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Leonid Knizhnerman

Leonid Knizhnerman
G. I. Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences

Dr. of Phys. and Math. Sci. (Comput. Mathematics)

About

107
Publications
10,780
Reads
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2,900
Citations
Additional affiliations
November 1980 - present
Central Geophysical Expedition
Position
  • A chief specialist
Description
  • Computational Mathematics, Computational Geophysics
Education
September 1972 - October 1980
Moscow State Pedagogical Institute
Field of study
  • Mathematics

Publications

Publications (107)
Preprint
Full-text available
We propose algorithms for efficient time integration of large systems of oscillatory second order ordinary differential equations (ODEs) whose solution can be expressed in terms of trigonometric matrix functions. Our algorithms are based on a residual notion for second order ODEs, which allows to extend the ``residual-time restarting'' Krylov subsp...
Article
Rational approximation recently emerged as an efficient numerical tool for the solution of exterior wave propagation problems. Currently, this technique is limited to wave media which are invariant along the main propagation direction. We propose a new model order reduction-based approach for compressing unbounded waveguides with layered inclusions...
Preprint
Full-text available
Rational approximation recently emerged as an efficient numerical tool for the solution of exterior wave propagation problems. Currently, this technique is limited to wave media which are invariant along the main propagation direction. We propose a new model order reduction-based approach for compressing unbounded waveguides with layered inclusions...
Preprint
An efficient Krylov subspace algorithm for computing actions of the $\varphi$ matrix function for large matrices is proposed. This matrix function is widely used in exponential time integration, Markov chains and network analysis and many other applications. Our algorithm is based on a reliable residual based stopping criterion and a new efficient...
Presentation
Full-text available
Title: Network realizations of data-driven reduced order models and remote sensing Location: 54-209 (Nafi Toksöz seminar room) Date: Friday - Nov 15, 2019 Time: 12:00 pm to 1 pm Abstract: Network synthesis was at the base of modern electronics design and consecutively of model order reduction that tremendously impacted many areas of engineering b...
Article
In this paper a new restarting method for Krylov subspace matrix exponential evaluations is proposed. Since our restarting technique essentially employs the residual, some convergence results for the residual are given. We also discuss how the restart length can be adjusted after each restart cycle, which leads to an adaptive restarting procedure....
Preprint
Full-text available
In this paper a new restarting method for Krylov subspace matrix exponential evaluations is proposed. Since our restarting technique essentially employs the residual, some convergence results for the residual are given. We also discuss how the restart length can be adjusted after each restart cycle, which leads to an adaptive restarting procedure....
Data
A new construction of an absorbing boundary condition for indefinite Helmholtz problems on unbounded domains is presented. This construction is based on a near-best uniform rational interpolant of the inverse square root function on the union of a negative and positive real interval, designed with the help of a classical result by Zolotarev. Using...
Article
Full-text available
Rational Arnoldi is a powerful method for approximating functions of large sparse matrices times a vector. The selection of asymptotically optimal parameters for this method is crucial for its fast convergence. We present and investigate a novel strategy for the automated parameter selection when the function to be approximated is of Cauchy–Stieltj...
Article
Rational Arnoldi is a powerful method for approximating functions of large sparse matrices times a vector. The selection of asymptotically optimal parameters for this method is crucial for its fast convergence. We present a heuristic for the automated pole selection when the function to be approximated is of Markov type, such as the matrix square r...
Article
Full-text available
Time-domain problems for controlled-source electromagnetic exploration require accurate discretization of the solution for multiple spacial and temporal scales. Therefore, forward simulation using conventional computational methods becomes computationally expensive, even without accounting for induced-polarization (IP) effects. These effects create...
Article
Full-text available
The extended Krylov subspace method has recently arisen as a competitive method for solving large-scale Lyapunov equations. Using the theoretical framework of orthogonal rational functions, in this paper we provide a general a priori error estimate when the known term has rank-one. Special cases, such as symmetric coefficient matrix, are also treat...
Article
Full-text available
The modeling of the controlled-source electromagnetic (CSEM) and single-well and crosswell electromagnetic (EM) configurations requires fine gridding to take into account the 3D nature of the geometries encountered in these applications that include geological structures with complicated shapes and exhibiting large variations in conductivities such...
