Numerical Analysis for Inverse Problems and Applications of Wavelet Analysis

1.小波分析在数据分析和数值分析中的应用首先，我们研究了作为函数（信号，图像）小波展开系数的双序列的弱lp准范数，并认为准范数适用于数据的估计。其次，我们改进了一个重要的结果，即在小波分析中，利用Coifman尺度函数的配点法可以对函数进行精细的逼近。然后将所得结果应用于非线性偏微分方程解的快速精确数值计算。最后，我们将小波展开的半离散化技术应用于解，证明了由变分变元导出的偏微分方程可以化为无穷维的Hamilton动力系统.偏微分方程的数值分析首先，我们将Furihata等人的数值格式应用到非线性偏微分方程的数值模拟中，并验证了其有效性。特别地，我们使用该格式研究了Fujita问题，即非线性热方程的爆破问题，并研究了无色散户田方程数值解的激波现象。其次，我们发现了新的可能的研究方向，在函数逼近的样条函数，这是许多应用科学和数学分析的研究已经开始从理论和应用的角度来看。

1. Application of Wavelets to Data Analysis and Numerical Analysis.First, we investigated weak lp quasi-norms for double sequences obtained as coefficients in the wavelet expansion of functions (signals, images) and argued that the quasi-norms are suited for the estimate of data. Next, we improved an important result which tells that a fine approximation of functions is possible using the collocation method by the so-called Coifman scaling function in the wavelet analysis. Then we applied the result to fast and accurate numerical computation of solutions to nonlinear partial differential equations. Finally, we applied the semi-discretization technique by means of wavelet expansion to solutions and proved that partial differential equations derived via variational arguments can be reduced to Hamiltonian dynamical systems of infinite dimension.2. Numerical analysis of Partial Differential EquationsFirst, we applied numerical schemes due to Furihata and co. which conserves a discretized version of energy to numerical simulation of nonlinear partial differential equations and confirmed its effectiveness. In particular, we used the scheme to study the Fujita problem, i.e. the blow-up problem of nonlinear heat equations and to investigate the shock phenomenon for numerical solutions to the dispersionless Toda equation. Next, we have found new possible direction of research in the approximation of functions by spline functions which are much used in applied sciences and have begun the research of mathematical analysis from theoretical as well as applied point of view.