Correctness and regularization of stochastic problems

IF 0.9 4区数学 Q2 MATHEMATICS

Journal of Inverse and Ill-Posed Problems Pub Date : 2023-10-04 DOI:10.1515/jiip-2023-0011

Irina V. Melnikova, Vadim A. Bovkun

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引用次数: 0

Abstract

Abstract The paper is devoted to the regularization of ill-posed stochastic Cauchy problems in Hilbert spaces: (0.1) d ⁢ u ⁢ ( t ) = A ⁢ u ⁢ ( t ) ⁢ d ⁢ t + B ⁢ d ⁢ W ⁢ ( t ) , t > 0 , u ⁢ ( 0 ) = ξ . du(t)=Au(t)dt+BdW(t),\quad t>0,\qquad u(0)=\xi. The need for regularization is connected with the fact that in the general case the operator A is not supposed to generate a strongly continuous semigroup and with the divergence of the series defining the infinite-dimensional Wiener process { W ⁢ ( t ) : t ≥ 0 } {\{W(t):t\geq 0\}} . The construction of regularizing operators uses the technique of Dunford–Schwartz operators, regularized semigroups, generalized Fourier transform and infinite-dimensional Q -Wiener processes.

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随机问题的正确性和正则化

研究Hilbert空间中不适定随机柯西问题的正则化问题:(0.1)d²u²(t) = A²u²(t)²d²(t) + B²W²(t)， t ＆gt;0 u²(0)= ξ。du(t)=Au(t)dt+BdW(t)， \quadt＆gt;0\qquad, u(0)= \xi。正则化的需要与以下事实有关:在一般情况下，算子A不应生成强连续半群，并与定义无限维维纳过程{W¹(t):t≥0 }｛W(t):t{\geq 0｝的级数的散度有关}。正则算子的构造利用了Dunford-Schwartz算子、正则半群、广义傅里叶变换和无穷维Q -Wiener过程等技术。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Inverse and Ill-Posed Problems MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

2.60

自引率

9.10%

发文量

审稿时长

>12 weeks

期刊介绍： This journal aims to present original articles on the theory, numerics and applications of inverse and ill-posed problems. These inverse and ill-posed problems arise in mathematical physics and mathematical analysis, geophysics, acoustics, electrodynamics, tomography, medicine, ecology, financial mathematics etc. Articles on the construction and justification of new numerical algorithms of inverse problem solutions are also published. Issues of the Journal of Inverse and Ill-Posed Problems contain high quality papers which have an innovative approach and topical interest. The following topics are covered: Inverse problems existence and uniqueness theorems stability estimates optimization and identification problems numerical methods Ill-posed problems regularization theory operator equations integral geometry Applications inverse problems in geophysics, electrodynamics and acoustics inverse problems in ecology inverse and ill-posed problems in medicine mathematical problems of tomography