A Stochastic Approximation for Fully Nonlinear Free Boundary Parabolic Problems
Erhan Bayraktar, Arash Fahim
本文提出了一种求解完全非线性抛物型自由边界问题的随机数值方法，并在合理的非线性条件下给出了收敛速率。
We present a stochastic numerical method for solving fully non-linear free boundary problems of parabolic type and provide a rate of convergence under reasonable conditions on the non-linearity.
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Higher order spatial approximations for degenerate parabolic stochastic partial differential equations
针对退化抛物型随机偏微分方程，采用隐式有限差分格式并应用外推技术，在适当正则性假设下实现空间逼近的任意高阶收敛加速。
Explicit stabilized integrators are an efficient alternative to implicit or semi-implicit methods to avoid the severe timestep restriction faced by standard explicit integrators applied to stiff diffusion problems. In this paper, we provide a fully discrete strong convergence analysis of a family of explicit stabilized methods coupled with finite element methods for a class of parabolic semilinear deterministic and stochastic partial differential equations. Numerical experiments including the semilinear stochastic heat equation with space-time white noise confirm the theoretical findings.
This paper investigates the influences of standard numerical discretizations on hitting probabilities for linear stochastic parabolic system driven by space-time white noises. We establish lower and upper bounds for hitting probabilities of the associated numerical solutions of both temporal and spatial semi-discretizations in terms of Bessel-Riesz capacity and Hausdorff measure, respectively. Moreover, the critical dimensions of both temporal and spatial semi-discretizations turn out to be half of those of the exact solution. This reveals that for a large class of Borel sets $A$, the probabil…
This paper studies the convergence of a spatial semi-discretization for a backward semilinear stochastic parabolic equation. The filtration is general, and the spatial semi-discretization uses the standard continuous piecewise linear element method. Firstly, higher regularity of the solution to the continuous equation is derived. Secondly, the first-order spatial accuracy is derived for the spatial semi-discretization. Thirdly, an application of the theoretical result to a stochastic linear quadratic control problem is presented.
We propose a two-point flux approximation finite-volume scheme for a stochastic non-linear parabolic equation with a multiplicative noise. The time discretization is implicit except for the stochastic noise term in order to be compatible with stochastic integration in the sense of Itô. We show existence and uniqueness of solutions to the scheme and the appropriate measurability for stochastic integration follows from the uniqueness of approximate solutions.
A discretization of an optimal control problem of a stochastic parabolic equation driven by multiplicative noise is analyzed. The state equation is discretized by the continuous piecewise linear element method in space and by the backward Euler scheme in time. The convergence rate $ O(τ^{1/2} + h^2) $ is rigorously derived.
We study a general class of singular degenerate parabolic stochastic partial differential equations (SPDEs) which include, in particular, the stochastic porous medium equations and the stochastic fast diffusion equation. We propose a fully discrete numerical approximation of the considered SPDEs based on the very weak formulation. By exploiting the monotonicity properties of the proposed formulation we prove the convergence of the numerical approximation towards the unique solution. Furthermore, we construct an implementable finite element scheme for the spatial discretization of the very weak…
In this paper, we prove a Carleman estimate for fully-discrete approximations of parabolic operators in which the discrete parameters $h$ and $\triangle t$ are connected to the large Carleman parameter. We use this estimate to obtain relaxed observability inequalities which yield, by duality, controllability results for fully-discrete linear and semilinear parabolic equations.
The aim of this work is to show an abstract framework to analyze the numerical approximation by using a finite element method in space and a Backward-Euler scheme in time of a family of degenerate parabolic problems. We deduce sufficient conditions to ensure that the fully-discrete problem has a unique solution and to prove quasi-optimal error estimates for the approximation. Finally, we show a degenerate parabolic problem which arises from electromagnetic applications and deduce its well-posedness and convergence by using the developed abstract theory, including numerical tests to illustrate …
This paper analyzes the discretization of a Neumann boundary control problem with a stochastic parabolic equation, where an additive noise occurs in the Neumann boundary condition. The convergence is established for general filtrations, and the convergence rate $ O(τ^{1/4-ε} + h^{1/2-ε}) $ is derived for the natural filtration of the Q-Wiener process.
