Computation and Learning in High Dimensions (hybrid meeting)

The most challenging problems in science often involve the learning and accurate computation of high dimensional functions. High-dimensionality is a typical feature for a multitude of problems in various areas of science. The so-called curse of dimensionality typically negates the use of traditional numerical techniques for the solution of high-dimensional problems. Instead, novel theoretical and computational approaches need to be developed to make them tractable and to capture fine resolutions and relevant features. Paradoxically, increasing computational power may even serve to heighten this demand, since the wealth of new computational data itself becomes a major obstruction. Extracting essential information from complex problem-inherent structures and developing rigorous models to quantify the quality of information in a high-dimensional setting pose challenging tasks from both theoretical and numerical perspective. This has led to the emergence of several new computational methodologies, accounting for the fact that by now well understood methods drawing on spatial localization and mesh-refinement are in their original form no longer viable. Common to these approaches is the nonlinearity of the solution method. For certain problem classes, these methods have drastically advanced the frontiers of computability. The most visible of these new methods is deep learning. Although the use of deep neural networks has been extremely successful in certain application areas, their mathematical understanding is far from complete. This workshop proposed to deepen the understanding of the underlying mathematical concepts that drive this new evolution of computational methods and to promote the exchange of ideas emerging in various disciplines about how to treat multiscale and high-dimensional problems

Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations

Computational costs of numerically solving multidimensional partial differential equations (PDEs) increase significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points. For multidimensional reaction-diffusion equations, stiffness of the system provides additional challenges for achieving efficient numerical simulations. In this paper, we propose a class of Krylov implicit integration factor (IIF) discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion equations on high spatial dimensions. The key ingredient of spatial DG discretization is the multiwavelet bases on nested sparse grids, which can significantly reduce the numbers of degrees of freedom. To deal with the stiffness of the DG spatial operator in discretizing reaction-diffusion equations, we apply the efficient IIF time discretization methods, which are a class of exponential integrators. Krylov subspace approximations are used to evaluate the large size matrix exponentials resulting from IIF schemes for solving PDEs on high spatial dimensions. Stability and error analysis for the semi-discrete scheme are performed. Numerical examples of both scalar equations and systems in two and three spatial dimensions are provided to demonstrate the accuracy and efficiency of the methods. The stiffness of the reaction-diffusion equations is resolved well and large time step size computations are obtained

Space-time residual minimization for parabolic partial differential equations

Many processes in nature and engineering are governed by partial differential equations (PDEs). We focus on parabolic PDEs, that describe time-dependent phenomena like heat conduction, chemical concentration, and fluid flow. Even if we know that a unique solution exists, we can express it in closed form only under very strict circumstances. So, to understand what it looks like, we turn to numerical approximation. Historically, parabolic PDEs are solved using time-stepping. One first discretizes the PDE in space as to obtain a system of coupled ordinary differential equations in time. This system is then solved using the vast theory for ODEs. While efficient in terms of memory and computational cost, time-stepping schemes take global time steps, which are independent of spatial position. As a result, these methods cannot efficiently resolve details in localized regions of space and time. Moreover, being inherently sequential, they have limited possibilities for parallel computation. In this thesis, we take a different approach and reformulate the parabolic evolution equation as an equation posed in space and time simultaneously. Space-time methods mitigate the aforementioned issues, and moreover produce approximations to the unknown solution that are uniformly quasi-optimal. The focal point of this thesis is the space-time minimal residual (MR) method introduced by R. Andreev, that finds the approximation that minimizes both PDE- and initial error. We discuss its theoretical properties, provide numerical algorithms for its computation, and discuss its applicability in data assimilation (the problem of fusing measured data to its underlying PDE)

Multiscale Method for Elastic Wave Propagation in the Heterogeneous, Anisotropic Media

Seismic wave simulation in realistic Earth media with full wavefield methods is a fundamental task in geophysical studies. Conventional approaches such as the finite-difference method and the finite-element method solve the wave equation in geological models represented with discrete grids and elements. When the Earth model includes complex heterogeneities at multiple spatial scales, the simulation requires fine discretization and therefore a system with many degrees of freedom, which often exceeds current computational abilities. In this dissertation, I address this problem by proposing new multiscale methods for simulating elastic wave propagation based on previously developed algorithms for solving the elliptic partial differential equations and the acoustic wave equation. The fundamental motivation for developing the multiscale method is that it can solve the wave equation on a coarsely discretized mesh by incorporating the effects of fine-scale medium properties using so-called multiscale basis functions. This can greatly reduce computation time and degrees of freedom compared with conventional methods. I first derive a numerical homogenization method for arbitrarily heterogeneous, anisotropic media that utilizes the multiscale basis functions determined from a local linear elasticity equation to compute effective, anisotropic properties, and these equivalent elastic medium parameters can be used directly in existing elastic modeling algorithms. Then I extend the approach by constructing multiple basis functions using two types of appropriately defined local spectral linear elasticity problems. Given the eigenfunctions determined from local spectral problems, I develop a generalized multiscale finite-element method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media in both continuous Galerkin (CG) and discontinuous Galerkin (DG) formulations. The advantage of the multiscale basis functions is they are model-dependent, unlike the predefined polynomial basis functions applied in conventional finite-element methods. For this reason, the GMsFEM can effectively capture the influence of fine-scale variation of the media. I present results for several numerical experiments to verify the effectiveness of both the numerical homogenization method and GMsFEM. These tests show that the effectiveness of the multiscale method relies on the appropriate choice of boundary conditions that are applied for the local problem in numerical homogenization method and on the selection of basis functions from a large set of eigenfunctions contained in local spectral problems in GMsFEM. I develop methods for solving both these problems, and the results confirm that the multiscale method can be powerful tool for providing accurate full wavefield solutions in heterogeneous, anisotropic media, yet with reduced computation time and degrees of freedom compared with conventional full wavefield modeling methods. Specially, I applied the DG-GMsFEM to the Marmousi-2 elastic model, and find that DG-GMsFEM can greatly reduce the computation time compared with continuous Galerkin (CG) FEM

Adaptive Algorithms

Overwhelming empirical evidence in computational science and engineering proved that self-adaptive mesh-generation is a must-do in real-life problem computational partial differential equations. The mathematical understanding of corresponding algorithms concerns the overlap of two traditional mathematical disciplines, numerical analysis and approximation theory, with computational sciences. The half workshop was devoted to the mathematics of optimal convergence rates and instance optimality of the Dörfler marking or the maximum strategy in various versions of space discretisations and time-evolution problems with all kind of applications in the efficient numerical treatment of partial differential equations

Cumulative reports and publications through December 31, 1990

This document contains a complete list of ICASE reports. Since ICASE reports are intended to be preprints of articles that will appear in journals or conference proceedings, the published reference is included when it is available

Research in progress in applied mathematics, numerical analysis, fluid mechanics, and computer science

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, fluid mechanics, and computer science during the period October 1, 1993 through March 31, 1994. The major categories of the current ICASE research program are: (1) applied and numerical mathematics, including numerical analysis and algorithm development; (2) theoretical and computational research in fluid mechanics in selected areas of interest to LaRC, including acoustics and combustion; (3) experimental research in transition and turbulence and aerodynamics involving LaRC facilities and scientists; and (4) computer science