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Computing Bounds for Linear Functionals of Exact Weak Solutions to Poisson’s EquationSauerBudge, A.M., Huerta, A., Bonet, J., Peraire, Jaime 01 1900 (has links)
We present a method for Poisson’s equation that computes guaranteed upper and lower bounds for the values of linear functional outputs of the exact weak solution of the infinite dimensional continuum problem using traditional finite element approximations. The guarantee holds uniformly for any level of refinement, not just in the asymptotic limit of refinement. Given a finite element solution and its output adjoint solution, the method can be used to provide a certificate of precision for the output with an asymptotic complexity which is linear in the number of elements in the finite element discretization. / SingaporeMIT Alliance (SMA)

2 
ReducedBasis Output Bound Methods for Parametrized Partial Differential EquationsPrud'homme, C., Rovas, D.V., Veroy, K., Machiels, L., Maday, Y., Patera, Anthony T., Turinici, G. 01 1900 (has links)
We present a technique for the rapid and reliable prediction of linearfunctional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reducedbasis 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 errorresidual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) offline/online computational procedures  methods which decouple the generation and projection stages of the approximation process. The operation count for the online 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 realtime control. / SingaporeMIT Alliance (SMA)

3 
Advanced Time Integration Methods with Applications to Simulation, Inverse Problems, and Uncertainty QuantificationNarayanamurthi, Mahesh 29 January 2020 (has links)
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 semidiscretized multiphysics partial differential equations (PDEs), and to estimate and control numerical errors via a goaloriented a posteriori error framework.
We extend exponential propagation iterative methods of RungeKutta type (EPIRK) by [Tokman, JCP 2011], to build EPIRKW and EPIRKK time integration methods that admit approximate Jacobians in the matrixexponential like operations. EPIRKW methods extend the Wmethod theory by [Steihaug and Wofbrandt, Math. Comp. 1979] to preserve their order of accuracy under arbitrary Jacobian approximations. EPIRKK methods extend the theory of Kmethods by [Tranquilli and Sandu, JCP 2014] to EPIRK and use a Krylovsubspace 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 Ttrees and corresponding Bseries. The new partitioned methods are found to perform better than traditional unpartitioned exponential methods for some problems in mildmedium stiffness regimes. Subsequently, partitioned stiff exponential RungeKutta (PEXPRK) methods  that extend stiffly accurate exponential RungeKutta 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 diffusionreaction system.
We address the problem of estimation of numerical errors in a multiphysics discretization by developing a goaloriented 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 reactiondiffusion 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 followup 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 openended questions and is a critical component in the progress of modern science and engineering.

4 
A Discontinuous Galerkin Method for HigherOrder Differential Equations Applied to the Wave EquationTemimi, Helmi 02 April 2008 (has links)
We propose a new discontinuous finite element method for higherorder initial value problems where the finite element solution exhibits an optimal convergence rate in the L2 norm. We further show that the qdegree discontinuous solution of a differential equation of order m and its first (m1)derivatives are strongly superconvergent at the end of each step. We also establish that the qdegree discontinuous solution is superconvergent at the roots of (q+1m)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 pdegree polynomials in space to obtain a system of secondorder 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 spacetime 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.

5 
RealTime Optimal Parametric Design of a Simple InfiltrationEvaporation Model Using the AssessPredictOptimize (APO) StrategyAli, S., Damodaran, Murali, Patera, Anthony T. 01 1900 (has links)
Optimal parametric design of a system must be able to respond quickly to short term needs as well as long term conditions. To this end, we present an AssessPredictOptimize (APO) strategy which allows for easy modification of a system’s characteristics and constraints, enabling quick design adaptation. There are three components to the APO strategy: Assess  extract necessary information from given data; Predict  predict future behavior of system; and Optimize – obtain optimal system configuration based on information from the other components. The APO strategy utilizes three key mathematical ingredients to yield realtime results which would certainly conform to given constraints: dimension reduction of the model, a posteriori error estimation, and optimization methods. The resulting formulation resembles a bilevel optimization problem with an inherent nonconvexity in the inner level. Using a simple infiltrationevaporation model to simulate an irrigation system, we demonstrate the APO strategy’s ability to yield realtime optimal results. The linearized model, described by a coercive elliptic partial differential equation, is discretized by the reducedbasis output bounds method. A primaldual interior point method is then chosen to solve the resulting APO problem. / SingaporeMIT Alliance (SMA)

6 
Reliable RealTime Solution of Parametrized Elliptic Partial Differential Equations: Application to ElasticityVeroy, K., Leurent, T., Prud'homme, C., Rovas, D.V., Patera, Anthony T. 01 1900 (has links)
The optimization, control, and characterization of engineering components or systems require fast, repeated, and accurate evaluation of a partialdifferentialequationinduced inputoutput relationship. We present a technique for the rapid and reliable prediction of linearfunctional outputs of elliptic partial differential equations with affine parameter dependence. The method has three components: (i) rapidly convergent reduced{basis approximations; (ii) a posteriori error estimation; and (iii) offline/online computational procedures. These components  integrated within a special network architecture  render partial differential equation solutions truly "useful": essentially real{time as regards operation count; "blackbox" as regards reliability; and directly relevant as regards the (limited) inputoutput data required. / SingaporeMIT Alliance (SMA)

