Inverse Problems - IOPscience

Inverse Problems - IOPscience
Inverse Problems
Purpose-led Publishing
is a coalition of three not-for-profit publishers in the field of physical sciences: AIP Publishing, the American Physical Society and IOP Publishing.
Together, as publishers that will always put purpose above profit, we have defined a set of industry standards that underpin high-quality, ethical scholarly communications.
We are proudly declaring that science is our only shareholder.
SUPPORTS OPEN ACCESS
An interdisciplinary journal combining mathematical and experimental papers on inverse problems with numerical and practical approaches to their solution.
We are pleased to welcome Fioralba Cakoni as the new Editor-in-Chief of
Inverse Problems
Submit
an article
opens in new tab
Track my article
opens in new tab
RSS
Sign up for new issue notifications
Median submission to first decision before peer review
12 days
Median submission to first decision after peer review
82 days
Impact factor
2.1
Citescore
3.3
Full list of journal metrics
The following article is
Open access
Deep learning methods for partial differential equations and related parameter identification problems
Derick Nganyu Tanyu
et al
2023
Inverse Problems
39
103001
View article
, Deep learning methods for partial differential equations and related parameter identification problems
PDF
, Deep learning methods for partial differential equations and related parameter identification problems
Recent years have witnessed a growth in mathematics for deep learning—which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust—and deep learning for mathematics, where deep learning algorithms are used to solve problems in mathematics. The latter has popularised the field of scientific machine learning where deep learning is applied to problems in scientific computing. Specifically, more and more neural network (NN) architectures have been developed to solve specific classes of partial differential equations (PDEs). Such methods exploit properties that are inherent to PDEs and thus solve the PDEs better than standard feed-forward NNs, recurrent NNs, or convolutional neural networks. This has had a great impact in the area of mathematical modelling where parametric PDEs are widely used to model most natural and physical processes arising in science and engineering. In this work, we review such methods as well as their extensions for parametric studies and for solving the related inverse problems. We also show their relevance in various industrial applications.
The following article is
Open access
A guide to stochastic optimisation for large-scale inverse problems
Matthias J Ehrhardt
et al
2025
Inverse Problems
41
053001
View article
, A guide to stochastic optimisation for large-scale inverse problems
PDF
, A guide to stochastic optimisation for large-scale inverse problems
Stochastic optimisation algorithms are the de facto standard for machine learning with large amounts of data. Handling only a subset of available data in each optimisation step dramatically reduces the per-iteration computational costs, while still ensuring significant progress towards the solution. Driven by the need to solve large-scale optimisation problems as efficiently as possible, the last decade has witnessed an explosion of research in this area. Leveraging the parallels between machine learning and inverse problems has allowed harnessing the power of this research wave for solving inverse problems. In this survey, we provide a comprehensive account of the state-of-the-art in stochastic optimisation from the viewpoint of variational regularisation for inverse problems where the solution is modelled as minimising an objective function. We cover topics such as variance reduction, acceleration and higher-order methods, and compare theoretical results with practical behaviour. We focus on the potential and the challenges for stochastic optimisation that are unique to variational regularisation for inverse imaging problems and are not commonly encountered in machine learning. We conclude the survey with illustrative examples on linear inverse problems in imaging to examine the advantages and disadvantages that this new generation of algorithms brings to the field of inverse problems.
The following article is
Open access
On the convergence of stochastic variance reduced gradient for linear inverse problems
Bangti Jin and Zehui Zhou 2026
Inverse Problems
42
045006
View article
, On the convergence of stochastic variance reduced gradient for linear inverse problems
PDF
, On the convergence of stochastic variance reduced gradient for linear inverse problems
Stochastic variance reduced gradient (SVRG) is an accelerated version of stochastic gradient descent based on variance reduction, and is promising for solving large-scale inverse problems. In this work, we analyze SVRG and a regularized version that incorporates
a priori
knowledge of the problem, for solving linear inverse problems in Hilbert spaces. We prove that, with suitable constant step size schedules and regularity conditions, the regularized SVRG can achieve optimal convergence rates in terms of the noise level without any early stopping rules, provided that the truncation level is chosen suitably, and standard SVRG is also optimal for problems with nonsmooth solutions under
a priori
stopping rules. The analysis is based on an explicit error recursion and suitable
a priori
estimates on the inner loop updates with respect to the anchor point. Numerical experiments are provided to complement the theoretical analysis.
The following article is
Open access
Elliptic Bayesian inverse problems on metric graphs
D Bolin
et al
2026
Inverse Problems
42
035012
View article
, Elliptic Bayesian inverse problems on metric graphs
PDF
, Elliptic Bayesian inverse problems on metric graphs
This paper studies the formulation, well-posedness, and numerical solution of Bayesian inverse problems on metric graphs, in which the edges represent one-dimensional wires connecting vertices. We focus on the inverse problem of recovering the diffusion coefficient of a (fractional) elliptic equation on a metric graph from noisy measurements of the solution. Well-posedness hinges on both stability of the forward model and an appropriate choice of prior. We establish the stability of elliptic and fractional elliptic forward models using recent regularity theory for differential equations on metric graphs. For the prior, we leverage modern Gaussian Whittle–Matérn process models on metric graphs with sufficiently smooth sample paths. Numerical results demonstrate accurate reconstruction and effective uncertainty quantification.
The following article is
Open access
NETT: solving inverse problems with deep neural networks
Housen Li
et al
2020
Inverse Problems
36
065005
View article
, NETT: solving inverse problems with deep neural networks
PDF
, NETT: solving inverse problems with deep neural networks
Recovering a function or high-dimensional parameter vector from indirect measurements is a central task in various scientific areas. Several methods for solving such inverse problems are well developed and well understood. Recently, novel algorithms using deep learning and neural networks for inverse problems appeared. While still in their infancy, these techniques show astonishing performance for applications like low-dose CT or various sparse data problems. However, there are few theoretical results for deep learning in inverse problems. In this paper, we establish a complete convergence analysis for the proposed NETT (network Tikhonov) approach to inverse problems. NETT considers nearly data-consistent solutions having small value of a regularizer defined by a trained neural network. We derive well-posedness results and quantitative error estimates, and propose a possible strategy for training the regularizer. Our theoretical results and framework are different from any previous work using neural networks for solving inverse problems. A possible data driven regularizer is proposed. Numerical results are presented for a tomographic sparse data problem, which demonstrate good performance of NETT even for unknowns of different type from the training data. To derive the convergence and convergence rates results we introduce a new framework based on the absolute Bregman distance generalizing the standard Bregman distance from the convex to the non-convex case.
