24 research outputs found

    Using Deep Neural Networks for Detecting Spurious Oscillations in Discontinuous Galerkin Solutions of Convection-Dominated Convection–Diffusion Equations

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    Standard discontinuous Galerkin finite element solutions to convection-dominated convection–diffusion equations usually possess sharp layers but also exhibit large spurious oscillations. Slope limiters are known as a post-processing technique to reduce these unphysical values. This paper studies the application of deep neural networks for detecting mesh cells on which slope limiters should be applied. The networks are trained with data obtained from simulations of a standard benchmark problem with linear finite elements. It is investigated how they perform when applied to discrete solutions obtained with higher order finite elements and to solutions for a different benchmark problem

    Using deep neural networks for detecting spurious oscillations in discontinuous Galerkin solutions of convection-dominated convection-diffusion equations

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    Standard discontinuous Galerkin (DG) finite element solutions to convection-dominated con- vection-diffusion equations usually possess sharp layers but also exhibit large spurious oscillations. Slope limiters are known as a post-processing technique to reduce these unphysical values. This paper studies the application of deep neural networks for detecting mesh cells on which slope limiters should be applied. The networks are trained with data obtained from simulations of a standard benchmark problem with linear finite elements. It is investigated how they perform when applied to discrete solutions obtained with higher order finite elements and to solutions for a different benchmark problem

    Doctor of Philosophy

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    dissertationPartial differential equations (PDEs) are widely used in science and engineering to model phenomena such as sound, heat, and electrostatics. In many practical science and engineering applications, the solutions of PDEs require the tessellation of computational domains into unstructured meshes and entail computationally expensive and time-consuming processes. Therefore, efficient and fast PDE solving techniques on unstructured meshes are important in these applications. Relative to CPUs, the faster growth curves in the speed and greater power efficiency of the SIMD streaming processors, such as GPUs, have gained them an increasingly important role in the high-performance computing area. Combining suitable parallel algorithms and these streaming processors, we can develop very efficient numerical solvers of PDEs. The contributions of this dissertation are twofold: proposal of two general strategies to design efficient PDE solvers on GPUs and the specific applications of these strategies to solve different types of PDEs. Specifically, this dissertation consists of four parts. First, we describe the general strategies, the domain decomposition strategy and the hybrid gathering strategy. Next, we introduce a parallel algorithm for solving the eikonal equation on fully unstructured meshes efficiently. Third, we present the algorithms and data structures necessary to move the entire FEM pipeline to the GPU. Fourth, we propose a parallel algorithm for solving the levelset equation on fully unstructured 2D or 3D meshes or manifolds. This algorithm combines a narrowband scheme with domain decomposition for efficient levelset equation solving

    On slope limiting and deep learning techniques for the numerical solution to convection-dominated convection-diffusion problems

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    As the first main topic, several slope-limiting techniques from the literature are presented, and various novel methods are proposed. These post-processing techniques aim to automatically detect regions where the discrete solution has unphysical values and approximate the solution locally by a lower degree polynomial. This thesis's first major contribution is that two novel methods can reduce the spurious oscillations significantly and better than the previously known methods while preserving the mass locally, as seen in two benchmark problems with two different diffusion coefficients. The second focus is showing how to incorporate techniques from machine learning into the framework of classical finite element methods. Hence, another significant contribution of this thesis is the construction of a machine learning-based slope limiter. It is trained with data from a lower-order DG method from a particular problem and applied to a higher-order DG method for the same and a different problem. It reduces the oscillations significantly compared to the standard DG method but is slightly worse than the classical limiters. The third main contribution is related to physics-informed neural networks (PINNs) to approximate the solution to the model problem. Various ways to incorporate the Dirichlet boundary data, several loss functionals that are novel in the context of PINNs, and variational PINNs are presented for convection-diffusion-reaction problems. They are tested and compared numerically. The novel loss functionals improve the error compared to the vanilla PINN approach. It is observed that the approximations are free of oscillations and can cope with interior layers but have problems capturing boundary layers

    Goal-oriented inference : theoretical foundations and application to carbon capture and storage

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    Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2013.This electronic version was submitted and approved by the author's academic department as part of an electronic thesis pilot project. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from department-submitted PDF version of thesis.Includes bibliographical references (p. 127-132).Many important engineering problems require computation of prediction output quantities of interest that depend on unknown distributed parameters of the governing partial differential equations. Examples include prediction of concentration levels in critical areas for contamination events in urban areas and prediction of trapped volume of supercritical carbon dioxide in carbon capture and storage. In both cases the unknown parameter is a distributed quantity that is to be inferred from indirect and sparse data in order to make accurate predictions of the quantities of interest. Traditionally parameter inference involves regularization in deterministic formulations or specification of a prior probability density in Bayesian statistical formulations to resolve the ill-posedness manifested in the many possible parameters giving rise to the same observed data. Critically, the final prediction requirements are not considered in the inference process. Goal-oriented inference, on the other hand, utilizes the prediction requirements to drive the inference process. Since prediction quantities of interest are often very low-dimensional, the same ill-posedness that stymies the inference process can be exploited when inference of the parameter is undertaken solely to obtain predictions. Many parameters give rise to the same predictions; as a result, resolving the parameter is not required in order to accurately make predictions. In goal-oriented inference, we exploit this fact to obtain fast and accurate predictions from experimental data by sacrificing accuracy in parameter estimation. When the governing models for experimental data and prediction quantities of interest depend linearly on the parameter, a linear algebraic analysis reveals a dimensionally-optimal parameter subspace within which inference proceeds. Parameter estimates are inaccurate but the resulting predictions are identical to those achieved by first performing inference in the full high-dimensional parameter space and then computing predictions. The analysis required to identify the parameter subspace reveals inefficiency in experiment and sources of uncertainty in predictions, which can also be utilized in experimental design. Linear goal-oriented inference is demonstrated on a model problem in contaminant source inversion and prediction. In the nonlinear setting, we focus on the Bayesian statistical inverse problem formulation where the target of our goal-oriented inference is the posterior predictive probability density function representing the relative likelihood of predictions given the observed experimental data. In many nonlinear settings, particularly those involving nonlinear partial differential equations, distributed parameter estimation remains an unsolved problem. We circumvent estimation of the parameter by establishing a statistical model for the joint density of experimental data and predictions using either a Gaussian mixture model or kernel density estimate derived from simulated experimental data and simulated predictions based on parameter samples from the prior distribution. When experiments are conducted and data are observed, the statistical model is conditioned on the observed data, and the posterior predictive probability density is obtained. Nonlinear goal-oriented inference is applied to a realistic application in carbon capture and storage.by Chad Lieberman.Ph.D

    Generalized averaged Gaussian quadrature and applications

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    A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

    MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

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    Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

    ISCR Annual Report: Fical Year 2004

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