Dynamic programming, reduction of dimensionality and eigenvalue problems-II. Multidimensional case

Dynamic programming as used in most optimal control applications relies heavily on the causal structure of the underlying dynamics. In this paper, the authors will show that noncausal problems, such as the Helmholtz equation, can be recast into causal form and then be handled as a vector multistage decision process using a modified version of dynamic programming. The proposed technique, documented in detail in a previous paper by Ng and Sancho, is based on an a priori deduction from Bellman’s principle of optimality. The significance of the method is that it provides a means of reducing Bellman’s ‘curse of dimensionality’ and broadens the scope of problems that can effectively be solved with the dynamic programming approach. The outstanding feature of the technique is illustrated by its application to the evaluation of the dominant TM-mode of rectangular and ridge electric waveguides. The latter problem is particularly challenging because it involves singularities in the solution. The modified dynamic programming solutions are found to be comparable in accuracy to analytical and other numerical solutions.