An improved quasi-reversibility method for a terminal-boundary value multi-species model with white Gaussian noise

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:384

Article

Nguyen Huy Tuan¹  Vo Anh Khoa^2,3  Phan Thi Khanh Van^1,4  Vo Van Au^5,6 

[1] Vietnam Natl Univ, Univ Sci, Fac Math & Comp Sci, 227 Nguyen Van Cu,Dist 5, Ho Chi Minh City, Vietnam

[2] Univ North Carolina Charlotte, Dept Math & Stat, Charlotte, NC 28223 USA

[3] Hasselt Univ, Fac Sci, Campus Diepenbeek,Agoralaan Bldg D, BE-3590 Diepenbeek, Belgium

[4] Ho Chi Minh City Univ Technol, Fac Appl Sci, Ho Chi Minh City, Vietnam

[5] Duy Tan Univ, Inst Fundamental & Appl Sci, Ho Chi Minh City 700000, Vietnam

[6] Duy Tan Univ, Fac Nat Sci, Da Nang 550000, Vietnam

关键词: Backward reaction-diffusion systems;    Quasi-reversibility method;    Gaussian white noise;    Weak solvability;    Global estimates;    Convergence rates;   

DOI  :  10.1016/j.cam.2020.113176

来源: Elsevier

【 摘 要 】

Upon the recent development of the quasi-reversibility method for terminal value parabolic problems in Nguyen et al. (2019), it is imperative to investigate the convergence analysis of this regularization method in the stochastic setting. In this paper, we positively unravel this open question by focusing on a coupled system of Dirichlet reaction-diffusion equations with additive white Gaussian noise on the terminal data. In this regard, the approximate problem is designed by adding the so-called perturbing operator to the original problem and by exploiting the Fourier reconstructed terminal data. By this way, Gevrey-type source conditions are included, while we successfully maintain the logarithmic stability estimate of the corresponding stabilized operator, which is necessary for the error analysis. As the main theme of this work, we prove the error bounds for the concentrations and for the concentration gradients, driven by a large amount of weighted energy-like controls involving the expectation operator. Compared to the classical error bounds in L-2 and H-1 that we obtained in the previous studies, our analysis here needs a higher smoothness of the true terminal data to ensure their reconstructions from the stochastic fashion. Two numerical examples are provided to corroborate the theoretical results. Published by Elsevier B.V.

【 授权许可】

Free