Algorithms For Inverse Problems Research Articles

Simulated Annealing-Based Image Reconstruction for Patients With COVID-19 as a Model for Ultralow-Dose Computed Tomography.

The proposed algorithm of inverse problem of computed tomography (CT), using limited views, is based on stochastic techniques, namely simulated annealing (SA). The selection of an optimal cost function for SA-based image reconstruction is of prime importance. It can reduce annealing time, and also X-ray dose rate accompanying better image quality. In this paper, effectiveness of various cost functions, namely universal image quality index (UIQI), root-mean-squared error (RMSE), structural similarity index measure (SSIM), mean absolute error (MAE), relative squared error (RSE), relative absolute error (RAE), and root-mean-squared logarithmic error (RMSLE), has been critically analyzed and evaluated for ultralow-dose X-ray CT of patients with COVID-19. For sensitivity analysis of this ill-posed problem, the stochastically estimated images of lung phantom have been reconstructed. The cost function analysis in terms of computational and spatial complexity has been performed using image quality measures, namely peak signal-to-noise ratio (PSNR), Euclidean error (EuE), and weighted peak signal-to-noise ratio (WPSNR). It has been generalized for cost functions that RMSLE exhibits WPSNR of 64.33 ± 3.98 dB and 63.41 ± 2.88 dB for 8 × 8 and 16 × 16 lung phantoms, respectively, and it has been applied for actual CT-based image reconstruction of patients with COVID-19. We successfully reconstructed chest CT images of patients with COVID-19 using RMSLE with eighteen projections, a 10-fold reduction in radiation dose exposure. This approach will be suitable for accurate diagnosis of patients with COVID-19 having less immunity and sensitive to radiation dose.

A Stochastic Algorithm for Fault Inverse Problems in Elastic Half Space with Proof of Convergence

A Stochastic Algorithm for Fault Inverse Problems in Elastic Half Space with Proof of Convergence

A Data-Driven Model Reduction Method for Parabolic Inverse Source Problems and its Convergence Analysis

In this paper, we propose a data-driven model reduction method to solve parabolic inverse source problems efficiently. Our method consists of offline and online stages. In the off-line stage, we explore the low-dimensional structures in the solution space of the parabolic partial differential equations (PDEs) in the forward problem with a given class of source functions and construct a small number of proper orthogonal decomposition (POD) basis functions to achieve significant dimension reduction. Equipped with the POD basis functions, we can solve the forward problem extremely fast in the online stage. Thus, we develop a fast algorithm to solve the optimization problem in the parabolic inverse source problems, which is referred to as the POD algorithm in this paper. Under a weak regularity assumption on the solution of the parabolic PDEs, we prove the convergence of the POD algorithm in solving the forward parabolic PDEs. In addition, we obtain the error estimate of the POD algorithm for parabolic inverse source problems. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method. Our numerical results show that the POD algorithm provides considerable computational savings over the finite element method.

Numerical Identification of the Filtration Capacitive Parameters in Two-Phase Petroleum Reservoirs

We study initial and boundary value problem for nonlinear three dimensional two phase nonlinear filtration problem in three dimensional bounded regions. The reservoir is a two phase and three dimensional oil-water system that is been implemented with typical parallelepiped model. The reservoir constructed with different number of grid blocks in x, y and z directions and initialized with initial pressure, water saturation, corresponding fluid and rock properties in every grid block. To find approximate solution of the above mentioned problem we use finite difference method. We form solution’s algorithm of inverse problem for numerical parameter identification of the petroleum reservoir.

A modified directional bat algorithm for extensive inverse p-facility maxian location problems on networks

A modified directional bat algorithm for extensive inverse p-facility maxian location problems on networks

Accelerated Information Gradient Flow

We present a framework for Nesterov’s accelerated gradient flows in probability space to design efficient mean-field Markov chain Monte Carlo algorithms for Bayesian inverse problems. Here four examples of information metrics are considered, including Fisher-Rao metric, Wasserstein-2 metric, Kalman-Wasserstein metric and Stein metric. For both Fisher-Rao and Wasserstein-2 metrics, we prove convergence properties of accelerated gradient flows. In implementations, we propose a sampling-efficient discrete-time algorithm for Wasserstein-2, Kalman-Wasserstein and Stein accelerated gradient flows with a restart technique. We also formulate a kernel bandwidth selection method, which learns the gradient of logarithm of density from Brownian-motion samples. Numerical experiments, including Bayesian logistic regression and Bayesian neural network, show the strength of the proposed methods compared with state-of-the-art algorithms.

