Algorithms For Inverse Problems Research Articles

A direct reconstruction algorithm for the anisotropic inverse conductivity problem based on Calderón’s method in the plane

A direct reconstruction algorithm based on Calderón’s linearization method for the reconstruction of isotropic conductivities is proposed for anisotropic conductivities in two-dimensions. To overcome the non-uniqueness of the anisotropic inverse conductivity problem, the entries of the unperturbed anisotropic tensors are assumed known a priori, and it remains to reconstruct the multiplicative scalar field. The quasi-conformal map in the plane facilitates the Calderón-based approach for anisotropic conductivities. The method is demonstrated on discontinuous radially symmetric conductivities of high and low contrast.

On the Convergence of Learning-Based Iterative Methods for Nonconvex Inverse Problems.

Numerous tasks at the core of statistics, learning and vision areas are specific cases of ill-posed inverse problems. Recently, learning-based (e.g., deep) iterative methods have been empirically shown to be useful for these problems. Nevertheless, integrating learnable structures into iterations is still a laborious process, which can only be guided by intuitions or empirical insights. Moreover, there is a lack of rigorous analysis about the convergence behaviors of these reimplemented iterations, and thus the significance of such methods is a little bit vague. This paper moves beyond these limits and proposes Flexible Iterative Modularization Algorithm (FIMA), a generic and provable paradigm for nonconvex inverse problems. Our theoretical analysis reveals that FIMA allows us to generate globally convergent trajectories for learning-based iterative methods. Meanwhile, the devised scheduling policies on flexible modules should also be beneficial for classical numerical methods in the nonconvex scenario. Extensive experiments on real applications verify the superiority of FIMA.

Bayesian optimization for inverse problems in time-dependent quantum dynamics.

We demonstrate an efficient algorithm for inverse problems in time-dependent quantum dynamics based on feedback loops between Hamiltonian parameters and the solutions of the Schrödinger equation. Our approach formulates the inverse problem as a target vector estimation problem and uses Bayesian surrogate models of the Schrödinger equation solutions to direct the optimization of feedback loops. For the surrogate models, we use Gaussian processes with vector outputs and composite kernels built by an iterative algorithm with the Bayesian information criterion (BIC) as a kernel selection metric. The outputs of the Gaussian processes are designed to model an observable simultaneously at different time instances. We show that the use of Gaussian processes with vector outputs and the BIC-directed kernel construction reduces the number of iterations in the feedback loops by, at least, a factor of 3. We also demonstrate an application of Bayesian optimization for inverse problems with noisy data. To demonstrate the algorithm, we consider the orientation and alignment of polyatomic molecules SO2 and propylene oxide (PPO) induced by strong laser pulses. We use simulated time evolutions of the orientation or alignment signals to determine the relevant components of the molecular polarizability tensors. We show that, for the five independent components of the polarizability tensor of PPO, this can be achieved with as few as 30 quantum dynamics calculations.

INVERSE PROBLEM ALGORITHM APPLICATION TO SEMI-QUANTITATIVE ANALYSIS OF 272 PATIENTS WITH ISCHEMIC STROKE SYMPTOMS: CAROTID STENOSIS RISK ASSESSMENT FOR FIVE RISK FACTORS

This study proposes the inverse problem algorithm (IPA) with five risk factors applied to the semi-quantitative analysis of carotid stenosis 272 patients with suspected ischemic stroke. The IPA is known to provide a substantiated machine learning-based prediction of the expected outcomes by solving an inverse matrix of variable coefficients. In case of carotid stenosis prediction, such risk factors as patient’s age, mean arterial pressure (MAP), glucose AC, low-density lipoprotein-cholesterol (LDL-C), and C-Reactive protein (CRP) were assessed for the main group of 217 patients. Their results were processed by the STATISTICA program with a customized loss function ([Formula: see text]), yielding the first-order nonlinear semi-empirical formula with 16 terms. The loss function was calculated via the total mismatch between the theoretical predictions and true carotid stenosis cases (%) for all 217 patients. Thus, the carotid stenosis (%) compromised solution array [[Formula: see text]] was optimized using [Formula: see text] individual data points via the proposed algorithm. The results showed a complete regression with loss function [Formula: see text]=2.3543, variance [Formula: see text]=87.46%, and correlation coefficient [Formula: see text]. The reference group of 55 more patients with the same preliminary diagnosis and symptoms was selected to validate the method predictive feasibility, which was found quite satisfactory. The decreasing order of three dominant risk factors was as follows: CRP, glucose AC, and MAP, whereas age and LDL-C weakly influenced the program computation results. The IPA showed a strong convergence by its default characteristic. The reduction of the number of variables in computation deteriorated the prediction accuracy, exhibiting the algorithm’s high sensitivity to the number of variables.

