Methods For Inverse Problems Research Articles

Numerical Verification of the Convexification Method for a Frequency-Dependent Inverse Scattering Problem with Experimental Data

Numerical Verification of the Convexification Method for a Frequency-Dependent Inverse Scattering Problem with Experimental Data

A projected gradient method for nonlinear inverse problems with 𝛼ℓ1 − 𝛽ℓ2 sparsity regularization

Abstract The non-convex α ⁢ ∥ ⋅ ∥ ℓ 1 − β ⁢ ∥ ⋅ ∥ ℓ 2 \alpha\lVert\,{\cdot}\,\rVert_{\ell_{1}}-\beta\lVert\,{\cdot}\,\rVert_{\ell_{2}} ( α ≥ β ≥ 0 \alpha\geq\beta\geq 0 ) regularization is a new approach for sparse recovery. A minimizer of the α ⁢ ∥ ⋅ ∥ ℓ 1 − β ⁢ ∥ ⋅ ∥ ℓ 2 \alpha\lVert\,{\cdot}\,\rVert_{\ell_{1}}-\beta\lVert\,{\cdot}\,\rVert_{\ell_{2}} regularized function can be computed by applying the ST-( α ⁢ ℓ 1 − β ⁢ ℓ 2 \alpha\ell_{1}-\beta\ell_{2} ) algorithm which is similar to the classical iterative soft thresholding algorithm (ISTA). Unfortunately, It is known that ISTA converges quite slowly, and a faster alternative to ISTA is the projected gradient (PG) method. Nevertheless, the current applicability of the PG method is limited to linear inverse problems. In this paper, we extend the PG method based on a surrogate function approach to nonlinear inverse problems with the α ⁢ ∥ ⋅ ∥ ℓ 1 − β ⁢ ∥ ⋅ ∥ ℓ 2 \alpha\lVert\,{\cdot}\,\rVert_{\ell_{1}}-\beta\lVert\,{\cdot}\,\rVert_{\ell_{2}} ( α ≥ β ≥ 0 \alpha\geq\beta\geq 0 ) regularization in the finite-dimensional space R n \mathbb{R}^{n} . It is shown that the presented algorithm converges subsequentially to a stationary point of a constrained Tikhonov-type functional for sparsity regularization. Numerical experiments are given in the context of a nonlinear compressive sensing problem to illustrate the efficiency of the proposed approach.

A combination of Kohn-Vogelius and DDM methods for a geometrical inverse problem

We consider the inverse geometrical problem of identifying the discontinuity curve of an electrical conductivity from boundary measurements. This standard inverse problem is used as a model to introduce and study a combined inversion algorithm coupling a gradient descent on the Kohn-Vogelius cost functional with a domain decomposition method that includes the unknown curve in the domain partitioning. We prove the local convergence of the method in a simplified case and numerically show its efficiency for some two dimensional experiments.

A multiple-data-based direct method for inverse problem in three-dimensional linear elasticity

Estimating the unknown Young’s modulus and boundary conditions of an elastic object from locally observed boundary data is an inverse problem which is usually solved by iterative methods. In this paper, a multiple-data-based direct (MD) method is proposed to solve this inverse problem without iterations. The proposed method applies a global normalized objective function and a global regularization coefficient τg for energy-like regularization to ensure the convergence of the unknown displacement boundary conditions. A coarse-mesh based preprocessing algorithm is proposed to speed up the determination of the regularization coefficient. A multi-force-position method is proposed to obtain observation data, which improves accuracy of the MD method when the number of experiments is limited. Both numerical and physical experiments were conducted. Compared with the single-data-based direct (SD) methods proposed in our previous work, the MD method yields more accurate estimation of Young’s modulus and boundary conditions in both numerical and physical experiments.