The following article is
Open access
On the intensity-based inversion method for quantitative quasi-static elastography
Ekaterina Sherina and Simon Hubmer 2026
Inverse Problems
42
045004
View article
, On the intensity-based inversion method for quantitative quasi-static elastography
PDF
, On the intensity-based inversion method for quantitative quasi-static elastography
In this paper, we consider the intensity-based inversion method (IIM) for quantitative material parameter estimation in quasi-static elastography. In particular, we consider the problem of estimating the material parameters of a given sample from two internal measurements, one obtained before and one after applying some form of deformation. These internal measurements can be obtained via any imaging modality of choice, for example ultrasound, optical coherence or photo-acoustic tomography. Compared to two-step approaches to elastography, which first estimate internal displacement fields or strains and then reconstruct the material parameters from them, the IIM is a one-step approach which computes the material parameters directly from the internal measurements. To do so, the IIM combines image registration together with a model-based, regularized parameter reconstruction approach. This combination has the advantage of avoiding some approximations and derivative computations typically found in two-step approaches, and results in the IIM being generally more stable to measurement noise. In the paper, we provide a full convergence analysis of the IIM within the framework of inverse problems, and detail its application to linear elastography. Furthermore, we discuss the numerical implementation of the IIM and provide numerical examples simulating an optical coherence elastography experiment.
The following article is
Open access
Bayesian inverse Navier–Stokes problems: joint flow field reconstruction and parameter learning
Alexandros Kontogiannis
et al
2025
Inverse Problems
41
015008
View article
, Bayesian inverse Navier–Stokes problems: joint flow field reconstruction and parameter learning
PDF
, Bayesian inverse Navier–Stokes problems: joint flow field reconstruction and parameter learning
We formulate and solve a Bayesian inverse Navier–Stokes (N–S) problem that assimilates velocimetry data in order to jointly reconstruct a 3D flow field and learn the unknown N–S parameters, including the boundary position. By hardwiring a generalised N–S problem, and regularising its unknown parameters using Gaussian prior distributions, we learn the most likely parameters in a collapsed search space. The most likely flow field reconstruction is then the N–S solution that corresponds to the learned parameters. We develop the method in the variational setting and use a stabilised Nitsche weak form of the N–S problem that permits the control of all N–S parameters. To regularise the inferred geometry, we use a viscous signed distance field as an auxiliary variable, which is given as the solution of a viscous Eikonal boundary value problem. We devise an algorithm that solves this inverse problem, and numerically implement it using an adjoint-consistent stabilised cut-cell finite element method. We then use this method to reconstruct magnetic resonance velocimetry (flow-MRI) data of a 3D steady laminar flow through a physical model of an aortic arch for two different Reynolds numbers and signal-to-noise ratio (SNR) levels (low/high). We find that the method can accurately (i) reconstruct the low SNR data by filtering out the noise/artefacts and recovering flow features that are obscured by noise, and (ii) reproduce the high SNR data without overfitting. Although the framework that we develop applies to 3D steady laminar flows in complex geometries, it readily extends to time-dependent laminar and Reynolds-averaged turbulent flows, as well as non-Newtonian (e.g. viscoelastic) fluids.
The following article is
Open access
Reconstructing wind fields from gravitational data on gas giants: an investigation of mathematical methods
Tim-Jonas Peter
et al
2026
Inverse Problems
42
035011
View article
, Reconstructing wind fields from gravitational data on gas giants: an investigation of mathematical methods
PDF
, Reconstructing wind fields from gravitational data on gas giants: an investigation of mathematical methods
The atmospheric structure of gas giants, especially those of Jupiter and Saturn, has been an object of scientific studies for a long time. The measurement of the gravitational fields by the Juno mission for Jupiter and the Cassini mission for Saturn offered new possibilities to study the interior structure of these planets. Accordingly, the reconstruction of the wind velocities from gravitational data on gas giants has been the subject of many research papers over the years, yet the mathematical foundations of this inverse problem and its numerical resolution have not been studied in detail. This article suggests a rigorous mathematical theory for inferring the wind fields of gas giants. In particular, an orthonormal basis is derived which can be associated to models of the gravitational potential and the interior wind velocity field. Moreover, this approach provides the foundations for existing resolution concepts of the inverse problem.
The following article is
Open access
The method of the approximate inverse for limited-angle CT
Bernadette N Hahn
et al
2026
Inverse Problems
42
045011
View article
, The method of the approximate inverse for limited-angle CT
PDF
, The method of the approximate inverse for limited-angle CT
Limited-angle computerized tomography stands for one of the most difficult challenges in imaging. Although it opens the way to faster data acquisition in industry and less dangerous scans in medicine, standard approaches, such as the filtered backprojection (FBP) algorithm or the widely used total-variation functional, often produce various artefacts that hinder the diagnosis. With the rise of deep learning, many modern techniques have proven themselves successful in removing such artefacts but at the cost of large datasets. In this paper, we propose a new model-driven approach based on the method of the approximate inverse, which could serve as new starting point for learning strategies in the future. In contrast to FBP-type approaches, our reconstruction step consists in evaluating linear functionals on the measured data using reconstruction kernels that are precomputed as solution of an auxiliary problem. With this problem being uniquely solvable, the derived limited-angle reconstruction kernel is able to fully reconstruct the object without the well-known streak artefacts, even for large limited angles. However, it inherits severe ill-conditioning which leads to a different kind of artefacts arising from the singular functions of the limited-angle Radon transform. The problem becomes particularly challenging when working on semi-discrete (real or analytical) measurements. We develop a general regularization strategy by combining spectral filter, the method of the approximate inverse and custom edge-preserving denoising in order to stabilize the whole process. We further derive and interpret error estimates for the application on real, i.e. semi-discrete, data and we validate our approach on synthetic and real data.
The following article is
Open access
Bayesian adaptive eigenspace inversion
Marie Graff
et al
2026
Inverse Problems
42
035002
View article
, Bayesian adaptive eigenspace inversion
PDF
, Bayesian adaptive eigenspace inversion
We develop a probabilistic analogue of the adaptive eigenspace inversion (AEI) method, an algorithm for the adapting regularisation terms towards the recovery of sharp interfaces, from the perspective of Bayesian inverse problems (BIPs) to enable the quantification of uncertainties in the reconstructions. We show that the adaptations of the regularisation terms in AEI is analogous to adapting the prior distributions in a sequence of BIPs. We present a theoretical result, showing that appropriate convergence of the AEI loop will lead to convergence of the resulting Bayesian posteriors in a Wasserstein distance. Furthermore, we present some numerical experiments to demonstrate the performance and wide applicability of the Bayesian extension of AEI in both linear and nonlinear inverse problems. In particular, we observe in our experiments that the posterior means recover sharp edges in the unknown fields, and the posterior variances are largely concentrated around sharp interfaces, a feature that is also observed in other models for recovery of piecewise constant fields.
The following article is
Open access
Sparse variational regularization with oversmoothing penalty term in the scale of sequence spaces
Robert Plato and Bernd Hofmann 2026
Inverse Problems
42
045012
View article
, Sparse variational regularization with oversmoothing penalty term in the scale of sequence spaces
PDF
, Sparse variational regularization with oversmoothing penalty term in the scale of sequence spaces
In this work, we consider a class of linear ill-posed problems with operators that map from the sequence space
) into a Banach space and in addition satisfy a conditional stability estimate in the scale of sequence spaces
. For the regularization of such problems in the presence of deterministic noise, we consider variational regularization with a penalty functional either of the form
for some
> 0 or in form of the counting measure
. The latter case guarantees sparsity of the corresponding regularized solutions. In this framework, we present first stability and then convergence rates for suitable
a priori
parameter choices. The results cover the oversmoothing situation, where the desired solution does not belong to the domain of definition of the considered penalty functional. The analysis of the oversmoothing case utilizes auxiliary elements that are defined by means of hard thresholding. Such technique can also be used for post processing to guarantee sparsity. Some numerical illustrations are included.
