On level set regularization for highly ill-posed distributed parameter estimation problems

Doel, Kees van den; Ascher, Uri M.

doi:10.1016/j.jcp.2006.01.022

Cited by 73 publications

(87 citation statements)

References 32 publications

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“…For better capturing of the irregular tumor shape, the order of the spherical harmonic expansion was increased leading to an increase in the computational time. The current work is not a parameter based optimization, as was the case in [6], but it is an implicit shape reconstruction method based on the level set technique that solves the Hamilton Jacobi equation with respect to the space and time [25][26][27].…”

Section: Introductionmentioning

confidence: 99%

Inverse Scattering of Three-Dimensional Pec Objects Using the Level-Set Method

Hajihashemi¹,

El-Shenawee²

2011

PIER

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Abstract-A 3D shape reconstruction algorithm for multiple PEC objects immersed in air is presented. The Hamilton-Jacobi PDE is solved in the entire computational domain with employing the marching cubes method to retrieve the unknown objects. The method of moment surface integral equation is used as the forward solver. An appropriate form of the deformation velocity, based on the forward and adjoint fields, is implemented to minimize the mismatch between the measurements and simulated scattered fields from the evolving objects. The inversion algorithm demonstrated good shape reconstruction results even when using limited view data or noisy corrupted data with SNR of 5 dB.

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Section: Introductionmentioning

confidence: 99%

Inverse Scattering of Three-Dimensional Pec Objects Using the Level-Set Method

Hajihashemi¹,

El-Shenawee²

2011

PIER

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“…Although the choice of time step τ is not obvious here, and the choice of M is not intuitive, this is an important interpretation, as some significant differences observed in [78] between dynamic regularization and the Tikhonov approach no longer seem major in the present setting.…”

Section: A Level Set Approachmentioning

confidence: 96%

“…As in Section 2.1 we can view (3.2) as the steady-state equations of a time-dependent differential equation [4,78],…”

Section: Distributed Parameter Estimation and The Exponential Filter mentioning

confidence: 99%

“…[3,4]). An approach considered by many for this purpose is to use level sets, and we proceed to investigate this further in the present context, following [78].…”

Section: Level Sets For Shape Optimizationmentioning

confidence: 99%

“…We refer to [78] for this and other implementation considerations, as well as numerical experiments that demonstrate that this method works very well, often delivering good reconstructions in less than 10 iterations. Here we concentrate just on deriving the method (4.4) from different points of view.…”

Section: A Level Set Approachmentioning

confidence: 99%

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Artificial time integration

2007

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Abstract.Many recent algorithmic approaches involve the construction of a differential equation model for computational purposes, typically by introducing an artificial time variable. The actual computational model involves a discretization of the now time-dependent differential system, usually employing forward Euler. The resulting dynamics of such an algorithm is then a discrete dynamics, and it is expected to be "close enough" to the dynamics of the continuous system (which is typically easier to analyze) provided that small -hence many -time steps, or iterations, are taken. Indeed, recent papers in inverse problems and image processing routinely report results requiring thousands of iterations to converge. This makes one wonder if and how the computational modeling process can be improved to better reflect the actual properties sought.In this article we elaborate on several problem instances that illustrate the above observations. Algorithms may often lend themselves to a dual interpretation, in terms of a simply discretized differential equation with artificial time and in terms of a simple optimization algorithm; such a dual interpretation can be advantageous. We show how a broader computational modeling approach may possibly lead to algorithms with improved efficiency.

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A computationally efficient parallel Levenberg‐Marquardt algorithm for highly parameterized inverse model analyses

Lin

O’Malley

Vesselinov

2016

Water Resources Research

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Inverse modeling seeks model parameters given a set of observations. However, for practical problems because the number of measurements is often large and the model parameters are also numerous, conventional methods for inverse modeling can be computationally expensive. We have developed a new, computationally efficient parallel Levenberg‐Marquardt method for solving inverse modeling problems with a highly parameterized model space. Levenberg‐Marquardt methods require the solution of a linear system of equations which can be prohibitively expensive to compute for moderate to large‐scale problems. Our novel method projects the original linear problem down to a Krylov subspace such that the dimensionality of the problem can be significantly reduced. Furthermore, we store the Krylov subspace computed when using the first damping parameter and recycle the subspace for the subsequent damping parameters. The efficiency of our new inverse modeling algorithm is significantly improved using these computational techniques. We apply this new inverse modeling method to invert for random transmissivity fields in 2‐D and a random hydraulic conductivity field in 3‐D. Our algorithm is fast enough to solve for the distributed model parameters (transmissivity) in the model domain. The algorithm is coded in Julia and implemented in the MADS computational framework (http://mads.lanl.gov). By comparing with Levenberg‐Marquardt methods using standard linear inversion techniques such as QR or SVD methods, our Levenberg‐Marquardt method yields a speed‐up ratio on the order of ∼101 to ∼102 in a multicore computational environment. Therefore, our new inverse modeling method is a powerful tool for characterizing subsurface heterogeneity for moderate to large‐scale problems.

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On level set regularization for highly ill-posed distributed parameter estimation problems

Cited by 73 publications

References 32 publications

Inverse Scattering of Three-Dimensional Pec Objects Using the Level-Set Method

Inverse Scattering of Three-Dimensional Pec Objects Using the Level-Set Method

Artificial time integration

A computationally efficient parallel Levenberg‐Marquardt algorithm for highly parameterized inverse model analyses

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