Inverse estimation of the sand concentration for sand-oil flow in a horizontal pipeline based on the Eulerian–Eulerian model

A method that predicts the sand concentration and volume fraction distribution of sand–oil flow in a horizontal pipeline was proposed based on the inverse heat transfer problem theory. The forward problem of the convective heat transfer process for the sand–oil flow was investigated using a computational fluid dynamics (CFD) simulation based on the Eulerian–Eulerian model. The inverse problem of sand concentration estimation was solved using the temperature distributions at different measurement locations. The sand volume fraction was controlled by an inverse problem algorithm and optimized by inputting the external experiment temperature. Experiments with different sand concentrations were carried out to verify the accuracy of the inverse problem approach. Due to the inherent error between the simulation results and the actual experimental measurements, appropriate modifications in the inverse procedure were made in the input information to ensure that the CFD calculation was more consistent with the actual concentration data. The relative error of the sand concentration estimations could be controlled within 6.0% when the modified measurements were considered the input information of the inverse problem. The volume fraction distributions were also investigated in this study. The important variables that can impact flow temperatures, such as flow velocities and sand diameters, were investigated, and the results showed the inverse method is feasible for estimating the sand concentration of sand–oil flow.

An inverse elastic method of crack identification based on sparse strain sensing sheet

Reliable damage detection over large areas of structures can be achieved by spatially quasi-continuous structural health monitoring enabled by two-dimensional sensing sheets. They contain dense arrays of short-gauge sensors, which increases the probability to have sensors in direct contact with damage (e.g. crack opening) and thus identify (i.e. detect, localize, and quantify) it at an early stage. This approach in damage identification is called direct sensing. Although the sensing sheet is a reliable and low-cost technology, the overall structural health monitoring system that is using it might become complex due to large number of sensors. Hence, intentional reduction in number of sensors might be desirable. In addition, malfunction of sensors can occur in real-life settings, which results in unintentional reduction in the number of functioning sensors. In both cases, reduction in the number of (functioning) sensors may lead to lack of performance of sensing sheet. Therefore, it is important to explore the performance of sparse arrays of sensors, in the cases where sensors are not necessarily in direct contact with damage (indirect sensing). The aim of this research is to create a method for optimizing the design of arrays of sensors, that is, to find the smallest number of sensors while maintaining a satisfactory reliability of crack detection and accuracy of damage localization and quantification. To achieve that goal, we first built a phase field finite element model of cracked structure verified by the analytical model to determine the crack existence (detection), and then we used the algorithm of inverse elastostatic problem combined with phase field finite element model to determine the crack length (quantification) and location (localization) by minimizing the difference between the sensor measurements and the phase field finite element model results. In addition, we experimentally validated the method by means of a reduced-scale laboratory test and assessed the accuracy and reliability of indirect sensing.

Machine learning and evolutionary algorithm studies of graphene metamaterials for optimized plasmon-induced transparency

Machine learning and optimization algorithms have been widely applied in the design and optimization for photonics devices. We briefly review recent progress of this field of research and show data-driven applications, including spectrum prediction, inverse design and performance optimization, for novel graphene metamaterials (GMs). The structure of the GMs is well-designed to achieve the wideband plasmon induced transparency (PIT) effect, which can be theoretically demonstrated by using the transfer matrix method. Some traditional machine learning algorithms, including k nearest neighbour, decision tree, random forest and artificial neural networks, are utilized to equivalently substitute the numerical simulation in the forward spectrum prediction and complete the inverse design for the GMs. The calculated results demonstrate that all algorithms are effective and the random forest has advantages in terms of accuracy and training speed. Moreover, evolutionary algorithms, including single-objective (genetic algorithm) and multi-objective optimization (NSGA-II), are used to achieve the steep transmission characteristics of PIT effect by synthetically taking many different performance metrics into consideration. The maximum difference between the transmission peaks and dips in the optimized transmission spectrum reaches 0.97. In comparison to previous works, we provide a guidance for intelligent design of photonics devices based on machine learning and evolutionary algorithms and a reference for the selection of machine learning algorithms for simple inverse design problems.

