# Browsing by Subject "Monte Carlo methods"

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Item Coupled flow systems, adjoint techniques and uncertainty quantification(2012-08) Garg, Vikram Vinod, 1985-; Carey, Graham F.; Prudhomme, Serge M.; Dawson, Clint N.; Gamba, Irene; Ghattas, Omar; Oden, J. Tinsley; Carey, VarisShow more Coupled systems are ubiquitous in modern engineering and science. Such systems can encompass fluid dynamics, structural mechanics, chemical species transport and electrostatic effects among other components, all of which can be coupled in many different ways. In addition, such models are usually multiscale, making their numerical simulation challenging, and necessitating the use of adaptive modeling techniques. The multiscale, multiphysics models of electrosomotic flow (EOF) constitute a particularly challenging coupled flow system. A special feature of such models is that the coupling between the electric physics and hydrodynamics is via the boundary. Numerical simulations of coupled systems are typically targeted towards specific Quantities of Interest (QoIs). Adjoint-based approaches offer the possibility of QoI targeted adaptive mesh refinement and efficient parameter sensitivity analysis. The formulation of appropriate adjoint problems for EOF models is particularly challenging, due to the coupling of physics via the boundary as opposed to the interior of the domain. The well-posedness of the adjoint problem for such models is also non-trivial. One contribution of this dissertation is the derivation of an appropriate adjoint problem for slip EOF models, and the development of penalty-based, adjoint-consistent variational formulations of these models. We demonstrate the use of these formulations in the simulation of EOF flows in straight and T-shaped microchannels, in conjunction with goal-oriented mesh refinement and adjoint sensitivity analysis. Complex computational models may exhibit uncertain behavior due to various reasons, ranging from uncertainty in experimentally measured model parameters to imperfections in device geometry. The last decade has seen a growing interest in the field of Uncertainty Quantification (UQ), which seeks to determine the effect of input uncertainties on the system QoIs. Monte Carlo methods remain a popular computational approach for UQ due to their ease of use and "embarassingly parallel" nature. However, a major drawback of such methods is their slow convergence rate. The second contribution of this work is the introduction of a new Monte Carlo method which utilizes local sensitivity information to build accurate surrogate models. This new method, called the Local Sensitivity Derivative Enhanced Monte Carlo (LSDEMC) method can converge at a faster rate than plain Monte Carlo, especially for problems with a low to moderate number of uncertain parameters. Adjoint-based sensitivity analysis methods enable the computation of sensitivity derivatives at virtually no extra cost after the forward solve. Thus, the LSDEMC method, in conjuction with adjoint sensitivity derivative techniques can offer a robust and efficient alternative for UQ of complex systems. The efficiency of Monte Carlo methods can be further enhanced by using stratified sampling schemes such as Latin Hypercube Sampling (LHS). However, the non-incremental nature of LHS has been identified as one of the main obstacles in its application to certain classes of complex physical systems. Current incremental LHS strategies restrict the user to at least doubling the size of an existing LHS set to retain the convergence properties of LHS. The third contribution of this research is the development of a new Hierachical LHS algorithm, that creates designs which can be used to perform LHS studies in a more flexibly incremental setting, taking a step towards adaptive LHS methods.Show more Item Hierarchical game-theoretic control for multi-agent autonomous racing(2022-04-28) Thakkar, Rishabh Saumil; Topcu, UfukShow more We develop a hierarchical control scheme for autonomous racing with realistic safety and fairness rules. The first part constructs a discrete game approximation with simplified dynamics and rules presented as temporal logic specifications. Using the discrete representation, we use a model checking tool to synthesize an optimal strategy in the form of a sequence of target waypoints. We apply the model to several case studies of common racing scenarios, and its resulting strategies are qualitatively verified against those executed by racing experts. This formulation is used as the high-level planner in the hierarchical controller but is solved using Monte Carlo tree search (MCTS) to run in real-time. In the next part, we integrate the high-level planner with a low-level controller to track the target waypoints. Two low-level approaches are considered: a multi-agent reinforcement learning (MARL) trained policy and a linear-quadratic Nash game (LQNG) formulation. As a result, we produce two hierarchical controllers, MCTS-RL and MCTS-LQNG, respectively. The hierarchical agents are tested against three baselines: an end-to-end MARL controller, a MARL controller tracking the optimal racing line, and an LQNG controller tracking the optimal racing line. The controllers compete head-to-head on an oval track and a complex track, and we count the number of wins and a safety score representing the number of rule violations. Our hierarchical controllers outperform their respective baseline methods in terms of wins, but only MCTS-RL is better than its baselines in terms of safety score. The MCTS-LQNG controller has a worse safety score, but this result is due to the simplicity and conservative nature of the fixed trajectory LQNG baseline. Overall, the MCTS-RL controller outperforms all of the other controllers across both metrics and executes maneuvers resembling those seen in real-life racing. In the final part, we extend the hierarchical controllers to team-based racing where they must consider a mixture of competitive and cooperative objectives. The formulations are generalized to consider these challenging objectives while still being required to adhere to the complex rules. We test our controllers against the previously discussed baselines in races where the agents compete in teams of two instead of head-to-head. In addition to counting the number of wins and the safety score, we introduce a third metric to measure the cooperative performance of the controllers. We allocate points based on the finishing position of each agent and aggregate them across the teams, which indicates how the team performed as a whole. The results show our hierarchical agents outperforming their baselines in terms of wins while maintaining similar safety scores. In addition, our controllers also have a higher number of average points per race indicating that they produce greater success as a team. Finally, we observe that the MCTS-RL controller continues to outperform all of the other implemented controllers across all metrics and exhibits tactics performed by expert human drivers. We show that hierarchical planning for game-theoretic reasoning produces competitive behavior even when challenged with complex objectives, rules, and constraints.Show more Item Working field theory problems with random walks(Emerald Group Publishing Limited, 2005) Davey, K.R.;Show more Abstract Purpose – The purpose of this paper is to demonstrate how Monte Carlo methods can be applied to the solution of field theory problems. Design – This objective is achieved by building insight from Laplacian field problems. The point solution of a Laplacian field problem can be viewed as the solid angle average of the Dirichlet potentials from that point. Alternatively it can be viewed as the average of the termination potential of a number of random walks. Poisson and Helmholtz equations add the complexity of collecting a number of packets along this walk, and noting the termination of a random walk at a Dirichlet boundary. Findings – When approached as a Monte Carlo problem, Poisson type problems can be interpreted as collecting and summing source packets representative of current or charge. Helmholtz problems involve the multiplication of packets of information modified by a multiplier reflecting the conductivity of the medium. Practical implications – This method naturally lends itself to parallel processing computers. Originality/value – This is the first paper to explore random walk solutions for all classes of eddy current problems, including those involving velocity. In problems involving velocity, the random walk direction enters depending on the walk direction with respect to the local velocity. Keywords Finite difference methods, Monte Carlo methods, Random functions Paper type Technical paperShow more