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PRODID:-//Microsoft Corporation//Outlook MIMEDIR//EN
VERSION:1.0
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DTSTART:20111117T221400Z
DTEND:20111117T223600Z
LOCATION:TCC 101
DESCRIPTION;ENCODING=QUOTED-PRINTABLE:ABSTRACT: Recent developments in Monte Carlo methods (MCMs) for the solution of partial differential equations (PDEs) provide new algorithms that are computationally competitive or faster than traditional deterministic methods for certain elliptic and parabolic PDEs.  In fact, there are a variety of new MCM alternatives to a variety of numerical computations traditionally solved with deterministic methods.  Besides the fact that in certain circumstances these new MCMs are faster than deterministic methods, MCMs have some generic properties that make them especially interesting from the high-performance computing point-of-view.=0A=0ARecall that MCMs are based on random sampling, and so given a suitable high-quality random number generation tool, the statistical sampling can be done in many different ways to optimally take advantage of the parallelism. This means that MCMs offer up tremendous amounts of independent and potentially parallel work that can be mapped onto many different architectures in a favorable way.  In addition, MCMs offer the following, well known, advantageous properties:=0A-MCMs often require less memory than deterministic methods, as the physical domain involved in the computation does not need to be discretized.=0A-In deterministic methods, the size of the error related to the spatial discretization (often not necessary in MCMs) depends on the level of spatial refinement.  Thus, deterministic methods require more memory to solve the same problem to a higher order of accuracy using the same underlying numerical approach.  In MCMs, the (sampling) error scales with the number of statistical samples for each numerical quantity.  Thus higher accuracy can be achieved without increasing the memory required to compute a single instance of the problem.=0A-Error estimates in MCMs are based the size of confidence intervals whose dimensions can be readily estimated by monitoring the variance of the MCM estimators.  This provides an a posteriori set of error estimates that readily permit adaptivity and accurate error estimates that are often hard to achieve with deterministic methods without refinement studies.=0A-MC estimates usually require only storing the mean and the variance of the numerical quantity of interest.  This not only minimizes memory requirements, but independent sampling of the same quantity can occur with very little information exchange.  In fact, when many independent samples are concurrently computed, one need only combine three numerical quantities from the different computations to complete the execution.  Moreover, due to the statistical nature of these computations, loosely coupled or completely asynchronous communication can be used do the final accumulation.=0AMCMs offer a very different computational paradigm for the computation of many numerical problems.  They offer many desirable numerical properties, and they are so compute intensive, that a distributed MC computation can be undertaken quite efficiently on a very loosely coupled collection of computing resources.  This means that MCMs can take advantage of relatively cheap and widely computational resources and can be expected to achieve high levels of parallel efficiency.  A development effort, on the scale of deterministic mathematical software development, over the course of decade will create a completely different computational environment in terms of computational complexity, scalability, and architectural efficiency.
SUMMARY:Stochastic Algorithms Change Complexity Estimates and Optimal Architectures
PRIORITY:3
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