Locally-averaged McCormick relaxations for discretization-regularized inverse problems
Barbara Kaltenbacher and Paul Manns 2026
Inverse Problems
42
045015
View article
, Locally-averaged McCormick relaxations for discretization-regularized inverse problems
PDF
, Locally-averaged McCormick relaxations for discretization-regularized inverse problems
In this paper, by means of a standard model problem, we devise an approach to computing approximate dual bounds for use in global optimization of coefficient identification in partial differential equations (PDEs) by, e.g. (spatial) branch-and-bound methods. Linearization is achieved by a McCormick relaxation (that is, replacing the bilinear PDE term by a linear one and adding inequality constraints), combined with local averaging to reduce the number of inequalities. Optimization-based bound tightening allows us to tighten the relaxation and thus reduce the induced error. Combining this with a quantification of the discretization error and the propagated noise, we prove that the resulting discretization regularizes the inverse problem, thus leading to an overall convergent scheme. Numerical experiments illustrate the theoretical findings.
Robust sparse phase retrieval: statistical guarantee, optimality theory and convergent algorithm
Jun Fan
et al
2026
Inverse Problems
42
045014
View article
, Robust sparse phase retrieval: statistical guarantee, optimality theory and convergent algorithm
PDF
, Robust sparse phase retrieval: statistical guarantee, optimality theory and convergent algorithm
Phase retrieval (PR) is a popular research topic in signal processing and machine learning. However, its performance degrades significantly when the measurements are corrupted by noise or outliers. To address this limitation, we propose a novel robust sparse PR method that covers both real- and complex-valued cases. The core is to leverage the Huber function to measure the loss and adopt the
-norm regularization to realize feature selection, thereby improving the robustness of PR. In theory, we establish statistical guarantees for such robustness and derive necessary optimality conditions for global minimizers. Particularly, for the complex-valued case, we provide a fixed point inclusion property inspired by Wirtinger derivatives. Furthermore, we develop an efficient optimization algorithm by integrating the gradient descent method into a majorization–minimization framework. It is rigorously proved that the whole generated sequence is convergent and also has a linear convergence rate under mild conditions, which has not been investigated before. Numerical examples under different types of noise validate the robustness and effectiveness of our proposed method.
Identification problems for anisotropic time-fractional subdiffusion equations
Simone Creo
et al
2026
Inverse Problems
42
045013
View article
, Identification problems for anisotropic time-fractional subdiffusion equations
PDF
, Identification problems for anisotropic time-fractional subdiffusion equations
We investigate the inverse problem consisting in the identification of constant coefficients appearing in a finite sum of positive self-adjoint operators governing a fractional-in-time partial differential equation on a Hilbert space under overdeterminating conditions. We prove the uniqueness of the solution to the inverse problem when the fractional order
of the derivative is in (0, 1). Also a conditioned existence result is provided. A suitable selection of numerical calculations complements the existence result by giving a visual description of the shape of some key sets related to our problem in special cases in dimension two. In addition, we prove that, as
, the solution corresponding to
tends to the classical one (
= 1). Applications to examples of heat diffusion and elasticity are presented.
The following article is
Open access
The method of the approximate inverse for limited-angle CT
Bernadette N Hahn
et al
2026
Inverse Problems
42
045011
View article
, The method of the approximate inverse for limited-angle CT
PDF
, The method of the approximate inverse for limited-angle CT
Limited-angle computerized tomography stands for one of the most difficult challenges in imaging. Although it opens the way to faster data acquisition in industry and less dangerous scans in medicine, standard approaches, such as the filtered backprojection (FBP) algorithm or the widely used total-variation functional, often produce various artefacts that hinder the diagnosis. With the rise of deep learning, many modern techniques have proven themselves successful in removing such artefacts but at the cost of large datasets. In this paper, we propose a new model-driven approach based on the method of the approximate inverse, which could serve as new starting point for learning strategies in the future. In contrast to FBP-type approaches, our reconstruction step consists in evaluating linear functionals on the measured data using reconstruction kernels that are precomputed as solution of an auxiliary problem. With this problem being uniquely solvable, the derived limited-angle reconstruction kernel is able to fully reconstruct the object without the well-known streak artefacts, even for large limited angles. However, it inherits severe ill-conditioning which leads to a different kind of artefacts arising from the singular functions of the limited-angle Radon transform. The problem becomes particularly challenging when working on semi-discrete (real or analytical) measurements. We develop a general regularization strategy by combining spectral filter, the method of the approximate inverse and custom edge-preserving denoising in order to stabilize the whole process. We further derive and interpret error estimates for the application on real, i.e. semi-discrete, data and we validate our approach on synthetic and real data.
The following article is
Open access
A guide to stochastic optimisation for large-scale inverse problems
Matthias J Ehrhardt
et al
2025
Inverse Problems
41
053001
View article
, A guide to stochastic optimisation for large-scale inverse problems
PDF
, A guide to stochastic optimisation for large-scale inverse problems
Stochastic optimisation algorithms are the de facto standard for machine learning with large amounts of data. Handling only a subset of available data in each optimisation step dramatically reduces the per-iteration computational costs, while still ensuring significant progress towards the solution. Driven by the need to solve large-scale optimisation problems as efficiently as possible, the last decade has witnessed an explosion of research in this area. Leveraging the parallels between machine learning and inverse problems has allowed harnessing the power of this research wave for solving inverse problems. In this survey, we provide a comprehensive account of the state-of-the-art in stochastic optimisation from the viewpoint of variational regularisation for inverse problems where the solution is modelled as minimising an objective function. We cover topics such as variance reduction, acceleration and higher-order methods, and compare theoretical results with practical behaviour. We focus on the potential and the challenges for stochastic optimisation that are unique to variational regularisation for inverse imaging problems and are not commonly encountered in machine learning. We conclude the survey with illustrative examples on linear inverse problems in imaging to examine the advantages and disadvantages that this new generation of algorithms brings to the field of inverse problems.
The following article is
Open access
Deep learning methods for partial differential equations and related parameter identification problems
Derick Nganyu Tanyu
et al
2023
Inverse Problems
39
103001
View article
, Deep learning methods for partial differential equations and related parameter identification problems
PDF
, Deep learning methods for partial differential equations and related parameter identification problems
Recent years have witnessed a growth in mathematics for deep learning—which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust—and deep learning for mathematics, where deep learning algorithms are used to solve problems in mathematics. The latter has popularised the field of scientific machine learning where deep learning is applied to problems in scientific computing. Specifically, more and more neural network (NN) architectures have been developed to solve specific classes of partial differential equations (PDEs). Such methods exploit properties that are inherent to PDEs and thus solve the PDEs better than standard feed-forward NNs, recurrent NNs, or convolutional neural networks. This has had a great impact in the area of mathematical modelling where parametric PDEs are widely used to model most natural and physical processes arising in science and engineering. In this work, we review such methods as well as their extensions for parametric studies and for solving the related inverse problems. We also show their relevance in various industrial applications.
