Related Inverse Problem Research Articles

A strong positivity property and a related inverse source problem for multi-term time-fractional diffusion equations

A strong positivity property and a related inverse source problem for multi-term time-fractional diffusion equations

Optimal Transport Map With Prescribed Direction Indicator for Seismic Full‐Waveform Inversion

AbstractThe quadratic Wasserstein (W2) metric has been proposed as a promising misfit function to mitigate cycle‐skipping phenomena in full‐waveform inversion. Mathematically, we demonstrate that the smoothness of the W2‐based adjoint source is two orders of magnitude higher than that based on L2‐norm, which guarantees a larger convergence radius of related inverse problems. However, the oscillatory characteristics of seismic signals and subsequent operations of transforming them into probability densities would decrease the accuracy of the optimal transport map T(t) and exacerbate the nonconvexity of the misfit function. To tackle these challenges, we propose the concept of prescribed direction indicator, which indicates the properly matching direction from predictions to observations, in order to correct inaccurate T(t). 1D synthetic examples suggest that reasonable bijection can be constructed through the proposed method. Numerical experiments demonstrate that it works well during optimization procedures, including enlarging the convergence radius of the inverse problem, improving the computational efficiency and enhancing the reliability of inversion results.

A robust alternative to examine data dependency of fixed points of quasi-contractive operators: an efficient approach that relies on the collage theorem

Usurelu et al. (Int J Comput Math 98:1049–1068, 2021) presented stability and data dependence results for a TTP (Thakur–Thakur–Postolache) iteration algorithm associated with quasi-strictly contractive mappings and contraction mappings, respectively, but these results were subject to strong conditions on the parametric control sequences used in the TTP iteration algorithm. This article aims to expand those results conducting a thorough analysis of the convergence of TTP and S iteration algorithms and improve those results by removing the restrictions on the parametric control sequences. Additionally, a data dependence result for the TTP iteration algorithm of quasi-strictly contractive mappings is established and several collage theorems are introduced to offer new insights into the data dependence of fixed points of quasi-strictly contractive mappings and to solve related inverse problems. In order to exhibit the dependability and effectiveness of all the results discussed in this work, a multitude of numerical examples are furnished, encompassing both linear and nonlinear differential equations (DEs) and partial differential equations (PDEs). This work can be viewed as an important refinement and complement to the study by Usurelu et al. (Int J Comput Math 98:1049–1068, 2021).

Single-stage approach for estimating optical parameters in spectral quantitative photoacoustictomography.

In quantitative photoacoustic tomography, the optical parameters of a target, most importantly the concentrations of chromophores such as deoxygenated and oxygenated hemoglobin, are estimated from photoacoustic data measured on the boundary of the target. In this work, a numerical approximation of a forward model for spectral quantitative photoacoustic tomography is constructed by utilizing the diffusion approximation for light propagation, the acoustic wave equation for ultrasound propagation, and spectral models of optical absorption and scattering to describe the wavelength dependence of the optical parameters. The related inverse problem is approached in the framework of Bayesian inverse problems. Concentrations of four chromophores (deoxygenated and oxygenated hemoglobin, water, and lipid), two scattering parameters (reference scattering and scattering power), and the Grüneisen parameter are estimated in a single-stage from photoacoustic data. The methodology is evaluated using numerical simulations in different full-view and limited-view imaging settings. The results show that, utilizing spectral data and models, the spectral optical parameters and the Grüneisen parameter can be simultaneously estimated. Furthermore, the approach can also be utilized in limited-view imaging situations.

Inverse problems for real principal type operators

abstract: We consider inverse boundary value problems for general real principal type differential operators. The first results state that the Cauchy data set uniquely determines the scattering relation of the operator and bicharacteristic ray transforms of lower order coefficients. We also give two different boundary determination methods for general operators, and prove global uniqueness results for determining coefficients in nonlinear real principal type equations. The article presents a unified approach for treating inverse boundary problems for transport and wave equations, and highlights the role of propagation of singularities in the solution of related inverse problems.

Inverse design of mechanical metamaterial achieving a prescribed constitutive curve

Besides exhibiting excellent capabilities such as energy absorption, phase-transforming metamaterials offer a vast design space for achieving nonlinear constitutive relations. This is facilitated by switching between different patterns under deformation. However, the related inverse design problem is quite challenging, due to the lack of appropriate mathematical formulation and the convergence issue in the post-buckling analysis of intermediate designs. In this work, periodic unit cells are explicitly described by the moving morphable voids method and effectively analyzed by eliminating the degrees of freedom in void regions. Furthermore, by exploring the Pareto frontiers between error and cost, an inverse design formulation is proposed for unit cells. This formulation aims to achieve a prescribed constitutive curve and is validated through numerical examples and experimental results. The design approach presented here can be extended to the inverse design of other types of mechanical metamaterials with prescribed nonlinear effective properties.

