scholarly journals Iterative solution of the Lippmann-Schwinger equation in strongly scattering acoustic media by randomized construction of preconditioners

2020 ◽  
Author(s):  
Kjersti Solberg Eikrem ◽  
Geir Nævdal ◽  
Morten Jakobsen
Keyword(s):  
Numerical Experiments ◽  
Seismic Waves ◽  
Iterative Solution ◽  
Matrix Inversion ◽  
Constant Density ◽  
Scalar Wave ◽  
Born Series ◽  
Acoustic Media ◽  
Discretized Problem ◽  
Number Of Iterations

Summary In this work the Lippmann-Schwinger equation is used to model seismic waves in strongly scattering acoustic media. We consider the Helmholtz equation, which is the scalar wave equation in the frequency domain with constant density and variable velocity, and transform it to an integral equation of the Lippmann-Schwinger type. To directly solve the discretized problem with matrix inversion is time-consuming, therefore we use iterative methods. The Born series is a well-known scattering series which gives the solution with relatively small cost, but it has limited use as it only converges for small scattering potentials. There exist other scattering series with preconditioners that have been shown to converge for any contrast, but the methods might require many iterations for models with high contrast. Here we develop new preconditioners based on randomized matrix approximations and hierarchical matrices which can make the scattering series converge for any contrast with a low number of iterations. We describe two different preconditioners; one is best for lower frequencies and the other for higher frequencies. We use the fast Fourier transform (FFT) both in the construction of the preconditioners and in the iterative solution, and this makes the methods efficient. The performance of the methods are illustrated by numerical experiments on two 2D models.

Reconstruction of the Reservoir Fine Structure by Using Scattering Attributes

2021 ◽  
Author(s):  
Vladimir Cheverda ◽  
Vadim Lisitsa ◽  
Maksim Protasov ◽  
Galina Reshetova ◽  
Andrey Ledyaev ◽  
...  
Keyword(s):  
Seismic Data ◽  
Numerical Experiments ◽  
Seismic Waves ◽  
Fractured Reservoirs ◽  
Three Dimensional ◽  
Geological Structure ◽  
Discrete Elements ◽  
Digital Twin ◽  
The North ◽  
Slip Surfaces

Abstract To develop the optimal strategy for developing a hydrocarbon field, one should know in fine detail its geological structure. More and more attention has been paid to cavernous-fractured reservoirs within the carbonate environment in the last decades. This article presents a technology for three-dimensional computing images of such reservoirs using scattered seismic waves. To verify it, we built a particular synthetic model, a digital twin of one of the licensed objects in the north of Eastern Siberia. One distinctive feature of this digital twin is the representation of faults not as some ideal slip surfaces but as three-dimensional geological bodies filled with tectonic breccias. To simulate such breccias and the geometry of these bodies, we performed a series of numerical experiments based on the discrete elements technique. The purpose of these experiments is the simulation of the geomechanical processes of fault formation. For the digital twin constructed, we performed full-scale 3D seismic modeling, which made it possible to conduct fully controlled numerical experiments on the construction of wave images and, on this basis, to propose an optimal seismic data processing graph.

A CONSTRAINT PRECONDITIONER FOR SOLVING SYMMETRIC POSITIVE DEFINITE SYSTEMS AND APPLICATION TO THE HELMHOLTZ EQUATIONS AND POISSON EQUATIONS

2010 ◽  
Vol 15 (3) ◽  
pp. 299-311 ◽  
Author(s):  
Zhuo-Hong Huang ◽  
Ting-Zhu Huang
Keyword(s):  
Numerical Experiments ◽  
Positive Definite ◽  
Cholesky Factorization ◽  
Helmholtz Equations ◽  
Poisson Equations ◽  
Symmetric Positive Definite ◽  
Incomplete Cholesky Factorization ◽  
Preconditioned Matrix ◽  
Factorization Methods ◽  
Number Of Iterations

In this paper, first, by using the diagonally compensated reduction and incomplete Cholesky factorization methods, we construct a constraint preconditioner for solving symmetric positive definite linear systems and then we apply the preconditioner to solve the Helmholtz equations and Poisson equations. Second, according to theoretical analysis, we prove that the preconditioned iteration method is convergent. Third, in numerical experiments, we plot the distribution of the spectrum of the preconditioned matrix M−1A and give the solution time and number of iterations comparing to the results of [5, 19].

