could lead to a local breakdown of fluctuating hydrodynamics.
The transition to turbulence in pipes is characterized by a coexistence of laminar and turbulent states. At the lower end of the transition, localized turbulent pulses, called puffs, can be excited. Puffs can decay when rare fluctuations drive them close to an edge state lying at the phase-space boundary with laminar flow. At higher Reynolds numbers, homogeneous turbulence can be sustained, and dominates over laminar flow. Here we complete this landscape of localized states, placing it within a unified bifurcation picture. We demonstrate our claims within the Barkley model, and motivate them generally. Specifically, we suggest the existence of an antipuff and a gap-edge-states which mirror the puff and related edge state. Previously observed laminar gaps forming within homogeneous turbulence are then naturally identified as antipuffs nucleating and decaying through the gap edge. We also discuss alternatives to the suggested bifurcation diagram, which could be relevant for wall-bounded flows other than straight pipes.
We revisit the issue of whether thermal fluctuations are relevant for incompressible fluid turbulence, and estimate the scale at which they become important. As anticipated by Betchov in a prescient series of works more than six decades ago, this scale is about equal to the Kolmogorov length, even though that is several orders of magnitude above the mean free path. This result implies that the deterministic version of the incompressible Navier-Stokes equation is inadequate to describe the dissipation range of turbulence in molecular fluids. Within this range, the fluctuating hydrodynamics equation of Landau and Lifschitz is more appropriate. In particular, our analysis implies that both the exponentially decaying energy spectrum and the far-dissipation range intermittency predicted by Kraichnan for deterministic Navier-Stokes will be generally replaced by Gaussian thermal equipartition at scales just below the Kolmogorov length. Stochastic shell model simulations at high Reynolds numbers verify our theoretical predictions and reveal furthermore that inertial-range intermittency can propagate deep into the dissipation range, leading to large fluctuations in the equipartition length scale. We explain the failure of previous scaling arguments for the validity of deterministic Navier-Stokes equations at any Reynolds number and we provide a mathematical interpretation and physical justification of the fluctuating Navier-Stokes equation as an ``effective field-theory'' valid below some high-wavenumber cutoff , rather than as a continuum stochastic partial differential equation. At Reynolds number around a million the strongest turbulent excitations observed in our simulation penetrate down to a length-scale of microns. However, for longer observation times or higher Reynolds numbers, more extreme turbulent events could lead to a local breakdown of fluctuating hydrodynamics.
In this paper, we present new types of exponential integrators for Stochastic Differential Equations (SDEs) that take advantage of the exact solution of (generalized) geometric Brownian motion. We examine both Euler and Milstein versions of the scheme and prove strong convergence, taking care to deal with the dependence on the noise in the solution operator. For the special case of linear noise we obtain an improved rate of convergence for the Euler version over standard integration methods. We investigate the efficiency of the methods compared with other exponential integrators for low dimensional SDEs and high dimensional SDEs arising from the discretization of stochastic partial differential equations. We show that, by introducing a suitable homotopy parameter, these schemes are competitive not only when the noise is linear, but also in the presence of nonlinear noise terms. Although our new schemes are derived and analysed under zero commutator conditions (1.2), our numerical investigations illustrate that the resulting methods rival traditional methods even when this does not hold.
A fully discrete approximation of the one-dimensional stochastic heat equation driven by multiplicative space–time white noise is presented. The standard finite difference approximation is used in space and a stochastic exponential method is used for the temporal approximation. Observe that the proposed exponential scheme does not suffer from any kind of CFL-type step size restriction. When the drift term and the diffusion coefficient are assumed to be globally Lipschitz this explicit time integrator allows for error bounds in , for all , improving some existing results in the literature. On top of this we also prove almost sure convergence of the numerical scheme. In the case of nonglobally Lipschitz coefficients, under a strong assumption about pathwise uniqueness of the exact solution, convergence in probability of the numerical solution to the exact solution is proved. Numerical experiments are presented to illustrate the theoretical results.
In this paper, the numerical method for semi-linear stochastic variable delay integro-differential equations is studied. The stability of analytic solutions of semi-linear stochastic variable delay integro-differential equations are studied first, some suitable conditions for the mean-square stability of the analytic solutions are also obtained. Then the numerical approximation of exponential integrators for semi-linear stochastic variable delay integro-differential equations are constructed, the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent with the strong order 1/2 and the exponential Euler method can keep the mean-square exponential stability of the analytical solution under some restrictions on the step size. In addition, numerical experiments are presented to confirm the theoretical results.
