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Study of algorithm for the construction of solutions in inverse problems and its visualization

反问题解的构造算法及其可视化研究

基本信息

批准号：
18540110
负责人：
MATSUURA Tsutomu
金额：
$ 2.44万
依托单位：
Gunma University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2006
资助国家：
日本
起止时间：
2006 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18540110/
关键词：
inverse problem reproducing kernel theory algorithm visualization real inversion of the Laplace transform Tikhonov regularization multiple-precision arithmetic 実ラプラス逆変換

项目摘要

We succeeded to put reproduction kernel theory and regularization theory together Using our new method we can solve some inverse problems which have not been able to be solved historically. (1) The inverse problem of heat conduction is known as a difficult problem historically. We applied Tikhonov regularization for this inverse problem and gave some concrete algorithms for solving it. And we confirmed the effectiveness of those algorithms by numerical experiments. In addition, I contrived visual of the solution to clarify utility of this manner of solving. We think that it may be said that the inverse problem of the heat conduction has been settled by these our contributions. (2) We made the calculation environment of the multiple-precision arithmetic from the both sides of hardware and software. And we applied these implements for development of computation algorithm on real inversion of Laplace transform. Furthermore we verified that we can obtain numerical solutions of this typical ill-posed problem with satisfactory accuracy by our new method. (3) We verified theoretically and numerically that the usage of sinc function and Fredholm integral equation of the second kind are very effective for real inversion of Laplace transform. And we demonstrated numerically that double exponential method is very powerful for solving this ill-posed problem. (4) We pointed out theoretically that singular value decomposition is available and effective for solving real inverse problem of Laplace transform. And we showed that multiple-precision arithmetic is essential for this calculation. Furthermore we established this method and necessary numerical tables and applied for a patent of this study in conjunction with Kyoto University.

我们成功地将再现核理论和正则化理论结合在一起，利用我们的新方法可以解决一些历史上无法解决的逆问题。(1)热传导逆问题历来被认为是一个难题。将Tikhonov正则化应用于该反问题，并给出了求解该反问题的具体算法。并通过数值实验验证了算法的有效性。此外，我设计了可视化的解决方案，以阐明这种解决方式的效用。我们认为，可以说，我们的这些贡献已经解决了热传导的逆问题。(2)从硬件和软件两方面给出了多精度算法的计算环境。并应用这些工具开发了拉普拉斯变换实逆变换的计算算法。进一步验证了该方法能以满意的精度得到这一典型病态问题的数值解。(3)从理论上和数值上验证了sinc函数和第二类Fredholm积分方程的使用对于拉普拉斯变换的实逆变换是非常有效的。我们用数值证明了双指数法对于求解这个不适定问题是非常有效的。(4)从理论上指出奇异值分解是求解拉普拉斯变换实反问题的有效方法。我们证明了多精度算法在这个计算中是必不可少的。此外，我们建立了该方法和必要的数值表，并与京都大学联合申请了本研究的专利。