Iterative solution of the interest rate in a financial transaction: the hyperbolic method

1981 ◽  
Vol 108 (1) ◽  
pp. 75-84
Author(s):  
P. L. Durbin
Keyword(s):  
Interest Rate ◽  
Iterative Method ◽  
Iterative Solution ◽  
Financial Contracts ◽  
Rates Of Return ◽  
Rectangular Hyperbola ◽  
Comparable Accuracy ◽  
The Interest Rate ◽  
Hyperbolic Method ◽  
Number Of Iterations

1.1 This note describes an iterative method for determining the root of an equation, based on the assumption that the curve representing the equation is a rectangular hyperbola near the root.1.2. The method was derived for use in calculating the interest rate under financial contracts. Despite having a theoretically slower rate of convergence than Newton's method, in practice the hyperbolic method seems to require a fewer number of iterations for comparable accuracy. Moreover, it applies where the form of the function being evaluated is incapable of explicit description, and hence its derivative cannot be defined, such as where reinvestment rates of return are assumed.

scholarly journals Implementation of TAGE Method Using Seikkala Derivatives Applied to Two-Point Fuzzy Boundary Value Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
A. A. Dahalan ◽  
J. Sulaiman
Keyword(s):  
Boundary Value Problems ◽  
Numerical Experiments ◽  
Boundary Value ◽  
Linear Equations ◽  
Alternating Group ◽  
System Of Linear Equations ◽  
Fuzzy Boundary Value Problems ◽  
Fuzzy Boundary ◽  
Two Parameter ◽  
Number Of Iterations

Iterative methods particularly the Two-Parameter Alternating Group Explicit (TAGE) methods are used to solve system of linear equations generated from the discretization of two-point fuzzy boundary value problems (FBVPs). The formulation and implementation of the TAGE method are also presented. Then numerical experiments are carried out onto two example problems to verify the effectiveness of the method. The results show that TAGE method is superior compared to GS method in the aspect of number of iterations, execution time, and Hausdorff distance.

Electromagnetic modeling of 3-D structures by the method of system iteration using integral equations

Geophysics ◽  
1992 ◽  
Vol 57 (12) ◽  
pp. 1556-1561 ◽  
Author(s):  
Zonghou Xiong
Keyword(s):  
Integral Equations ◽  
Three Dimensional ◽  
Computation Time ◽  
Iterative Solution ◽  
Matrix Inversion ◽  
Electromagnetic Modeling ◽  
Direct Inversion ◽  
Conductivity Structure ◽  
The Matrix ◽  
Earth Conductivity

A new approach for electromagnetic modeling of three‐dimensional (3-D) earth conductivity structures using integral equations is introduced. A conductivity structure is divided into many substructures and the integral equation governing the scattering currents within a substructure is solved by a direct matrix inversion. The influence of all other substructures are treated as external excitations and the solution for the whole structure is then found iteratively. This is mathematically equivalent to partitioning the scattering matrix into many block submatrices and solving the whole system by a block iterative method. This method reduces computer memory requirements since only one submatrix at a time needs to be stored. The diagonal submatrices that require direct inversion are defined by local scatterers only and thus are generally better conditioned than the matrix for the whole structure. The block iterative solution requires much less computation time than direct matrix inversion or conventional point iterative methods as the convergence depends on the number of the submatrices, not on the total number of unknowns in the solution. As the submatrices are independent of each other, this method is suitable for parallel processing.

scholarly journals Iterative regularization for constrained minimization formulations of nonlinear inverse problems

2021 ◽  
Author(s):  
Barbara Kaltenbacher ◽  
Kha Van Huynh
Keyword(s):  
Inverse Problems ◽  
Numerical Experiments ◽  
Hybrid Imaging ◽  
Iterative Solution ◽  
Constrained Minimization ◽  
Acoustic Tomography ◽  
Iterative Regularization ◽  
Minimization Problems ◽  
Constrained Minimization Problems ◽  
Spatially Varying

AbstractIn this paper we study the formulation of inverse problems as constrained minimization problems and their iterative solution by gradient or Newton type methods. We carry out a convergence analysis in the sense of regularization methods and discuss applicability to the problem of identifying the spatially varying diffusivity in an elliptic PDE from different sets of observations. Among these is a novel hybrid imaging technology known as impedance acoustic tomography, for which we provide numerical experiments.

