Multiscale inversion of elliptic operators

A fast adaptive algorithm for the solution of elliptic partial differential equations is presented. It is applied here to the Poisson equation with periodic boundary conditions. The extension to more complicated equations and boundary conditions is outlined. The purpose is to develop algorithms requiring a number of operations proportional to the number of significant coefficients in the representation of the r.h.s. of the equation. This number is related to the specified accuracy, but independent of the resolution. The wavelet decomposition and the conjugate gradient iteration serve as the basic elements of the present approach. The main difficulty in solving such equations stems from the inherently largecondition number of the matrix representing the linear system that result from the discretization. However, it is known that periodized differential operators have an effective diagonal preconditioner in the wavelet system of coordinates. The condition number of the preconditioned matrix is O(1) and, thus, depends only weakly on the size of the linear system. The nonstandard form (nsf) is preferable in multiple dimensions since it requires O(1) elements to represent the operator on all scales. Unfortunately, the preconditioned nsf turns out to be dense. This obstacle can be avoided if in the process of solving the linear system, the preconditioner is applied separately before and after the operator (to maintain sparsity). A constrained version of the preconditioned conjugate gradient algorithm is developed in wavelet coordinates. Only those entries of the conjugate directions which are in the set of significant indices are used. The combination of the above-mentioned elements yields an algorithm wherethe number of operations at each iteration is proportional to the number of elements. At the same time, the number of iterations is bounded by a constant.