Synthetic approach for new developments of self-validating numerics

在这项研究中，我们发展了新的自验证数值方法，可以应用于广泛的数学和分析问题，并扩展或改进了现有的技术，并将这些方法实际应用于数学流体力学中的方程和振动问题等特殊问题。主要研究成果如下：1.Nakao，N.Yamamoto，Watanabe对椭圆问题解的数值验证方法进行了改进和推广。也就是说，他们成功地完成了二阶椭圆型算子特征值反问题的数值计算，并保证了误差界。他们还得到了一些关于椭圆型变分方程解的封闭结果。此外，他们还计算了Poisson问题有限元投影的先验误差估计中出现的具有保证精度的最优常数，这是对非线性椭圆问题的数值验证的重要贡献。Nagatou和Minamoto分别为Kolmogorov问题和摄动的Gelfand方程获得了有趣的计算机辅助证明。Oishi为线性方程组解的基本验证计算建立了一些快速算法。西田等人。对热对流问题的非平凡解进行了有保证的误差界计算，这是流体力学中计算机辅助证明的一个重要结果。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。

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.