We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced-basis approximations -- Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation -- relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures -- methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage -- in which, given a new parameter value, we calculate the output of interest and associated error bound -- depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. / Singapore-MIT Alliance (SMA)

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Computational Fluid Dynamics (CFD) has been widely used as a tool to analyze physical phenomena involving fluids. To perform a CFD simulation, the governing equations are discretized to formulate a set of nonlinear algebraic equations. Typical spatial discretization schemes include finite-difference methods, finite-volume methods, and finite-element methods. Error introduced in the discretization process is called discretization error and defined as the difference between the exact solution to the discrete equations and the exact solution to the partial differential or integral equations. For most CFD simulations, discretization error accounts for the largest portion of the numerical error in the simulation. Discretization error has a complicated nonlinear relationship with the computational grid and the discretization scheme, which makes it especially difficult to estimate. Thus, it is important to study the discretization error to characterize numerical errors in a CFD simulation. Discretization error estimation is performed using the Error Transport Equations (ETE) in this work. The original nonlinear form of the ETE can be linearized to formulate the linearized ETE. Results of the two types of the ETE are compared. This work implements the ETE for finite-volume methods and Discontinuous Galerkin (DG) finite-element methods. For finite volume methods, discretization error estimates are obtained for both steady state problems and unsteady problems. The work on steady-state problems focuses on turbulent flow modelled by the Spalart-Allmaras (SA) model and Menter's $k-omega$ SST model. Higher-order discretization error estimates are obtained for both the mean variables and the turbulence working variables. The type of pseudo temporal discretization used for the steady-state problems does not have too much influence on the final converged solution. However, the temporal discretization scheme makes a significant difference for unsteady problems. Different temporal discretizations also impact the ETE implementation. This work discusses the implementation of the ETE for the 2-step Backward Difference Formula (BDF2) and the Singly Diagonally Implicit Runge-Kutta (SDIRK) methods. Most existing work on the ETE focuses on finite-volume methods. This work also extends ETE to work with the DG methods and tests the implementation with steady state inviscid test cases. The discretization error estimates for smooth test cases achieve the expected order of accuracy. When the test case is non-smooth, the truncation error estimation scheme fails to generate an accurate truncation error estimate. This causes a reduction of the discretization error estimate to first-order accuracy. Discussions are made on how accurate truncation error estimates can be found for non-smooth test cases. / Doctor of Philosophy / For a general practical fluid flow problem, the governing equations can not be solved analytically. Computational Fluid Dynamics (CFD) approximates the governing equations by a set of algebraic equations that can be solved directly by the computer. Compared to experiments, CFD has certain advantages. The cost for running a CFD simulation is typically much lower than performing an experiment. Changing the conditions and geometry is usually easier for a CFD simulation than for an experiment. A CFD simulation can obtain information of the entire flow field for all field variables, which is nearly impossible for a single experiment setup. However, numerical errors are inherently persistent in CFD simulations due to the approximations made in CFD and finite precision arithmetic of the computer. Without proper characterization of errors, the accuracy of the CFD simulation can not be guaranteed. Numerical errors can even result in false flow features in the CFD solution. Thus, numerical errors need to be carefully studied so that the CFD simulation can provide useful information for the chosen application. The focus of this work is on numerical error estimation for the finite-volume method and the Discontinuous Galerkin (DG) finite-element method. In general, discretization error makes the most significant contribution to the numerical error of a CFD simulation. This work estimates discretization error by solving a set of auxiliary equations derived for the discretization error of a CFD solution. Accurate discretization error estimates are obtained for different test cases. The work on the finite-volume method focus on discretization error estimation for steady state turbulent test cases and unsteady test cases. To the best of the author's knowledge, the implementation of the current discretization error estimation scheme has only been applied as an intermediate step for the error estimation of functionals for the DG method in the literature. Results for steady-state inviscid test cases for the DG method are presented.