Chapter
We developed a 2.5D finite-difference (FD) code for modeling EM tool responses for 2D formation conductivity distributions and 3D well trajectories. The code is primarily developed for well placement and for 3D formation evaluation applications to interpret responses of the new generation LWD deep directional EM tools and tensor induction tools in...
Article
Full-text available
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...
Article
Stability of passing from Gaussian quadrature data to the Lanczos recurrence coefficients is considered. Special attention is paid to estimates explicitly expressed in terms of quadrature data and not having weights in denominators. It has been shown that the recent approach, exploiting integral representation of Hankel determinants, implies quanti...
Article
Study of Padé-Faber approximation (generalization of Padéapproximation and Padé-Chebyshev approximation) of Markov functions is important not only from the point of view of mathematical analysis, but also of computational mathematics. The theorem on the existence of subdiagonal approximants is constructively proved. Various estimates of the approxi...
Article
In this abstract we present a fast and robust algorithm for solution of the 3D time-domain Maxwell's equations. We combine the rational Krylov subspace reduction ideas, the rational Arnoldi method and optimization of the subspace choice. We reduce the problem to the frequency domain and construct our rational Krylov subspace based on the Laplace fr...
Article
Full-text available
We solve an electromagnetic frequency domain induction problem in ℝ 3 for a frequency interval using rational Krylov subspace (RKS) approximation. The RKS is constructed by spanning on the solutions for a certain a priori chosen set of frequencies. We reduce the problem of the optimal choice of these frequencies to the third Zolotaryov problem in t...
Article
Full-text available
We consider the computation of u(t)=exp(-tA)φ using rational Krylov subspace reduction for 0≤t<∞, where u(t),φ∈ℝ N and 0<A=A * ∈ℝ N×N . The objective of this work is the optimization of the shifts for the rational Krylov subspace (RKS). We consider this problem in the frequency domain and reduce it to a classical Zolotaryov problem. The latter yiel...
Article
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...
Article
Full-text available
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 simulati...
Article
The efficiency of Gauss-Arnoldi quadrature for the calculation of the quantity is studied, where is a bounded operator in a Hilbert space and is a non-trivial vector in this space. A necessary and a sufficient conditions are found for the efficiency of the quadrature in the case of a normal operator. An example of a non-normal operator for which th...
Article
Full-text available
We developed two algorithms for solving the nonlinear electromagnetic inversion problem in the Earth. To achieve a balance between efficiency and robustness, both algorithms employ the Gauss-Newton inversion method. Moreover, to speed up the inversion's computational time, the so-called optimal grid technique is utilized. The first algorithm uses a...
Article
Full-text available
We present two-and-half-dimensional (2.5D) forward and inversion algorithms for the interpretation of Marine controlled-source electromagnetic (CSEM) data. The forward algorithm employs a frequency domain finite difference solution of Maxwell equation. Fast computational times are achieved through the use of a) the optimal grid techniques to extend...
Article
Full-text available
We develop a parametric inversion algorithm to determine simultaneously the horizontal and vertical resistivities of both the formation and invasion zones, invasion radius, bed boundary upper location and thickness, and relative dip angle from electromagnetic triaxial induction logging data. This is a full 3D inverse scattering problem in transvers...
Article
We present two-dimensional forward and inversion algorithms for the interpretation of marine controlled-source electromagnetic data. The forward algorithm employs a staggered-grid finite difference solution to the total-electricfield Helmholtz equation. Quick solution times are achieved through a) an optimal grid technique that extends the boundari...
Article
Full-text available
The modelling of the marine electromagnetic (EM) problem requires fine gridding to account for the seafloor bathymetry and to model complicated targets. This makes the computational cost of the problem large by using conventional Finite-Difference (FD) solvers. To circumvent these problems, we employ a volume Integral Equation (IE) approach to arri...
Conference Paper
We present a 2.5D inversion algorithm for the interpretation of electromagnetic data collected in a cross‐well configuration. Some inversion results from simulated data as well as from field measurements are presented in order to show the efficiency and the robustness of the algorithm.
