Second-order Stochastic Partial Differential Equation Research Articles

Solving nonlinear filtering problems with correlated noise based on Hermite–Galerkin spectral method

Nonlinear filtering problem has important applications in various fields. One of the core issues in nonlinear filtering is to numerically solve the Duncan–Mortensen–Zakai (DMZ) equation, which is an evolution equation satisfied by the unnormalized conditional density of state process under noisy observations, in a real-time and memoryless manner. When the noise in observations is correlated to the state process, the DMZ equation we need to deal with is a second-order stochastic partial differential equation. In this paper, we will propose an algorithm to solve the DMZ equation in this case, based on Hermite–Galerkin spectral method. According to this method, the DMZ equation is converted into a system of linear stochastic differential equations generated by the observation process. The effects of different discretization schemes on this stochastic differential system will also be discussed. Moreover, rigorous convergence analysis of the algorithm is given under mild conditions. Numerical results show that the method proposed in this paper can provide an instantaneous and accurate estimation to the state process of the system.

A sharp Lp-regularity result for second-order stochastic partial differential equations with unbounded and fully degenerate leading coefficients

We present existence, uniqueness, and sharp regularity results of solution to the stochastic partial differential equation (SPDE)(0.1)du=(aij(ω,t)uxixj+f)dt+(σik(ω,t)uxi+gk)dwtk,u(0,x)=u0, where {wtk:k=1,2,⋯} is a sequence of independent Brownian motions. The coefficients are merely measurable in (ω,t) and can be unbounded and fully degenerate, that is, coefficients aij, σik merely satisfy(0.2)(αij(ω,t))d×d:=(aij(ω,t)−12∑k=1∞σik(ω,t)σjk(ω,t))≥0. In this article, we prove that there exists a unique solution u to (0.1), and(0.3)‖uxx‖Hpγ(τ,δ)≤N(d,p)(‖u0‖Bpγ+2(1−1/p)+‖f‖Hpγ(τ,δ1−p)+‖gx‖Hpγ(τ,|σ|pδ1−p,l2)p+‖gx‖Hpγ(τ,δ1−p/2,l2)), where p≥2, γ∈R, τ is an arbitrary stopping time, δ(ω,t) is the smallest eigenvalue of αij(ω,t), Hpγ(τ,δ) is a weighted stochastic Sobolev space, and Bpγ+2(1−1/p) is a stochastic Besov space.

Parameter estimation for linear parabolic SPDEs in two space dimensions based on high frequency data

AbstractWe consider parameter estimation for a linear parabolic second‐order stochastic partial differential equation (SPDE) in two space dimensions driven by two types of ‐Wiener processes based on high frequency data in time and space. We first estimate the parameters which appear in the eigenfunctions of the differential operator of the SPDE using the minimum contrast estimator based on the thinned data with respect to space, and then construct an approximate coordinate process of the SPDE. Furthermore, we propose estimators of the coefficient parameters of the SPDE utilizing the approximate coordinate process based on the thinned data with respect to time. We also give some simulation results.

A stochastic regularized second-order iterative scheme for optimal control and inverse problems in stochastic partial differential equations.

Numerous applied models used in the study of optimal control problems, inverse problems, shape optimization, machine learning, fractional programming, neural networks, image registration and so on lead to stochastic optimization problems in Hilbert spaces. Under a suitable convexity assumption on the objective function, a necessary and sufficient optimality condition for stochastic optimization problems is a stochastic variational inequality. This article presents a new stochastic regularized second-order iterative scheme for solving a variational inequality in a stochastic environment where the primary operator is accessed by employing sampling techniques. The proposed iterative scheme, which fits within the general framework of the stochastic approximation approach, has its almost-sure convergence analysis given in a Hilbert space. We test the feasibility and the efficacy of the proposed stochastic approximation approach for a stochastic optimal control problem and a stochastic inverse problem, both associated with a second-order stochastic partial differential equation. This article is part of the theme issue 'Non-smooth variational problems and applications'.

