Synthetic approach for new developments of self-validating numerics

自验证数值新发展的综合方法

基本信息

批准号：
13440035
负责人：
NAKAO Mitsuhiro
金额：
$ 10.88万
依托单位：
Kyushu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-13440035/
关键词：
Numerical analysis Validated computation Numerical verification Computer assisted proof

项目摘要

In this research, we newly developed the self-validating numerical methods which can be applied to wide mathematical and analytical problems as well as extended or improved the existing techniques.And we actually applied these methods to particular problems such as equations in the mathematical fluid mechanics and oscillation problems. The important research results obtained by investigators and co-investigators are as follows :1. Nakao, N.Yamamoto, Watanabe established several refinements and extensions for the numerical verification methods of solutions for elliptic problems. Namely, they succeeded the numerical computation with guaranteed error bounds for the inverse eigenvalue problems of second order elliptic operator. They also obtained some results for enclosing the solutions for elliptic variational inequlities. Moreover, they computed an optimal constant with guaranteed accuracy appearing in the a priori error estimates for the finite element projection of the Poisson problem, which is an important contribution for the numerical verification for nonlinear elliptic problems.2. Nagatou and Minamoto obtained interesting computer assisted proofs for the Kolmogorov problem and for the perturbed Gelfand equation, respectively.3. Oishi established some fast algorithms for the fundamental validated computations for the solutions of linear equations.4. Nishida et al. computed with guaranteed error bounds for the non-trivial solution of heat convection problems, which is an important result for a computer assisted proof in the fluid mechanics.5. T. Yamamoto obtained some convergence results of the finite difference scheme for the singular solutions of two point boundary value problems.

在这项研究中，我们新开发了自验证数值方法，该方法可应用于广泛的数学和分析问题，并扩展或改进了现有技术。并且我们实际上将这些方法应用于数学流体力学中的方程等特定问题和振荡问题。研究者和合作研究者取得的重要研究成果如下： 1． Nakao、N.Yamamoto、Watanabe 对椭圆问题解的数值验证方法进行了一些改进和扩展。也就是说，他们成功地对二阶椭圆算子的反特征值问题进行了有保证误差范围的数值计算。他们还获得了一些椭圆变分不等式的封闭解的结果。此外，他们还计算了泊松问题有限元投影的先验误差估计中出现的保证精度的最优常数，这为非线性椭圆问题的数值验证做出了重要贡献。 2． Nagatou 和 Minamoto 分别对 Kolmogorov 问题和扰动 Gelfand 方程获得了有趣的计算机辅助证明。 3．大石建立了一些快速算法，用于线性方程组解的基本验证计算。 4．西田等人。计算出热对流问题的非平凡解的保证误差范围，这是流体力学计算机辅助证明的重要结果。 5． T. Yamamoto 得到了两点边值问题奇异解的有限差分格式的一些收敛结果。