Optimal experimental design for infinite-dimensional Bayesian inverse problems governed by PDEs: a review
Alen Alexanderian 2021
Inverse Problems
37
043001
View article
, Optimal experimental design for infinite-dimensional Bayesian inverse problems governed by PDEs: a review
PDF
, Optimal experimental design for infinite-dimensional Bayesian inverse problems governed by PDEs: a review
We present a review of methods for optimal experimental design (OED) for Bayesian inverse problems governed by partial differential equations with infinite-dimensional parameters. The focus is on problems where one seeks to optimize the placement of measurement points, at which data are collected, such that the uncertainty in the estimated parameters is minimized. We present the mathematical foundations of OED in this context and survey the computational methods for the class of OED problems under study. We also outline some directions for future research in this area.
The following article is
Open access
Higher-order total variation approaches and generalisations
Kristian Bredies and Martin Holler 2020
Inverse Problems
36
123001
View article
, Higher-order total variation approaches and generalisations
PDF
, Higher-order total variation approaches and generalisations
Over the last decades, the total variation (TV) has evolved to be one of the most broadly-used regularisation functionals for inverse problems, in particular for imaging applications. When first introduced as a regulariser, higher-order generalisations of TV were soon proposed and studied with increasing interest, which led to a variety of different approaches being available today. We review several of these approaches, discussing aspects ranging from functional-analytic foundations to regularisation theory for linear inverse problems in Banach space, and provide a unified framework concerning well-posedness and convergence for vanishing noise level for respective Tikhonov regularisation. This includes general higher orders of TV, additive and infimal-convolution multi-order total variation, total generalised variation, and beyond. Further, numerical optimisation algorithms are developed and discussed that are suitable for solving the Tikhonov minimisation problem for all presented models. Focus is laid in particular on covering the whole pipeline starting at the discretisation of the problem and ending at concrete, implementable iterative procedures. A major part of this review is finally concerned with presenting examples and applications where higher-order TV approaches turned out to be beneficial. These applications range from classical inverse problems in imaging such as denoising, deconvolution, compressed sensing, optical-flow estimation and decompression, to image reconstruction in medical imaging and beyond, including magnetic resonance imaging, computed tomography, magnetic-resonance positron emission tomography, and electron tomography.
The D-bar method for electrical impedance tomography—demystified
J L Mueller and S Siltanen 2020
Inverse Problems
36
093001
View article
, The D-bar method for electrical impedance tomography—demystified
PDF
, The D-bar method for electrical impedance tomography—demystified
Electrical impedance tomography (EIT) is an imaging modality where a patient or object is probed using harmless electric currents. The currents are fed through electrodes placed on the surface of the target, and the data consists of voltages measured at the electrodes resulting from a linearly independent set of current injection patterns. EIT aims to recover the internal distribution of electrical conductivity inside the target. The inverse problem underlying the EIT image formation task is nonlinear and severely ill-posed, and hence sensitive to modeling errors and measurement noise. Therefore, the inversion process needs to be regularized. However, traditional variational regularization methods, based on optimization, often suffer from local minima because of nonlinearity. This is what makes regularized direct (non-iterative) methods attractive for EIT. The most developed direct EIT algorithm is the D-bar method, based on complex geometric optics solutions and a nonlinear Fourier transform. Variants and recent developments of D-bar methods are reviewed, and their practical numerical implementation is explained.
Recovery Performance of PhaseLift for Phase Retrieval from Coded Diffraction Patterns
Huang et al
View accepted manuscript
, Recovery Performance of PhaseLift for Phase Retrieval from Coded Diffraction Patterns
PDF
, Recovery Performance of PhaseLift for Phase Retrieval from Coded Diffraction Patterns
The PhaseLift algorithm is an effective convex method for solving the phase retrieval problem from Fourier measurements with coded diffraction patterns (CDP). While exact reconstruction guarantees are well-established in the noiseless case, the stability of recovery under noise remains less well understood. In particular, when the measurements are corrupted by an additive noise vector $\vw \in \R^m$, existing recovery bounds scale on the order of $\norm{\vw}$, which is conjectured to be suboptimal. More recently, Soltanolkotabi conjectured that the optimal PhaseLift recovery bound should scale with the average noise magnitude, that is, on the order of $\norm{\vw}/\sqrt m$. However, establishing this theoretically is considerably more challenging and has remained an open problem. In this paper, we focus on this conjecture and prove that under adversarial noise, the recovery error of PhaseLift is bounded by $O\xkh{ \sqrt{\frac{\norm{\vw}\log n }{\sqrt m}}}\norm{\vx_0}$. Here, $\vx_0 \in \C^n$ is the signals we aim to recover. Moreover, for mean-zero sub-Gaussian noise vector $\vw \in \R^m$, a upper error bound and its corresponding minimax lower bound are also provided. Our results represent a significant step toward Soltanolkotabi's conjecture, offering new insights into the stability of PhaseLift under noisy CDP measurements.
The following article is
Open access
Taylor Approximation Variance Reduction for Approximation Errors in PDE-constrained Bayesian Inverse Problems *
Nicholson et al
View accepted manuscript
, Taylor Approximation Variance Reduction for Approximation Errors in PDE-constrained Bayesian Inverse Problems *
PDF
, Taylor Approximation Variance Reduction for Approximation Errors in PDE-constrained Bayesian Inverse Problems *
In numerous applications, surrogate models are used as a replacement for accurate parameter-to-observable mappings when solving large-scale inverse problems governed by partial differential equations (PDEs). The surrogate model may be a computationally cheaper alternative to the accurate parameter-to-observable mappings and/or may ignore additional unknowns or sources of uncertainty. The Bayesian approximation error (BAE) approach provides a means to account for the induced uncertainties and approximation errors, i.e., the errors between the accurate parameter-to-observable mapping and the surrogate. The statistics of these errors are, however, in general unknown a priori, and are thus calculated using Monte Carlo sampling. Although the sampling is typically carried out offline, i.e., before considering the data, the process can still represent a computational bottleneck. In this work, we develop a scalable computational approach for reducing the costs associated with the sampling stage of the BAE approach. Specifically, we consider the Taylor expansion of the accurate and surrogate forward models with respect to the uncertain parameter fields either as a control variate for variance reduction or as a means to directly and efficiently approximate the mean and covariance of the approximation errors. We propose efficient methods for evaluating the expressions for the mean and covariance of the Taylor approximations based on linear(-ized) PDE solves. Furthermore, the proposed approach is independent of the dimension of the uncertain parameter, depending instead on the intrinsic dimension of the data, ensuring scalability to high-dimensional problems. The potential benefits of the proposed approach are demonstrated for two high-dimensional inverse problems governed by PDE examples, namely for the estimation of a distributed Robin boundary coefficient in a linear diffusion problem, and for a coefficient estimation problem governed by a nonlinear diffusion problem.