An improved grey wolf optimizer for welding shop inverse scheduling

The welding is widely used in many manufacturing industries. In real-life welding production shop, because of a variety of dynamic events, the original schedule maybe not optimal or even unfeasible. However, if the scheduling scheme is rescheduled blindly, the stability of production and the overall production efficiency will be greatly impacted. To deal with this problem, this paper proposes the model and algorithm of the welding shop inverse scheduling problem (WSISP) for the dynamic events according to welding shop characteristics. Minimizing the parameter adjustment is regarded as the objective. And an improved grey wolf optimizer (IGWO) algorithm is proposed to solve this problem. First, for typical discrete optimization problems, the encoding based on a matrix and the initialization method by the critical path are designed. These methods can avoid unnecessary search and improve search efficiency. Second, a variable neighborhood search method is used to enrich the local search ability of the IGWO algorithm. Afterwards, with the objective of minimizing the parameter adjustment, three kinds of instances are selected to verify the performance of the proposed algorithm, and the comparisons with the state-of-art methods are also conducted. The experimental results show the superiority of the proposed method for solving WSISIP. Finally, a case study considering a real-life welding shop is conducted to validate the proposed method.

Simulation research on magneto-acoustic concentration tomography of magnetic nanoparticles based on truncated singular value decomposition (TSVD).

The existing magneto-acoustic concentration tomography with magnetic induction (MACT-MI) inverse problem algorithm has some problems such as the singularity of reconstructed boundary and poor anti-noise performance, which make it difficult to be applied to recognition of early breast cancer tumor. Therefore, a system matrix linking the concentration distribution information of magnetic nanoparticles (MNPs) to the ultrasonic signal was built in this paper, and a truncated singular value decomposition (TSVD) based MNPS concentration reconstruction algorithm was proposed. Firstly, a simulation model was established. Secondly, the magnetic field and acoustic field simulation data were substituted into the inverse problem algorithm based on TSVD for concentration reconstruction. Finally, the effects of the number of singular values, SNR and radius of MNPs on the reconstruction results were studied. The simulation results show that, the inverse problem algorithm based on TSVD proposed in this paper can maximize the use of ultrasonic signals, and has a good reconstruction effect on 1mm small-radius MNPs, high resolution reconstructed images can also be obtained under the condition of low SNR, which can effectively promote the clinical application of this imaging method.

TIMING OPTIMIZATION OF HEAD AND NECK CT ANGIOGRAPHY VIA THE INVERSE PROBLEM ALGORITHM: in vivo SURVEY FOR 1001 PATIENTS IN 2020–2021

This study processed the recent in vivo survey results for over a thousand patients and optimized their neck and head CT angiography triggered timing (CTA-TT) via the inverse problem algorithm, which ensured the maximal ratio of both left and right arterial to upper sinuses (LRA/US). These results are instrumental in examining the ischemic stroke syndromes along the neck and head. These 1001 patients were randomly categorized into test surveyed (802 patients) and verification group (199 patients), then a six factors semi-empirical formula was constructed by the STATISTICA program. The six factors were assigned a patient’s biological data and preset of the CTA facility; namely Age, mean arterial pressure (MAP), heart rate (HR), contrast media dose (CMD), Pre (injected pressure of CMD), and body surface area (BSA). Each factor was normalized into dimensionless values and incorporated into the dataset matrix [Formula: see text] to analyze the coefficient matrix [Formula: see text]. The derived semi-empirical formula closely correlated with experimental data, according to the loss function [Formula: see text], correlation coefficient [Formula: see text], and variance of 0.8965. The formula verification for 199 more patients (verification group) yielded a correlation coefficient [Formula: see text]. Thus, it can be used for the CTA-TT estimation of patients without their preliminary tests, avoiding unnecessary irradiation. The estimated LRA/US was [Formula: see text] for the verification group in this study. A simplified three-factor formula, featuring only age, MAP, and BSA, was also proposed.

A comprehensive experimental and numerical estimation of thermal contact conductance

Realizing perfect contact between any two surfaces is impossible due to the presence of microscopic irregularities in material surfaces. Therefore, for true thermal analysis of multiple components systems knowledge of thermal contact conductance (TCC) between two solid bodies is essential. Estimation of TCC for different applications has been carried out in this paper by inverse heat transfer technique. A novel stochastic Jaya algorithm has been developed for this purpose and its performance has been evaluated for different heat conduction problems like the estimation of TCC with two metals contact (case-1), periodically contacting surfaces (case-2), multilayer wall surfaces (case-3), and simultaneous estimation of TCC and boundary heat flux (case-4). The experiment has also been conducted to determine the efficacy of this new algorithm for real-life situations. Different parameters like population size, sensor location from the interface, and errors in measured temperature data are taken into consideration. The population size of 5 candidate solutions for case-1, 60 candidate solutions for case-2, and 50 candidate solutions for case-3 and case-4 is found to be suitable for estimation. The results are accurate with the root mean square (RMS) error of 12.62 W/m2K even when the sensors are located 50 mm far away from the interface. Although errors in measurements deteriorate the performance of the Jaya algorithm, it still gives stable results (RMS = 2.90 W/m2K at σ = 0.3). Jaya algorithm is found to be an accurate and promising algorithm for inverse problems.