An $$L_p$$-maximal regularity estimate of moments of solutions to second-order stochastic partial differential equations

We obtain uniqueness and existence of a solution u to the following second-order stochastic partial differential equation: $$\begin{aligned} du= \left( {\bar{a}}^{ij}(\omega ,t)u_{x^ix^j}+ f \right) dt + g^k dw^k_t, \quad t \in (0,T); \quad u(0,\cdot )=0, \end{aligned}$$ (1) where \(T \in (0,\infty )\), \(w^k\) \((k=1,2,\ldots )\) are independent Wiener processes, \(({\bar{a}}^{ij}(\omega ,t))\) is a (predictable) nonnegative symmetric matrix valued stochastic process such that $$\begin{aligned} \kappa |\xi |^2 \le {\bar{a}}^{ij}(\omega ,t) \xi ^i \xi ^j \le K |\xi |^2 \quad \forall \;(\omega ,t,\xi ) \in \Omega \times (0,T) \times {\mathbf {R}}^d \end{aligned}$$for some \(\kappa , K \in (0,\infty )\), $$\begin{aligned} f \in L_p\left( (0,T) \times {\mathbf {R}}^d, dt \times dx ; L_r(\Omega , {\mathscr {F}} ,dP) \right) , \end{aligned}$$and $$\begin{aligned} g, g_x \in L_p\left( (0,T) \times {\mathbf {R}}^d, dt \times dx ; L_r(\Omega , {\mathscr {F}} ,dP; l_2) \right) \end{aligned}$$with \(2 \le r \le p < \infty \) and appropriate measurable conditions. Moreover, for the solution u, we obtain the following maximal regularity moment estimate $$\begin{aligned}&\int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |u(t,x)|^r\right] \right) ^{p/r} dx dt + \int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |u_{xx}(t,x)|^r\right] \right) ^{p/r} dx dt \nonumber \\&\le N \bigg (\int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |f(t,x)|^r\right] \right) ^{p/r} dx dt + \int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |g(t,x)|_{l_2}^r\right] \right) ^{p/r} dx dt \nonumber \\&\quad + \int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |g_x(t,x)|_{l_2}^r\right] \right) ^{p/r} dx dt \bigg ), \end{aligned}$$ (2) where N is a positive constant depending only on d, p, r, \(\kappa \), K, and T. As an application, for the solution u to (1), the rth moment \(m^r(t,x):=\mathbb {E}|u(t,x)|^r\) is in the parabolic Sobolev space \(W_{p/r}^{1,2}\left( (0,T) \times \mathbf {R}^d\right) \).

Adaptive estimator for a parabolic linear SPDE with a small noise

We deal with parametric estimation for a parabolic linear second-order stochastic partial differential equation (SPDE) with a small dispersion parameter based on high-frequency data which are observed in time and space. Using the thinned data with respect to space obtained from the high-frequency data, the minimum contrast estimators of two coefficient parameters of the SPDE are proposed. With these estimators and the thinned data with respect to time obtained from the high-frequency data, we construct an approximation of the coordinate process of the SPDE. Using the approximate coordinate process, we obtain the adaptive estimator of a coefficient parameter of the SPDE. Moreover, we give simulation results of the proposed estimators of the SPDE.

Diffusion-approximation in stochastically forced kinetic equations

We derive the hydrodynamic limit of a kinetic equation where the interactions in velocity are modelled by a linear operator (Fokker-Planck or Linear Boltzmann) and the force in the Vlasov term is a stochastic process with high amplitude and short-range correlation. In the scales and the regime we consider, the hydrodynamic equation is a scalar second-order stochastic partial differential equation. Compared to the deterministic case, we also observe a phenomenon of enhanced diffusion.

