Choice Of Regularization Parameter Research Articles

The quasi-reversibility method for an inverse source problem for time-space fractional parabolic equations

In this paper, we apply the quasi-reversibility method to solve an inverse source problem for a time-space fractional parabolic equation. Hölder-type error estimates for the regularized solutions are proved for both a priori and a posteriori regularization parameter choice rules. The theoretical error estimates are confirmed with numerical tests for one and two dimensional equations.

Fractional Landweber Iterative Regularization Method for Solving the Inverse Problem of Time-Fractional Schrödinger Equation

In this paper, we consider the inverse problem for the time-fractional Schrödinger equation. This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. We use the fractional Landweber iterative regularization method to solve this inverse problem and obtain the regularization solution. Under a priori and a posteriori regularization parameter choices, the error estimates are all obtained, respectively.

Recovering a Space-Dependent Source Term in the Fractional Diffusion Equation with the Riemann–Liouville Derivative

This research determines an unknown source term in the fractional diffusion equation with the Riemann–Liouville derivative. This problem is ill-posed. Conditional stability for the inverse source problem can be given. Further, a fractional Tikhonov regularization method was applied to regularize the inverse source problem. In the theoretical results, we propose a priori and a posteriori regularization parameter choice rules and obtain the convergence estimates.

Learned Block Iterative Shrinkage Thresholding Algorithm for Photothermal Super Resolution Imaging.

Block-sparse regularization is already well known in active thermal imaging and is used for multiple-measurement-based inverse problems. The main bottleneck of this method is the choice of regularization parameters which differs for each experiment. We show the benefits of using a learned block iterative shrinkage thresholding algorithm (LBISTA) that is able to learn the choice of regularization parameters, without the need to manually select them. In addition, LBISTA enables the determination of a suitable weight matrix to solve the underlying inverse problem. Therefore, in this paper we present LBISTA and compare it with state-of-the-art block iterative shrinkage thresholding using synthetically generated and experimental test data from active thermography for defect reconstruction. Our results show that the use of the learned block-sparse optimization approach provides smaller normalized mean square errors for a small fixed number of iterations. Thus, this allows us to improve the convergence speed and only needs a few iterations to generate accurate defect reconstruction in photothermal super-resolution imaging.

A fast iterative method for identifying the radiogenic source for the helium production-diffusion equation

ABSTRACT In this paper, we consider an inverse source problem for the helium production-diffusion equation. That is to determine a space-dependent source term in the helium production-diffusion equation from a noisy final data. Since the problem is ill-posed, we propose an effective iterative scheme to deal with the problem. More specifically, we use an acceleration technique to speed up a simple iterative scheme. We also give two kinds of convergence rates by using an a priori and a posteriori regularization parameter choice rule, respectively. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method.

Identification of the zeroth‐order coefficient and fractional order in a time‐fractional reaction‐diffusion‐wave equation

In this paper, we investigate an inverse problem of recovering the zeroth‐order coefficient and fractional order simultaneously in a time‐fractional reaction‐diffusion‐wave equation by using boundary measurement data from both of uniqueness and numerical method. We prove the uniqueness of the considered inverse problem and the Lipschitz continuity of the forward operator. Then the inverse problem is formulated into a variational problem by the Tikhonov‐type regularization. Based on the continuity of the forward operator, we prove that the minimizer of the Tikhonov‐type functional exists and converges to the exact solution under an a priori choice of regularization parameter. The steepest descent method combined with Nesterov acceleration is adopted to solve the variational problem. Three numerical examples are presented to support the efficiency and rationality of our proposed method.

Identifying the Unknown Source in Linear Parabolic Equation by a Convoluting Equation Method

This article is devoted to identifying a space-dependent source term in linear parabolic equations. Such a problem is ill posed, i.e., a small perturbation in the input data may cause a dramatically large error in the solution (if it exists). The conditional stability of the solution is analyzed. Based on a convoluting equation method, we can deal with the problem under the a priori parameter choice rule. Meanwhile, a modified version of Morozov’s discrepancy principle is provided to decide on an a posteriori regularization parameter choice strategy and a log-type error estimate is obtained. Two numerical results show that our proposed method works well.

Inverse Source Problem for Sobolev Equation with Fractional Laplacian

In this paper, we are interested in the problem of determining the source function for the Sobolev equation with fractional Laplacian. This problem is ill-posed in the sense of Hadamard. In order to edit the instability instability of the solution, we applied the fractional Landweber method. In the theoretical analysis results, we show the error estimate between the exact solution and the regularized solution by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Finally, we investigate the convergence of the source function when fractional order β ⟶ 1 + .

