Multiresolution strategies for the numerical solution of optimal control problems

Optimal control problems are often characterized by discontinuities or switchings in the control variables. One way of accurately capturing the irregularities in the solution is to use a high resolution (dense) uniform grid. This requires a large amount of computational resources both in terms of CPU time and memory. Hence, in order to accurately capture any irregularities in the solution using a few computational resources, one can refine the mesh locally in the region close to an irregularity instead of refining the mesh uniformly over the whole domain. Therefore, a novel multiresolution scheme for data compression has been designed which is shown to outperform similar data compression schemes. Specifically, we have shown that the proposed approach results in fewer grid points in the grid compared to a common multiresolution data compression scheme. The validity of the proposed mesh refinement algorithm has been verified by solving several challenging initial-boundary value problems for evolution equations in 1D. The examples have demonstrated the stability and robustness of the proposed algorithm. Next, a direct multiresolution-based approach for solving trajectory optimization problems is developed. The original optimal control problem is transcribed into a nonlinear programming (NLP) problem that is solved using standard NLP codes. The novelty of the proposed approach hinges on the automatic calculation of a suitable, nonuniform grid over which the NLP problem is solved, which tends to increase numerical efficiency and robustness. Control and/or state constraints are handled with ease, and without any additional computational complexity. The proposed algorithm is based on a simple and intuitive method to balance several conflicting objectives, such as accuracy of the solution, convergence, and speed of the computations. The benefits of the proposed algorithm over uniform grid implementations are demonstrated with the help of several nontrivial examples. Furthermore, two sequential multiresolution trajectory optimization algorithms for solving problems with moving targets and/or dynamically changing environments have been developed.Ph.D.Committee Chair: Tsiotras, Panagiotis; Committee Member: Calise, Anthony J.; Committee Member: Egerstedt, Magnus; Committee Member: Prasad, J. V. R.; Committee Member: Russell, Ryan P.; Committee Member: Zhou, Hao-Mi

Multilevel Preconditioning of Discontinuous-Galerkin Spectral Element Methods, Part I: Geometrically Conforming Meshes

This paper is concerned with the design, analysis and implementation of preconditioning concepts for spectral Discontinuous Galerkin discretizations of elliptic boundary value problems. While presently known techniques realize a growth of the condition numbers that is logarithmic in the polynomial degrees when all degrees are equal and quadratic otherwise, our main objective is to realize full robustness with respect to arbitrarily large locally varying polynomial degrees degrees, i.e., under mild grading constraints condition numbers stay uniformly bounded with respect to the mesh size and variable degrees. The conceptual foundation of the envisaged preconditioners is the auxiliary space method. The main conceptual ingredients that will be shown in this framework to yield "optimal" preconditioners in the above sense are Legendre-Gauss-Lobatto grids in connection with certain associated anisotropic nested dyadic grids as well as specially adapted wavelet preconditioners for the resulting low order auxiliary problems. Moreover, the preconditioners have a modular form that facilitates somewhat simplified partial realizations. One of the components can, for instance, be conveniently combined with domain decomposition, at the expense though of a logarithmic growth of condition numbers. Our analysis is complemented by quantitative experimental studies of the main components.Comment: 41 pages, 11 figures; Major revision: rearrangement of the contents for better readability, part on wavelet preconditioner adde

Multiresolution analysis as an approach for tool path planning in NC machining

Wavelets permit multiresolution analysis of curves and surfaces. A complex curve can be decomposed using wavelet theory into lower resolution curves. The low-resolution (coarse) curves are similar to rough-cuts and high-resolution (fine) curves to finish-cuts in numerical controlled (NC) machining.;In this project, we investigate the applicability of multiresolution analysis using B-spline wavelets to NC machining of contoured 2D objects. High-resolution curves are used close to the object boundary similar to conventional offsetting, while lower resolution curves, straight lines and circular arcs are used farther away from the object boundary.;Experimental results indicate that wavelet-based multiresolution tool path planning improves machining efficiency. Tool path length is reduced, sharp corners are smoothed out thereby reducing uncut areas and larger tools can be selected for rough-cuts

MURPHY -- A scalable multiresolution framework for scientific computing on 3D block-structured collocated grids

We present the derivation, implementation, and analysis of a multiresolution adaptive grid framework for numerical simulations on octree-based 3D block-structured collocated grids with distributed computational architectures. Our approach provides a consistent handling of non-lifted and lifted interpolating wavelets of arbitrary order demonstrated using second, fourth, and sixth order wavelets, combined with standard finite-difference based discretization operators. We first validate that the wavelet family used provides strict and explicit error control when coarsening the grid, and show that lifting wavelets increase the grid compression rate while conserving discrete moments across levels. Further, we demonstrate that high-order PDE discretization schemes combined with sufficiently high order wavelets retain the expected convergence order even at resolution jumps. We then simulate the advection of a scalar to analyze convergence for the temporal evolution of a PDE. The results shows that our wavelet-based refinement criterion is successful at controlling the overall error while the coarsening criterion is effective at retaining the relevant information on a compressed grid. Our software exploits a block-structured grid data structure for efficient multi-level operations, combined with a parallelization strategy that relies on a one-sided MPI-RMA communication approach with active PSCW synchronization. Using performance tests up to 16,384 cores, we demonstrate that this leads to a highly scalable performance. The associated code is available under a BSD-3 license at https://github.com/vanreeslab/murphy.Comment: submitted to SIAM Journal of Scientific Computing (SISC) on Dec 1

Self-Adaptive Methods for PDE

[no abstract available

Wavelet based Adaptive RBF Method for Nearly Singular Poisson-Type Problems on Irregular Domains

We present a wavelet based adaptive scheme and investigate the efficiency of this scheme for solving nearly singular potential PDEs over irregularly shaped domains. For a problem defined over Ω ∈ ℜd, the boundary of an irregularly shaped domain, Γ, is defined as a boundary curve that is a product of a Heaviside function along the normal direction and a piecewise continuous tangential curve. The link between the original wavelet based adaptive method presented in Libre, Emdadi, Kansa, Shekarchi, and Rahimian (2008, 2009) or LEKSR method and the generalized one is given through the use of simple Heaviside masking procedure. In addition level dependent thresholding were introduced to improve the efficiency and convergence rate of the solution. We will show how the generalized wavelet based adaptive method can be applied for detecting nearly singularities in Poisson type PDEs over irregular domains. The numerical examples have illustrated that the proposed method is powerful to analyze the Poisson type PDEs with rapid changes in gradients and nearly singularities