This paper is concerned with the delay dependent stability of the stochastic exponential Euler method for stochastic delay differential equations and stochastic delay partial differential equations. By using root locus technique, the necessary and sufficient condition of the numerical delay dependent stability of the method is derived for a class of stochastic delay differential equations and it is shown that the stochastic exponential Euler method can fully preserve the asymptotic mean square stability of the underlying system. Furthermore, we investigate the delay dependent stability of the semidiscrete and fully discrete systems for a linear stochastic delay partial differential equation. The necessary and sufficient condition for the delay dependent stability of the semidiscrete system based on the standard central difference scheme in space is given. Based on this condition, the delay dependent stability of the fully discrete system by using the stochastic exponential Euler method in time is studied. It is shown that the fully discrete scheme can inherit the delay dependent stability of the semidiscrete system completely. At last, some numerical experiments are given to validate our theoretical results.
A crucial point for the computational efficiency of the exponential-type integrators for differential equations is the effective evaluation of the integrals involving matrix exponential called phi-functions. Focusing on exponential integrators for differential equations of small dimensions, for which the Krylov-based approximations for the phi-functions are not viable, in this article refined algorithms are introduced for speeding up both, the evaluation of several phi-functions simultaneously and the computation of a linear combination of phi-functions times matrices. The algorithms are derived from a meticulous approximation to the exponential of certain partitioned matrix by adapting the conventional Padé method to the special structure of this matrix. Numerical simulations are provided to evaluate the performance of the proposed algorithms and compare them with others in the literature.
We consider the numerical solution, by finite differences, of second-order-in-time stochastic partial differential equations (SPDEs) in one space dimension. New timestepping methods are introduced by generalising recently-introduced methods for second-order-in-time stochastic differential equations to multidimensional systems. These stochastic methods, based on leapfrog and Runge–Kutta methods, are designed to give good approximations to the stationary variances and the correlations in the position and velocity variables. In particular, we introduce the reverse leapfrog method and stochastic Runge–Kutta Leapfrog methods, analyse their performance applied to linear SPDEs and perform numerical experiments to examine their accuracy applied to a type of nonlinear SPDE.
The aim of this paper is the derivation of an a-posteriori error estimate for the numerical method based on an exponential scheme in time and spectral Galerkin methods in space. We obtain analytically a rigorous bound on the conditional mean square error, which is conditioned to the given realization of the data calculated by a numerical method. This bound is explicitly computable and uses only the computed numerical approximation. Thus one can check a-posteriori the error for a given numerical computation for a fixed discretization without relying on an asymptotic result. All estimates are only based on the numerical data and the structure of the equation, but they do not use any a-priori information of the solution, which makes the approach applicable to equations where global existence and uniqueness of solutions is not known. For simplicity of presentation, we develop the method here in a relatively simple situation of a stable one-dimensional Allen-Cahn equation with additive forcing.
The explicit Euler scheme and similar explicit approximation schemes (such as the Milstein scheme) are known to diverge strongly and numerically weakly in the case of one-dimensional stochastic ordinary differential equations with superlinearly growing nonlinearities. It remained an open question whether such a divergence phenomenon also holds in the case of stochastic partial differential equations with superlinearly growing nonlinearities such as stochastic Allen-Cahn equations. In this work we solve this problem by proving that full-discrete exponential Euler and full-discrete linear-implicit Euler approximations diverge strongly and numerically weakly in the case of stochastic Allen-Cahn equations. This article also contains a short literature overview on existing numerical approximation results for stochastic differential equations with superlinearly growing nonlinearities.
The stationary density of the overdamped SPDE corresponds to a mean number of kinks and antikinks that is maintained by a balance between nucleation of new kink-antikink pairs and annihilation whenever a kink and antikink collide. We consider numerical methods for solution of the SPDE, and a definition of the location of the centre of a kink using a smoothing function that is the derivative of the function describing the shape and energy of an isolated kink. This definition allows calculation of the diffusivity of a kink as a function of noise intensity or “temperature” and defines the parameter characterising the “small-noise” régime. In the reaction-diffusion description of the dynamics (where kink-antikink pairs are nucleated with rate , diffuse with diffusivity, D and annihilate on collision) the number of kinks per unit length in the steady-state has a simple exact expression.
A new barycentric spectral domain decomposition methods algorithm for solving partial integro-differential models is described. The method is applied to European and butterfly call option pricing problems under a class of infinite activity Lévy models. It is based on the barycentric spectral domain decomposition methods which allows the implementation of the boundary conditions in an efficient way. After the approximation of the spatial derivatives, we obtained the semi-discrete equations. The computation of these equations is performed by using the barycentric spectral domain decomposition method. This is achieved with the implementation of an exponential time integration scheme. Several numerical tests for the pricing of European and butterfly options are given to illustrate the efficiency and accuracy of this new algorithm. We also show that Greek options, such as Delta and Gamma sensitivity, are computed with no spurious oscillation.