A Tikhonov regularization method for solving a backward time–space fractional diffusion problem

In this paper, a backward problem for a time–space fractional diffusion equation is considered, which is to determine the initial data from a noisy final data. To deal with this ill-posed problem, combined the ideas of the Tikhonov regularization in Hilbert Schales proposed by Natterer in 1984 and the fractional Tikhonov method given by Hochstenbach–Reichel in 2011, a Tikhonov regularization method is constructed. Based on the Fourier transform, an a-priori and an a-posteriori regularization parameter choice rules are used to guarantee the order optimal convergence rates. By the finite difference methods and the Discrete Fourier transform, numerical examples in one-dimensional and two-dimensional cases are given for the a-posteriori regularization parameter choice rule. Theoretical and numerical results show that the proposed method works well for both the smooth and the non-smooth functions.

Bootstrap confidence interval of ridge regression in linear regression model: A comparative study via a simulation study

It is well known that the variances of the least squares estimates are large and they can be far away from their true values in the case of multicollinearity. Therefore, the ridge regression method can be used as an alternative to the least squares method. However, the ridge estimator has a disadvantage that its distribution is unknown, so only asymptotic confidence intervals are obtained. The purpose of this paper is to study the impact of several ridge regularization parameters on the mean interval lengths of the confidence intervals and coverage probabilities constructed by the ridge estimator. A bootstrap method for the selection of ridge regularization parameter is used as well as the parametric methods. In order to compare the confidence intervals, standard normal approximation, student-t approximation and bootstrap methods are used and comparison is illustrated via real data and simulation study. The simulation study shows that the bootstrap choice of ridge regularization parameter yields narrower standard normal approximated confidence intervals than the PRESS choice of ridge regularization parameter but wider standard normal approximated, student-t approximated and bootstrap confidence intervals than the GCV choice of ridge regularization parameter.

A generalized quasi-boundary value method for recovering a source in a fractional diffusion-wave equation

This paper is devoted to identifying a space-dependent source in a time-fractional diffusion-wave equation by using the final time data. By the series expression of the solution of the direct problem, the inverse source problem can be formulated by a first kind of Fredholm integral equation. The existence and uniqueness, ill-posedness and a conditional stability in Hilbert scale for the considered inverse problem are provided. We propose a generalized quasi-boundary value regularization method to solve the inverse source problem and also prove that the regularized problem is well-posed. Further, two kinds of convergence rates in Hilbert scale for the regularized solution can be obtained by using an a priori and an a posteriori regularization parameter choice rule, respectively. The numerical examples in one-dimensional case and two-dimensional case are given to confirm our theoretical results for the constant coefficients problem. We also propose a finite difference method based on a variant of L1 scheme to solve the regularized problem for the variable coefficients problem and give its convergence rate. One finite difference method based on a convolution quadrature is provided to solve the regularized problem for comparison. The numerical results for three examples by two algorithms are provided to show the effectiveness and stability of the proposed algorithms.

Validating EEG, MEG and Combined MEG and EEG Beamforming for an Estimation of the Epileptogenic Zone in Focal Cortical Dysplasia.

MEG and EEG source analysis is frequently used for the presurgical evaluation of pharmacoresistant epilepsy patients. The source localization of the epileptogenic zone depends, among other aspects, on the selected inverse and forward approaches and their respective parameter choices. In this validation study, we compare the standard dipole scanning method with two beamformer approaches for the inverse problem, and we investigate the influence of the covariance estimation method and the strength of regularization on the localization performance for EEG, MEG, and combined EEG and MEG. For forward modelling, we investigate the difference between calibrated six-compartment and standard three-compartment head modelling. In a retrospective study, two patients with focal epilepsy due to focal cortical dysplasia type IIb and seizure freedom following lesionectomy or radiofrequency-guided thermocoagulation (RFTC) used the distance of the localization of interictal epileptic spikes to the resection cavity resp. RFTC lesion as reference for good localization. We found that beamformer localization can be sensitive to the choice of the regularization parameter, which has to be individually optimized. Estimation of the covariance matrix with averaged spike data yielded more robust results across the modalities. MEG was the dominant modality and provided a good localization in one case, while it was EEG for the other. When combining the modalities, the good results of the dominant modality were mostly not spoiled by the weaker modality. For appropriate regularization parameter choices, the beamformer localized better than the standard dipole scan. Compared to the importance of an appropriate regularization, the sensitivity of the localization to the head modelling was smaller, due to similar skull conductivity modelling and the fixed source space without orientation constraint.

