含乘性白噪声随机扩散方程的一个反势问题

IF 1.2 4区 数学 Q2 MATHEMATICS, APPLIED
Xiaoli Feng, Peijun Li, Xu Wang
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引用次数: 0

摘要

这项工作涉及乘性含时白噪声驱动的随机扩散方程的正势和反势问题。直接问题是检验随机扩散方程对给定势的适定性,而反问题是根据在空间域内固定观测点的解的期望来确定势。如果初始值是非负的,则直接问题可以得到唯一的正温和解。此外,还推导了一个重构势平方的显式公式,从而得到了非负势函数反问题的唯一性。利用两种正则化方法来克服重建公式中数值微分的不稳定性。数值结果表明,该方法对光滑和非光滑势函数的重构都是有效的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
An inverse potential problem for the stochastic diffusion equation with a multiplicative white noise
This work concerns the direct and inverse potential problems for the stochastic diffusion equation driven by a multiplicative time-dependent white noise. The direct problem is to examine the well-posedness of the stochastic diffusion equation for a given potential, while the inverse problem is to determine the potential from the expectation of the solution at a fixed observation point inside the spatial domain. The direct problem is shown to admit a unique and positive mild solution if the initial value is nonnegative. Moreover, an explicit formula is deduced to reconstruct the square of the potential, which leads to the uniqueness of the inverse problem for nonnegative potential functions. Two regularization methods are utilized to overcome the instability of the numerical differentiation in the reconstruction formula. Numerical results show that the methods are effective to reconstruct both smooth and nonsmooth potential functions.
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来源期刊
Inverse Problems and Imaging
Inverse Problems and Imaging 数学-物理:数学物理
CiteScore
2.50
自引率
0.00%
发文量
55
审稿时长
>12 weeks
期刊介绍: Inverse Problems and Imaging publishes research articles of the highest quality that employ innovative mathematical and modeling techniques to study inverse and imaging problems arising in engineering and other sciences. Every published paper has a strong mathematical orientation employing methods from such areas as control theory, discrete mathematics, differential geometry, harmonic analysis, functional analysis, integral geometry, mathematical physics, numerical analysis, optimization, partial differential equations, and stochastic and statistical methods. The field of applications includes medical and other imaging, nondestructive testing, geophysical prospection and remote sensing as well as image analysis and image processing. This journal is committed to recording important new results in its field and will maintain the highest standards of innovation and quality. To be published in this journal, a paper must be correct, novel, nontrivial and of interest to a substantial number of researchers and readers.
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