Almost all mathematical problems in science and engineering are approximated by linear and non-linear systems of integral and/or partial differential equations, which vary widely in number, form, and character. Since analytical solutions do not exist for most problems of interest, numerical approximation techniques, such as finite element and finite difference methods, are widely used, ultimately resulting in a system of linear or non-linear algebraic equations, which must then be solved. In the realistic representation of practical applications over large and/or domains, such systems may contain m=104 to 107 unknowns. For non-linear or transient problems, the system requires the repetitive solution of corresponding linearized problems for changing parameters. Direct solution methods often cannot be used because of excessive storage and time requirements (the latter are O(m3)). Iterative methods (such as multigrid, conjugate gradients, SSOR, orthomin, and GMRES), which take time O(m2) per iteration, are almost universally applied. Another candidate for solving very large linear systems is the Monte Carlo method, which takes time O(m) per iteration; but the high variances of the estimates and the very slow convergence has tended to eliminate it from consideration. The sequential Monte Carlo method is a promising modification which shows guaranteed convergence. This method is applied to broad classes of large systems of linear and non- linear equations. Preliminary computations show promisingly large improvements in speed and efficiency over both plain Monte Carlo and deterministic methods.
