An Efficient Policy Iteration Algorithm for Dynamic Programming Equations

We present an accelerated algorithm for the solution of static Hamilton-Jacobi-Bellman equations related to optimal control problems. Our scheme is based on a classic policy iteration procedure, which is known to have superlinear convergence in many relevant cases provided the initial guess is sufficiently close to the solution. In many cases, this limitation degenerates into a behavior similar to a value iteration method, with an increased computation time. The new scheme circumvents this problem by combining the advantages of both algorithms with an efficient coupling. The method starts with a value iteration phase and then switches to a policy iteration procedure when a certain error threshold is reached. A delicate point is to determine this threshold in order to avoid cumbersome computation with the value iteration and, at the same time, to be reasonably sure that the policy iteration method will finally converge to the optimal solution. We analyze the methods and efficient coupling in a number of examples in dimension two, three and four illustrating its properties

Hidden Beauty in Multiloop Amplitudes

Planar L-loop maximally helicity violating amplitudes in N = 4 supersymmetric Yang-Mills theory are believed to possess the remarkable property of satisfying iteration relations in L. We propose a simple new method for studying the iteration relations for four-particle amplitudes which involves the use of certain linear differential operators and eliminates the need to fully evaluate any loop integrals. We carry out this procedure in explicit detail for the two-loop amplitude and argue that this method can be used to prove the iteration relations to all loops up to polynomials in logarithms.Comment: 21 pages, harvmac; v2: minor change

Fine grain software pipelining of non-vectorizable nested loops

This paper presents a new technique to parallelize nested loops at the statement level. It transforms sequential nested loops, either vectorizable or not, into parallel ones. Previously, the wavefront method was used to parallelize non-vectorizable nested loops. However, in order to reduce the complexity of parallelization, the wavefront method regards an iteration as an unbreakable scheduling unit and draws parallelism through iteration overlapping. Our technique takes a statement rather than an iteration as the scheduling unit and exploits parallelism by overlapping the statements in all dimensions. In this paper, we show how this finer grain parallelization can be achieved with reasonable computational complexity, and the effectiveness of the resulting method in exploiting parallelism