Numerical multi-scaling method to solve the linear stochastic partial differential equations

In this work, we develop a method for dynamic output feedback covariance control of the state covariance of linear dissipative stochastic partial differential equations (PDEs) using spatially distributed control actuation and sensing with noise. Such stochastic PDEs arise naturally in the modeling of surface height profile evolution in thin film growth and sputtering processes. We begin with the formulation of the stochastic PDE into a system of infinite stochastic ordinary differential equations (ODEs) by using modal decomposition. A finite-dimensional approximation is then obtained to capture the dominant mode contribution to the surface roughness profile (i.e., the covariance of the surface height profile). Subsequently, a state feedback controller and a Kalman-Bucy filter are designed on the basis of the finite-dimensional approximation. The dynamic output feedback covariance controller is subsequently obtained by combining the state feedback controller and the state estimator. The steady-state expected surface covariance under the dynamic output feedback controller is then estimated on the basis of the closed-loop finite-dimensional system. An analysis is performed to obtain a theoretical estimate of the expected surface covariance of the closed-loop infinite-dimensional system. Applications of the linear dynamic output feedback controller to the linearized stochastic Kuramoto-Sivashinsky equation are presented.

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.

Convergence analysis of constraint energy minimizing generalized multiscale finite element method for a linear stochastic parabolic partial differential equation driven by additive noises

A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations

This paper is concerned with the numerical approximation of some linear stochastic partial differential equations with additive noises. A special representation of the noise is considered, and it is compared with general representations of noises in the infinite dimensional setting. Convergence analysis and error estimates are presented for the numerical solution based on the standard finite difference and finite element methods. The effects of the noises on the accuracy of the approximations are illustrated. Results of the numerical experiments are provided.

In this paper, we consider several stochastic models arising from environmental problems. First, we study pollution in a domain where undesired chemicals are deposited at random times and locations according to Poisson streams. The chemical concentration can be modeled by a linear stochastic partial differential equation (SPDE) which is solved by applying a general result. Various properties, especially the limit behavior of the pollution process, are discussed. Secondly, we consider the pollution problem when a tolerance level is imposed. The chemical concentration can still be modeled by a SPDE which is no longer linear. Its properties are investigated in this paper. When the leakage rate is positive, it is shown that the pollution process has an equilibrium state given by the deterministic model treated in [2]. Finally, the linear filtering problem is considered based on the data of several observation stations.

The article considers second-order system of linear stochastic partial differential equations of hyperbolic type with Goursat boundary conditions. Earlier, in a number of papers, representations of the solution Goursat problem for linear stochastic equations of hyperbolic type in the classical way under the assumption of sufficient smoothness of the coefficients of the terms included in the right-hand side of the equation were obtained. Meanwhile, study of many stochastic applied optimal control problems described by linear or nonlinear second-order stochastic differential equations, in partial derivatives hyperbolic type, the assumptions of sufficient smoothness of these equations are not natural. Proceeding from this, in the considered Goursat problem, in contrast to the known works, the smoothness of the coefficients of the terms in the right-hand side of the equation is not assumed. They are considered only measurable and bounded matrix functions. These assumptions, being natural, allow us to further investigate a wide class of optimal control problems described by systems of second-order stochastic hyperbolic equations. In this work, a stochastic analogue of the Riemann matrix is introduced, an integral representation of the solution of considered boundary value problem in explicit form through the boundary conditions is obtained. An analogue of the Riemann matrix was introduced as a solution of a two-dimensional matrix integral equation of the Volterra type with one-dimensional terms, a number of properties of an analogue of the Riemann matrix were studied.

We study the semidiscrete Galerkin approximation of a stochastic parabolic partial differential equation forced by an additive space-time noise. The discretization in space is done by a piecewise linear finite element method. The space-time noise is approximated by using the generalized L2 projection operator. Optimal strong convergence error estimates in the L2 and \(\dot{H}^{-1}\) norms with respect to the spatial variable are obtained. The proof is based on appropriate nonsmooth data error estimates for the corresponding deterministic parabolic problem. The error estimates are applicable in the multi-dimensional case.