A non-conservation stochastic partial differential equation driven by anisotropic fractional Lévy random field

We study a non-conservation second-order stochastic partial differential equation (SPDE) driven by multi-parameter anisotropic fractional Lévy noise (AFLN) and under different initial and/or boundary conditions. It includes the time-dependent linear heat equation and quasi-linear heat equation under the fractional noise as special cases. Unique existence and expressions of solution to the equation are proved and constructed. An AFLN is defined as the derivative of an anisotropic fractional Lévy random field (AFLRF) in certain sense. Comparing with existing noise systems, our non-Gaussian fractional noises are essentially observed from random disturbances on system accelerations rather than from those on system moving velocities. In the process of proving our claims, there are three folds. First, we consider the AFLRF as the generalized functional of sample paths of a pure jump Lévy process. Second, we build Skorohod integration with respect to the AFLN by white noise approach. Third, by combining this noise analysis method with the conventional PDE solution techniques, we provide solid proofs for our claims.

A Feedback Nash Equilibrium for Affine-Quadratic Zero-Sum Stochastic Differential Games With Random Coefficients

We study the affine-quadratic zero-sum stochastic differential game with random coefficients, where the coefficients of the stochastic differential equation (SDE) are random processes and both additive and state multiplicative noise are included in the diffusion term of the corresponding SDE. By applying Ito-Kunita’s formula to the quadratic random field, we develop a direct approach, also known as the completion of squares method, to characterize the explicit (feedback) Nash equilibrium and obtain the optimal game value. The characterized Nash equilibrium depends linearly on the state and the additional linear backward SDE. We also verify the optimality of the Nash equilibrium by characterizing the smooth solution of the stochastic Hamilton-Jacobi-Isaacs equation that is the second-order stochastic partial differential equation obtained from dynamic programming.

Constructing continuous stationary covariances as limits of the second-order stochastic difference equations

In Bayesian statistical inverse problems the a priori probability distributions are often given as stochastic difference equations. We derive a certain class of stochastic partial difference equations by starting from second-order stochastic partial differential equations in one and two dimensions. We discuss discretisation schemes on uniform lattices of these stationary continuous-time stochastic processes and convergence of the discrete-time processes to the continuous-time processes. A special emphasis is given to an analytical calculation of the covariance kernels of the processes. We find a representation for the covariance kernels in a simple parametric form with controllable parameters: correlation length and variance. In the discrete-time processes the discretisation step is also given as a parameter. Therefore, the discrete-time covariances can be considered as discretisation-invariant. In the two-dimensional cases we find rotation-invariant and anisotropic representations of the difference equations and the corresponding continuous-time covariance kernels.

Dynamics and control of aggregate thin film surface morphology for improved light trapping: Implementation on a large-lattice kinetic Monte Carlo model

This work demonstrates the use of feedback control, coupled with a suitable actuator design, in manufacturing thin films whose surface morphology is characterized by a desired visible light reflectance/transmittance level. The problem is particularly important in the context of thin film manufacturing for thin film solar cells where it is desirable to produce thin films with precisely tailored light trapping characteristics. Initially, a thin film deposition process involving atom adsorption and surface migration is considered and is modeled using a large-lattice (lattice size=40,000) kinetic Monte Carlo simulation. Subsequently, thin film surface morphology characteristics like roughness and slope are computed with respect to different characteristic length scales ranging from atomic to the ones corresponding to visible light wavelength and it is found that a patterned actuator design is needed to induce thin film surface roughness and slope at visible light wavelength spatial scales, which lead to desired thin film reflectance and transmittance levels. Then, an Edwards–Wilkinson-type equation (a second-order stochastic partial differential equation) is used to model the surface evolution at the visible light wavelength spatial scale and form the basis for the design of a feedback controller whose objective is to manipulate the deposition rate across the spatial domain of the process. The model parameters of the Edwards–Wilkinson equation are estimated from kinetic Monte Carlo simulations and their dependence on the deposition rate is used in the formulation of the predictive controller to predict the influence of the control action on the surface roughness and slope throughout the thin film growth process. Analytical solutions of the expected surface roughness and surface slope at the visible light wavelength spatial scale are obtained by solving the Edwards–Wilkinson equation and are used in the control action calculation. The cost function of the controller involves penalties on both surface roughness and slope from set-point values as well as constraints on the magnitude and rate of change of the control action. The controller is applied to the large-lattice kinetic Monte Carlo simulation. Simulation results demonstrate that the proposed controller and patterned actuator design successfully regulate aggregate surface roughness and slope to set-point values at the end of the deposition that yield desired levels of thin film reflectance and transmittance.