This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part, usually called stochastic dominated transport equations. Most standard numerical schemes lose their good stability properties on such equations, including the current linear implicit Euler method. We discretise the SPDE in space by the finite element method and propose a new scheme in time appropriate for such equations, called stochastic Rosenbrock-Type scheme, which is based on the local linearisation of the semi-discrete problem obtained after space discretisation. We provide a strong convergence of the new fully discrete scheme toward the exact solution for multiplicative and additive noise. Our convergence rates are in agreement with results in the literature. Numerical experiments to sustain our theoretical results are provided.
This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part, usually called stochastic dominated transport equations. Most standard numerical schemes lose their good stability properties on such equations, including the current linear implicit Euler method. We discretise the SPDE in space by the finite element method and propose a new scheme in time appropriate for such equations , called stochastic Rosenbrock-type scheme, which is based on the local linearisation of the semi-discrete problem obtained after space discretisation. We provide a strong convergence of the new fully discrete scheme toward the exact solution for multiplicative and additive noise. Our convergence rates are in agreement with results in the literature. Numerical experiments to sustain our theoretical results are provided.
This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise under more relaxed conditions. The SPDE is discretized in space by the finite element method and in time by the linear implicit Euler method. This extends the current results in the literature to not necessary self-adjoint operator with more general boundary conditions. As a consequence key part of the proof does not rely on the spectral decomposition of the linear operator. We achieve optimal convergence orders which depend on the regularity of the noise and the initial data. In particular, for multiplicative noise we achieve optimal order and for additive noise, we achieve optimal order . In contrast to current work in the literature, where the optimal convergence orders are achieved for additive noise by incorporating further regularity assumptions on the nonlinear drift function, our optimal convergence orders are obtained under only the standard Lipschitz condition of the nonlinear drift term. Numerical experiments to sustain our theoretical results are provided.
This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE)
driven by multiplicative or additive noise. The linear operator is not necessary self-adjoint, so more useful in concrete applications.
The SPDE is discretized in space by the finite element method and in time by the linear implicit Euler method. The corresponding scheme is more stable and efficient to solve stochastic advection-dominated reactive transport in porous media. This extends the current results in the literature to not necessary self-adjoint operator.
As a challenge, key part of the proof does not rely anymore on the spectral decomposition of the linear operator.
The results reveal how the convergence orders depend on the regularity of the noise and the initial data.
In particular for multiplicative trace class noise, we achieve optimal convergence order
and for additive trace class noise, we achieve optimal convergence order in space and sup-optimal convergence order in time of the form ,
for an arbitrarily small .
Numerical experiments to sustain our theoretical results are provided.
In this chapter, we discuss some basic aspects of stochastic differential equations (SDEs) including stochastic ordinary (SODEs) and partial differential equations (SPDEs).
The simulation of the expectation of a stochastic quantity E[Y] by Monte Carlo methods is known to be computationally expensive especially if the stochastic quantity or its approximation . Yn is expensive to simulate, e.g., the solution of a stochastic partial differential equation. If the convergence of Yn to Y in terms of the error |E[Y-Yn]| is to be simulated, this will typically be done by a Monte Carlo method, i.e., |E[Y]-EN[Yn]| is computed. In this article upper and lower bounds for the additional error caused by this are determined and compared to those of |EN[Y-Yn]|, which are found to be smaller. Furthermore, the corresponding results for multilevel Monte Carlo estimators, for which the additional sampling error converges with the same rate as |E[Y-Yn]|, are presented. Simulations of a stochastic heat equation driven by multiplicative Wiener noise and a geometric Brownian motion are performed which confirm the theoretical results and show the consequences of the presented theory for weak error simulations. © 2017 International Association for Mathematics and Computers in Simulation (IMACS).
Existence and uniqueness for semilinear stochastic evolution equations with additive noise by means of finite-dimensional Galerkin approximations is established and the convergence rate of the Galerkin approximations to the solution of the stochastic evolution equation is estimated. These abstract results are applied to several examples of stochastic partial differential equations (SPDEs) of evolutionary type including a stochastic heat equation, a stochastic reaction diffusion equation, and a stochastic Burgers equation. The estimated convergence rates are illustrated by numerical simulations. The main novelty in this article is the estimation of the difference of the finite-dimensional Galerkin approximations and of the solution of the infinite-dimensional SPDE uniformly in space, i.e., in the -topology, instead of the usual Hilbert space estimates in the -topology, that were shown before.
We study approximation methods for stochastic differential equations and point out a simple relation between their order of convergence in the pth mean and their order of convergence in the pathwise sense: Convergence in the pth mean of order α for all p 1 implies pathwise convergence of order α − ε for arbitrary ε > 0. We apply this result to several one-step and multi-step approximation schemes for stochastic differen-tial equations and stochastic delay differential equations. In addition, we give some numerical examples.