Convergence rates for a finite volume scheme of a stochastic non-linear parabolic equation
本文研究了随机非线性抛物方程有限体积方案的收敛速率，通过半隐式欧拉方案和两个点通量近似方案，给出了空间时间离散误差的估计。
A stochastic collocation approach for parabolic PDEs with random domain deformations
本文研究了如何通过随机置信法近似依赖于线性抛物型偏微分方程解的量的统计矩，其中几何形状被随机参数化，并通过等价的确定性域和随机系数进行重新映射，推导了随机矩的收敛率并与数值实验进行比较。
Convergence of meshfree collocation methods for fully nonlinear parabolic equations
该研究在粘性解框架下证明了全非线性抛物型偏微分方程终值问题的无网格配点法的收敛性，条件为终端条件与非线性项的基函数逼近在各时间步成立，并通过径向基函数数值实验验证了该结论。
Stability and regularization for ill-posed Cauchy problem of a stochastic parabolic differential equation
针对随机抛物型方程的不适定 Cauchy 问题，论文建立 Carleman 估计，证明条件稳定性并给出 Tikhonov 正则化的收敛率，且结合基于核的学习理论完成数值实现。
The aim of this work is the numerical homogenization of a parabolic problem with several time and spatial scales using the heterogeneous multiscale method. We replace the actual cell problem with an alternate one, using Dirichlet boundary and initial values instead of periodic boundary and time conditions. Further, we give a detailed a priori error analysis of the fully discretized, i.e., in space and time for both the macroscopic and the cell problem, method. Numerical experiments illustrate the theoretical convergence rates.
A regularity result for quasilinear stochastic partial differential equations of parabolic type
本文研究了由乘性噪声驱动的准线性抛物型随机偏微分方程弱解的正则性，确定了系数和初始数据的条件，使得弱解在时间上Holder连续且空间正则性仅受给定数据限制。
Fully-Discrete Finite Element Approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise: II. 2D and 3D Case
针对二维和三维空间中由加性时空白噪声驱动的四阶线性随机抛物方程，通过离散化时空白噪声引入建模误差并得到正则化问题，采用基于C1分片多项式的标准Galerkin有限元方法进行空间离散和向后欧拉方法进行时间步进，推导了建模误差和正则化解逼近误差的强先验估计。
This article extends the semidiscrete maximal $L^p$-regularity results in [27] to multistep fully discrete finite element methods for parabolic equations with more general diffusion coefficients in $W^{1,d+β}$, where $d$ is the dimension of space and $β>0$. The maximal angles of $R$-boundedness are characterized for the analytic semigroup $e^{zA_h}$ and the resolvent operator $z(z-A_h)^{-1}$, respectively, associated to an elliptic finite element operator $A_h$. Maximal $L^p$-regularity, optimal $\ell^p(L^q)$ error estimate, and $\ell^p(W^{1,q})$ estimate are established for fully discrete fin…
This paper studies the inverse problem of determination the history for a stochastic diffusion process, by means of the value at the final time $T$. By establishing a new Carleman estimate, the conditional stability of the problem is proven. Based on the idea of Tikhonov method, a regularized solution is proposed. The analysis of the existence and uniqueness of the regularized solution, and proof for error estimate under an a-proior assumption are present. Numerical verification of the regularization, including numerical algorithm and examples are also illustrated.
Strong convergence of finite element approximations for a fourth-order stochastic pseudo-parabolic equation with additive noise
本文研究了四阶随机伪抛物方程在加性噪声下的有限元近似强收敛性，采用有限元和半隐式方法，验证了理论收敛率。
Localized orthogonal decomposition for a multiscale parabolic stochastic partial differential equation
本文提出基于 LOD 的多尺度方法求解带加性噪声和强振荡扩散的抛物型随机偏微分方程，证明最优阶强收敛并结合多层 Monte-Carlo 分析弱误差，数值实验验证其效率。
Immersed finite element (IFE) methods are a group of long-existing numerical methods for solving interface problems on unfitted meshes. A core argument of the methods is to avoid mesh regeneration procedure when solving moving interface problems. Despite the various applications in moving interface problems, a complete theoretical study on the convergence behavior is still missing. This research is devoted to close the gap between numerical experiments and theory. We present the first fully discrete analysis including the stability and optimal error estimates for a backward Euler IFE method fo…
Convergence of Rothe's method for fully nonlinear parabolic equations
本文在连续性假设下证明Rothe方法对全非线性抛物方程收敛，解在时间上Lipschitz、空间上Holder连续且满足粘性解，并推论粘性解的时间Lipschitz性。
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