Generalized spherical mean associated with the generalized Darboux equation and its applications
Moon et al
View accepted manuscript
, Generalized spherical mean associated with the generalized Darboux equation and its applications
PDF
, Generalized spherical mean associated with the generalized Darboux equation and its applications
This paper introduces and analyzes a generalized spherical mean operator that unifies the classical spherical mean and the wave forward operator. It is shown that this generalized transform satisfies a generalized Euler–Poisson–Darboux (EPD) equation, and its analytical structure is investigated through Fourier techniques and explicit representation formulas. Several inversion formulas are derived for different cases: when the spatial variable lies on a hyperplane, through an analogue of the Fourier slice theorem; when it lies on the unit sphere, through spherical harmonics; and when it lies on the boundary of an arbitrary bounded domain, through the Kirchhoff integral representation for the wave forward operator. In each case, the derivation extends methods from earlier works on the classical spherical mean or the wave forward operator, and the resulting formulas unify previously known inversion results. Together, these results form a unified analytical foundation for inversion in wave-based imaging across diverse geometric configurations.
A localized consensus-based sampling algorithm
Bouillon et al
View accepted manuscript
, A localized consensus-based sampling algorithm
PDF
, A localized consensus-based sampling algorithm
We propose a localized consensus-based method for sampling from non-Gaussian distributions, a task that frequently arises when solving Bayesian inverse problems. Our method arises from an alternative derivation of consensus-based sampling (CBS). Starting from ensemble-preconditioned Langevin dynamics, we replace the potential by its Moreau envelope—a smoother approximation—in order to replace the gradient in the Langevin equation with a proximal operator. We then approximate this operator by a weighted mean. In the limit of infinitely smoothing the potential to a quadratic function, this procedure recovers the standard CBS dynamics. In addition, outside this limit, we retrieve a refined variant of polarized CBS. We call the resulting algorithm localized consensus-based sampling, since particles interact more with nearby particles than with faraway ones. Our method is affine-invariant, exact for Gaussian targets in the mean-field limit, and demonstrates improved robustness over polarized CBS in numerical experiments. Like other consensus-based methods, localized CBS is gradient-free and easily parallelizable.
Rothe's method in direct and time-dependent inverse source problems for a semilinear pseudo-parabolic equation
Van Bockstal et al
View accepted manuscript
, Rothe's method in direct and time-dependent inverse source problems for a semilinear pseudo-parabolic equation
PDF
, Rothe's method in direct and time-dependent inverse source problems for a semilinear pseudo-parabolic equation
In this paper, we investigate the inverse problem of determining an unknown time-dependent source term in a semilinear pseudo-parabolic equation with variable coefficients and a Dirichlet boundary condition. The unknown source term is recovered from additional measurement data expressed as a weighted spatial average of the solution. By employing Rothe's time-discretisation method, we prove the existence and uniqueness of a weak solution under a smallness condition on the problem data. We also provide a numerical scheme based on a perturbation approach, which reduces the solution of the resulting discrete problem to solving two standard variational problems and evaluating a scalar coefficient, and we demonstrate its accuracy and stability through numerical experiments.
More Accepted manuscripts
The following article is
Open access
Sparse variational regularization with oversmoothing penalty term in the scale of sequence spaces
Robert Plato and Bernd Hofmann 2026
Inverse Problems
42
045012
View article
, Sparse variational regularization with oversmoothing penalty term in the scale of sequence spaces
PDF
, Sparse variational regularization with oversmoothing penalty term in the scale of sequence spaces
In this work, we consider a class of linear ill-posed problems with operators that map from the sequence space
) into a Banach space and in addition satisfy a conditional stability estimate in the scale of sequence spaces
. For the regularization of such problems in the presence of deterministic noise, we consider variational regularization with a penalty functional either of the form
for some
> 0 or in form of the counting measure
. The latter case guarantees sparsity of the corresponding regularized solutions. In this framework, we present first stability and then convergence rates for suitable
a priori
parameter choices. The results cover the oversmoothing situation, where the desired solution does not belong to the domain of definition of the considered penalty functional. The analysis of the oversmoothing case utilizes auxiliary elements that are defined by means of hard thresholding. Such technique can also be used for post processing to guarantee sparsity. Some numerical illustrations are included.
The following article is
Open access
The method of the approximate inverse for limited-angle CT
Bernadette N Hahn
et al
2026
Inverse Problems
42
045011
View article
, The method of the approximate inverse for limited-angle CT
PDF
, The method of the approximate inverse for limited-angle CT
Limited-angle computerized tomography stands for one of the most difficult challenges in imaging. Although it opens the way to faster data acquisition in industry and less dangerous scans in medicine, standard approaches, such as the filtered backprojection (FBP) algorithm or the widely used total-variation functional, often produce various artefacts that hinder the diagnosis. With the rise of deep learning, many modern techniques have proven themselves successful in removing such artefacts but at the cost of large datasets. In this paper, we propose a new model-driven approach based on the method of the approximate inverse, which could serve as new starting point for learning strategies in the future. In contrast to FBP-type approaches, our reconstruction step consists in evaluating linear functionals on the measured data using reconstruction kernels that are precomputed as solution of an auxiliary problem. With this problem being uniquely solvable, the derived limited-angle reconstruction kernel is able to fully reconstruct the object without the well-known streak artefacts, even for large limited angles. However, it inherits severe ill-conditioning which leads to a different kind of artefacts arising from the singular functions of the limited-angle Radon transform. The problem becomes particularly challenging when working on semi-discrete (real or analytical) measurements. We develop a general regularization strategy by combining spectral filter, the method of the approximate inverse and custom edge-preserving denoising in order to stabilize the whole process. We further derive and interpret error estimates for the application on real, i.e. semi-discrete, data and we validate our approach on synthetic and real data.
The following article is
Open access
Taylor Approximation Variance Reduction for Approximation Errors in PDE-constrained Bayesian Inverse Problems *
Ruanui Nicholson
et al
2026
Inverse Problems
View article
, Taylor Approximation Variance Reduction for Approximation Errors in PDE-constrained Bayesian Inverse Problems *
PDF
, Taylor Approximation Variance Reduction for Approximation Errors in PDE-constrained Bayesian Inverse Problems *
In numerous applications, surrogate models are used as a replacement for accurate parameter-to-observable mappings when solving large-scale inverse problems governed by partial differential equations (PDEs). The surrogate model may be a computationally cheaper alternative to the accurate parameter-to-observable mappings and/or may ignore additional unknowns or sources of uncertainty. The Bayesian approximation error (BAE) approach provides a means to account for the induced uncertainties and approximation errors, i.e., the errors between the accurate parameter-to-observable mapping and the surrogate. The statistics of these errors are, however, in general unknown a priori, and are thus calculated using Monte Carlo sampling. Although the sampling is typically carried out offline, i.e., before considering the data, the process can still represent a computational bottleneck. In this work, we develop a scalable computational approach for reducing the costs associated with the sampling stage of the BAE approach. Specifically, we consider the Taylor expansion of the accurate and surrogate forward models with respect to the uncertain parameter fields either as a control variate for variance reduction or as a means to directly and efficiently approximate the mean and covariance of the approximation errors. We propose efficient methods for evaluating the expressions for the mean and covariance of the Taylor approximations based on linear(-ized) PDE solves. Furthermore, the proposed approach is independent of the dimension of the uncertain parameter, depending instead on the intrinsic dimension of the data, ensuring scalability to high-dimensional problems. The potential benefits of the proposed approach are demonstrated for two high-dimensional inverse problems governed by PDE examples, namely for the estimation of a distributed Robin boundary coefficient in a linear diffusion problem, and for a coefficient estimation problem governed by a nonlinear diffusion problem.
