Leonid Knizhnerman

2.5D forward and inverse modeling for interpreting low-frequency electromagnetic measurements

We present 2.5D fast and rigorous forward and inversion algorithms for deep electromagnetic (EM) applications that include crosswell and controlled-source EM measurements. The forward algorithm is based on a finite-difference approach in which a multifrontal LU decomposition algorithm simulates multisource experiments at nearly the cost of simulating one single-source experiment for each frequency of operation. When the size of the linear system of equations is large, the use of this noniterative solver is impractical. Hence, we use the optimal grid technique to limit the number of unknowns in the forward problem. The inversion algorithm employs a regularized Gauss-Newton minimization approach with a multiplicative cost function. By using this multiplicative cost function, we do not need a priori data to determine the so-called regularization parameter in the optimization process, making the algorithm fully automated. The algorithm is equipped with two regularization cost functions that allow us to reconstruct either a smooth or a sharp conductivity image. To increase the robustness of the algorithm, we also constrain the minimization and use a line-search approach to guarantee the reduction of the cost function after each iteration. To demonstrate the pros and cons of the algorithm, we present synthetic and field data inversion results for crosswell and controlled-source EM measurements.

Extended Krylov Subspaces: Approximation of the Matrix Square Root and Related Functions

We introduce an economical Gram--Schmidt orthogonalization on the extended Krylov subspace originated by actions of a symmetric matrix and its inverse. An error bound for a family of problems arising from the elliptic method of lines is derived. The bound shows that, for the same approximation quality, the diagonal variant of the extended subspaces requires about the square root of the dimension of the standard Krylov subspaces using only positive or negative matrix powers. An example of an application to the solution of a 2.5-D elliptic problem attests to the computational efficiency of the method for large-scale problems.

Spectral approach to solving three‐dimensional Maxwell's diffusion equations in the time and frequency domains

We describe a new explicit three-dimensional solver for the diffusion of electromagnetic fields in arbitrarily heterogeneous conductive media. The proposed method is based on a global Krylov subspace (Lanczos) approximation of the solution in the time and frequency domains. We derive solutions stable to spurious curl-free modes and provide estimates of the computer complexity involved in the calculations. Such estimates together with numerical experiments attest to a computationally efficient method suitable for large-scale problems. Also included are modeling examples drawn from practical geophysical applications.

New spectral Lanczos decomposition method for induction modeling in arbitrary 3-D geometry

Traditional resistivity tools are designed to function in vertical wells. In horizontal well environments, the interpretation of resistivity logs becomes much more difficult because of the nature of 3-D effects such as highly deviated bed boundaries and invasion. The ability to model these 3-D effects numerically can greatly facilitate the understanding of tool response in different formation geometries. Three‐dimensional modeling of induction tools requires solving Maxwell’s equations in a discrete setting, either finite element or finite difference. The solutions of resulting discretized equations are computationally expensive, typically on the order of 30 to 60 minutes per log point on a workstation. This is unacceptable if the 3-D modeling code is to be used in interpreting induction logs. In this paper we propose a new approach for solutions to Maxwell’s equations. The new method is based on the spectral Lanczos decomposition method (SLDM) with Krylov subspaces generated from the inverse powers of th...

Krylov subspace approximation of eigenpairs and matrix functions in exact and computer arithmetic

Many researchers are now working on computing the product of a matrix function and a vector,using approximations in a Krylov subspace. We review our results on the analysis of one implemen-tation of that approach for symmetric matrices, which we call the Spectral Lanczos DecompositionMethod (SLDM).We have proved a general convergence estimate, relating SLDM error bounds to those obtainedthrough approximation of the matrix function by a part of its Chebyshev series. Thus, we arrivedat e ective estimates for matrix functions arising when solving parabolic, hyperbolic and ellipticpartial di erential equations. We concentrate on the parabolic case, where we obtain estimatesthat indicate superconvergence of SLDM. For this case a combination of SLDM and splittingmethods is also considered and some numerical results are presented.We implement our general estimates to obtain convergence bounds of Lanczos approximationsto eigenvalues in the internal part of the spectrum. Unlike Kaniel-Saad estimates, our estimatesare independent of the set of eigenvalues between the required one and the nearest spectrumbound.We consider an extension of our general estimate to the case of the simple Lanczos method(without reorthogonalization) in nite computer arithmetic which shows that for a moderatedimension of the Krylov subspace the results, proved for the exact arithmetic, are stable up toroundo .

A new investigation of the extended Krylov subspace method for matrix function evaluations

For large square matrices A and functions f, the numerical approximation of the action of f(A) to a vector v has received considerable attention in the last two decades. In this paper we investigate theextended Krylov subspace method, a technique that was recently proposed to approximate f(A)v for A symmetric. We provide a new theoretical analysis of the method, which improves the original result for A symmetric, and gives a new estimate for A nonsymmetric. Numerical experiments confirm that the new error estimates correctly capture the linear asymptotic convergence rate of the approximation. By using recent algorithmic improvements, we also show that the method is computationally competitive with respect to other enhancement techniques. Copyright

Analysis of the Rational Krylov Subspace and ADI Methods for Solving the Lyapunov Equation

For large scale problems, an effective approach for solving the algebraic Lyapunov equation consists of projecting the problem onto a significantly smaller space and then solving the reduced order matrix equation. Although Krylov subspaces have been used for a long time, only more recent developments have shown that rational Krylov subspaces can be a competitive alternative to the classical and very popular alternating direction implicit (ADI) recurrence. In this paper we develop a convergence analysis of the rational Krylov subspace method (RKSM) based on the Kronecker product formulation and on potential theory. Moreover, we propose new enlightening relations between this approach and the ADI method. Our results provide solid theoretical ground for recent numerical evidence of the superiority of RKSM over ADI when the involved parameters cannot be computed optimally, as is the case in many practical application problems.

On Optimal Finite-Difference Approximation of PML

A technique derived from two related methods suggested earlier by some of the authors for optimization of finite-difference grids and absorbing boundary conditions is applied to discretization of perfectly matched layer (PML) absorbing boundary conditions for wave equations in Cartesian coordinates. We formulate simple sufficient conditions for optimality and implement them. It is found that the minimal error can be achieved using pure imaginary coordinate stretching. As such, the PML discretization is algebraically equivalent to the rational approximation of the square root on [0,1] conventionally used for approximate absorbing boundary conditions. We present optimal solutions for two cost functions, with exponential (and exponential of the square root) rates of convergence with respect to the number of the discrete PML layers using a second order finite-difference scheme with optimal grids. Results of numerical calculations are presented.

Using Nonorthogonal Lanczos Vectors in the Computation of Matrix Functions

The Lanczos algorithm uses a three-term recurrence to construct an orthonormal basis for the Krylov space corresponding to a symmetric matrix

Optimal finite difference grids and rational approximations of the square root I. Elliptic problems

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