Quantum Algorithms for Solving Ordinary Differential Equations via Classical Integration Methods

Identifying computational tasks suitable for (future) quantum computers is an active field of research. Here we explore utilizing quantum computers for the purpose of solving differential equations. We consider two approaches: (i) basis encoding and fixed-point arithmetic on a digital quantum computer, and (ii) representing and solving high-order Runge-Kutta methods as optimization problems on quantum annealers. As realizations applied to two-dimensional linear ordinary differential equations, we devise and simulate corresponding digital quantum circuits, and implement and run a 6th order Gauss-Legendre collocation method on a D-Wave 2000Q system, showing good agreement with the reference solution. We find that the quantum annealing approach exhibits the largest potential for high-order implicit integration methods. As promising future scenario, the digital arithmetic method could be employed as an "oracle" within quantum search algorithms for inverse problems.

New method to interpret the ‘canister test’ data for determining kinetic parameters of coalbed gas: theory and experiment

The nonlinear model has been developed and implemented to describe gas emission from coal slack placed in a sealed container (‘canister test’). The model accounts for initial gas content S, coefficients of diffusion D, mass transfer β and desorption kinetics γ, as well as for fractional composition of the sample. Using the developed analytical method of the initial boundary value problem solution, it is found that the pressure of gas in the container tends to a constant value with time. The inverse problem of determining the kinetic parameters S, D, β and γ of gas by the pressure readouts in the experiments has been formulated and subjected to solvability tests. The introduced objective function (discrepancy between and the calculated pressure) is almost independent of β. Based on the modification of the conjugate gradient method, the inverse problem algorithm is proposed. Using the data of in situ measurements of pressure in three containers, the authors perform quantification of gas content, as well as diffusion and desorption kinetics coefficients in terms of a coal bed in Berezovskaya Mine (Kuznetsk Coal Basin). The range of the mass transfer coefficient is estimated by the results of the grain-size analysis of the test samples.

Randomized approaches to accelerate MCMC algorithms for Bayesian inverse problems

Markov chain Monte Carlo (MCMC) approaches are traditionally used for uncertainty quantification in inverse problems where the physics of the underlying sensor modality is described by a partial differential equation (PDE). However, the use of MCMC algorithms is prohibitively expensive in applications where each log-likelihood evaluation may require hundreds to thousands of PDE solves corresponding to multiple sensors; i.e., spatially distributed sources and receivers perhaps operating at different frequencies or wavelengths depending on the precise application. We show how to mitigate the computational cost of each log-likelihood evaluation by using several randomized techniques and embed these randomized approximations within MCMC algorithms. The resulting MCMC algorithms are computationally efficient methods for quantifying the uncertainty associated with the reconstructed parameters. We demonstrate the accuracy and computational benefits of our proposed algorithms on a model application from diffuse optical tomography where we invert for the spatial distribution of optical absorption.

Learning physics-based models from data: perspectives from inverse problems and model reduction

This article addresses the inference of physics models from data, from the perspectives of inverse problems and model reduction. These fields develop formulations that integrate data into physics-based models while exploiting the fact that many mathematical models of natural and engineered systems exhibit an intrinsically low-dimensional solution manifold. In inverse problems, we seek to infer uncertain components of the inputs from observations of the outputs, while in model reduction we seek low-dimensional models that explicitly capture the salient features of the input–output map through approximation in a low-dimensional subspace. In both cases, the result is a predictive model that reflects data-driven learning yet deeply embeds the underlying physics, and thus can be used for design, control and decision-making, often with quantified uncertainties. We highlight recent developments in scalable and efficient algorithms for inverse problems and model reduction governed by large-scale models in the form of partial differential equations. Several illustrative applications to large-scale complex problems across different domains of science and engineering are provided.

A stable non-iterative reconstruction algorithm for the acoustic inverse boundary value problem

<p style='text-indent:20px;'>We present a non-iterative algorithm to reconstruct the isotropic acoustic wave speed from the measurement of the Neumann-to-Dirichlet map. The algorithm is designed based on the boundary control method and involves only computations that are stable. We prove the convergence of the algorithm and present its numerical implementation. The effectiveness of the algorithm is validated on both constant speed and variable speed, with full and partial boundary measurement as well as different levels of noise.</p>