The following article is
Open access
Hybrid CG-Tikhonov is a filtration of the CG Lanczos vectors
Daniel Gerth and Kirk M Soodhalter 2026
Inverse Problems
42
045008
View article
, Hybrid CG-Tikhonov is a filtration of the CG Lanczos vectors
PDF
, Hybrid CG-Tikhonov is a filtration of the CG Lanczos vectors
We consider iterative methods for solving linear ill-posed problems with compact operator and right-hand side only available via noise-polluted measurements. Conjugate gradients (CGs) applied to the normal equations with an appropriate stopping rule and CG applied to the system solving for a Tikhonov-regularized solution (
CGtikh
are closely related regularization methods that build iterates from the same Krylov subspaces. In this work, we show that the
CGtikh
iterate can be expressed as
where
are functions of the Tikhonov parameter
and
is the
th CG iterate. We call these functions
Lanczos filters
, and they can be shown to have decay properties as
with the speed of decay increasing with
. This has the effect of filtering out the contribution of the later terms of the CG iterate. The filters can be constructed using quantities defined via recursions at each iteration. We demonstrate with numerical experiments that good parameter choices correspond to appropriate damping of the Lanczos vectors. The filtration approach also provides a platform for further development of parameter choice rules, and similar representations may hold for other hybrid iterative schemes.
The following article is
Open access
A two-point phase recovering from holographic data on a single plane
R G Novikov and V N Sivkin 2026
Inverse Problems
42
045009
View article
, A two-point phase recovering from holographic data on a single plane
PDF
, A two-point phase recovering from holographic data on a single plane
We consider a plane wave, a radiation solution, and the sum of these solutions (total solution) for the Helmholtz equation in an exterior region in
. In this region, we consider a hyperplane
with sufficiently large distance
from the origin in
. We give two-point local formulas for approximate recovering the radiation solution restricted to the plane
from the intensity of the total solution at
, that is, from holographic data. The recovering is given in terms of the far-field pattern of the radiation solution with a decaying error term as
. A numerical implementation is also presented.
The following article is
Open access
On the convergence of stochastic variance reduced gradient for linear inverse problems
Bangti Jin and Zehui Zhou 2026
Inverse Problems
42
045006
View article
, On the convergence of stochastic variance reduced gradient for linear inverse problems
PDF
, On the convergence of stochastic variance reduced gradient for linear inverse problems
Stochastic variance reduced gradient (SVRG) is an accelerated version of stochastic gradient descent based on variance reduction, and is promising for solving large-scale inverse problems. In this work, we analyze SVRG and a regularized version that incorporates
a priori
knowledge of the problem, for solving linear inverse problems in Hilbert spaces. We prove that, with suitable constant step size schedules and regularity conditions, the regularized SVRG can achieve optimal convergence rates in terms of the noise level without any early stopping rules, provided that the truncation level is chosen suitably, and standard SVRG is also optimal for problems with nonsmooth solutions under
a priori
stopping rules. The analysis is based on an explicit error recursion and suitable
a priori
estimates on the inner loop updates with respect to the anchor point. Numerical experiments are provided to complement the theoretical analysis.
The following article is
Open access
On the intensity-based inversion method for quantitative quasi-static elastography
Ekaterina Sherina and Simon Hubmer 2026
Inverse Problems
42
045004
View article
, On the intensity-based inversion method for quantitative quasi-static elastography
PDF
, On the intensity-based inversion method for quantitative quasi-static elastography
In this paper, we consider the intensity-based inversion method (IIM) for quantitative material parameter estimation in quasi-static elastography. In particular, we consider the problem of estimating the material parameters of a given sample from two internal measurements, one obtained before and one after applying some form of deformation. These internal measurements can be obtained via any imaging modality of choice, for example ultrasound, optical coherence or photo-acoustic tomography. Compared to two-step approaches to elastography, which first estimate internal displacement fields or strains and then reconstruct the material parameters from them, the IIM is a one-step approach which computes the material parameters directly from the internal measurements. To do so, the IIM combines image registration together with a model-based, regularized parameter reconstruction approach. This combination has the advantage of avoiding some approximations and derivative computations typically found in two-step approaches, and results in the IIM being generally more stable to measurement noise. In the paper, we provide a full convergence analysis of the IIM within the framework of inverse problems, and detail its application to linear elastography. Furthermore, we discuss the numerical implementation of the IIM and provide numerical examples simulating an optical coherence elastography experiment.
The following article is
Open access
Numerical analysis of simultaneous reconstruction of initial condition and potential in subdiffusion
Xu Wu
et al
2026
Inverse Problems
View article
, Numerical analysis of simultaneous reconstruction of initial condition and potential in subdiffusion
PDF
, Numerical analysis of simultaneous reconstruction of initial condition and potential in subdiffusion
This paper investigates the simultaneous identification of a spatially dependent potential and the initial condition in a subdiffusion model based on two terminal observations. The existence, uniqueness, and conditional stability of the inverse problem are established under weak regularity assumptions through a constructive fixed-point iteration approach. The theoretical analysis further inspires the development of an easy-to-implement iterative algorithm. A fully discrete scheme is then proposed, combining the finite element method for spatial discretization, convolution quadrature for temporal discretization, and the quasi-boundary value method to handle the ill-posedness of recovering the initial condition. Inspired by the conditional stability estimate, we demonstrate the linear convergence of the iterative algorithm and provide a detailed error analysis for the reconstructed initial condition and potential. The derived a priori error estimate offers a practical guide for selecting regularization parameters and discretization mesh sizes based on the noise level. Numerical experiments are provided to illustrate and support our theoretical findings.
The following article is
Open access
Elliptic Bayesian inverse problems on metric graphs
D Bolin
et al
2026
Inverse Problems
42
035012
View article
, Elliptic Bayesian inverse problems on metric graphs
PDF
, Elliptic Bayesian inverse problems on metric graphs
This paper studies the formulation, well-posedness, and numerical solution of Bayesian inverse problems on metric graphs, in which the edges represent one-dimensional wires connecting vertices. We focus on the inverse problem of recovering the diffusion coefficient of a (fractional) elliptic equation on a metric graph from noisy measurements of the solution. Well-posedness hinges on both stability of the forward model and an appropriate choice of prior. We establish the stability of elliptic and fractional elliptic forward models using recent regularity theory for differential equations on metric graphs. For the prior, we leverage modern Gaussian Whittle–Matérn process models on metric graphs with sufficiently smooth sample paths. Numerical results demonstrate accurate reconstruction and effective uncertainty quantification.