This study seeks to reduce the degree of uncertainty that often arises in computational fluid dynamics simulations about the computed accuracy of functional outputs. An error estimation methodology based on discrete adjoint sensitivity analysis is developed to provide a quantitative measure of the error in computed outputs. The developed procedure relates the local residual errors to the global error in output function via adjoint variables as weight functions. The three major steps in the error estimation methodology are: (1) development of adjoint sensitivity analysis capabilities; (2) development of an efficient error estimation procedure; (3) implementation of an output-based grid adaptive scheme. Each of these steps are investigated. For the first step, parallel discrete adjoint capabilities are developed for the variable Mach version of the U2NCLE flow solver. To compare and validate the implementation of adjoint solver, this study also develops direct sensitivity capabilities. A modification is proposed to the commonly used unstructured flux-limiters, specifically, those of Barth-Jespersen and Venkatakrishnan, to make them piecewise continuous and suitable for sensitivity analysis. A distributed-memory message-passing model is employed for the parallelization of sensitivity analysis solver and the consistency of linearization is demonstrated in sequential and parallel environments. In the second step, to compute the error estimates, the flow and adjoint solutions are prolongated from a coarse-mesh to a fine-mesh using the meshless Moving Least Squares (MLS) approximation. These error estimates are used as a correction to obtain highlyurate functional outputs and as adaptive indicators in an iterative grid adaptive scheme to enhance the accuracy of the chosen output to a prescribed tolerance. For the third step, an output-based adaptive strategy that takes into account the error in both the primal (flow) and dual (adjoint) solutions is implemented. A second adaptive strategy based on physics-based feature detection is implemented to compare and demonstrate the robustness and effectiveness of the output-based adaptive approach. As part of the study, a general-element unstructured mesh adaptor employing h-refinement is developed using Python and C++. Error estimation and grid adaptation results are presented for inviscid, laminar and turbulent flows.

We propose a new discontinuous finite element method for higher-order initial value problems where the finite element solution exhibits an optimal convergence rate in the L2- norm. We further show that the q-degree discontinuous solution of a differential equation of order m and its first (m-1)-derivatives are strongly superconvergent at the end of each step. We also establish that the q-degree discontinuous solution is superconvergent at the roots of (q+1-m)-degree Jacobi polynomial on each step. Furthermore, we use these results to construct asymptotically correct a posteriori error estimates. Moreover, we design a new discontinuous Galerkin method to solve the wave equation by using a method of lines approach to separate the space and time where we first apply the classical finite element method using p-degree polynomials in space to obtain a system of second-order ordinary differential equations which is solved by our new discontinuous Galerkin method. We provide an error analysis for this new method to show that, on each space-time cell, the discontinuous Galerkin finite element solution is superconvergent at the tensor product of the shifted roots of the Lobatto polynomials in space and the Jacobi polynomial in time. Then, we show that the global L2 error in space and time is convergent. Furthermore, we are able to construct asymptotically correct a posteriori error estimates for both spatial and temporal components of errors. We validate our theory by presenting several computational results for one, two and three dimensions. / Ph. D.

Simulation and optimization of complex physical systems are an integral part of modern science and engineering. The systems of interest in many fields have a multiphysics nature, with complex interactions between physical, chemical and in some cases even biological processes. This dissertation seeks to advance forward and adjoint numerical time integration methodologies for the simulation and optimization of semi-discretized multiphysics partial differential equations (PDEs), and to estimate and control numerical errors via a goal-oriented a posteriori error framework. We extend exponential propagation iterative methods of Runge-Kutta type (EPIRK) by [Tokman, JCP 2011], to build EPIRK-W and EPIRK-K time integration methods that admit approximate Jacobians in the matrix-exponential like operations. EPIRK-W methods extend the W-method theory by [Steihaug and Wofbrandt, Math. Comp. 1979] to preserve their order of accuracy under arbitrary Jacobian approximations. EPIRK-K methods extend the theory of K-methods by [Tranquilli and Sandu, JCP 2014] to EPIRK and use a Krylov-subspace based approximation of Jacobians to gain computational efficiency. New families of partitioned exponential methods for multiphysics problems are developed using the classical order condition theory via particular variants of T-trees and corresponding B-series. The new partitioned methods are found to perform better than traditional unpartitioned exponential methods for some problems in mild-medium stiffness regimes. Subsequently, partitioned stiff exponential Runge-Kutta (PEXPRK) methods -- that extend stiffly accurate exponential Runge-Kutta methods from [Hochbruck and Ostermann, SINUM 2005] to a multiphysics context -- are constructed and analyzed. PEXPRK methods show full convergence under various splittings of a diffusion-reaction system. We address the problem of estimation of numerical errors in a multiphysics discretization by developing a goal-oriented a posteriori error framework. Discrete adjoints of GARK methods are derived from their forward formulation [Sandu and Guenther, SINUM 2015]. Based on these, we build a posteriori estimators for both spatial and temporal discretization errors. We validate the estimators on a number of reaction-diffusion systems and use it to simultaneously refine spatial and temporal grids. / Doctor of Philosophy / The study of modern science and engineering begins with descriptions of a system of mathematical equations (a model). Different models require different techniques to both accurately and effectively solve them on a computer. In this dissertation, we focus on developing novel mathematical solvers for models expressed as a system of equations, where only the initial state and the rate of change of state as a function are known. The solvers we develop can be used to both forecast the behavior of the system and to optimize its characteristics to achieve specific goals. We also build methodologies to estimate and control errors introduced by mathematical solvers in obtaining a solution for models involving multiple interacting physical, chemical, or biological phenomena. Our solvers build on state of the art in the research community by introducing new approximations that exploit the underlying mathematical structure of a model. Where it is necessary, we provide concrete mathematical proofs to validate theoretically the correctness of the approximations we introduce and correlate with follow-up experiments. We also present detailed descriptions of the procedure for implementing each mathematical solver that we develop throughout the dissertation while emphasizing on means to obtain maximal performance from the solver. We demonstrate significant performance improvements on a range of models that serve as running examples, describing chemical reactions among distinct species as they diffuse over a surface medium. Also provided are results and procedures that a curious researcher can use to advance the ideas presented in the dissertation to other types of solvers that we have not considered. Research on mathematical solvers for different mathematical models is rich and rewarding with numerous open-ended questions and is a critical component in the progress of modern science and engineering.