Controlling aggregate thin film surface morphology for improved light trapping using a patterned deposition rate profile

This work focuses on modeling and control of aggregate thin film surface morphology for improved light trapping using a patterned deposition rate profile. The dynamics of the evolution of the thin film surface height profile are modeled by an Edwards–Wilkinson-type equation (a second-order stochastic partial differential equation) in two spatial dimensions. The thin film surface morphology is characterized in terms of aggregate surface roughness and surface slope. These variables are computed with respect to appropriate visible light-relevant characteristic length scales and defined as the root-mean-squares of height deviation and slope of aggregate surface height profiles, respectively. Analytical solutions of the expected aggregate surface roughness and surface slope are obtained by solving the Edwards–Wilkinson equation and are used in the controller design. The model parameters of the Edwards–Wilkinson equation are estimated from kinetic Monte-Carlo simulations using a novel parameter estimation procedure. This parameter dependence on the deposition rate is used in the formulation of the predictive controller to predict the influence of the control action on the surface roughness and slope at the end of the growth process. The cost function of the controller involves penalties on both aggregate surface roughness and mean slope from set-point values as well as constraints on the magnitude and rate of change of the control action. The controller is applied to the two-dimensional Edwards–Wilkinson equation. Simulation results show that the proposed controller successfully regulates aggregate surface roughness and slope to set-point values at the end of the deposition that yield desired levels of thin film reflectance and transmittance.

Multivariable Model Predictive Control of Thin Film Surface Roughness and Slope for Light Trapping Optimization

This work focuses on the development of a multivariable model predictive controller that simultaneously regulates thin film surface roughness and mean slope to optimize light reflectance and transmittance during thin film manufacturing by manipulating substrate temperature and deposition rate. Surface roughness and surface slope are defined as the root-mean-squares of the surface height profile and the surface slope profile, respectively. The dynamics of the evolution of the thin film surface height profile are assumed to be described by an Edwards-Wilkinson-type equation (a second-order stochastic partial differential equation) in two spatial dimensions. Analytical solutions of the expected surface roughness and surface slope are obtained on the basis of the Edwards-Wilkinson equation and are used in the controller design. The model parameters of the Edwards-Wilkinson equation depend on the substrate temperature and deposition rate. This dependence is used in the formulation of the predictive controller to predict the influence of the control action on the surface roughness and slope at the end of the growth process. The model predictive controller involves constraints on the magnitude and rate of change of the control action and optimizes a cost that involves penalty on both surface roughness and mean slope from the set-point values. The controller is applied to the two-dimensional Edwards-Wilkinson equation and is shown to successfully regulate surface roughness and mean slope to set-point values at the end of the deposition that yield desired film reflectance and transmittance.

Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations

In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E: $$({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], U(0) = u_0.\end{array}\right.$$ Here, \({(A(t))_{t\in [0,T]}}\) are unbounded operators with domains \({(D(A(t)))_{t\in [0,T]}}\) which may be time dependent. We assume that \({(A(t))_{t\in [0,T]}}\) satisfies the conditions of Acquistapace and Terreni. The functions F and B are nonlinear functions defined on certain interpolation spaces and \({u_0\in E}\) is the initial value. W H is a cylindrical Brownian motion on a separable Hilbert space H. We assume that the Banach space E is a UMD space with type 2. Under locally Lipschitz conditions we show that there exists a unique local mild solution of (SE). If the coefficients also satisfy a linear growth condition, then it is shown that the solution exists globally. Under assumptions on the interpolation spaces we extend the factorization method of Da Prato, Kwapien, and Zabczyk, to obtain space-time regularity results for the solution U of (SE). For Hilbert spaces E we obtain a maximal regularity result. The results improve several previous results from the literature. The theory is applied to a second-order stochastic partial differential equation which has been studied by Sanz-Sole and Vuillermot. This leads to several improvements of their result.