The following article is
Open access
Reconstructing wind fields from gravitational data on gas giants: an investigation of mathematical methods
Tim-Jonas Peter
et al
2026
Inverse Problems
42
035011
View article
, Reconstructing wind fields from gravitational data on gas giants: an investigation of mathematical methods
PDF
, Reconstructing wind fields from gravitational data on gas giants: an investigation of mathematical methods
The atmospheric structure of gas giants, especially those of Jupiter and Saturn, has been an object of scientific studies for a long time. The measurement of the gravitational fields by the Juno mission for Jupiter and the Cassini mission for Saturn offered new possibilities to study the interior structure of these planets. Accordingly, the reconstruction of the wind velocities from gravitational data on gas giants has been the subject of many research papers over the years, yet the mathematical foundations of this inverse problem and its numerical resolution have not been studied in detail. This article suggests a rigorous mathematical theory for inferring the wind fields of gas giants. In particular, an orthonormal basis is derived which can be associated to models of the gravitational potential and the interior wind velocity field. Moreover, this approach provides the foundations for existing resolution concepts of the inverse problem.
More Open Access articles
A unified treatment of some iterative algorithms in signal processing and image reconstruction
Charles Byrne 2004
Inverse Problems
20
103
View article
, A unified treatment of some iterative algorithms in signal processing and image reconstruction
PDF
, A unified treatment of some iterative algorithms in signal processing and image reconstruction
Let
be a (possibly nonlinear) continuous operator on Hilbert space
. If, for some starting vector
, the orbit sequence {
= 0,1,...}
converges, then the limit
is a fixed point of
; that is,
Tz
An operator
on a Hilbert space
is nonexpansive (ne) if, for each
and
in
Even when
has fixed
points the orbit sequence {
need not converge; consider the example
= −
where
denotes the identity operator. However, for any
the iterative procedure defined by
converges (weakly) to a fixed point of
whenever such points exist. This is the Krasnoselskii–Mann (KM) approach to finding fixed
points of ne operators.
A wide variety of iterative procedures used in signal processing and image reconstruction and
elsewhere are special cases of the KM iterative procedure, for particular choices of the ne operator
. These include the Gerchberg–Papoulis method for bandlimited extrapolation, the SART
algorithm of Anderson and Kak, the Landweber and projected Landweber algorithms,
simultaneous and sequential methods for solving the convex feasibility problem, the ART
and Cimmino methods for solving linear systems of equations, the CQ algorithm for solving
the split feasibility problem and Dolidze’s procedure for the variational inequality problem
for monotone operators.
Iterative oblique projection onto convex sets and the split feasibility problem
Charles Byrne 2002
Inverse Problems
18
441
View article
, Iterative oblique projection onto convex sets and the split feasibility problem
PDF
, Iterative oblique projection onto convex sets and the split feasibility problem
Let
and
be nonempty closed
convex sets in
and
respectively, and
an
by
real matrix. The
split feasibility problem
(SFP) is to find
with
Ax
if such
exist. An iterative method for solving the SFP, called the
CQ
algorithm,
has the following iterative step:
+1
γA
Ax
), where
∊ (0, 2∖
) with
the largest eigenvalue
of the matrix
and
and
denote the orthogonal
projections onto
and
, respectively; that
is,
minimizes
||
||,
over all
The
CQ
algorithm converges to a solution of the SFP, or, more generally, to a minimizer of ||
Ac
Ac
||
over
in
, whenever such exist.
The
CQ
algorithm involves only the orthogonal projections onto
and
which we shall assume are easily calculated, and involves no matrix inverses. If
is normalized so that each row has length one, then
does
not exceed the maximum number of nonzero entries in any column of
, which provides a
helpful estimate of
for sparse matrices.
Particular cases of the
CQ
algorithm are the Landweber and projected Landweber methods for
obtaining exact or approximate solutions of the linear equations
Ax
the
algebraic reconstruction technique
of Gordon, Bender and Herman
is a particular case of a block-iterative version of the
CQ
algorithm.
One application of the
CQ
algorithm that is the subject of ongoing
work is dynamic emission tomographic image reconstruction, in which
the vector
is the
concatenation of several images corresponding to successive discrete times. The matrix
and
the set
can then be selected to impose constraints on the behaviour over time of the
intensities at fixed voxels, as well as to require consistency (or near consistency)
with measured data.
Synthetic aperture radar interferometry
Richard Bamler and Philipp Hartl 1998
Inverse Problems
14
R1
View article
, Synthetic aperture radar interferometry
PDF
, Synthetic aperture radar interferometry
Synthetic aperture radar (SAR) is a coherent active microwave imaging method. In remote sensing it is used for mapping the scattering properties of the Earth's surface in the respective wavelength domain. Many physical and geometric parameters of the imaged scene contribute to the grey value of a SAR image pixel. Scene inversion suffers from this high ambiguity and requires SAR data taken at different wavelength, polarization, time, incidence angle, etc.
Interferometric SAR (InSAR) exploits the phase differences of at least two complex-valued SAR images acquired from different orbit positions and/or at different times. The information derived from these interferometric data sets can be used to measure several geophysical quantities, such as topography, deformations (volcanoes, earthquakes, ice fields), glacier flows, ocean currents, vegetation properties, etc.
This paper reviews the technology and the signal theoretical aspects of InSAR. Emphasis is given to mathematical imaging models and the statistical properties of the involved quantities. Coherence is shown to be a useful concept for system description and for interferogram quality assessment. As a key step in InSAR signal processing two-dimensional phase unwrapping is discussed in detail. Several interferometric configurations are described and illustrated by real-world examples. A compilation of past, current and future InSAR systems concludes the paper.
Optical tomography in medical imaging
S R Arridge 1999
Inverse Problems
15
R41
View article
, Optical tomography in medical imaging
PDF
, Optical tomography in medical imaging
We present a review of methods for the forward and inverse problems in optical tomography. We limit ourselves to the highly scattering case found in applications in medical imaging, and to the problem of absorption and scattering reconstruction. We discuss the derivation of the diffusion approximation and other simplifications of the full transport problem. We develop sensitivity relations in both the continuous and discrete case with special concentration on the use of the finite element method. A classification of algorithms is presented, and some suggestions for open problems to be addressed in future research are made.
