Less is often more: Applied inverse problems using hp-forward models

Smyl, Danny; Liu, Dong

doi:10.1016/j.jcp.2019.108949

Cited by 21 publications

(21 citation statements)

References 100 publications

(123 reference statements)

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“…We may conclude that NN-QN is comparable to GN, even though GN is based on the more accurate J and H. Meanwhile, it is also apparent that the background water conductivity estimated using Broyden's method has significant fluctuations near the electrodes and is significantly lower than the accurate GN estimate. This observation is likely owed to roundoff errors accumulating in successive iterations, consistent with findings in [4]. On the other hand, such roundoff errors do not occur in the proposed NN-QN method, resulting in a more accurate estimation of the background conductivity.…”

Section: Resultssupporting

confidence: 79%

“…However, perturbation methods require repetitive function evaluations which can be extremely demanding. Therefore, when semi-analytical gradients are not available, conventional numerical differentiation approaches may be infeasible in solving high-dimensional nonlinear inverse problems -if not intractable on accessible computing resources [3], [4]. To address this challenge, a number of researchers have proposed methods for updating gradient terms, rather than, e.g., computing finite differences at each iteration; these approaches are referred to as Quasi-Newton (QN) methods.…”

Section: Introductionmentioning

confidence: 99%

“…Since then, a number of successful updating methods have been proposed including the ever popular Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm [6]- [9], the limited memory BFGS algorithm [10], and a suite of other QN-family derivative approximators [11]. Nevertheless, incorporation of prior information to regularize and stabilize the reconstruction for ill-posed problems is often either limited or not possible for QN methods [4], [12], thus effectively limiting applicability in practice. This is exacerbated in the case of nonlinear problems, where prior information is essential to stabilize reconstructions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An Efficient Quasi-Newton Method for Nonlinear Inverse Problems via Learned Singular Values

Smyl

Tallman

Liu

et al. 2021

IEEE Signal Process. Lett.

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Solving complex optimization problems in engineering and the physical sciences requires repetitive computation of multi-dimensional function derivatives, which commonly require computationally-demanding numerical differentiation such as perturbation techniques. In particular, Gauss-Newton methods are used for nonlinear inverse problems that require iterative updates to be computed from the Jacobian and allow for flexible incorporation of prior knowledge. Computationally more efficient alternatives are Quasi-Newton methods, where the repeated computation of the Jacobian is replaced by an approximate update, but unfortunately are often too restrictive for highly ill-posed problems. To overcome this limitation, we present a highly efficient data-driven Quasi-Newton method applicable to nonlinear inverse problems, by using the singular value decomposition and learning a mapping from model outputs to the singular values to compute the updated Jacobian. Enabling time-critical applications and allowing for flexible incorporation of prior knowledge necessary to solve ill-posed problems. We present results for the highly non-linear inverse problem of electrical impedance tomography with experimental data.

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Section: Resultssupporting

confidence: 79%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An Efficient Quasi-Newton Method for Nonlinear Inverse Problems via Learned Singular Values

Smyl

Tallman

Liu

et al. 2021

IEEE Signal Process. Lett.

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“…we immediately observe that solving the EIT inverse problem, estimating σ from V , is highly dependent on the properties of this numerical forward model, U . Importantly, the nonlinearity of U results in a severely ill-conditioned sensitivity matrix J = ∂U (σ) ∂σ and Hessian approximation H = J T J [16]. In particular, the ill-conditioning of H has a large impact on the ill-posedness of least-squares based solutions to the EIT reconstruction problem, which is usually addressed using regularization techniques.…”

Section: Background Nonlinearity Ill-conditioning and Ill-posednessmentioning

confidence: 99%

Optimizing Electrode Positions in 2-D Electrical Impedance Tomography Using Deep Learning

Smyl

Dong

2020

IEEE Trans. Instrum. Meas.

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Electrical Impedance Tomography (EIT) is a powerful tool for non-destructive evaluation, state estimation, process tomography, and much more. For these applications, and in order to reliably reconstruct images of a given process using EIT, we must obtain high-quality voltage measurements from the EIT sensor (or structure) of interest. Inasmuch, it is no surprise that the locations of electrodes used for measuring plays a key role in this task. Yet, to date, methods for optimally placing electrodes either require knowledge on the EIT target (which is, in practice, never fully known), are computationally difficult to implement numerically, or require electrode segmentation. In this paper, we circumvent these challenges and present a straightforward deep learning based approach for optimizing electrodes positions. It is found that the optimized electrode positions outperformed "standard" uniformly-distributed electrode layouts in all test cases using a metric independent from the optimization parameters.

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“…The researchers concluded that optimized electrode positions can reduce the error in EIT reconstructions; as regards the experimental input, however, they employed only synthetic data, meaning that factors accompanying real measurement, such as interference, uncertainties of the instruments, and limited resolution, were not assumed. Further, the size and polynomial degree of the mesh elements gained attention in report [ 32 ], which examined the relationship between the mesh density and the resulting error. The authors proposed that a higher density does not necessarily yield a lower reconstruction error, and they also specified the key variables causing unexpected behavior in the inverse task.…”

Section: Introductionmentioning

confidence: 99%

Measurement-Based Domain Parameter Optimization in Electrical Impedance Tomography Imaging

Dušek

Mikulka

2021

Sensors

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This paper discusses the optimization of domain parameters in electrical impedance tomography-based imaging. Precise image reconstruction requires accurate, well-correlated physical and numerical finite element method (FEM) models; thus, we employed the Nelder–Mead algorithm and a complete electrode model to evaluate the individual parameters, including the initial conductivity, electrode misplacement, and shape deformation. The optimization process was designed to calculate the parameters of the numerical model before the image reconstruction. The models were verified via simulation and experimental measurement with single source current patterns. The impact of the optimization on the above parameters was reflected in the applied image reconstruction process, where the conductivity error dropped by 6.16% and 11.58% in adjacent and opposite driving, respectively. In the shape deformation, the inhomogeneity area ratio increased by 11.0% and 48.9%; the imprecise placement of the 6th electrode was successfully optimized with adjacent driving; the conductivity error dropped by 12.69%; and the inhomogeneity localization exhibited a rise of 66.7%. The opposite driving option produces undesired duality resulting from the measurement pattern. The designed optimization process proved to be suitable for correlating the numerical and the physical models, and it also enabled us to eliminate imaging uncertainties and artifacts.

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Less is often more: Applied inverse problems using hp-forward models

Cited by 21 publications

References 100 publications

An Efficient Quasi-Newton Method for Nonlinear Inverse Problems via Learned Singular Values

An Efficient Quasi-Newton Method for Nonlinear Inverse Problems via Learned Singular Values

Optimizing Electrode Positions in 2-D Electrical Impedance Tomography Using Deep Learning

Measurement-Based Domain Parameter Optimization in Electrical Impedance Tomography Imaging

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