Convergence and Complexity Analysis of a Levenberg–Marquardt Algorithm for Inverse Problems

The Levenberg-Marquardt algorithm is one of the most popular algorithms for finding the solution of nonlinear least squares problems. Across different modified variations of the basic procedure, the algorithm enjoys global convergence, a competitive worst case iteration complexity rate, and a guaranteed rate of local convergence for both zero and nonzero small residual problems, under suitable assumptions. We introduce a novel Levenberg-Marquardt method that matches, simultaneously, the state of the art in all of these convergence properties with a single seamless algorithm. Numerical experiments confirm the theoretical behavior of our proposed algorithm.

ASSESSMENT OF EFFECTIVE BLOOD CONCENTRATION READINGS FROM CLINICAL DATA ON PATIENTS WITH HEART FAILURE DISEASES AFTER DIGOXIN INTAKE: A PROJECTION BASED ON THE INVERSE PROBLEM ALGORITHM

In this study, a projection of effective blood concentration (EBC) readings of digoxin is made using the inverse problem algorithm based on clinical data for patients with heart failure diseases. Seven factors, including body surface area (BSA), blood urine nitrogen (BUN), creatinine, sodium (Na), potassium (K), magnesium (Mg) ion readings, and mean arterial pressure (MAP) were compiled with nonlinear regression fit to develop a projection function having 29 terms obtained from an inverse problem algorithm via the default function run in STATISTICA. Accordingly, data collected from the clinical 168 heart failure patients were normalized to be included in same domain range ([Formula: see text]1 to +1), and then calculated by the specific algorithm to optimize the numerical solution to evaluate EBC readings of digoxin. The evaluated first-order regression fit owned an optimal loss function ([Formula: see text]) coupled with correlation coefficient [Formula: see text] = 0.892 and variance of 89.20%. Furthermore, 45 patients having similar clinical syndromes were also adopted to verify the projection and implied with high agreement. The BUN factor dominated the projection and defined as the most significant coefficient in the analysis, and K ion, MAP, BSA, and Mg ion factors exhibited minor contributions to the projection. The repeated trials to lower number of factors from seven to a smaller number (namely 6, 5, 4, 3, 2, and 1) for simplifying method but resulting with unaccepted outcomes, with high loss function values and low linearity. However, the algorithm held its accuracy to handle the verified data that were out of the original bounds. The proposed algorithm demonstrated a useful analysis to handle the drug administration in pharmaceutical field.

Image Restoration and Reconstruction using Targeted Plug-and-Play Priors

Leveraging current state-of-the-art denoisers to tackle other inverse problems in imaging is a challenging task, which has recently been the topic of significant research effort. In this paper, we present several contributions to this research front, based on two fundamental building blocks: 1) the recently proposed plug-and-play framework, which allows combining iterative algorithms for imaging inverse problems with state-of-the-art image denoisers, used in black-box fashion; and 2) patch-based denoisers, using Gaussian mixture models (GMM). We exploit the adaptability of GMM to learn class-adapted denoisers, which opens the door to embedding a patch classification step in the algorithmic loop, yielding simultaneous restoration and semantic segmentation. We apply the proposed approach to several standard imaging inverse problems (deblurring, compressive sensing reconstruction, and super-resolution), obtaining results that are competitive with the state of the art.

Application of the steady-state and unsteady-state lasers in reconstructing the geometrical characteristic parameters of soot fractal aggregates

The unsteady-state laser, i.e. time-domain (TD) and frequency-domain (FD) lasers, and the steady-state (SS) laser are employed as the measurement lasers to reconstruct the geometrical characteristic parameters of soot fractal aggregates, i.e. the root mean square radius Rg and fractal dimension Df. The direct problems are solved by the finite volume method and Rayleigh-Debye-Gans fractal aggregate method, while an improved artificial bee colony algorithm is used as the inverse problem algorithm. The sensitivity analysis is employed to study the effective sampling region of the TD and FD laser measurement signals, and the results show that the time-resolved signals selected within [L, 0.5L + ctp] for the TD laser and the dimensionless frequency selected within [0.5, 5] for the FD laser are conducive to improving the retrieval accuracy. Then, the geometrical characteristic parameters of soot fractal aggregates are retrieved by using different measurement lasers, and the results reveal that the retrieval accuracy can be ensured by using the TD laser measurement technique. Moreover, the retrieval accuracy and robustness of Rg are better than those of Df. All the studies in the present manuscript supply an application plat of the laser measurement techniques in reconstructing the geometrical characteristic parameters of soot fractal aggregates.