Ensemble Kalman methods for inverse problems
Marco A Iglesias
et al
2013
Inverse Problems
29
045001
View article
, Ensemble Kalman methods for inverse problems
PDF
, Ensemble Kalman methods for inverse problems
The ensemble Kalman filter (EnKF) was introduced by Evensen in 1994 (Evensen 1994
J. Geophys. Res.
99
10143–62) as a novel method for
data assimilation
: state estimation for noisily observed time-dependent problems. Since that time it has had enormous impact in many application domains because of its robustness and ease of implementation, and numerical evidence of its accuracy. In this paper we propose the application of an iterative ensemble Kalman method for the solution of a wide class of
inverse problems
. In this context we show that the estimate of the unknown function that we obtain with the ensemble Kalman method lies in a subspace
spanned by the initial ensemble. Hence the resulting error may be bounded above by the error found from the best approximation in this subspace. We provide numerical experiments which compare the error incurred by the ensemble Kalman method for inverse problems with the error of the best approximation in
, and with variants on traditional least-squares approaches, restricted to the subspace
. In so doing we demonstrate that the ensemble Kalman method for inverse problems provides a derivative-free optimization method with comparable accuracy to that achieved by traditional least-squares approaches. Furthermore, we also demonstrate that the accuracy is of the same order of magnitude as that achieved by the best approximation. Three examples are used to demonstrate these assertions: inversion of a compact linear operator; inversion of piezometric head to determine hydraulic conductivity in a Darcy model of groundwater flow; and inversion of Eulerian velocity measurements at positive times to determine the initial condition in an incompressible fluid.
The multiple-sets split feasibility problem and its applications for inverse problems
Yair Censor
et al
2005
Inverse Problems
21
2071
View article
, The multiple-sets split feasibility problem and its applications for inverse problems
PDF
, The multiple-sets split feasibility problem and its applications for inverse problems
The multiple-sets split feasibility problem requires finding a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. It can be a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in the operator's range. It generalizes the convex feasibility problem as well as the two-sets split feasibility problem. We propose a projection algorithm that minimizes a proximity function that measures the distance of a point from all sets. The formulation, as well as the algorithm, generalize earlier work on the split feasibility problem. We offer also a generalization to proximity functions with Bregman distances. Application of the method to the inverse problem of intensity-modulated radiation therapy treatment planning is studied in a separate companion paper and is here only described briefly.
Adaptive tempered reversible jump algorithm for Bayesian curve fitting
Zhiyao Tian
et al
2024
Inverse Problems
40
045024
View article
, Adaptive tempered reversible jump algorithm for Bayesian curve fitting
PDF
, Adaptive tempered reversible jump algorithm for Bayesian curve fitting
Bayesian curve fitting plays an important role in inverse problems, and is often addressed using the reversible jump Markov chain Monte Carlo (RJMCMC) algorithm. However, this algorithm can be computationally inefficient without appropriately tuned proposals. As a remedy, we present an adaptive RJMCMC algorithm for the curve fitting problems by extending the adaptive Metropolis sampler from a fixed-dimensional to a trans-dimensional case. In this presented algorithm, both the size and orientation of the proposal function can be automatically adjusted in the sampling process. Specifically, the curve fitting setting allows for the approximation of the posterior covariance of the
a priori
unknown function on a representative grid of points. This approximation facilitates the definition of efficient proposals. In addition, we introduce an auxiliary-tempered version of this algorithm via non-reversible parallel tempering. To evaluate the algorithms, we conduct numerical tests involving a series of controlled experiments. The results demonstrate that the adaptive algorithms exhibit significantly higher efficiency compared to the conventional ones. Even in cases where the posterior distribution is highly complex, leading to ineffective convergence in the auxiliary-tempered conventional RJMCMC, the proposed auxiliary-tempered adaptive RJMCMC performs satisfactorily. Furthermore, we present a realistic inverse example to test the algorithms. The successful application of the adaptive algorithm distinguishes it again from the conventional one that fails to converge effectively even after millions of iterations.
Stable architectures for deep neural networks
Eldad Haber and Lars Ruthotto 2018
Inverse Problems
34
014004
View article
, Stable architectures for deep neural networks
PDF
, Stable architectures for deep neural networks
Deep neural networks have become invaluable tools for supervised machine learning, e.g. classification of text or images. While often offering superior results over traditional techniques and successfully expressing complicated patterns in data, deep architectures are known to be challenging to design and train such that they generalize well to new data. Critical issues with deep architectures are numerical instabilities in derivative-based learning algorithms commonly called exploding or vanishing gradients. In this paper, we propose new forward propagation techniques inspired by systems of ordinary differential equations (ODE) that overcome this challenge and lead to well-posed learning problems for arbitrarily deep networks.
The backbone of our approach is our interpretation of deep learning as a parameter estimation problem of nonlinear dynamical systems. Given this formulation, we analyze stability and well-posedness of deep learning and use this new understanding to develop new network architectures. We relate the exploding and vanishing gradient phenomenon to the stability of the discrete ODE and present several strategies for stabilizing deep learning for very deep networks. While our new architectures restrict the solution space, several numerical experiments show their competitiveness with state-of-the-art networks.
Learning about physical parameters: the importance of model discrepancy
Jenný Brynjarsdóttir and Anthony OʼHagan 2014
Inverse Problems
30
114007
View article
, Learning about physical parameters: the importance of model discrepancy
PDF
, Learning about physical parameters: the importance of model discrepancy
Science-based simulation models are widely used to predict the behavior of complex physical systems. It is also common to use observations of the physical system to solve the inverse problem, that is, to learn about the values of parameters within the model, a process which is often called
calibration
. The main goal of calibration is usually to improve the predictive performance of the simulator but the values of the parameters in the model may also be of intrinsic scientific interest in their own right. In order to make appropriate use of observations of the physical system it is important to recognize
model discrepancy
, the difference between reality and the simulator output. We illustrate through a simple example that an analysis that does not account for model discrepancy may lead to biased and over-confident parameter estimates and predictions. The challenge with incorporating model discrepancy in statistical inverse problems is being confounded with calibration parameters, which will only be resolved with meaningful priors. For our simple example, we model the model-discrepancy via a Gaussian process and demonstrate that through accounting for model discrepancy our prediction within the range of data is correct. However, only with realistic priors on the model discrepancy do we uncover the true parameter values. Through theoretical arguments we show that these findings are typical of the general problem of learning about physical parameters and the underlying physical system using science-based mechanistic models.
Separable nonlinear least squares: the variable projection method and its applications
Gene Golub and Victor Pereyra 2003
Inverse Problems
19
R1
View article
, Separable nonlinear least squares: the variable projection method and its applications
PDF
, Separable nonlinear least squares: the variable projection method and its applications
In this paper we review 30 years of developments and applications of the variable
projection method for solving separable nonlinear least-squares problems. These
are problems for which the model function is a linear combination of
nonlinear functions. Taking advantage of this special structure, the method of
variable projections eliminates the linear variables obtaining a somewhat
more complicated function that involves only the nonlinear parameters.
This procedure not only reduces the dimension of the parameter space
but also results in a better-conditioned problem. The same optimization
method applied to the original and reduced problems will always converge
faster for the latter. We present first a historical account of the basic
theoretical work and its various computer implementations, and then
report on a variety of applications from electrical engineering, medical and
biological imaging, chemistry, robotics, vision, and environmental sciences.
An extensive bibliography is included. The method is particularly well
suited for solving real and complex exponential model fitting problems,
which are pervasive in their applications and are notoriously hard to solve.
Journal links
Submit an article
About the journal
Editorial Board
Author guidelines
Review for this journal
Publication charges
Awards
Journal collections
Pricing and ordering
Journal information
1985-present
Inverse Problems
doi: 10.1088/issn.0266-5611
Online ISSN: 1361-6420
Print ISSN: 0266-5611