Covariance models and Gaussian process regression for the wave equation. Application to related inverse problems

In this article, we consider the general task of performing Gaussian process regression (GPR) on pointwise observations of solutions of the 3 dimensional homogeneous free space wave equation. In a recent article, we obtained promising covariance expressions tailored to this equation: we now explore the potential applications of these formulas. We first study the particular cases of stationarity and radial symmetry, for which significant simplifications arise. We next show that the true-angle multilateration method for point source localization, as used in GPS systems, is naturally recovered by our GPR formulas in the limit of the small source radius. Additionally, we show that this GPR framework provides a new answer to the ill-posed inverse problem of reconstructing initial conditions for the wave equation from a limited number of sensors, and simultaneously enables the inference of physical parameters from these data. We finish by illustrating this “physics informed” GPR on a number of practical examples.

The inverse Rytov series for diffuse optical tomography

The Rytov approximation is known in near-infrared spectroscopy including diffuse optical tomography. In diffuse optical tomography, the Rytov approximation often gives better reconstructed images than the Born approximation. Although related inverse problems are nonlinear, the Rytov approximation is almost always accompanied by the linearization of nonlinear inverse problems. In this paper, we will develop nonlinear reconstruction with the inverse Rytov series. By this, linearization is not necessary and higher order terms in the Rytov series can be used for reconstruction. The convergence and stability are discussed. We find that the inverse Rytov series has a recursive structure similar to the inverse Born series.

Iterative schemes for linear equations of the second kind and related inverse problems

This paper consists of two parts. The first one deals with the generation of an iterative algorithm to obtain an approximate solution of a linear equation of the second kind in a Banach space. This generation is based on a perturbed version of the geometric series theorem which, in particular, allows us to find a family of unisolvent linear Fredholm integral equations of the second kind, even when the associated linear operator has norm greater than or equal to 1. When we consider Fredholm equations of this type and linear Volterra integral equations of second kind, the numerical schemes obtained when appropriate Schauder bases are also introduced in the spaces where the equations operate, enable us to approximate their respective solutions iteratively. The second part of this work focuses on the design of a numerical method for solving an inverse problem associated with a linear equation of the second kind in a Banach space, a method which we apply to problems of parameter estimation related to the two classes of integral equations mentioned above.

Thermal conductivity reconstruction method with application in a face milling operation

PurposeThis paper aims to reconstruct the spatially varying orthotropic conductivity based on a two-dimensional inverse heat conduction problem described by a partial differential equation (PDE) model with mixed boundary conditions. The proposed discretization uses a highly accurate technique and allows simple implementations. Also, the authors solve the related inverse problem in such a way that smoothness is enforced on the iterations, showing promising results in synthetic examples and real problems with moving heat source.Design/methodology/approachThe discretization procedure applied to the model for the direct problem uses a pseudospectral collocation strategy in the spatial variables and Crank–Nicolson method for the time-dependent variable. Then, the related inverse problem of recovering the conductivity from temperature measurements is solved by a modified version of Levenberg–Marquardt method (LMM) which uses singular scaling matrices. Problems where data availability is limited are also considered, motivated by a face milling operation problem. Numerical examples are presented to indicate the accuracy and efficiency of the proposed method.FindingsThe paper presents a discretization for the PDEs model aiming on simple implementations and numerical performance. The modified version of LMM introduced using singular scaling matrices shows the capabilities on recovering quantities with precision at a low number of iterations. Numerical results showed good fit between exact and approximate solutions for synthetic noisy data and quite acceptable inverse solutions when experimental data are inverted.Originality/valueThe paper is significant because of the pseudospectral approach, known for its high precision and easy implementation, and usage of singular regularization matrices on LMM iterations, unlike classic implementations of the method, impacting positively on the reconstruction process.

On the crack inverse problem for pressure waves in half-space

After formulating the pressure wave equation in half-space minus a crack with a zero Neumann condition on the top plane, we introduce a related inverse problem. That inverse problem consists of identifying the crack and the unknown forcing term on that crack from overdetermined boundary data on a relatively open set of the top plane. This inverse problem is not uniquely solvable unless some additional assumption is made. However, we show that we can differentiate two cracks Γ1 and Γ2 under the assumption that R3∖Γ1∪Γ2‾ is connected. If that is not the case we provide counterexamples that demonstrate non-uniqueness, even if Γ1 and Γ2 are smooth and “almost” flat. Finally, we show in the case where R3∖Γ1∪Γ2‾ is not necessarily connected that after excluding a discrete set of frequencies, Γ1 and Γ2 can again be differentiated from overdetermined boundary data.