We consider the pathwise numerical approximation of nonlinear parabolic stochastic partial differential equations (SPDEs)
driven by additive white noise under local assumptions on the coefficients only. We avoid the standard global Lipschitz assumption
in the literature on the coefficients by first showing convergence under global Lipschitz coefficients but with a strong error
criteria and then by applying a localization technique for one sample path on a bounded set.
The method of characteristics (the averaging over the characteristic
formula) and the weak-sense numerical integration of ordinary stochastic
differential equations together with the Monte Carlo technique are used
to propose numerical methods for linear stochastic partial differential
equations (SPDEs). Their orders of convergence in the mean-square sense
and in the sense of almost sure convergence are obtained. A variance
reduction technique for the Monte Carlo procedures is considered. Layer
methods for linear and semilinear SPDEs are constructed and the
corresponding convergence theorems are proved. The approach developed is
supported by numerical experiments.
We investigate the strong approximation of stochastic parabolic
partial differential equations with additive noise. We introduce
post-processing in the context of a standard Galerkin approximation,
although other spatial discretisations are possible. In time, we
use an
exponential integrator. We prove strong error estimates
and discuss the best number of post-processing terms to take.
Numerically, we evaluate the efficiency of the methods and observe
rates of convergence. Some experiments with the implicit
Euler--Maruyama method are described
We prove that Euler's approximations for stochastic differential equations on domains of R-d converge almost surely if the drift satisfies the monotonicity condition and the diffusion coefficient is Lipschitz continuous.
We consider strong approximations to parabolic stochastic PDEs. We assume the noise lies in a Gevrey space of analytic functions. This type of stochastic forcing includes the case of forcing in a finite number of Fourier modes. We show that with Gevrey noise our numerical scheme has solutions in a discrete equivalent of this space and prove a strong error estimate. Finally we present some numerical results for a stochastic PDE with a Ginzburg-Landau nonlinearity and compare this to the more standard implicit Euler-Maruyama scheme.
Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.
In this paper, LpLp convergence and almost sure convergence of the Milstein approximation of a partial differential equation of advection–diffusion type driven by a multiplicative continuous martingale is proven. The (semidiscrete) approximation in space is a projection onto a finite dimensional function space. The considered space approximation has to have an order of convergence fitting to the order of convergence of the Milstein approximation and the regularity of the solution. The approximation of the driving noise process is realized by the truncation of the Karhunen–Loève expansion of the driving noise according to the overall order of convergence. Convergence results in LpLp and almost sure convergence bounds for the semidiscrete approximation as well as for the fully discrete approximation are provided.
We approximate quasi-linear parabolic SPDEs substituting the derivatives with finite differences. We investigate the resulting implicit and explicit schemes. For the implicit scheme we estimate the rate of Lp convergence of the approximations and we also prove their almost sure convergence when the nonlinear terms are Lipschitz continuous. When the nonlinear terms are not Lipschitz continuous we obtain convergence in probability provided pathwise uniqueness for the equation holds. For the explicit scheme we get these results under an additional condition on the mesh sizes in time and space.
Contents. 2.1 Infinite Dimensional Brownian Motion 2.2 The Stochastic Integral 2.3 Fundamental Tools 2.3.1 Itô’s Formula 2.3.2 The Stochastic Fubini Theorem 2.3.3 Girsanov’s Theorem 2.4 Stochastic Equations 2.4.1 Mild, Weak and Strong Solutions 2.4.2 Existence and Uniqueness
The main objective of this work is to describe a Galerkin ap-proximation for stochastic partial differential equations driven by square– integrable martingales. Error estimates in the semidiscrete case, where dis-cretization is only done in space, and in the fully discrete case are derived. Parabolic as well as transport equations are studied.
We approximate quasi-linear parabolic SPDEs substituting the derivatives with finite differences. We investigate the resulting implicit and explicit schemes. For the implicit scheme we estimate the rate of Lp convergence of the approximations and we also prove their almost sure convergence when the nonlinear terms are Lipschitz continuous. When the nonlinear terms are not Lipschitz continuous we obtain convergence in probability provided pathwise uniqueness for the equation holds. For the explicit scheme we get these results under an additional condition on the mesh sizes in time and space.
We prove that Euler''s approximations for stochastic differential equations on domains of
d
converge almost surely if the drift satisfies the monotonicity condition and the diffusion coefficient is Lipschitz continuous.
In this paper we consider an implicit approximation scheme for the heat equation with a nonlinear term and an additive space-time white noise. Assuming that the nonlinear drift is measurable and verifies a one-sided linear growth condition we show that the approximation scheme converges to the unique solution in probability, uniformly in space and time.
Incluye bibliografía e índice
A note on Euler’s approximations, Potential anal
- I Gyöngy