This article shows that increasing the observation variance at small scales can reduce the ensemble size required to avoid collapse in particle filtering of spatially extended dynamics and improve the resulting uncertainty quantification at large scales. Particle filter weights depend on how well ensemble members agree with observations, and collapse occurs when a few ensemble members receive most of the weight. Collapse causes catastrophic variance underestimation. Increasing small-scale variance in the observation error model reduces the incidence of collapse by de-emphasizing small-scale differences between the ensemble members and the observations. Doing so smooths the posterior mean, though it does not smooth the individual ensemble members. Two options for implementing the proposed observation error model are described. Taking a discretized elliptic differential operator as an observation error covariance matrix provides the desired property of a spectrum that grows in the approach to small scales. This choice also introduces structure exploitable by scalable computation techniques, including multigrid solvers and multiresolution approximations to the corresponding integral operator. Alternatively the observations can be smoothed and then assimilated under the assumption of independent errors, which is equivalent to assuming large errors at small scales. The method is demonstrated on a linear stochastic partial differential equation, where it significantly reduces the occurrence of particle filter collapse while maintaining accuracy. It also improves continuous ranked probability scores by as much as 25%, indicating that the weighted ensemble more accurately represents the true distribution. The method is compatible with other techniques for improving the performance of particle filters.

We study linear stochastic partial differential equations of parabolic type with special boundary conditions in time. The standard Cauchy condition at the initial time is replaced by a condition that mixes the values of the solution at different times, including the terminal time and continuously distributed times. Uniqueness, solvability and regularity results for the solutions are obtained.

We study linear stochastic partial differential equations ofparabolic type with non-local in time or mixed in time boundaryconditions. The standard Cauchy condition at the terminal time isreplaced by a condition that mixes the random values of thesolution at different times, including the terminal time, initialtime and continuously distributed times. For the case of backwardequations, this setting covers almost surely periodicity.Uniqueness, solvability and regularity results for the solutions areobtained. Some possible applications to portfolio selection arediscussed.

We study linear backward stochastic partial differential equations (BSPDEs) of parabolic type. We consider a new boundary value problem where a Cauchy condition is replaced by a prescribed average of the solution over time. We establish well-posedness, existence, uniqueness, and regularity, for the solutions of this new problem. In particular, this can be considered as a possibility to recover a solution of a BSPDE in a setting where its values at the terminal time are unknown, and where the average of the solution over time is preselected.

In this work, we focus on a thin film deposition process which takes place on a two-dimensional (2D) lattice and is governed by three microscopic processes including molecule adsorption, surface migration, and desorption. A 2D linear stochastic partial differential equation (PDE) model is initially constructed which describes the spatio-temporal evolution of the film surface. Then, the control problem is formulated as the one of regulating the thin film thickness and surface roughness by manipulating the substrate temperature and adsorption rate. Subsequently, a computationally efficient multivariable predictive control algorithm is developed which uses a finite-dimensional approximation of the stochastic PDE model to regulate the thin film thickness and surface roughness at desired levels at the end of the deposition. The predictive controller is then applied to the kinetic Monte Carlo simulation of the deposition process. Closed-loop system simulation results demonstrate that the model is adequately accurate and that the controller is effective in enforcing the desired control objectives and reducing film variance.

We develop criteria for hitting probabilities of anisotropic Gaussian random fields with associated canonical pseudo-metric given by a class of gauge functions. This yields lower and upper bounds in terms of general notions of capacity and Hausdorff measure, respectively, therefore extending the classical estimates with the Bessel-Riesz capacity and the $\gamma$-dimensional Hausdorff measure. We apply the criteria to a system of linear stochastic partial differential equations driven by space-time noises that are fractional in time and either white or colored in space.

This note reviews and discusses some recent results on linear stochastic partial differential equations with multiplicative Gaussian noise colored in time.