Multiphysics systems with interface coupling are used to model a variety of physical phenomena, such as arterial blood flow, air flow around aeroplane wings, or interactions between surface and ground water flows. Numerical methods enable the practical application of these models through computer simulations. Specifically a high level of detail and accuracy is achieved in finite element methods by discretisations which use extremely large numbers of degrees of freedom, rendering the solution process challenging from the computational perspective. In this thesis we address this challenge by developing a twofold strategy for improving the efficiency of standard finite element coupled solvers. First, we propose to solve a monolithic coupled problem using block-preconditioned GMRES with a new Schur complement approximation. This results in a modular and robust method which significantly reduces the computational cost of solving the system. In particular, numerical tests show mesh-independent convergence of the solver for all the considered problems, suggesting that the method is well-suited to solving large-scale coupled systems. Second, we derive an adjoint-based formula for goal-oriented a posteriori error estimation, which leads to a time-space mesh refinement strategy. The strategy produces a mesh tailored to a given problem and quantity of interest. The monolithic formulation of the coupled problem allows us to obtain expressions for the error in the Lagrange multiplier, which often represents a physically relevant quantity, such as the normal stress on the interface between the problem components. This adaptive refinement technique provides an effective tool for controlling the error in the quantity of interest and/or the size of the discrete system, which may be limited by the available computational resources. The solver and the mesh refinement strategy are both successfully employed to solve a coupled Stokes-Darcy-Stokes problem modelling flow through a cartridge filter.

This thesis consists of four papers that concern error estimates for the Born-Oppenheimer molecular dynamics, and adaptive algorithms for the Car-Parrinello and Ehrenfest molecular dynamics. In Paper I, we study error estimates for the Born-Oppenheimer molecular dynamics with nearly crossing potential surfaces. The paper first proves an error estimate showing that the difference of the values of observables for the time-independent Schrödinger equation, with matrix valued potentials, and the values of observables for the ab initio Born-Oppenheimer molecular dynamics of the ground state depends on the probability to be in the excited states and the nuclei/electron mass ratio. Then we present a numerical method to determine the probability to be in the excited states, based on the Ehrenfest molecular dynamics, and stability analysis of a perturbed eigenvalue problem. In Paper II, we present an approach, motivated by the so called Landau-Zener probability estimation, to systematically choose the artificial electron mass parameters appearing in the Car-Parrinello and Ehrenfest molecular dynamics methods to approximate the Born-Oppenheimer molecular dynamics solutions. In Paper III, we extend the work presented in Paper II for a set of more general problems with more than two electron states. A main conclusion of Paper III is that it is necessary to resolve the near avoided conical intersections between all electron eigenvalue gaps, including gaps between the occupied states. In Paper IV, we numerically compare, using simple model problems, the Ehrenfest molecular dynamics using the adaptive mass algorithm proposed in Paper II and III and the Born-Oppenheimer molecular dynamics based on the so called purification of the density matrix method concluding that the Born-Oppenheimer molecular dynamics based on purification of density matrix method performed better in terms of computational efficiency. / <p>QC 20161102</p>