A variable projection method for large-scale inverse problems with ℓ1 regularization

Inference by means of mathematical modeling from a collection of observations remains a crucial tool for scientific discovery and is ubiquitous in application areas such as signal compression, imaging restoration, and supervised machine learning. With ever-increasing model complexities and larger data sets, new specially designed methods are urgently needed to recover meaningful quantities of interest. We consider the broad spectrum of linear inverse problems where the aim is to reconstruct quantities with a sparse representation on some vector space. We provide a new variable projection augmented Lagrangian algorithm to solve the underlying ℓ1 regularized inverse problem that is both efficient and effective. We present the proof of convergence for an algorithm using an inexact step for the projected problem at each iteration. The performance and convergence properties for various imaging problems are investigated. The efficiency of the algorithm makes it feasible to automatically find the regularization parameter, here illustrated using an argument based on the degrees of freedom of the objective function equipped with a bisection algorithm for root-finding.

Variable projection methods for separable nonlinear inverse problems with general-form Tikhonov regularization

The variable projection (VarPro) method is an efficient method to solve separable nonlinear least squares problems. In this paper, we propose a modified VarPro method for solving separable nonlinear least squares problems with general-form Tikhonov regularization. In particular, we apply the Gauss–Newton method to the corresponding reduced problem and investigate its convergence when different approximations of the Jacobian matrix are used. For special cases when computing the generalized singular value decomposition is feasible or a joint spectral decomposition of both forward and regularization operators exists, we provide efficient ways to compute the Jacobians and the solution of the linear subproblems. For large-scale problems, where matrix decompositions are not an option, we compute a reduced Jacobian and apply projection-based iterative methods and generalized Krylov subspace methods to solve the linear subproblems. In all cases, the regularization parameter can be computed automatically at each iteration using generalized cross validation. Several numerical examples highlight the proposed approach’s performance in the quality of the reconstructed image and the reconstructed forward operator, including large-scale two-dimensional imaging problems arising from semi-blind deblurring.

Filtering Methods for Coupled Inverse Problems

In many applications it is required to determine a parameters set which is suitable for competing models. To this end, we are interested in ensemble methods to solve multiobjective optimization problems. Here, the ensemble Kalman Filter method is applied and adapted in order to solve coupled inverse nonlinear problems using a weighted function approach. An analysis of the mean field limit of the ensemble method yields an explicit update formula for the weights. Numerical examples show the improved performance of the proposed method.

A data-driven and model-based accelerated Hamiltonian Monte Carlo method for Bayesian elliptic inverse problems

A data-driven and model-based accelerated Hamiltonian Monte Carlo method for Bayesian elliptic inverse problems

On unifying randomized methods for inverse problems

This work unifies the analysis of various randomized methods for solving linear and nonlinear inverse problems with Gaussian priors by framing the problem in a stochastic optimization setting. By doing so, we show that many randomized methods are variants of a sample average approximation (SAA). More importantly, we are able to prove a single theoretical result that guarantees the asymptotic convergence for a variety of randomized methods. Additionally, viewing randomized methods as an SAA enables us to prove, for the first time, a single non-asymptotic error result that holds for randomized methods under consideration. Another important consequence of our unified framework is that it allows us to discover new randomization methods. We present various numerical results for linear, nonlinear, algebraic, and PDE-constrained inverse problems that verify the theoretical convergence results and provide a discussion on the apparently different convergence rates and the behavior for various randomized methods.

A majorization–minimization-based method for nonconvex inverse rig problems in facial animation: algorithm derivation

Automated methods for facial animation are a necessary tool in the modern industry since the standard blendshape head models consist of hundreds of controllers, and a manual approach is painfully slow. Different solutions have been proposed that produce output in real-time or generalize well for different face topologies. However, all these prior works consider a linear approximation of the blendshape function and hence do not provide a high-enough level of detail for modern realistic human face reconstruction. A second-order blendshape approximation leads to higher fidelity facial animation but generates a non-linear least squares optimization problem with high dimensionality. We derive a method for solving the inverse rig in blendshape animation using quadratic corrective terms, which increases accuracy. At the same time, due to the proposed construction of the objective function, it yields a sparser estimated weight vector compared to the state-of-the-art methods. The former feature means lower demand for subsequent manual corrections of the solution, while the latter indicates that the manual modifications are also easier to include. Our algorithm is iterative and employs a Majorization–Minimization paradigm to cope with the increased complexity produced by adding corrective terms. The surrogate function is easy to solve and allows for further parallelization on the component level within each iteration. This paper is complementary to an accompanying paper (Racković et al. arxiv preprint. https://arxiv.org/abs/2302.04843, 2023) where we provide detailed experimental results and discussion, including highly-realistic animation data, and show a clear superiority of the results compared to the state-of-the-art methods.