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Adaptive finite element methods for multiphysics problemsBengzon, Fredrik January 2009 (has links)
In this thesis we develop and analyze the performance ofadaptive finite element methods for multiphysics problems. Inparticular, we propose a methodology for deriving computable errorestimates when solving unidirectionally coupled multiphysics problemsusing segregated finite element solvers. The error estimates are of a posteriori type and are derived using the standard frameworkof dual weighted residual estimates. A main feature of themethodology is its capability of automatically estimating thepropagation of error between the involved solvers with respect to anoverall computational goal. The a posteriori estimates are used todrive local mesh refinement, which concentrates the computationalpower to where it is most needed. We have applied and numericallystudied the methodology to several common multiphysics problems usingvarious types of finite elements in both two and three spatialdimensions. Multiphysics problems often involve convectiondiffusion equations for whichstandard finite elements can be unstable. For such equations we formulatea robust discontinuous Galerkin method of optimal order with piecewiseconstant approximation. Sharp a priori and a posteriori error estimatesare proved and verified numerically. Fractional step methods are popular for simulating incompressiblefluid flow. However, since they are not genuine Galerkin methods, butrather based on operator splitting, they do not fit into the standardframework for a posteriori error analysis. We formally derive an aposteriori error estimate for a prototype fractional step method byseparating the error in a functional describing the computational goalinto a finite element discretization residual, a time steppingresidual, and an algebraic residual.

8 
RealTime Reliable Prediction of LinearElastic ModeI Stress Intensity Factors for Failure AnalysisHuynh, Dinh Bao Phuong, Peraire, Jaime, Patera, Anthony T., Liu, Guirong 01 1900 (has links)
Modern engineering analysis requires accurate, reliable and efficient evaluation of outputs of interest. These outputs are functions of "input" parameter that serve to describe a particular configuration of the system, typical input geometry, material properties, or boundary conditions and loads. In many cases, the inputoutput relationship is a functional of the field variable  which is the solution to an inputparametrized partial differential equations (PDE). The reducedbasis approximation, adopting offline/online computational procedures, allows us to compute accurate and reliable functional outputs of PDEs with rigorous error estimations. The operation count for the online stage depends only on a small number N and the parametric complexity of the problem, which make the reducedbasis approximation especially suitable for complex analysis such as optimizations and designs. In this work we focus on the development of finiteelement and reducedbasis methodology for the accurate, fast, and reliable prediction of the stress intensity factors or strainenergy release rate of a modeI linear elastic fracture problem. With the use of offline/online computational strategy, the stress intensity factor for a particular problem can be obtained in miliseconds. The method opens a new promising prospect: not only are the numerical results obtained only in miliseconds with great savings in computational time; the results are also reliable  thanks to the rigorous and sharp a posteriori error bounds. The practical uses of our prediction are presented through several example problems. / SingaporeMIT Alliance (SMA)

9 
Robust local problem error estimation for a singularly perturbed reactiondiffusion problem on anisotropic finite element meshesGrosman, Serguei 05 April 2006 (has links) (PDF)
Singularly perturbed reactiondiffusion problems exhibit in general solutions with anisotropic features, e.g. strong boundary and/or interior layers. This anisotropy is reflected in the discretization by using meshes with anisotropic elements. The quality of the numerical solution rests on the robustness of the a posteriori error estimator with respect to both the perturbation parameters of the problem and the anisotropy of the mesh. An estimator that has shown to be one of the most reliable for reactiondiffusion problem is the <i>equilibrated residual method</i> and its modification done by Ainsworth and Babuška for singularly perturbed problem. However, even the modified method is not robust in the case of anisotropic meshes. The present work modifies the equilibrated residual method for anisotropic meshes. The resulting error estimator is equivalent to the equilibrated residual method in the case of isotropic meshes and is proved to be robust on anisotropic meshes as well. A numerical example confirms the theory.

10 
A Posteriori Error Analysis for a Discontinuous Galerkin Method Applied to Hyperbolic Problems on Tetrahedral MeshesMechaii, Idir 26 April 2012 (has links)
In this thesis, we present a simple and efficient \emph{a posteriori} error estimation procedure for a discontinuous finite element method applied to scalar firstorder hyperbolic problems on structured and unstructured tetrahedral meshes. We present a local error analysis to derive a discontinuous Galerkin orthogonality condition for the leading term of the discretization error and find basis functions spanning the error for several finite element spaces. We describe an implicit error estimation procedure for the leading term of the discretization error by solving a local problem on each tetrahedron. Numerical computations show that the implicit \emph{a posteriori} error estimation procedure yields accurate estimates for linear and nonlinear problems with smooth solutions. Furthermore, we show the performance of our error estimates on problems with discontinuous solutions.
We investigate pointwise superconvergence properties of the discontinuous Galerkin (DG) method using enriched polynomial spaces. We study the effect of finite element spaces on the superconvergence properties of DG solutions on each class and type of tetrahedral elements. We show that, using enriched polynomial spaces, the discretization error on tetrahedral elements having one inflow face, is O(h^{p+2}) superconvergent on the three edges of the inflow face, while on elements with one inflow and one outflow faces the DG solution is O(h^{p+2}) superconvergent on the outflow face in addition to the three edges of the inflow face. Furthermore, we show that, on tetrahedral elements with two inflow faces, the DG solution is O(h^{p+2}) superconvergent on the edge shared by two of the inflow faces. On elements with two inflow and one outflow faces and on elements with three inflow faces, the DG solution is O(h^{p+2}) superconvergent on two edges of the inflow faces. We also show that using enriched polynomial spaces lead to a simpler{a posterior error estimation procedure.
Finally, we extend our error analysis for the discontinuous Galerkin method applied to linear threedimensional hyperbolic systems of conservation laws with smooth solutions. We perform a local error analysis by expanding the local error as a series and showing that its leading term is O( h^{p+1}). We further simplify the leading term and express it in terms of an optimal set of polynomials which can be used to estimate the error. / Ph. D.

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