scholarly journals A Weak Space-Time Formulation for the Linear Stochastic Heat Equation

2016 ◽  
Vol 3 (2) ◽  
pp. 787-806 ◽  
Author(s):  
Stig Larsson ◽  
Matteo Molteni
Keyword(s):  
Heat Equation ◽  
Space Time ◽  
Stochastic Heat Equation ◽  
Weak Space ◽  
Time Formulation

scholarly journals Numerical Solution of Parabolic Problems Based on a Weak Space-Time Formulation

2017 ◽  
Vol 17 (1) ◽  
pp. 65-84 ◽  
Author(s):  
Stig Larsson ◽  
Matteo Molteni
Keyword(s):  
Heat Equation ◽  
Numerical Experiments ◽  
Numerical Scheme ◽  
A Priori ◽  
Space Time ◽  
Parabolic Problems ◽  
Weak Space ◽  
Time Formulation ◽  
The Difference

AbstractWe investigate a weak space-time formulation of the heat equation and its use for the construction of a numerical scheme. The formulation is based on a known weak space-time formulation, with the difference that a pointwise component of the solution, which in other works is usually neglected, is now kept. We investigate the role of such a component by first using it to obtain a pointwise bound on the solution and then deploying it to construct a numerical scheme. The scheme obtained, besides being quasi-optimal in the ${L^{2}}$ sense, is also pointwise superconvergent in the temporal nodes. We prove a priori error estimates and we present numerical experiments to empirically support our findings.

scholarly journals Spatial Moduli of Non-Differentiability for Linearized Kuramoto–Sivashinsky SPDEs and Their Gradient

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1251
Author(s):  
Wensheng Wang
Keyword(s):  
Heat Equation ◽  
White Noise ◽  
Fourth Order ◽  
Gaussian Processes ◽  
Three Dimensional ◽  
Space Time ◽  
Symmetry Analysis ◽  
Stochastic Heat Equation ◽  
Moduli Of Continuity

We investigate spatial moduli of non-differentiability for the fourth-order linearized Kuramoto–Sivashinsky (L-KS) SPDEs and their gradient, driven by the space-time white noise in one-to-three dimensional spaces. We use the underlying explicit kernels and symmetry analysis, yielding spatial moduli of non-differentiability for L-KS SPDEs and their gradient. This work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. Moreover, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of L-KS SPDEs and their gradient.

scholarly journals Strong rates of convergence for a space-time discretization of the backward stochastic heat equation, and of a linear-quadratic control problem for the stochastic heat equation

2021 ◽  
Author(s):  
Andreas Prohl ◽  
Yanqing Wang
Keyword(s):  
Heat Equation ◽  
Control Problem ◽  
Time Discretization ◽  
Rates Of Convergence ◽  
Space Time ◽  
Stochastic Heat Equation ◽  
Linear Quadratic Control ◽  
Linear Quadratic ◽  
The Stochastic Heat Equation ◽  
Linear Quadratic Control Problem

We verify strong rates of convergence for a time-implicit, finite-element based space-time discretization of the backward stochastic heat equation, and the forward-backward stochastic heat equation from stochastic optimal control. The fully discrete version of the forward-backward stochastic heat equation is then used within a gradient descent algorithm to approximately solve the linear-quadratic control problem for the stochastic heat equation driven by additive noise. This work is thus giving a theoretical foundation for the computational findings in [ 14 ].

Large deviation principle for a space-time fractional stochastic heat equation with fractional noise

2018 ◽  
Vol 21 (2) ◽  
pp. 462-485 ◽  
Author(s):  
Litan Yan ◽  
Xiuwei Yin
Keyword(s):  
Heat Equation ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Fractional Integration ◽  
Fractional Power ◽  
Space Time ◽  
Stochastic Heat Equation ◽  
Fractional Noise ◽  
Deviation Principle ◽  
Fractional Integration Operator

Abstract In this paper, we consider the large deviation principle for a class of space-time fractional stochastic heat equation $$\begin{array}{} \displaystyle \partial^\beta_tu^\varepsilon(t,x)=-\nu(-\Delta)^{\frac\alpha 2}u^\varepsilon(t,x)+I_t^{1-\beta}f(u^\varepsilon(t,x))+ \sqrt{\varepsilon}I^{1-\beta}_t[\dot{W}^H(t,x)], \end{array}$$ where ẆH is a fractional white noise, ν > 0, β ∈ (0, 1), α ∈ (0, 2]. The operator $\begin{array}{} \displaystyle \partial^\beta_t \end{array}$ is the Caputo fractional integration operator, and $\begin{array}{} \displaystyle -(-\Delta)^{\frac\alpha 2} \end{array}$ is the fractional power of Laplacian. Our proof is based on the weak convergence approach.

A parallel space–time boundary element method for the heat equation

2019 ◽  
Vol 78 (9) ◽  
pp. 2852-2866 ◽  
Author(s):  
Stefan Dohr ◽  
Jan Zapletal ◽  
Günther Of ◽  
Michal Merta ◽  
Michal Kravčenko
Keyword(s):  
Boundary Element Method ◽  
Heat Equation ◽  
Boundary Element ◽  
Space Time ◽  
Time Boundary ◽  
Element Method
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