Estimating adsorption isotherm parameters in chromatography via a virtual injection promoting double feed-forward neural network

Abstract The means to obtain the adsorption isotherms is a fundamental open problem in competitive chromatography. A modern technique of estimating adsorption isotherms is to solve a nonlinear inverse problem in a partial differential equation so that the simulated batch separation coincides with actual experimental results. However, this identification process is usually ill-posed in the sense that the uniqueness of adsorption isotherms cannot be guaranteed, and moreover, the small noise in the measured response can lead to a large fluctuation in the traditional estimation of adsorption isotherms. The conventional mathematical method of solving this problem is the variational regularization, which is formulated as a non-convex minimization problem with a regularized objective functional. However, in this method, the choice of regularization parameter and the design of a convergent solution algorithm are quite difficult in practice. Moreover, due to the restricted number of injection profiles in experiments, the types of measured data are extremely limited, which may lead to a biased estimation. In order to overcome these difficulties, in this paper, we develop a new inversion method – the virtual injection promoting double feed-forward neural network (VIP-DFNN). In this approach, the training data contain various types of artificial injections and synthetic noisy measurement at outlet, generated by a conventional physics model – a time-dependent convection-diffusion system. Numerical experiments with both artificial and real data from laboratory experiments show that the proposed VIP-DFNN is an efficient and robust algorithm.

A numerical comparison of some heuristic stopping rules for nonlinear Landweber iteration

The choice of a suitable regularization parameter is an important part of most regularization methods for inverse problems. In the absence of reliable estimates of the noise level, heuristic parameter choice rules can be used to accomplish this task. While they are already fairly well understood and tested in the case of linear problems, not much is known about their behaviour for nonlinear problems and even less in the respective case of iterative regularization. Hence, in this paper, we numerically study the performance of some of these rules when used to determine a stopping index for Landweber iteration for various nonlinear inverse problems. These are chosen from different practically relevant fields such as integral equations, parameter estimation, and tomography.

Source Identification for the Time-Fractional Diffusion Equation With Robin Boundary Conditions

The source term identification of the time-fractional diffusion equation with Robin boundary conditions is studied. This problem is ill-posed. Therefore, an iterative regularization method is constructed to calculate the regularized approximate solution of the source term. The error estimates between the regularized approximate solution and the exact solution are given under the priori and the posteriori regularization parameter choice rules. Finally, numerical examples verify the effectiveness of the method.

Phase-field modeling and peridynamics for defect dynamics, and an augmented phase-field model with viscous stresses

This work begins by applying peridynamics and phase-field modeling to predict 1-d interface motion with inertia in an elastic solid with a non-monotone stress–strain response. In classical nonlinear elasticity, it is known that subsonic interfaces require a kinetic law, in addition to momentum balance, to obtain unique solutions; in contrast, for supersonic interfaces, momentum balance alone is sufficient to provide unique solutions. This work finds that peridynamics agrees with this classical result, in that different choices of regularization parameters provide different kinetics for subsonic motion but the same kinetics for supersonic motion. In contrast, conventional phase-field models coupled to elastodynamics are unable to model, even qualitatively, the supersonic motion of interfaces. This work identifies the shortcomings in the physics of standard phase-field models to be: (1) the absence of higher-order stress to balance unphysical stress singularities, and (2) the ability of the model to access unphysical regions of the energy landscape.Based on these observations, this work proposes an augmented phase-field model to introduce the missing physics. The augmented model adds: (1) a viscous stress to the momentum balance, in addition to the dissipative phase-field evolution, to regularize singularities; and (2) an augmented driving force that models the physical mechanism that keeps the system out of unphysical regions of the energy landscape. When coupled to elastodynamics, the augmented model correctly describes both subsonic and supersonic interface motion. The augmented model has essentially the same computational expense as conventional phase-field models and requires only minor modifications of numerical methods, and is therefore proposed as a replacement to the conventional phase-field models.

Landweber Iterative Regularization Method for Identifying the Initial Value Problem of the Rayleigh–Stokes Equation

In this paper, we study an inverse problem to identify the initial value problem of the homogeneous Rayleigh–Stokes equation for a generalized second-grade fluid with the Riemann–Liouville fractional derivative model. This problem is ill posed; that is, the solution (if it exists) does not depend continuously on the data. We use the Landweber iterative regularization method to solve the inverse problem. Based on a conditional stability result, the convergent error estimates between the exact solution and the regularization solution by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule are given. Some numerical experiments are performed to illustrate the effectiveness and stability of this method.

