Optimal Pointwise Approximation of a Linear Stochastic Heat Equation with Additive Space-Time White Noise

2008 ◽  
pp. 577-589 ◽  
Author(s):  
Thomas Müller-Gronbach ◽  
Klaus Ritter ◽  
Tim Wagner
Keyword(s):  
Heat Equation ◽  
White Noise ◽  
Space Time ◽  
Stochastic Heat Equation ◽  
Pointwise Approximation

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 Spatial Moduli of Non-Differentiability for Time-Fractional SPIDEs and Their Gradient

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 380
Author(s):  
Wensheng Wang
Keyword(s):  
Differential Equations ◽  
Heat Equation ◽  
White Noise ◽  
Harmonic Analysis ◽  
Fourth Order ◽  
Gaussian Processes ◽  
Space Time ◽  
The Other ◽  
Moduli Of Continuity ◽  
Fractional Pdes

High order and fractional PDEs have become prominent in theory and in modeling many phenomena. In this paper, we study spatial moduli of non-differentiability for the fourth order time fractional stochastic partial integro-differential equations (SPIDEs) and their gradient, driven by space-time white noise. We use the underlying explicit kernels and spectral/harmonic analysis, yielding spatial moduli of non-differentiability for time fractional SPIDEs and their gradient. On one hand, 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. On the other hand, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of time fractional SPIDEs 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 ].

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