Artificial boundary conditions and domain truncation in electrical impedance tomography. Part II: Stochastic extension of the boundary map

Calvetti, Daniela; Hadwin, Paul J.; Huttunen, Janne M. J.; Kaipio, Jari P.; Somersalo, Erkki

doi:10.3934/ipi.2015.9.767

Cited by 15 publications

(17 citation statements)

References 11 publications

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“…A review of EIT lung applications is timely since there is a profusion of theoretical and technological advances that are increasing the spatial resolution and the accuracy of the estimated electrical material properties [16,17,18,19,20,21,22,23,24,25,26,27]. Examples of these include the parallelization of the forward problem [17], the use of algorithms that do not require derivatives of the performance index [18,19], the regularizations based on anatomy and physiology [23,28], and the correction of the inversion problem caused by a reduced or simplified model [22].…”

Section: Introductionmentioning

confidence: 99%

A review of electrical impedance tomography in lung applications: Theory and algorithms for absolute images

Martins

Sato

Moura

et al. 2019

Annual Reviews in Control

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Electrical Impedance Tomography (EIT) is under fast development, the present paper is a review of some procedures that are contributing to improve spatial resolution and material properties accuracy, admitivitty or impeditivity accuracy. A review of EIT medical applications is presented and they were classified into three broad categories: ARDS patients, obstructive lung diseases and perioperative patients. The use of absolute EIT image may enable the assessment of absolute lung volume, which may significantly improve the clinical acceptance of EIT. The Control Theory, the State Observers more specifically, have a developed theory that can be used for the design and operation of EIT devices. Electrode placement, current injection strategy and electrode electric potential measurements strategy should maximize the number of observable and controllable directions of the state vector space. A non-linear stochastic state observer, the Unscented Kalman Filter, is used directly for the reconstruction of absolute EIT images. Historically, difference images were explored first since they are more stable in the presence of modelling errors. Absolute images require more detailed models of contact impedance, stray capacitance and properly refined finite element mesh where the electric potential gradient is high. Parallelization of the forward *

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Section: Introductionmentioning

confidence: 99%

A review of electrical impedance tomography in lung applications: Theory and algorithms for absolute images

Martins

Sato

Moura

et al. 2019

Annual Reviews in Control

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“…Low-rank approximations-matrix factorizations of the form A ≈ BC, where B is N × r (tall), and C is r × N (wide)-can be efficiently constructed in a matrix-free setting by using Krylov methods (Lanczos or Arnoldi), randomized SVD [38] or CUR decomposition/skeletonization [24,34,45,57]. Although low-rank approximations have been used for Dirichlet-to-Neumann maps [19,20], full-rank or high-rank operators typically still retain a high rank after being restricted to a boundary as a Schur complement. Likewise, although low-rank approximations have been used to approximate the (prior preconditioned) Hessian of the data misfit term in PDE-constrained inverse problems [18,26,30,50,54], the numerical rank of this term grows as the informativeness of the data in the inverse problem grows [4], making low-rank approximation inefficient for highly informative data.…”

Section: Low-rank Approximationmentioning

confidence: 99%

“…Using (19) we can compute individual matrix entries of A in O(1) time even though A is not stored in memory in the conventional sense.…”

Section: Computing Matrix Entries Of Amentioning

confidence: 99%

“…Whenever the Krylov method or randomized SVD requires the application of a block or its adjoint to a vector, we perform this computation using the method in Section 3.3. Alternatively, formula (19) for the matrix entries of A allows us to construct a low-rank approximation of a block by forming a CUR approximation, as is done in [9,14,57]. Whenever the CUR approximation algorithm requires a row, column, or entry of the block, we access it using (19).…”

Section: Conversion To Hierarchical Matrix Formatmentioning

confidence: 99%

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Scalable Matrix-Free Adaptive Product-Convolution Approximation for Locally Translation-Invariant Operators

Alger¹,

Rao²,

Myers³

et al. 2019

SIAM J. Sci. Comput.

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We present an adaptive grid matrix-free operator approximation scheme based on a "product-convolution" interpolation of convolution operators. This scheme is appropriate for operators that are locally translation-invariant, even if these operators are high-rank or full-rank. Such operators arise in Schur complement methods for solving partial differential equations (PDEs), as Hessians in PDE-constrained optimization and inverse problems, as integral operators, as covariance operators, and as Dirichlet-to-Neumann maps. Constructing the approximation requires computing the impulse responses of the operator to point sources centered on nodes in an adaptively refined grid of sample points. A randomized a-posteriori error estimator drives the adaptivity. Once constructed, the approximation can be efficiently applied to vectors using the fast Fourier transform. The approximation can be efficiently converted to hierarchical matrix (H-matrix) format, then inverted or factorized using scalable H-matrix arithmetic. The quality of the approximation degrades gracefully as fewer sample points are used, allowing cheap lower quality approximations to be used as preconditioners. This yields an automated method to construct preconditioners for locally translationinvariant Schur complements. We directly address issues related to boundaries and prove that our scheme eliminates boundary artifacts. We test the scheme on a spatially varying blurring kernel, on the non-local component of an interface Schur complement for the Poisson operator, and on the data misfit Hessian for an advection dominated advection-diffusion inverse problem. Numerical results show that the scheme outperforms existing methods.

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“…To construct the linear model, the derivative may be computed inexpensively using an adjoint method. The computation of the full Jacobian DF (u 0 ) requires J + 1 numerical solutions of a PDE of the form (13), and needs to be performed only once.…”

Section: Steady State Darcy Flowmentioning

confidence: 99%

Iterative updating of model error for Bayesian inversion

et al. 2018

Self Cite

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In computational inverse problems, it is common that a detailed and accurate forward model is approximated by a computationally less challenging substitute. The model reduction may be necessary to meet constraints in computing time when optimization algorithms are used to find a single estimate, or to speed up Markov chain Monte Carlo (MCMC) calculations in the Bayesian framework. The use of an approximate model introduces a discrepancy, or modeling error, that may have a detrimental effect on the solution of the ill-posed inverse problem, or it may severely distort the estimate of the posterior distribution. In the Bayesian paradigm, the modeling error can be considered as a random variable, and by using an estimate of the probability distribution of the unknown, one may estimate the probability distribution of the modeling error and incorporate it into the inversion. We introduce an algorithm which iterates this idea to update the distribution of the model error, leading to a sequence of posterior distributions that are demonstrated empiricially to capture the underlying truth with increasing accuracy. Since the algorithm is not based on rejections, it requires only limited full model evaluations.We show analytically that, in the linear Gaussian case, the algorithm converges geometrically fast with respect to the number of iterations. For more general models, we introduce particle approximations of the iteratively generated sequence of distributions; we also prove that each element of the sequence converges in the large particle limit. We show numerically that, as in the linear case, rapid convergence occurs with respect to the number of iterations. Additionally, we show through computed examples that point estimates obtained from this iterative algorithm are superior to those obtained by neglecting the model error.

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Artificial boundary conditions and domain truncation in electrical impedance tomography. Part II: Stochastic extension of the boundary map

Cited by 15 publications

References 11 publications

A review of electrical impedance tomography in lung applications: Theory and algorithms for absolute images

A review of electrical impedance tomography in lung applications: Theory and algorithms for absolute images

Scalable Matrix-Free Adaptive Product-Convolution Approximation for Locally Translation-Invariant Operators

Iterative updating of model error for Bayesian inversion

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