A parallel method for large scale time domain electromagnetic inverse problems

Haber, Eldad

doi:10.1016/j.apnum.2007.01.017

Cited by 9 publications

(5 citation statements)

References 20 publications

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“…(5) The final step is to obtain μ a ¼ ffiffiffi ffi D p ∕μ 1 from the first of Eqs. (48). The key to this method is part (2), finding a solution to Eq.…”

Section: Direct "Zero-divergence" Methodsmentioning

confidence: 99%

“…For example, a Hessianbased inversion would need to compute, store, and invert a Hessian matrix with perhaps 10 16 elements on each iteration, which is currently impractical. Clearly, techniques to reduce the size of the problem and inversion procedures that are computationally light 47,48 are required. One way to reduce the number of unknowns is to divide the domain into a few regions on which the optical coefficients are assumed constant.…”

Section: Large-scale Inversions and Domain Parameterizationmentioning

confidence: 99%

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Quantitative spectroscopic photoacoustic imaging: a review

et al. 2012

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Abstract. Obtaining absolute chromophore concentrations from photoacoustic images obtained at multiple wavelengths is a nontrivial aspect of photoacoustic imaging but is essential for accurate functional and molecular imaging. This topic, known as quantitative photoacoustic imaging, is reviewed here. The inverse problems involved are described, their nature (nonlinear and ill-posed) is discussed, proposed solution techniques and their limitations are explained, and the remaining unsolved challenges are introduced.

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“…(5) The final step is to obtain μ a ¼ ffiffiffi ffi D p ∕μ 1 from the first of Eqs. (48). The key to this method is part (2), finding a solution to Eq.…”

Section: Direct "Zero-divergence" Methodsmentioning

confidence: 99%

Section: Large-scale Inversions and Domain Parameterizationmentioning

confidence: 99%

Quantitative spectroscopic photoacoustic imaging: a review

et al. 2012

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“…However, we are able to iteratively change m 0 in the course of the inversion process. We have proved in Haber (2006) that such a process is a convergent one.…”

Section: Corrective Sourcesmentioning

confidence: 92%

Inversion of time domain three-dimensional electromagnetic data

Haber¹,

Oldenburg²,

Shekhtman

2007

Geophysical Journal International

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106

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S U M M A R YWe present a general formulation for inverting time domain electromagnetic data to recover a 3-D distribution of electrical conductivity. The forward problem is solved using finite volume methods in the spatial domain and an implicit method (Backward Euler) in the time domain. A modified Gauss-Newton strategy is employed to solve the inverse problem. The modifications include the use of a quasi-Newton method to generate a pre-conditioner for the perturbed system, and implementing an iterative Tikhonov approach in the solution to the inverse problem. In addition, we show how the size of the inverse problem can be reduced through a corrective source procedure. The same procedure can correct for discretization errors that inevidably arise. We also show how the inverse problem can be efficiently carried out even when the decay time for the conductor is significantly larger than the repetition time of the transmitter wave form. This requires a second processor to carry an additional forward modelling. Our inversion algorithm is general and is applicable for any electromagnetic field (E, H, d B/dt) measured in the air, on the ground, or in boreholes, and from an arbitrary grounded or ungrounded source.Three synthetic examples illustrate the basic functionality of the algorithm, and a result from a field example shows applicability in a larger-scale field example.

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“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php B995 It is also possible to employ multigrid approaches to such saddle point problems. This class of methods has previously been shown to demonstrate good performance when applied to solve a number of PDE-constrained optimization problems, subject to both steady and transient PDEs [1,2,8,9,23,24,28,29,57].…”

Section: Problem Formulation and Discretizationmentioning

confidence: 98%

Fast Iterative Solution of Reaction-Diffusion Control Problems Arising from Chemical Processes

Pearson

Stoll

2013

SIAM J. Sci. Comput.

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PDE-constrained optimization problems, and the development of preconditioned iterative methods for the efficient solution of the arising matrix systems, is a field of numerical analysis that has recently been attracting much attention. In this paper, we analyze and develop preconditioners for matrix systems that arise from the optimal control of reaction-diffusion equations, which themselves result from chemical processes. Important aspects of our solvers are saddle point theory, mass matrix representation, and effective Schur complement approximation, as well as the incorporation of control constraints and application of the outer (Newton) iteration to take into account the nonlinearity of the underlying PDEs. Introduction.A class of problems which has numerous applications within mathematical and physical problems is that of PDE-constrained optimization problems. One field in which these problems can be posed is that of chemical processes [4,19,20,21,22]. In this case the underlying PDEs are reaction-diffusion equations, and therefore the PDE constraints in our formulation are nonlinear PDEs.When solving such reaction-diffusion control problems using a finite element method, and employing a Lagrange-Newton iteration to take account of the nonlinearity involved in the PDEs, the resulting matrix system for each Newton iteration will be large, sparse, and of saddle point structure. It is therefore desirable to devise preconditioned iterative methods to solve these systems efficiently and in such a way that the structure of the matrix is exploited. Work in constructing preconditioners for PDE-constrained optimization problems has been considered for simpler problems previously, for instance, Poisson control [46,47,53], convection-diffusion control [45], Stokes control [36,50,56], and heat equation control [44,54].In this paper, we will consider an optimal control formulation of a reactiondiffusion problem, which generates a symmetric matrix system upon each Newton iteration. (Such an iteration is required to take into account the nonlinear terms within the underlying PDEs.) We will generally search for block triangular preconditioners for the matrix systems we examine, to be used in conjunction with a suitable iterative solver. In order to do this, we will need to approximate the (1, 1)-block by accurately representing the inverse of mass matrices amongst other things, as well as

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A parallel method for large scale time domain electromagnetic inverse problems

Cited by 9 publications

References 20 publications

Quantitative spectroscopic photoacoustic imaging: a review

Quantitative spectroscopic photoacoustic imaging: a review

Inversion of time domain three-dimensional electromagnetic data

Fast Iterative Solution of Reaction-Diffusion Control Problems Arising from Chemical Processes

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