Applications of the Multigrid Algorithm to Solving the Zakai Equation of Nonlinear Filtering with VLSI Implementation

Kevin Holley

Doctoral Dissertation, Year: 1986, Advisor: John S. Baras

Abstract

The feasiblity of designing nonlinear filtering algorithms for implementation via special purpose VLSI systems is explored. The filtering equations are dx_t = f(x_t)dt + g(x_t)dw_t, dy_t = h(x_t)dt + dv_t, where x_t E Rⁿ and dw_t and dv_t are white noise of appropriate dimension. Other conditions are required and explained in this paper.

The filtering algorithms are to be executed in real-time and in parallel, while the resulting filter can be used in various simultaneous estimation and detection systems.

We examine existence, uniqueness and asymptotic behavior of the stochastic partial differential equation governing the filter and show that it is amenable to numerical analysis, using a technique known as the Multigrid method.

The stochastic PDE is called the Zakai equation and its solutions is, when normalized, the probability density of the state x_t conditioned on the observations {y_s: 0 <= s <= t}. Given this density, all statistical information regarding x_t is obtainable.

When defined in n dimensions the Zakai equation is of the form:

dU_t = [sum a_ij (x) U_(x_i*x_j) + sum b_i (x) U_x_i + c(x)U]dt + U < h(x), dy_t > where the coefficients are defined in the text. This equation can be shown to have a stable, implicit finite difference scheme whose solution converges weakly to the solution of the PDE. We examine some properties of this equation that are relevant to its numerical analysis, such as how it is sometimes possible to approximate its solution, originally defined on R^n, on a compact set so as to meet with finite computational requirements.

The Multigrid method involves the use of nested grids in which an original, rather complcated, linear system can be solved by approximating it on coarser grids, and exploiting the resulting reduction is problem size. The error that is incurred can be smoothed out using relaxation techniques. This method can be shown to be very efficient as a parallel solver and can be extended without a dramatic increase in computing time to relatively high dimensions. We give an indepth analysis of this algorithm, demonstrating its performance and capabilities, together with an identification of its pertinent aspects regarding our problem.

We also describe some techniques in VLSI architectural analyis that will also prove useful in our work. These include design strategies for both systolic arrays, which are synchronous array machines, and more general asynchronous systems. These concepts are utilized in our own custom-made design for real-time processing with the Zakai equation.

An important question for us is: If the Zakai equation is defined in R^n, what is the maximum dimension n we can expect to allow, for real-time signal processing? We conduct an analysis of this question using the methods of R. W. Hockney for estimating performance of general computing systems. We find that dimensions no higher than about six or seven can be reasonably treated in conventional real-time signal processing environments, which is usually on the order of about one millisec, a time bound suggested by research on signal processing for the one-dimensional Zakai equation.