A data-driven model reduction method for parabolic inverse source problems and its convergence analysis

In this paper, we propose a data-driven model reduction method to solve parabolic inverse source problems with uncertain data efficiently. Our method consists of offline and online stages. In the offline stage, we explore the low-dimensional structures in the solution space of parabolic partial differential equations (PDEs) in the forward problems with a given class of source functions and construct a small number of proper orthogonal decomposition (POD) basis functions to achieve significant dimension reduction. Equipped with the POD basis functions, we can solve the forward problems extremely fast in the online stage. Thus, we develop a fast algorithm to solve the optimization problem in parabolic inverse source problems, which is referred to as the POD method. Moreover, we design an a posteriori algorithm to find the optimal regularization parameter in the optimization problem using the proposed POD method without knowing the noise level. Under a weak regularity assumption on the solution of the parabolic PDEs, we prove the convergence of the POD method in solving the forward parabolic PDEs. In addition, we obtain the error estimate of the POD method for parabolic inverse source problems. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method. Numerical results show that the POD method provides considerable computational savings over the finite element method while maintaining the same accuracy.

Hybrid Method for Inverse Elastic Obstacle Scattering Problems

The problem of determining the shape of an object from knowledge of the far-field of a single incident wave in two-dimensional elasticity was considered. We applied an iterative hybrid method to tackle this problem. An advantage of this method is that it does not need a forward solver, and therefore, the exact boundary condition is not essential. By deriving the Fréchet derivatives of two boundary operators, we established reconstruction algorithms for objects with Dirichlet, Neumann, and Robin boundary conditions; by introducing a general boundary condition, we also established the reconstruction algorithm for objects with unknown physical properties. Numerical experiments showed the effectiveness of the proposed method.

On finding a penetrable obstacle using a single electromagnetic wave in the time domain

Abstract The time domain enclosure method is one of the analytical methods for inverse obstacle problems governed by partial differential equations in the time domain. This paper considers the case when the governing equation is given by the Maxwell system and consists of two parts. The first part establishes the base of the time domain enclosure method for the Maxwell system using a single set of the solutions over a finite time interval for a general (isotropic) inhomogeneous medium in the whole space. It is a system of asymptotic inequalities for the indicator function which may enable us to apply the time domain enclosure method to the problem of finding unknown penetrable obstacles embedded in various background media. As a first step of its expected applications, the case when the background medium is homogeneous and isotropic is considered and the time domain enclosure method is realized. This is the second part.

A projected homotopy perturbation method for nonlinear inverse problems in Banach spaces

Abstract In this paper, we propose and analyze a projected homotopy perturbation method based on sequential Bregman projections for nonlinear inverse problems in Banach spaces. To expedite convergence, the approach uses two search directions given by homotopy perturbation iteration, and the new iteration is calculated as the projection of the current iteration onto the intersection of stripes decided by above directions. The method allows to use L 1 {L^{1}} -like penalty terms, which is significant to reconstruct sparsity solutions. Under reasonable conditions, we establish the convergence and regularization properties of the method. Finally, two parameter identification problems are presented to indicate the effectiveness of capturing the property of the sparsity solutions and the acceleration effect of the proposed method.

