Numerical Analysis for Inverse Problems and Applications of Wavelet Analysis

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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.