MULTILEVEL MATRIX MULTIPLICATION AND FAST SOLUTION OF INTEGRAL-EQUATIONS

A fast multigrid approach is described for the task of calculating ∫Ω K(x, y) u(y)dy for each x ε{lunate} ω ⊆ Rd. Discretizing Ω by an equidistant grid with n points and meshsize h, and approximating the integrations to O(h2s) accuracy, it is shown that the complexity of this calculation can be reduced from O(n2) to O(sn), provided the kernel K is sufficiently smooth. For potential-type kernels, the complexity is reduced to O(sn log n). Corresponding integral equations can be solved to a similar accuracy in basically the same amount of work, using a special kind of distributed relaxation in a multigrid algorithm. One- and two-dimensional numerical tests, and theoretical derivations of optimal strategies, are reported. The method is applicable to the task of multiplying by any matrix with appropriate smoothness properties, including most types of many body interactions. © 1990.