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<p>We develop algorithms to approximately count perfect matchings in bipartite graphs (or permanents of the corresponding adjacency matrices), perfect matchings in nonbipartite graphs (or hafnians), and general matchings in bipartite and nonbipartite graphs (or matching generating polynomials). </p><p>First, we study the problem of approximating the permanent and generating weighted perfect matchings in bipartite graphs from their correct distribution. We present a perfect sampling algorithm using self-reducible acceptance/rejection and an upper bound for the permanent. It has a polynomial expected running time for a class of dense problems, and it gives an improvement in running time by a factor of $n^3$ for matrices that are (.6)-dense. </p><p>Next, we apply the similar approach to study perfect matchings in nonbipartite graphs and also general matchings in general graphs. Our algorithms here have a subexponential expected running time for some classes of nontrivial graphs and are competitive with other Markov chain Monte Carlo methods.</p> / Dissertation

The performance of a program can sometimes greatly improve if it was known in advance the features of the input the program is supposed to process, the actual operating parameters it is supposed to work with, or the specific environment it is to run on. However, this information is typically not available until too late in the program’s operation to take advantage of it. This is especially true for simulation algorithms, which are sensitive to this late-arriving information, and whose role in the solution of decision-making, inference and valuation problems is crucial. To overcome this limitation we need to provide the flexibility for a program to adapt its behaviour to late-arriving information once it becomes available. In this thesis, I study three adaptation mechanisms: run-time code generation, model-specific (quasi) Monte Carlo sampling and dynamic computation offloading, and evaluate their benefits on Monte Carlo algorithms. First, run-time code generation is studied in the context of Monte Carlo algorithms for time-series filtering in the form of the Input-Adaptive Kalman filter, a dynamically generated state estimator for non-linear, non-Gaussian dynamic systems. The second adaptation mechanism consists of the application of the functional-ANOVA decomposition to generate model-specific QMC-samplers which can then be used to improve Monte Carlo-based integration. The third adaptive mechanism treated here, dynamic computation offloading, is applied to wireless communication management, where network conditions are assessed via option valuation techniques to determine whether a program should offload computations or carry them out locally in order to achieve higher run-time (and correspondingly battery-usage) efficiency. This ability makes the program well suited for operation in mobile environments. At their core, all these applications carry out or make use of (quasi) Monte Carlo simulations on dynamic Bayesian networks (DBNs). The DBN formalism and its associated simulation-based algorithms are of great value in the solution to problems with a large uncertainty component. This characteristic makes adaptation techniques like those studied here likely to gain relevance in a world where computers are endowed with perception capabilities and are expected to deal with an ever-increasing stream of sensor and time-series data.

The performance of a program can sometimes greatly improve if it was known in advance the features of the input the program is supposed to process, the actual operating parameters it is supposed to work with, or the specific environment it is to run on. However, this information is typically not available until too late in the program’s operation to take advantage of it. This is especially true for simulation algorithms, which are sensitive to this late-arriving information, and whose role in the solution of decision-making, inference and valuation problems is crucial. To overcome this limitation we need to provide the flexibility for a program to adapt its behaviour to late-arriving information once it becomes available. In this thesis, I study three adaptation mechanisms: run-time code generation, model-specific (quasi) Monte Carlo sampling and dynamic computation offloading, and evaluate their benefits on Monte Carlo algorithms. First, run-time code generation is studied in the context of Monte Carlo algorithms for time-series filtering in the form of the Input-Adaptive Kalman filter, a dynamically generated state estimator for non-linear, non-Gaussian dynamic systems. The second adaptation mechanism consists of the application of the functional-ANOVA decomposition to generate model-specific QMC-samplers which can then be used to improve Monte Carlo-based integration. The third adaptive mechanism treated here, dynamic computation offloading, is applied to wireless communication management, where network conditions are assessed via option valuation techniques to determine whether a program should offload computations or carry them out locally in order to achieve higher run-time (and correspondingly battery-usage) efficiency. This ability makes the program well suited for operation in mobile environments. At their core, all these applications carry out or make use of (quasi) Monte Carlo simulations on dynamic Bayesian networks (DBNs). The DBN formalism and its associated simulation-based algorithms are of great value in the solution to problems with a large uncertainty component. This characteristic makes adaptation techniques like those studied here likely to gain relevance in a world where computers are endowed with perception capabilities and are expected to deal with an ever-increasing stream of sensor and time-series data.

Multi-core, multi-processor, and Graphics Processing Unit (GPU) computer architectures pose significant challenges with respect to the efficient exploitation of parallelism for large-scale, scientific computing simulations. For example, a simulation of the human tonsil at the cellular level involves the computation of the motion and interaction of millions of cells over extended periods of time. Also, the simulation of Radiative Heat Transfer (RHT) effects by the Photon Monte Carlo (PMC) method is an extremely computationally demanding problem. The PMC method is example of the Monte Carlo simulation method—an approach extensively used in wide of application areas. Although the basic algorithmic framework of these Monte Carlo methods is simple, they can be extremely computationally intensive. Therefore, an efficient parallel realization of these simulations depends on a careful analysis of the nature these problems and the development of an appropriate software framework. The overarching goal of this dissertation is develop and understand what the appropriate parallel programming model should be to exploit these disparate architectures, both from the metric of efficiency, as well as from a software engineering perspective. In this dissertation we examine these issues through a performance study of PathSim2, a software framework for the simulation of large-scale biological systems, using two different parallel architectures’ distributed and shared memory. First, a message-passing implementation of a multiple germinal center simulation by PathSim2 is developed and analyzed for distributed memory architectures. Second, a germinal center simulation is implemented on shared memory architecture with two parallelization strategies based on Pthreads and OpenMP. Finally, we present work targeting a complete hybrid, parallel computing architecture. With this work we develop and analyze a software framework for generic Monte Carlo simulations implemented on multiple, distributed memory nodes consisting of a multi-core architecture with attached GPUs. This simulation framework is divided into two asynchronous parts: (a) a threaded, GPU-accelerated pseudo-random number generator (or producer), and (b) a multi-threaded Monte Carlo application (or consumer). The advantage of this approach is that this software framework can be directly used within any Monte Carlo application code, without requiring application-specific programming of the GPU. We examine this approach through a performance study of the simulation of RHT effects by the PMC method on a hybrid computing architecture. We present a theoretical analysis of our proposed approach, discuss methods to optimize performance based on this analysis, and compare this analysis to experimental results obtained from simulations run on two different hybrid, parallel computing architectures. / Ph. D.