Article
We consider finite difference approximations of solutions of inverse Sturm-Liouville problems in bounded intervals. Using three-point finite difference schemes, we discretize the equations on so-called optimal grids constructed as follows: For a staggered grid with 2 k points, we ask that the finite difference operator (a k × k Jacobi matrix) and t...
Article
We obtain novel, explicit formulas for the sensitivity of Jacobi matrices to small pertur-bations of their spectra. Our derivation is based on the connection between Lanczos's algo-rithm and the discrete Gel'fand–Levitan inverse spectral method. We prove uniform stability of Lanczos recursions in discrete primitive norms, for perturbations of the e...
Article
To minimize acoustic noise, designers of sonic logging tools often consider coatings of viscoelastic materials with very high attenuation properties. Efficient finite‐difference modeling of viscoelastic materials is a topic of current research. To model viscoelastic materials in the time domain through finite differences efficiently, one needs to r...
Article
Perturbation bounds for the Jacobi inverse eigenvalue problem (JIEP), which are more realistic than the earlier ones, are proved and illustrated by numerical experiments. The technique of orthonormal polynomials and integral representation of Hankel determinants is used. The same technique is then applied to the unitary Hessenberg inverse eigenvalu...
Conference Paper
Full-text available
We present a parametric inversion algorithm to determine formation resistivity anisotropy, invasion zone anisotropy, invasion radius, bed boundary location, relative dip and azimuth angles from multi‐component mult spacing induction logging data. This is a full vectorial three‐dimensional inverse scattering problem in anisotropic medium. In order t...
Article
Full-text available
We develop and validate a novel numerical algorithm for the simulation of axisymmetric single-phase fluid flow phenomena in porous and permeable media. In this new algorithm, the two-dimensional parabolic partial differential equation for fluid flow is transformed into an explicit finite-difference operator problem. The latter is solved by making u...
Article
Full-text available
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 opt...
Article
This work is the sequel to S. Asvadurov et al. (2000, J. Comput. Phys.158, 116), where we considered a grid refinement approach for second-order finite-difference time domain schemes. This approach permits one to compute solutions of certain wave equations with exponential superconvergence. An algorithm was presented that generates a special sequen...
Article
We develop a solution to the nonlinear inverse problem via a cascade sequence of auxiliary least-squares minimizations. The auxiliary minimizations are nonlinear inverse problems themselves, except that they are implemented with an approximate forward problem that is at least an order of magnitude faster to solve than the algorithm used to simulate...
Article
Full-text available
Earlier the authors suggested an algorithm of grid optimization for a second order finite-difference approximation of a two-point problem. The algorithm yields exponential superconvergence of the Neumann-to-Dirichlet map (or the boundary impedance). Here we extend that approach to PDEs with piecewise-constant coefficients and rectangular homogeneou...
Article
The main objective of this paper is optimization of second-order finite difference schemes for elliptic equations, in particular, for equations with singular solutions and exterior problems. A model problem corresponding to the Laplace equation on a semi-infinite strip is considered. The boundary impedance (Neumann-to-Dirichlet map) is computed as...
Article
The main objective of this paper is optimization of second-order finite difference schemes for elliptic equations, in particular, for equations with singular solutions and exterior problems. A model problem corresponding to the Laplace equation on a semi-infinite strip is considered. The boundary impedance (Neumann-to-Dirichlet map) is computed as...
Article
Given a bounded linear operator A in a Hilbert space $\calH$ and a nonzero vector $\mbox{\eufm r}\in\calH$, we construct a unitary operator U and (under some conditions) bounded self-adjoint operators P and T (nonnegative definite and indefinite, respectively) such that all the residual Krylov subspaces of $(A,\mbox{\eufm r})$, $(U,\mbox{\eufm r})$...
Article
Two of the authors earlier suggested a method of calculating special grid steps for three point finite-difference schemes which yielded exponential superconvergence of the Neumann-to-Dirichlet map. We apply this approach to solve the two-dimensional time-domain wave problem and the 2.5-D elasticity system in cylindrical coordinates. Our numerical e...
