不相交区间上回溯反抛物问题的数值解

IF 1.9 Q2 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Computation Pub Date : 2023-10-16 DOI:10.3390/computation11100204
Miglena N. Koleva, Lubin G. Vulkov
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引用次数: 0

摘要

演化方程的回溯反问题是用给定的最终解对未知的初始数据进行重构。研究了一类具有弱解的一维抛物型方程在加权Sobolev空间中的逆回溯问题。这两种解都与非标准界面条件相联系,从而在整个空间区域内解决了这一问题。这样的问题,和其他逆问题一样,是不适定的,对于它的数值解,必须使用特定的技术。直接问题首先用差分格式离散,差分格式在空间上提供二阶近似。对于所得到的常微分方程组,建立了正矫顽力。其次,我们提出了一种迭代共轭梯度法来求解逆问题的加权时间离散后得到的差分方程的病态系统。讨论了带有噪声输入数据的测试实例。
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Numerical Solution of the Retrospective Inverse Parabolic Problem on Disjoint Intervals
The retrospective inverse problem for evolution equations is formulated as the reconstruction of unknown initial data by a given solution at the final time. We consider the inverse retrospective problem for a one-dimensional parabolic equation in two disconnected intervals with weak solutions in weighted Sobolev spaces. The two solutions are connected with nonstandard interface conditions, and thus this problem is solved in the whole spatial region. Such a problem, as with other inverse problems, is ill-posed, and for its numerical solution, specific techniques have to be used. The direct problem is first discretized by a difference scheme which provides a second order of approximation in space. For the resulting ordinary differential equation system, the positive coerciveness is established. Next, we develop an iterative conjugate gradient method to solve the ill-posed systems of the difference equations, which are obtained after weighted time discretization, of the inverse problem. Test examples with noisy input data are discussed.
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来源期刊
Computation
Computation Mathematics-Applied Mathematics
CiteScore
3.50
自引率
4.50%
发文量
201
审稿时长
8 weeks
期刊介绍: Computation a journal of computational science and engineering. Topics: computational biology, including, but not limited to: bioinformatics mathematical modeling, simulation and prediction of nucleic acid (DNA/RNA) and protein sequences, structure and functions mathematical modeling of pathways and genetic interactions neuroscience computation including neural modeling, brain theory and neural networks computational chemistry, including, but not limited to: new theories and methodology including their applications in molecular dynamics computation of electronic structure density functional theory designing and characterization of materials with computation method computation in engineering, including, but not limited to: new theories, methodology and the application of computational fluid dynamics (CFD) optimisation techniques and/or application of optimisation to multidisciplinary systems system identification and reduced order modelling of engineering systems parallel algorithms and high performance computing in engineering.
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