HIPPYlib

We present an extensible software framework, hIPPYlib, for solution of large-scale deterministic and Bayesian inverse problems governed by partial differential equations (PDEs) with (possibly) infinite-dimensional parameter fields (which are high-dimensional after discretization). hIPPYlib overcomes the prohibitively expensive nature of Bayesian inversion for this class of problems by implementing state-of-the-art scalable algorithms for PDE-based inverse problems that exploit the structure of the underlying operators, notably the Hessian of the log-posterior. The key property of the algorithms implemented in hIPPYlib is that the solution of the inverse problem is computed at a cost, measured in linearized forward PDE solves, that is independent of the parameter dimension. The mean of the posterior is approximated by the MAP point, which is found by minimizing the negative log-posterior with an inexact matrix-free Newton-CG method. The posterior covariance is approximated by the inverse of the Hessian of the negative log posterior evaluated at the MAP point. The construction of the posterior covariance is made tractable by invoking a low-rank approximation of the Hessian of the log-likelihood. Scalable tools for sample generation are also discussed. hIPPYlib makes all of these advanced algorithms easily accessible to domain scientists and provides an environment that expedites the development of new algorithms.

Design and Optimization of Numerical Methods for Solving Inverse Problems

. The purpose of inverse problems is to identify the unknown causes or parameters based on observable effects or measurements in a variety of scientific and technical domains. To arrive at precise and trustworthy solutions, numerical algorithms for inverse problems must be designed and optimised. The creation and optimisation of numerical algorithms specifically created for solving inverse issues are presented in detail in this abstract.The formulation of inverse issues and the mathematical models that explain the underlying processes are the primary topics of the first part of this work. We examine many inverse issues, such as ill-posed, parameter estimation, and linear and nonlinear ones. Prior information and regularisation methods must be used to increase the stability and uniqueness of the answers.The creation and application of numerical algorithms for the solution of inverse problems is the focus of the second component. We study in depth a number of iterative techniques, including the conjugate gradient method, Levenberg-Marquardt algorithm, and Gauss-Newton method. There is also discussion of sophisticated methods such variational methods, Bayesian inference, and approaches based on optimisation.The third consideration focuses on optimising numerical techniques to raise their effectiveness and precision. To hasten convergence and lower computational costs, methods like adaptive mesh refinement, parallel processing, and model reduction are examined. Additionally, methods for handling noisy or missing data are looked at, as well as methods for choosing the right regularisation settings.

Learning-based Algorithms for Inverse Problems in MR Image Reconstruction and Quantitative Perfusion Imaging

Learning-based Algorithms for Inverse Problems in MR Image Reconstruction and Quantitative Perfusion Imaging

Algorithms for solving scattering problems for the Manakov model of nonlinear Schrödinger equations

Abstract We introduce numerical algorithms for solving the inverse and direct scattering problems for the Manakov model of vector nonlinear Schrödinger equation. We have found an algebraic group of 4-block matrices with off-diagonal blocks consisting of special vector-like matrices for generalizing the scalar problem’s efficient numerical algorithms to the vector case. The inversion of block matrices of the discretized system of Gelfand–Levitan–Marchenko integral equations solves the inverse scattering problem using the vector variant the Toeplitz Inner Bordering algorithm of Levinson’s type. The reversal of steps of the inverse problem algorithm gives the solution of the direct scattering problem. Numerical tests confirm the proposed vector algorithms’ efficiency and stability. We also present an example of the algorithms’ application to simulate the Manakov vector solitons’ collision.

A SIX-PARAMETER SEMI-QUANTITATIVE ANALYSIS OF 251 PATIENTS FOR THE ENHANCED TRIGGERED TIMING OF HEAD AND NECK CT ANGIOGRAPHY SCANNING VIA THE INVERSE PROBLEM ALGORITHM

In this study, the correlation between triggered timing for head and neck CT angiography (TT CTA) scanning and the average of CT values of both left and right arterial to upper sinuses (LRA/US) reaching a maximal ratio was surveyed and explored using the inverse problem algorithm according to a six-factor semi-quantitative analysis of 251 patients. Six risk factors, namely TT CTA, mean arterial pressure (MAP), heart rate (HR), contrast media solution (CMS), given pressure (Pre), and body surface area (BSA) were used to identify a nonlinear first-order regression correlation between projected and actual LRA/US values. The respective 22 terms were derived via the STATISTICA program. In doing so, a customized loss function ([Formula: see text]) was defined according to the total fluctuation between theoretically projected and actual LRA/US values for all 216 patients. Thus, [Formula: see text] individual data points were included in the algorithm to optimize the compromised solution array [[Formula: see text]] of LRA/US values. The results exhibited a close correlation with loss function [Formula: see text], correlation coefficient [Formula: see text], and a 93.13% variance. Another group of 35 patients with similar symptoms was selected to verify the prediction accuracy and exhibited a high coincidence, although the reverse calculation-based correlation between CC CTA and LRA/US was still controversial from a clinical viewpoint. The proposed algorithm is considered quite instrumental in predicting the LRA/US with ischemic stroke symptoms in the CTA examination.