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

我们成功地把再生核理论和正则化理论结合起来，用我们的新方法解决了一些历史上无法解决的反问题。(1)热传导反问题历来被认为是一个难题。将Tikhonov正则化方法应用于该反问题，给出了具体的求解算法，并通过数值实验验证了算法的有效性。此外，我还设计了解决方案的可视化，以阐明这种解决方式的实用性。我们认为，可以说，热传导的反问题已经解决了我们的这些贡献。(2)从硬件和软件两方面构建了多精度算法的计算环境。并将这些实现应用于拉普拉斯变换的真实的逆运算算法的开发。此外，我们验证了我们的新方法，我们可以获得令人满意的精度，这一典型的不适定问题的数值解。(3)从理论上和数值上证明了sinc函数和第二类Fredholm积分方程的使用对于拉普拉斯变换的真实的反演是非常有效的。数值结果表明，双指数方法是解决这类不适定问题的有效方法。(4)从理论上指出奇异值分解是求解拉普拉斯变换的真实的反问题的有效方法。我们证明了多精度算法对于这种计算是必不可少的。此外，我们建立了这种方法和必要的数值表，并与京都大学联合申请了这项研究的专利。

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.