Analysis of High-Throughput Electrochemical Impedance Spectroscopy via Hierarchical Bayesian Models

Electrochemical impedance spectroscopy (EIS) is a powerful characterization technique that enables in-situ investigation of electrochemical processes over a wide range of timescales, making it particularly effective in analysis of electrochemical energy conversion devices such as fuel cells and batteries. The advent of automated high-throughput EIS systems presents an exciting opportunity for materials discovery, rapidly providing detailed information about candidate materials. However, the utility of EIS as an electrochemical method is paralleled by the complexity of interpreting the resulting data: overlapping features, noise, and measurement artifacts make EIS spectra notoriously difficult to analyze. Equivalent circuit models (ECMs) are widely used to deconvolute EIS data, but there is significant ambiguity in model selection; meanwhile, the distribution of relaxation times (DRT) has become increasingly popular as an alternative/complementary deconvolution method, but faces similar uncertainty in the choice of regularization parameters. The sheer volume of data produced by high-throughput methods amplifies these challenges, necessitating automated, self-consistent analysis procedures.Here, we discuss high-throughput EIS characterization of protonic ceramic electrode thin films and demonstrate the application of hierarchical Bayesian models for robust analysis of the data. The calibrated hierarchical Bayesian DRT resolves much of the ambiguity in regularization parameters by encoding desired properties of the DRT in prior distributions [1]. Meanwhile, the versatility of the DRT enables appropriate deconvolution of thousands of spectra without predetermined model selection. The DRT may be translated to an ECM for further quantitative analysis of individual relaxation timescales and magnitudes by leveraging a flexible semi-empirical model. In addition, advanced Bayesian DRT models can be implemented to address outliers, sample drift, and multi-spectrum deconvolution. These tools enable conversion of raw spectra to a dataset amenable to further interpretation via machine learning and other methods, thereby expanding opportunities to distill meaningful insight from high-throughput electrochemical studies.

Flexible Krylov Methods for Edge Enhancement in Imaging.

Many successful variational regularization methods employed to solve linear inverse problems in imaging applications (such as image deblurring, image inpainting, and computed tomography) aim at enhancing edges in the solution, and often involve non-smooth regularization terms (e.g., total variation). Such regularization methods can be treated as iteratively reweighted least squares problems (IRLS), which are usually solved by the repeated application of a Krylov projection method. This approach gives rise to an inner–outer iterative scheme where the outer iterations update the weights and the inner iterations solve a least squares problem with fixed weights. Recently, flexible or generalized Krylov solvers, which avoid inner–outer iterations by incorporating iteration-dependent weights within a single approximation subspace for the solution, have been devised to efficiently handle IRLS problems. Indeed, substantial computational savings are generally possible by avoiding the repeated application of a traditional Krylov solver. This paper aims to extend the available flexible Krylov algorithms in order to handle a variety of edge-enhancing regularization terms, with computationally convenient adaptive regularization parameter choice. In order to tackle both square and rectangular linear systems, flexible Krylov methods based on the so-called flexible Golub–Kahan decomposition are considered. Some theoretical results are presented (including a convergence proof) and numerical comparisons with other edge-enhancing solvers show that the new methods compute solutions of similar or better quality, with increased speedup.

Optimizing constrained reconstruction in magnetic resonance imaging for signal detection

Constrained reconstruction in magnetic resonance imaging (MRI) allows the use of prior information through constraints to improve reconstructed images. These constraints often take the form of regularization terms in the objective function used for reconstruction. Constrained reconstruction leads to images which appear to have fewer artifacts than reconstructions without constraints but because the methods are typically nonlinear, the reconstructed images have artifacts whose structure is hard to predict. In this work, we compared different methods of optimizing the regularization parameter using a total variation (TV) constraint in the spatial domain and sparsity in the wavelet domain for one-dimensional (2.56×) undersampling using variable density undersampling. We compared the mean squared error (MSE), structural similarity (SSIM), L-curve and the area under the receiver operating characteristic (AUC) using a linear discriminant for detecting a small and a large signal. We used a signal-known-exactly task with varying backgrounds in a simulation where the anatomical variation was the major source of clutter for the detection task. Our results show that the AUC dependence on regularization parameters varies with the imaging task (i.e. the signal being detected). The choice of regularization parameters for MSE, SSIM, L-curve and AUC were similar. We also found that a model-based reconstruction including TV and wavelet sparsity did slightly better in terms of AUC than just enforcing data consistency but using these constraints resulted in much better MSE and SSIM. These results suggest that the increased performance in MSE and SSIM over-estimate the improvement in detection performance for the tasks in this paper. The MSE and SSIM metrics show a big difference in performance where the difference in AUC is small. To our knowledge, this is the first time that signal detection with varying backgrounds has been used to optimize constrained reconstruction in MRI.