Theory of a 2.5-D acoustic wave equation for constant density media

Geophysics ◽  
1991 ◽  
Vol 56 (12) ◽  
pp. 2114-2117 ◽  
Author(s):  
Christopher L. Liner
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Acoustic Wave ◽  
Symmetry Axis ◽  
Constant Density ◽  
Efficient Computation ◽  
Acoustic Wave Equation ◽  
Out Of Plane ◽  
Acoustic Media ◽  
D Wave

The theory of 2.5-dimensional (2.5-D) wave propagation (Bleistein, 1986) allows efficient computation of 3-D wavefields in c(x, z) acoustic media when the source and receivers lie in a common y-plane (assumed to be y = 0 in this paper). It is really a method of efficiently computing an inplane 3-D wavefield in media with one symmetry axis. The idea is to raytrace the wavefield in the (x, z)-plane while allowing for out‐of‐plane spreading. In this way 3-D amplitude decay is honored without 3-D ray tracing. This theory has its conceptual origin in work by Ursin (1978) and Hubral (1978). Bleistein (1986) gives an excellent overview and detailed reference to earlier work.

scholarly journals Equivalent Boundary Conditions for Heterogeneous Acoustic Media

2014 ◽  
Vol 15 (3) ◽  
pp. 301
Author(s):  
Manuela Longoni De Castro ◽  
Julien Diaz ◽  
Victor Perón
Keyword(s):  
Boundary Conditions ◽  
Seismic Waves ◽  
Heterogeneous Media ◽  
Absorbing Boundary Conditions ◽  
Computational Domain ◽  
Absorbing Boundary ◽  
Equivalent Boundary ◽  
Artificial Boundaries ◽  
Acoustic Media ◽  
Propagation Of Waves

In this work, we have worked on possibilities to model artificial boundaries needed in the simulation of wave propagation in acoustic heterogeneous media.  Our motivation is to restrict the computational domain in the simulation of seismic waves that are propagated from the earth and transmitted to the stratified heterogeneous media composed by ocean and atmosphere. Two possibilities were studied and compared in computational tests: the use of absorbing boundary conditions on an artificial boundary in the atmosphere layer and the elimination of the atmosphere layer using an equivalent boundary condition that mimics the propagation of waves through the atmosphere. <br />

Numerical modeling of seismic waves with a 3-D, anisotropic scalar-wave equation

1991 ◽  
Vol 81 (3) ◽  
pp. 769-780
Author(s):  
Zhengxin Dong ◽  
George A. McMechan
Keyword(s):  
Wave Equation ◽  
Seismic Waves ◽  
Layered Medium ◽  
Grid Point ◽  
Scalar Wave ◽  
Scalar Wave Equation ◽  
Velocity Anisotropy ◽  
Isotropic Media ◽  
Isotropic Structure ◽  
Stress Orientations

Abstract By systematically defining the orientation and amount of velocity anisotropy at every point in a computational grid, and modifying the scalar-wave equation to accommodate directionally dependent velocity coefficients, scalar waves may be numerically synthesized in heterogeneous anisotropic 3-D structure by finite-differencing. The use of an intermediate, local, rotated coordinate system associated with each grid point allows the anisotropy orientation to conform spatially with 3-D structure, stress orientations, or any other correlate of the anisotropy. Both travel times and amplitudes in anisotropic media may differ significantly from those in the corresponding isotropic media. Under some conditions, the seismic response of an anisotropic flat-layered medium is nearly identical to, and may be confused with, that of a symmetrical isotropic structure. In general, these alternate interpretations can be evaluated by obtaining independent data from different recording configurations.

Sign in / Sign up

Export Citation Format

Share Document