Reliability and computational efficiency of classification error estimators are critical factors in classifier design. In a high-dimensional data setting where data is scarce, the conventional method of error estimation, cross-validation, can be very computationally expensive. In this thesis, we consider a particular discriminant analysis type classifier, the Randomly-Projected RLDA ensemble classifier, which operates under the assumption of such a ‘small sample’ regime. We conduct an asymptotic study of the generalization error of this classifier under this regime, which necessitates the use of tools from the field of random matrix theory. The main outcome of this study is a deterministic function of the true statistics of the data and the problem dimension that approximates the generalization error well for large enough dimensions. This is demonstrated by simulation on synthetic data. The main advantage of this approach is that it is computationally efficient. It also constitutes a major step towards the construction of a consistent estimator of the error that depends on the training data and not the true statistics, and so can be applied to real data. An analogous quantity for the Randomly-Projected LDA ensemble classifier, which appears in the literature and is a special case of the former, is also derived. We motivate its use for tuning the parameter of this classifier by simulation on synthetic data.

O Método dos Elementos Finitos Generalizados (MEFG) propõe, basicamente, uma ampliação no espaço de aproximação do Método dos Elementos Finitos (MEF) convencional por meio de funções de enriquecimento que representem bem comportamentos locais da solução do problema. Ele tem se apresentado como uma alternativa eficaz para a obtenção de soluções numéricas com boa precisão para problemas nos quais o MEF convencional requer custo computacional bastante elevado. Em relação ao controle sobre a precisão da resposta numérica obtida, o estudo e análise de erros de discretização, assim como a implementação de estratégias adaptativas, são temas que já foram amplamente abordados para o MEF e recentemente vêm sendo explorados no contexto do MEFG e suas versões estáveis. Neste trabalho, trata-se do tema de adaptatividade para o MEFG, objetivando melhor avaliar a precisão das soluções encontradas assim como garantir que elas atendam a limitações pré-especificadas para medidas dos erros. Em primeiro lugar, avalia-se a utilização de um estimador de erro a posteriori, recentemente proposto, como indicador de regiões onde a adaptatividade h ou p possa ser aplicada. Com o indicador adotado, estende-se para o MEFG estratégias h-adaptativas comumente utilizadas para o MEF, realizadas a partir de sucessivas gerações da malha. Além disso, explora-se neste trabalho uma técnica de agrupamento de partições da unidade, específica do MEFG, para tratar problemas de malhas irregulares e possibilitar análises h-adaptativas realizadas sobre sub-regiões do domínio do problema. Já no que se refere às análises p-adaptativas, a estratégia consiste em definir regiões de interesse para ativar o enriquecimento polinomial da solução aproximada. Exemplos numéricos ilustram a efetividade de todas as análises adaptativas implementadas, propostas para o MEFG e suas versões estáveis, as quais proporcionam respostas que atendem a limites de tolerância previamente estabelecidos. / The Generalized Finite Element Method (GFEM) proposes the generation of numerical approximations that belong to an space obtained by augmenting low-order standard finite element approximation spaces by enrichment functions that well represent local behaviours of the problem solution. The method has become an efficient alternative to obtain solutions with good accuracy for problems in which the standard Finite Element Method (FEM) would require excessively high computational cost. Regarding the control over the numerical solutions\' accuracy, discretization error analysis and study, as well as the implementation of adaptive strategies, are subjects largely studied for the FEM and they are recently being exploited in the GFEM and its stable versions context. In this work, adaptivity for the GFEM is addressed, looking for better evaluate the solutions\' accuracy and ensure that they meet users\' pre-specified limits for error measures. Firstly, the use of a recently proposed a posteriori error estimator as an indicator of the regions where h- or p-adaptivity can be performed is evaluated. With this chosen indicator, h-adaptive strategies commonly used for the FEM are extended to the GFEM by performing successive remeshings. Moreover, a partition of unity clustering technique is also exploited in order to treat nonmatching meshes and to enable h-adaptive analysis to be performed over some pre-defined domain subregions. Regarding the p-adaptive analysis, the basic strategy consists of defining some regions over which it will be set polynomial enrichments for the approximate solution using a particular GFEM stable version. Numerical examples show the effectiveness of all performed adaptive analysis, proposed for conventional and stable GFEMs. All implementations provide responses that can meet the users\' pre-specified tolerance.