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Uncertainty Quantification of Geophysical Inversion Using Stochastic Partial Differential Equations (LDRD #218329)
This report summarizes work completed under the Laboratory Directed Research and Development (LDRD) project "Uncertainty Quantification of Geophysical Inversion Using Stochastic Differential Equations." Geophysical inversions often require computationally expensive algorithms to find even one solution, let alone propagating uncertainties through to the solution domain. The primary purpose of this project was to find more computationally efficient means to approximate solution uncertainty in geophysical inversions. We found multiple computationally efficient methods of propagating Earth model uncertainty into uncertainties in solutions of full waveform seismic moment tensor inversions. However, the optimum method of approximating the uncertainty in these seismic source solutions was to use the Karhunen-Love theorem with data misfit residuals. This method was orders of magnitude more computationally efficient than traditional Monte Carlo methods and yielded estimates of uncertainty that closely approximated those of Monte Carlo. We will summarize the various methods we evaluated for estimating uncertainty in seismic source inversions as well as work toward this goal in the realm of 3-D seismic tomographic inversion uncertainty.
The Mallat Scattering Transform fro reduced order modeling of partial differential equations.
Abstract not provided.
Mesh Optimization and Basis Compression]{Mesh Optimization and Basis Compression for Extreme-Scale Solution of Partial Differential Equations
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Basis Compression for Extreme-Scale Solution of Partial Differential Equations
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Hybrid computer techniques for solving partial differential equations
Techniques overcome equipment limitations that restrict other computer techniques in solving trivial cases. The use of curve fitting by quadratic interpolation greatly reduces required digital storage space.
Similarity analysis of partial differential equations Final report
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On the numerical solution of elliptic partial differential equations by the method of lines.
Abstract missing.
Numerical solution of nonlinear partial differential equations of mixed type
A review is presented of some recently developed numerical methods for the solution of nonlinear equations of mixed type. The methods considered use finite difference approximations to the differential equation. Central difference formulas are employed in the subsonic zone and upwind difference formulas are used in the supersonic zone. The relaxation method for the small disturbance equation is discussed and a description is given of difference schemes for the potential flow equation in quasi-linear form. Attention is also given to difference schemes for the potential flow equation in conservation form, the analysis of relaxation schemes by the time dependent analogy, the accelerated iterative method, and three-dimensional calculations.
Simplified clustering of nonorthogonal grids generated by elliptic partial differential equations
A simple clustering transformation is combined with the Thompson, Thames, and Mastin (TTM) method of generating computational grids to produce controlled mesh spacings. For various practical grids, the resulting hybrid scheme is easier to apply than the inhomogeneous clustering terms included in the TTM method for this purpose. The technique is illustrated in application to airfoil problems, and listings of a FORTRAN computer code for this usage are included.
On several aspects and applications of the multigrid method for solving partial differential equations
Several aspects of multigrid methods are briefly described. The main subjects include the development of very efficient multigrid algorithms for systems of elliptic equations (Cauchy-Riemann, Stokes, Navier-Stokes), as well as the development of control and prediction tools (based on local mode Fourier analysis), used to analyze, check and improve these algorithms. Preliminary research on multigrid algorithms for time dependent parabolic equations is also described. Improvements in existing multigrid processes and algorithms for elliptic equations were studied.
Use of hyperbolic partial differential equations to generate body fitted coordinates
The hyperbolic scheme is used to efficiently generate smoothly varying grids with good step size control near the body. Although only two dimensional applications are presented, the basic concepts are shown to extend to three dimensions.
Finite difference numerical methods for boundary control problems governed by hyperbolic partial differential equations
This paper briefly reviews convergent finite difference schemes for hyperbolic initial boundary value problems and their applications to boundary control systems of hyperbolic type which arise in the modelling of vibrations. These difference schemes are combined with the primal and the dual approaches to compute the optimal control in the unconstrained case, as well as the case when the control is subject to inequality constraints. Some of the preliminary numerical results are also presented.
Spectral methods for partial differential equations
Origins of spectral methods, especially their relation to the Method of Weighted Residuals, are surveyed. Basic Fourier, Chebyshev, and Legendre spectral concepts are reviewed, and demonstrated through application to simple model problems. Both collocation and tau methods are considered. These techniques are then applied to a number of difficult, nonlinear problems of hyperbolic, parabolic, elliptic, and mixed type. Fluid dynamical applications are emphasized.
Spectral methods for partial differential equations
Origins of spectral methods, especially their relation to the Method of Weighted Residuals, are surveyed. Basic Fourier, Chebyshev, and Legendre spectral concepts are reviewed, and demonstrated through application to simple model problems. Both collocation and tau methods are considered. These techniques are then applied to a number of difficult, nonlinear problems of hyperbolic, parabolic, elliptic, and mixed type. Fluid-dynamical applications are emphasized.
Construction of finite difference schemes having special properties for ordinary and partial differential equations
Work on the construction of finite difference models of differential equations having zero truncation errors is summarized. Both linear and nonlinear unidirectional wave equations are discussed. Results regarding the construction of zero truncation error schemes for the full wave equation and Burger's equation are also briefly reported.
Generation of Surface Coordinates by Elliptic Partial Differential Equations
The problem of generating spatial coordinates by numerical methods through carefully selected mathematical models is of current interest both in mechanics and physics. The problem of generation of a desired system of coordinates in a given surface was considered, which essentially is an effort directed to the problem of grid generation in a two-dimensional non-Euclidean space. The mathematical model selected for this purpose is based on the formulae of Gauss for a surface. The proposed equations can be used to generate a new coordinate system from the data of an already given coordinate system in a surface. If the coefficients of the first and second fundamental forms have been given, then the proposed equations can be used to generate a surface satisfying the given data (surface fitting). The proposed equations can also be used to generate surfaces in the space between two arbitrary given surfaces, thus providing 3D grids in an Euclidean space.