Wavelet methods -- elliptic boundary value problems and control problems (Q2715802)

The author presents the general concept of a minimization problem consisting in three steps:NEWLINENEWLINENEWLINE(1) well-posedness, i.e. formulate the problem such that one can identify a space \(\mathcal H\) and a linear operator \(\mathcal L\) such that \(\mathcal L:\mathcal H\rightarrow\mathcal H'\) is an isomorphism;NEWLINENEWLINENEWLINE(2) shifting, i.e. use wavelets to transform the problem into an equivalent infinite linear system of equations;NEWLINENEWLINENEWLINE(3) stability of the discrete systems, i.e. Galerkin stability of the discretization in the finite dimensional case.NEWLINENEWLINENEWLINEBiorthogonal wavelets on an interval or on a manifold are obtained by the lifting scheme, free of Fourier techniques. General saddle point problems are presented in an abstract framework following the above general concepts. Elliptic boundary value problems are treated as saddle point problems to lay the foundation for the control problems where the boundary control plays an essential role. A wavelet representation of least squares formulation of general saddle point problems is presented with truncation, preconditioning, computational work and numerical experiments. Lastly, control problems are considered in the continuous case for two coupled saddle point problems, including discretization, preconditioning and discrete finite dimensional problems, and iterative methods for weakly coupled saddle point problems. Other alternative iterative methods for the coupled systems are presented, such as semi-iterative methods, block Kaczmarz iteration and fully iterative method.NEWLINENEWLINENEWLINEThe book attempts to be a complete and self-contained exposition, although at a level which may require some background in some or all of the fields involved as a knowledge of two key references, \textit{F. Brezzi} and \textit{M. Fortin} [Mixed and hybrid finite element methods, Springer (1991; Zlb 0788.73002)] and \textit{W. Dahmen} [Wavelets and multiscale methods for operator equations, Acta Numerica Vol. 6, 1997, 55-228 (1997; Zbl 0884.65106)]. A ten-page up-to-date bibliography is included.