Replacement of numerical simulations with machine learning in the inverse problem of two-phase flow in porous medium

A possibility to replace full-physics numerical simulations with machine-learning-based algorithms in inverse problems of multiphase flow in porous media is studied on an example of specialized oil-well tests. The performance function is computed at each iteration of the inverse problem solution using an artificial neural network (ANN) instead of forward numerical simulations. Proper ANN structure, training set, and learning algorithm are established and implemented. Achieved approximation quality for the performance function on a test sample is well suitable for the inverse problem solution. Further improvements and alternative ANN formulations are proposed for practical applications.

New over‐relaxed monotone fast iterative shrinkage‐thresholding algorithm for linear inverse problems

The over-relaxed monotone fast iterative shrinkage-thresholding algorithm (OMFISTA) needs to satisfy a complex convergence condition with respect to its additional parameters. To simplify the convergence condition, this study proposes a new OMFISTA, termed OMFISTAv2, using a parameter setting strategy which will derive a simple sufficient condition with respect to the additional parameters to guarantee the convergence of OMFISTAv2. Moreover, the authors find experimentally that OMFISTAv2 can accelerate MFISTA in some cases where the system matrix is ill-conditioned or rank-deficient, while OMFISTA cannot.

A Convex Optimization Algorithm for Compressed Sensing in a Complex Domain: The Complex-Valued Split Bregman Method

The Split Bregman method (SBM), a popular and universal CS reconstruction algorithm for inverse problems with both l1-norm and TV-norm regularization, has been extensively applied in complex domains through the complex-to-real transforming technique, e.g., MRI imaging and radar. However, SBM still has great potential in complex applications due to the following two points; Bregman Iteration (BI), employed in SBM, may not make good use of the phase information for complex variables. In addition, the converting technique may consume more time. To address that, this paper presents the complex-valued Split Bregman method (CV-SBM), which theoretically generalizes the original SBM into the complex domain. The complex-valued Bregman distance (CV-BD) is first defined by replacing the corresponding regularization in the inverse problem. Then, we propose the complex-valued Bregman Iteration (CV-BI) to solve this new problem. How well-defined and the convergence of CV-BI are analyzed in detail according to the complex-valued calculation rules and optimization theory. These properties prove that CV-BI is able to solve inverse problems if the regularization is convex. Nevertheless, CV-BI needs the help of other algorithms for various kinds of regularization. To avoid the dependence on extra algorithms and simplify the iteration process simultaneously, we adopt the variable separation technique and propose CV-SBM for resolving convex inverse problems. Simulation results on complex-valued l1-norm problems illustrate the effectiveness of the proposed CV-SBM. CV-SBM exhibits remarkable superiority compared with SBM in the complex-to-real transforming technique. Specifically, in the case of large signal scale n = 512, CV-SBM yields 18.2%, 17.6%, and 26.7% lower mean square error (MSE) as well as takes 28.8%, 25.6%, and 23.6% less time cost than the original SBM in 10 dB, 15 dB, and 20 dB SNR situations, respectively.

A re-weighted smoothed -norm regularized sparse reconstructed algorithm for linear inverse problems

This paper addresses the problems of sparse signal and image recovery using compressive sensing (CS), especially in the case of Gaussian noise. The main contribution of this paper is the proposal of the regularization re-weighted Composite Sine function smoothed L 0-norm minimization (RRCSFSL0) algorithm where the Composite Sine function (CSF), the iteratively re-weighted scheme and the regularization mechanism represent the core of an approach to the solution of the problem. Compared with other state-of-the-art functions, the CSF we proposed can better approximate the L 0-norm and improve the reconstruction accuracy, the new re-weighted scheme we adopted can promote sparsity and speed up convergence. Moreover, the use of the regularization mechanism makes the RRCSFSL0 algorithm more robust against noise. The performance of the proposed algorithm is verified via numerical experiments in the noise environment. Furthermore, experiments and comparisons demonstrate the superiority of the RRCSFSL0 algorithm in image restoration.

On Single-Image Super-Resolution in 3D Brain Magnetic Resonance Imaging.

