Weak error estimates for trajectories of SPDEs for Spectral Galerkin discretization

We consider stochastic semi-linear evolution equations which are driven by additive, spatially correlated, Wiener noise, and in particular consider problems of heat equation (analytic semigroup) and damped-driven wave equations (bounded semigroup) type. We discretize these equations by means of a spectral Galerkin projection, and we study the approximation of the probability distribution of the trajectories: test functions are regular, but depend on the values of the process on the interval $[0,T]$. We introduce a new approach in the context of quantative weak error analysis for discretization of SPDEs. The weak error is formulated using a deterministic function (It\^o map) of the stochastic convolution found when the nonlinear term is dropped. The regularity properties of the It\^o map are exploited, and in particular second-order Taylor expansions employed, to transfer the error from spectral approximation of the stochastic convolution into the weak error of interest. We prove that the weak rate of convergence is twice the strong rate of convergence in two situations. First, we assume that the covariance operator commutes with the generator of the semigroup: the first order term in the weak error expansion cancels out thanks to an independence property. Second, we remove the commuting assumption, and extend the previous result, thanks to the analysis of a new error term depending on a commutator

Approximation of the invariant measure with an Euler scheme for Stochastic PDE's driven by Space-Time White Noise

In this article, we consider a stochastic PDE of parabolic type, driven by a space-time white-noise, and its numerical discretization in time with a semi-implicit Euler scheme. When the nonlinearity is assumed to be bounded, then a dissipativity assumption is satisfied, which ensures that the SDPE admits a unique invariant probability measure, which is ergodic and strongly mixing - with exponential convergence to equilibrium. Considering test functions of class $\mathcal{C}^2$, bounded and with bounded derivatives, we prove that we can approximate this invariant measure using the numerical scheme, with order 1/2 with respect to the time step

Solving optimal control problems governed by random Navier-Stokes equations using low-rank methods

Many problems in computational science and engineering are simultaneously characterized by the following challenging issues: uncertainty, nonlinearity, nonstationarity and high dimensionality. Existing numerical techniques for such models would typically require considerable computational and storage resources. This is the case, for instance, for an optimization problem governed by time-dependent Navier-Stokes equations with uncertain inputs. In particular, the stochastic Galerkin finite element method often leads to a prohibitively high dimensional saddle-point system with tensor product structure. In this paper, we approximate the solution by the low-rank Tensor Train decomposition, and present a numerically efficient algorithm to solve the optimality equations directly in the low-rank representation. We show that the solution of the vorticity minimization problem with a distributed control admits a representation with ranks that depend modestly on model and discretization parameters even for high Reynolds numbers. For lower Reynolds numbers this is also the case for a boundary control. This opens the way for a reduced-order modeling of the stochastic optimal flow control with a moderate cost at all stages.Comment: 29 page

Numerics for Stochastic Distributed Parameter Control Systems: a Finite Transposition Method

In this chapter, we present some recent progresses on the numerics for stochastic distributed parameter control systems, based on the \emph{finite transposition method} introduced in our previous works. We first explain how to reduce the numerics of some stochastic control problems in this respect to the numerics of backward stochastic evolution equations. Then we present a method to find finite transposition solutions to such equations. At last, we give an illuminating example

Weak convergence and invariant measure of a full discretization for non-globally Lipschitz parabolic SPDE

Approximating the invariant measure and the expectation of the functionals for parabolic stochastic partial differential equations (SPDEs) with non-globally Lipschitz coefficients is an active research area and is far from being well understood. In this article, we study such problem in terms of a full discretization based on the spectral Galerkin method and the temporal implicit Euler scheme. By deriving the a priori estimates and regularity estimates of the numerical solution via a variational approach and Malliavin calculus, we establish the sharp weak convergence rate of the full discretization. When the SPDE admits a unique $V$-uniformly ergodic invariant measure, we prove that the invariant measure can be approximated by the full discretization. The key ingredients lie on the time-independent weak convergence analysis and time-independent regularity estimates of the corresponding Kolmogorov equation. Finally, numerical experiments confirm the theoretical findings.Comment: 47 page

An efficient DP algorithm on a tree-structure for finite horizon optimal control problems