Terrain profile modelling using stochastic partial differential equations

For realistic and efficient predictions of vehicle performance in off-road conditions, it is critical to incorporate in the simulation realistic representations of the variability of the terrain profile. Many studies using dedicated modelling techniques and computational methods have been recently conducted to identify effective ways to reduce experimental data in order to preserve only essential information needed to recreate the main terrain characteristics. The method we propose in this paper is to model non-stationary terrain profile in two dimensions using the finite difference approach for solving linear second-order stochastic partial differential equations. Certain assumptions are made for the values of the model coefficients to obtain the terrain profile while preserving the stochastic properties of the original terrain topology. Additional techniques are currently being explored to further refine the estimation of the model coefficients to realistically model the terrain profile.

Feedback Control of Surface Roughness in a Deposition Process Using a Stochastic PDE

This paper focuses on feedback control of surface roughness in a deposition process, based on a stochastic partial differential equation (PDE) which describes the fluctuation of surface height in the spatial domain. Specifically, we focus on control of surface roughness in a deposition process on a 1-dimensional lattice, whose fluctuation of surface height can be described by the Edwards-Wilkinson equation, a secondorder stochastic PDE. We initially refonnulate the stochastic PDE into a system of infinite stochastic ordinary differential equations by using modal decomposition. A finite-dimensional approximation of the Edwards-Wilkinson equation is then derived that captures the dominant mode contribution to the surface roughness. A state feedback controller is designed based on the finite-dimensional approximation to control the surface roughness. Analysis of the closed-loop system shows that the controller can drive the surface roughness governed by the infinite-dimensional system to desired levels. The effectiveness of the proposed method is demonstrated by numerical simulations.

Holder Estimates for Solutions of Parabolic SPDEs

This paper considers second-order stochastic partial differential equations with additive noise given in a bounded domain of~${\bf R}^n$. We suppose that the coefficients of the noise are $L^p$-functions with sufficiently large~p. We prove that the solutions are H\older-continuous functions almost surely (a.s.) and that the respective H\older norms have finite momenta of any order.

On stochastic partial differential equations with Unbounded coefficients

Second-order stochastic partial differential equations of parabolic type are considered. Generalizations of known theorems on existence, uniqueness and on approximations are presented. Thus, in particular the case of unbounded coefficients in investigated. Some examples illustrating the usefulness of the results are also given.

On a class of stochastic partial differential equations

Recently, the Cauchy problem for the second-order stochastic partial differential equations of parabolic type has been investigated in [6, 7j. The present paper considers the same problem for a wider class of equations of hypoelliptic type. We obtain a theorem on the existence and uniqueness of solutions (Theorem 3.1) by making use of the methods developed in [S, 63, where the fundamental solutions constructed in [5] play an essential role. Now we state some general definitions and formulate the problem of the paper. Let (0, F,p) be a probability space, endowed with a right-continuous increasing family F,(d), and let w(t) = w(t, o), t > 0, cM2, be a standard one-dimensional F,-Brownian motion. Let R” be the n-dimensional Euclidean space, and set Q, = (0, t) x R”, 0 < t Q T. Consider the differential problem

ON STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

In this paper we consider the Cauchy problem for second-order stochastic partial differential equations of parabolic type. We study linear and nonlinear equations for filtering Markov diffusion processes. Theorems on the existence, uniqueness and smoothness of solutions are proved.Bibliography: 21 items.