The objective of this work is to apply 3D super resolution (SR) techniques to brain magnetic resonance (MR) image restoration. Two 3D SR methods are considered following different trends: one recently proposed tensor-based approach and one inverse problem algorithm based on total variation and low rank regularization. The evaluation of their effectiveness is assessed through the segmentation of brain compartments: gray matter, white matter and cerebrospinal fluid. The two algorithms are qualitatively and quantitatively evaluated on simulated images with ground truth available and on experimental data. The originality of this work is to consider the SR methods as an initial step towards the final segmentation task. The results show the ability of both methods to overcome the loss of spatial resolution and to facilitate the segmentation of brain structures with improved accuracy compared to native low-resolution MR images. Both algorithms achieved almost equivalent results with a highly reduced computational time cost for the tensor-based approach.

Recovering Differential Operators with a Retarded Argument

We consider second-order differential operators with a constant delay. The properties of their spectral characteristics are established, and the inverse problem of recovering the operators from their spectra is studied. We develop constructive algorithms for inverse problems and prove the uniqueness of the solution.

Depth-resolved local atomic structure of Fe/Cr multilayer film with GMR effect: Experimental results

A subnanometric resolution method for studying the local atomic structure of interface and surface of low contrast multilayered nanoheterostructure thin films is applied to the Fe/Cr multilayer sample with GMR (Giant Magnetoresistance) effect. We consider combination of the X-ray reflectivity (XRR) and the Extended X-ray absorption Fine Structure (EXAFS) spectroscopy with angular resolution. The XRR experiment has been carried out according to standard procedure using a specialized X-ray diffractometer «Empyrean» at the Institute of Metal Physics (Ekaterinburg). We have applied a measurement method proposed earlier by V. P. Romanov et al. This approach allowed us to separate the diffuse and pure specular contributions. To determine the concentration depth-profile of element from XRR data, we have implemented the Levenberg-Marquardt (L-M) algorithm for nonlinear inverse problem. Thus, the phase problem for X-ray specular reflectivity has been solving. The EXAFS measurements has been performed using synchrotron facilities (National Research Centre “Kurchatov Institute”). {The Fe and Cr K absorption spectra} were recorded in fluorescence mode with angular resolution. The experimental results for depth-resolved local atomic structure of Fe/Cr multilayer with GMR effect has been obtained. A depth resolution of 2-3 Å was reached. Atomic structure of Fe and Cr atoms located at seven depth points including interface for the Al2O3/Cr(100 Å)/[Fe(8 Å)/Cr(10,5 Å)]2/Cr(20 Å) multilayered sample was obtained. It was shown what chromium oxides were present on the surface of the sample..

Localization for MCMC: sampling high-dimensional posterior distributions with local structure

We investigate how ideas from covariance localization in numerical weather prediction can be used in Markov chain Monte Carlo (MCMC) sampling of high-dimensional posterior distributions arising in Bayesian inverse problems. To localize an inverse problem is to enforce an anticipated “local” structure by (i) neglecting small off-diagonal elements of the prior precision and covariance matrices; and (ii) restricting the influence of observations to their neighborhood. For linear problems we can specify the conditions under which posterior moments of the localized problem are close to those of the original problem. We explain physical interpretations of our assumptions about local structure and discuss the notion of high dimensionality in local problems, which is different from the usual notion of high dimensionality in function space MCMC. The Gibbs sampler is a natural choice of MCMC algorithm for localized inverse problems and we demonstrate that its convergence rate is independent of dimension for localized linear problems. Nonlinear problems can also be tackled efficiently by localization and, as a simple illustration of these ideas, we present a localized Metropolis-within-Gibbs sampler. Several linear and nonlinear numerical examples illustrate localization in the context of MCMC samplers for inverse problems.

A Nash game algorithm for the solution of coupled conductivity identification and data completion in cardiac electrophysiology

We consider the identification problem of the conductivity coefficient for an elliptic operator using an incomplete over specified measures on the surface. Our purpose is to introduce an original method based on a game theory approach, and design a new algorithm for the simultaneous identification of conductivity coefficient and data completion process. We define three players with three corresponding criteria. The two first players use Dirichlet and Neumann strategies to solve the completion problem, while the third one uses the conductivity coefficient as strategy, and uses a cost which basically relies on an identifiability theorem. In our work, the numerical experiments seek the development of this algorithm for the electrocardiography imaging inverse problem, dealing with inhomogeneities in the torso domain. Furthermore, in our approach, the conductivity coefficients are known only by an approximate values. We conduct numerical experiments on a 2D torso case including noisy measurements. Results illustrate the ability of our computational approach to tackle the difficult problem of joint identification and data completion.