The classical Dynamic Programming (DP) approach to optimal control problems is based on the characterization of the value function as the unique viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation. The DP scheme for the numerical approximation of viscosity solutions of Bellman equations is typically based on a time discretization which is projected on a fixed state-space grid. The time discretization can be done by a one-step scheme for the dynamics and the projection on the grid typically uses a local interpolation. Clearly the use of a grid is a limitation with respect to possible applications in high-dimensional problems due to the curse of dimensionality. Here, we present a new approach for finite horizon optimal control problems where the value function is computed using a DP algorithm on a tree structure algorithm (TSA) constructed by the time discrete dynamics. In this way there is no need to build a fixed space triangulation and to project on it. The tree will guarantee a perfect matching with the discrete dynamics and drop off the cost of the space interpolation allowing for the solution of very high-dimensional problems. Numerical tests will show the effectiveness of the proposed method

PDE Acceleration: A convergence rate analysis and applications to obstacle problems

This paper provides a rigorous convergence rate and complexity analysis for a recently introduced framework, called PDE acceleration, for solving problems in the calculus of variations, and explores applications to obstacle problems. PDE acceleration grew out of a variational interpretation of momentum methods, such as Nesterov's accelerated gradient method and Polyak's heavy ball method, that views acceleration methods as equations of motion for a generalized Lagrangian action. Its application to convex variational problems yields equations of motion in the form of a damped nonlinear wave equation rather than nonlinear diffusion arising from gradient descent. These accelerated PDE's can be efficiently solved with simple explicit finite difference schemes where acceleration is realized by an improvement in the CFL condition from $dt\sim dx^2$ for diffusion equations to $dt\sim dx$ for wave equations. In this paper, we prove a linear convergence rate for PDE acceleration for strongly convex problems, provide a complexity analysis of the discrete scheme, and show how to optimally select the damping parameter for linear problems. We then apply PDE acceleration to solve minimal surface obstacle problems, including double obstacles with forcing, and stochastic homogenization problems with obstacles, obtaining state of the art computational results

Piecewise Constant Policy Approximations to Hamilton-Jacobi-Bellman Equations

An advantageous feature of piecewise constant policy timestepping for Hamilton-Jacobi-Bellman (HJB) equations is that different linear approximation schemes, and indeed different meshes, can be used for the resulting linear equations for different control parameters. Standard convergence analysis suggests that monotone (i.e., linear) interpolation must be used to transfer data between meshes. Using the equivalence to a switching system and an adaptation of the usual arguments based on consistency, stability and monotonicity, we show that if limited, potentially higher order interpolation is used for the mesh transfer, convergence is guaranteed. We provide numerical tests for the mean-variance optimal investment problem and the uncertain volatility option pricing model, and compare the results to published test cases

Spatio-Temporal Stochastic Optimization: Theory and Applications to Optimal Control and Co-Design

There is a rising interest in Spatio-temporal systems described by Partial Differential Equations (PDEs) among the control community. Not only are these systems challenging to control, but the sizing and placement of their actuation is an NP-hard problem on its own. Recent methods either discretize the space before optimziation, or apply tools from linear systems theory under restrictive linearity assumptions. In this work we consider control and actuator placement as a coupled optimization problem, and derive an optimization algorithm on Hilbert spaces for nonlinear PDEs with an additive spatio-temporal description of white noise. We study first and second order systems and in doing so, extend several results to the case of second order PDEs. The described approach is based on variational optimization, and performs joint RL-type optimization of the feedback control law and the actuator design over episodes. We demonstrate the efficacy of the proposed approach with several simulated experiments on a variety of SPDEs.Comment: 19 pages, 5 figure

Weak convergence rates for an explicit full-discretization of stochastic Allen-Cahn equation with additive noise

We discretize the stochastic Allen-Cahn equation with additive noise by means of a spectral Galerkin method in space and a tamed version of the exponential Euler method in time. The resulting error bounds are analyzed for the spatio-temporal full discretization in both strong and weak senses. Different from existing works, we develop a new and direct approach for the weak error analysis, which does not rely on the use of the associated Kolmogorov equation or It\^{o}'s formula and is therefore non-Markovian in nature. Such an approach thus has a potential to be applied to non-Markovian equations such as stochastic Volterra equations or other types of fractional SPDEs, which suffer from the lack of Kolmogorov equations. It turns out that the obtained weak convergence rates are, in both spatial and temporal direction, essentially twice as high as the strong convergence rates. Also, it is revealed how the weak convergence rates depend on the regularity of the noise. Numerical experiments are finally reported to confirm the theoretical conclusion.Comment: 28 page