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，我们新开发了可以应用于广泛的数学和分析问题以及扩展或改进现有技术的自我验证数值方法。实际上，我们将这些方法应用于特定问题，例如数学流体机制和振动问题中的方程。研究人员和共同评估者获得的重要研究结果如下：1。 Nakao，N.Yamamoto，Watanabe为椭圆问题的溶液的数值验证方法建立了几种改进和扩展。也就是说，他们成功地计算了数值计算，并保证了二阶椭圆操作员的逆特征值问题的误差界限。他们还获得了一些结果，以封闭椭圆形变分无率的解决方案。此外，他们计算了一个最佳常数，并在泊松问题的有限元投影的先验误差估计中显示出确保准确性，这是非线性椭圆问题的数值验证的重要贡献。2。 Nagatou和Minamoto分别为Kolmogorov问题和扰动的Gelfand方程获得了有趣的计算机辅助证明。3。 Oishi为线性方程解决方案的基本验证计算建立了一些快速算法。4。 Nishida等。用保证的误差界限计算出热流问题的非平凡解决方案，这对于流体力学中的计算机辅助证明是一个重要的结果。5。 Yamamoto T.获得了两个点边界值问题的单数解的有限差异方案的一些收敛结果。