Periodic solutions of the modified Korteweg–de Vries equation in hemodynamics

In this paper, we consider the periodic solution of the modified Korteweg-de Vries equation used for studies of hemodynamic processes. It is shown that the modified Korteweg-de Vries equation can be integrated by the inverse spectral problem method. The evolution of the spectral data of the Dirac operator with a periodic potential associated with the solution of the modified Korteweg-de Vries equation is determined. The obtained results substantiate the applicability of the inverse problem method for solving the modified Korteweg-de Vries equation for studying the laws of hemodynamics.

Numerical studies of domain sampling methods for inverse boundary value problems by one measurement

We consider an inverse boundary value problem for the Laplace equation, which discusses the reconstruction of an unknown target inside the background medium from one boundary measurement. We are interested in two domain sampling methods, i.e., the range test and no-response test, whose convergences are justified theoretically in previous work [17]. As a continuation of this work, we study the numerical realizations of these methods. Some new techniques are proposed to set up efficient algorithms, which yield reasonably good numerical reconstructions. To demonstrate the performance of proposed algorithms, we show several numerical examples for different shapes of unknown targets with noisy measurement data. Some key ingredients of numerical implementations are discussed in detail.

Advancements in Numerical Methods for Forward and Inverse Problems in Functional near Infra-Red Spectroscopy: A Review

In the field of biomedical image reconstruction, functional near infra-red spectroscopy (fNIRs) is a promising technology that uses near infra-red light for non-invasive imaging and reconstruction. Reconstructing an image requires both forward and backward problem-solving in order to figure out what the image’s optical properties are from the boundary data that has been measured. Researchers are using a variety of numerical methods to solve both the forward and backward problems in depth. This study will show the latest improvements in numerical methods for solving forward and backward problems in fNIRs. The physical interpretation of the forward problem is described, followed by the explanation of the state-of-the-art numerical methods and the description of the toolboxes. A more in-depth discussion of the numerical solution approaches for the inverse problem for fNIRs is also provided.

RETRACTED ARTICLE: A heat polynomials method for the two-phase inverse Stefan problem

RETRACTED ARTICLE: A heat polynomials method for the two-phase inverse Stefan problem

An effective nonlinear interval sequential quadratic programming method for uncertain inverse problems

An effective nonlinear interval sequential quadratic programming method is proposed to provide an efficient tool for uncertain inverse problems. Assisted by the ideology of sequential quadratic programming and dimension-reduction analysis theory, the interval inverse problem is transformed into several interval arithmetic and deterministic optimizations, which could enhance computational efficiency without losing much accuracy. The novelty of the proposed method lies in two main aspects. First, an alternate updating strategy is proposed to identify the radii and midpoints of the interval inputs in each cycle, which could reduce the number of iterative steps. Second, the standard quadratic models are constructed based on the dimension-reduction analysis results, rather than the second-order Taylor expansion. Therefore, the interval arithmetic can be applied to efficiently calculate the interval response, which avoids the inner optimization. Moreover, a novel iterative mechanism is developed to accelerate the convergence rate of the proposed method. Finally, two numerical examples and an engineering application are adopted to verify its feasibility, accuracy and efficiency.

A new regularized stochastic approximation framework for stochastic inverse problems

We study the nonlinear inverse problem of estimating stochastic parameters in the fourth-order partial differential equation with random data. The primary focus is on developing a novel stochastic approximation framework for inverse problems consisting of three key components. As a first step, we reformulate the inverse problem into a stochastic convex optimization problem. The second step includes developing a new regularized stochastic extragradient framework for a nonlinear variational inequality, which subsumes the optimality conditions for the optimization formulation of the inverse problem. The third step involves modeling random variables by a Karhunen–Loève type finite-dimensional noise representation, allowing the direct and the inverse problems to be conveniently discretized. We show that the regularized extragradient methods are strongly convergent in a Hilbert space setting, and we also provide several auxiliary results for the inverse problem, including Lipschitz continuity and a derivative characterization of the solution map. We provide the outcome of computational experiments to estimate stochastic and deterministic parameters. The numerical results demonstrate the feasibility and effectiveness of the developed framework and validate stochastic approximation as an effective method for stochastic inverse problems.