Article
Exponential estimates are proved for the error of computing a finite set of extreme eigenvalues and corresponding eigenvectors of a bounded operator by means of the Arnoldi method. An analogous estimate for computation of an operator function is given. Numerical examples are presented.
Article
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 understa...
Article
Full-text available
We suggest an approach to grid optimization for a second order finite-difference scheme for elliptic equations. A model problem corresponding to the three-point finite-difference semidiscretization of the Laplace equation on a semi-infinite strip is considered. We relate the approximate boundary Neumann-to-Dirichlet map to a rational function and c...
Article
A new result on the distribution of Ritz values at two consecutive steps of a simple Lanczos process is proved. It is used in showing why the Lanczos method effectively solves indefinite symmetric linear systems in computer arithmetic. The result of a numerical experiment is presented.
Article
Full-text available
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...
Article
Full-text available
The Lanczos algorithm uses a three-term recurrence to construct an orthonormal basis for the Krylov space corresponding to a symmetric matrix $A$ and a nonzero starting vector $\varphi$. The vectors and recurrence coefficients produced by this algorithm can be used for a number of purposes, including solving linear systems $Au= \varphi$ and computi...
Article
Full-text available
Computationally expensive time-stepping is the bottleneck of finite-difference methods used for valuing multi-asset options. The authors consider a novel algorithm with radically accelerated convergence, which is based on an optimal approximation of the matrix exponential. This algorithm is modified to compute the price of American options. A reduc...
Article
Estimates of the error of the Gauss-type quadrature formula generated by m steps of the simple Lanczos process are proved. Some applications are described. A numerical example is given.
Article
Full-text available
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 implementation of that approach for symmetric matrices, which we call the spectral lanczos decomposition method (SLDM). We have proved a general convergence estimate, re...
Article
A new error estimate for the approximation of an isolated eigenvalue in the Lanczos procedure is proved. It is confirmed by numerical experiment that no qualitative improvement of this estimate is possible. Results on the partitioning of the Ritz numbers into narrow clusters are established.
Article
A three-dimensional well model (r − θ − z) for the simulation of single-phase fluid flow in porous media is developed. Rather than directly solving the 3-D parabolic PDE (partial differential equation) for fluid flow, the PDE is transformed to a linear operator problem that is defined as u = f(A)σ, where A is a real symmetric square matrix and σ is...
Article
Full-text available
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 estimate...
Article
Let A be a square symmetric n × n matrix, φ be a vector from Rn, and f be a function defined on the spectral interval of A. The problem of computation of the vector u = f(A)φ arises very often in mathematical physics. We propose the following method to compute u. First, perform m steps of the Lanczos method with A and φ. Define the spectral Lanczos...
Article
Let the Cauchy problem for a symmetrical homogeneous ODE system be solved by a difference scheme and let s be the required number of matrix-vector operations with the finite-difference matrix. In classical schemes s is proportional to the number of time steps. The Lanczos method is used to decrease s without essential increase of error. A theoretic...
Article
The interpretation of long-offset transient electromagnetic (LOTEM) data is usually based on layered earth models. Effects of lateral conductivity variations are commonly explained qualitatively, because three-dimensional (3-D) numerical modeling is not readily available for complex geology. One of the first quantitative 3-D interpretations of LOTE...
Article
Two aspects of Arnoldi's method are examined: the convergence of the "outermost" eigenpairs of a normal matrix and the use of the method to compute matrix-valued functions with singularities lying among the outermost eigenvalues. Error bounds are established and results of numerical experiments are described and a theoretical example confirms that...
Article
Arnoldi's method for the approximate computation of the spectrum of an unsymmetric matrix consists of projecting the problem onto a Krylov subspace, constructing an orthonormal basis for the latter and solving the resulting spectral problem for a Hessenberg matrix. The method is used to compute the product of a function of matrices and a vector. An...
Article
Full-text available
Error bounds when evaluating functions of matrices by the simple Lanczos procedure, under conditions of machine arithmetic are derived. These bounds in turn yield estimates for the convergence of the computed eigenvalues, which explain the Lanczos phenomenon. The applicability of the bounds is ascertained.