Spectral methods for solving elliptic PDEs on unknown manifolds

In this paper, we propose a mesh-free numerical method for solving elliptic PDEs on unknown manifolds, identified with randomly sampled point cloud data. The PDE solver is formulated as a spectral method where the test function space is the span of the leading eigenfunctions of the Laplacian operator, which are approximated from the point cloud data. While the framework is flexible for any test functional space, we will consider the eigensolutions of a weighted Laplacian obtained from a symmetric Radial Basis Function (RBF) method induced by a weak approximation of a weighted Laplacian on an appropriate Hilbert space. In this paper, we consider a test function space that encodes the geometry of the data yet does not require us to identify and use the sampling density of the point cloud. To attain a more accurate approximation of the expansion coefficients, we adopt a second-order tangent space estimation method to improve the RBF interpolation accuracy in estimating the tangential derivatives. This spectral framework allows us to efficiently solve the PDE many times subjected to different parameters, which reduces the computational cost in the related inverse problem applications. In a well-posed elliptic PDE setting with randomly sampled point cloud data, we provide a theoretical analysis to demonstrate the convergence of the proposed solver as the sample size increases. We also report some numerical studies that show the convergence of the spectral solver on simple manifolds and unknown, rough surfaces. Our numerical results suggest that the proposed method is more accurate than a graph Laplacian-based solver on smooth manifolds. On rough manifolds, these two approaches are comparable. Due to the flexibility of the framework, we empirically found improved accuracies in both smoothed and unsmoothed Stanford bunny domains by blending the graph Laplacian eigensolutions and RBF interpolator.

On Global Solvability and Regularity for Generalized Rayleigh–Stokes Equations with History-Dependent Nonlinearities

We are concerned with the initial value problem governed by generalized Rayleigh–Stokes equations, where the nonlinearity depends on history states and takes values in Hilbert scales of negative order. The solvability and Hölder regularity of solutions are proved using fixed point arguments and embeddings of fractional Sobolev spaces. An application to a related inverse source problem is given.

Compressed Sensing With Upscaled Vector Approximate Message Passing

The Recently proposed Vector Approximate Message Passing (VAMP) algorithm demonstrates a great reconstruction potential at solving compressed sensing related linear inverse problems. VAMP provides high per-iteration improvement, can utilize powerful denoisers like BM3D, has rigorously defined dynamics and is able to recover signals measured by highly undersampled and ill-conditioned linear operators. Yet, its applicability is limited to relatively small problem sizes due to the necessity to compute the expensive LMMSE estimator at each iteration. In this work we consider the problem of upscaling VAMP by utilizing Conjugate Gradient (CG) to approximate the intractable LMMSE estimator. We propose a rigorous method for correcting and tuning CG withing CG-VAMP to achieve a stable and efficient reconstruction. To further improve the performance of CG-VAMP, we design a warm-starting scheme for CG and develop theoretical models for the Onsager correction and the State Evolution of Warm-Started CG-VAMP (WS-CG-VAMP). Additionally, we develop robust and accurate methods for implementing the WS-CG-VAMP algorithm. The numerical experiments on large-scale image reconstruction problems demonstrate that WS-CG-VAMP requires much fewer CG iterations compared to CG-VAMP to achieve the same or superior level of reconstruction.

Inverse Eigenvalue Problem and Least‐Squares Problem for Skew‐Hermitian {P,K + 1}‐Reflexive Matrices

This paper involves related inverse eigenvalue problem and least‐squares problem of skew‐Hermitian {P,k + 1}‐reflexive(antireflexive) matrices and their optimal approximation problems. The above problems are studied by converting them into two simpler cases: k = 1 and k = 2. Firstly, with some special properties of skew‐Hermitian {P,k + 1}‐reflexive(antireflexive) matrices, the necessary and sufficient conditions for the solvability and the general solution are presented, and the solution of corresponding optimal approximation problems also given, respectively. Then, we give the least‐squares solution of AX = B satisfying the special condition by the singular value decomposition. Finally, we give an algorithm and an example to illustrate our results.

Evaluating selection criteria for optimized excitation coils in magnetorelaxometry imaging

Objective. Magnetorelaxometry imaging (MRXI) is an experimental imaging technique applicable for noninvasive, qualitative and quantitative imaging of magnetic nanoparticles (MNPs). Accurate reconstructions of nanoparticle distributions are crucial for several novel treatment methods employing MNPs such as magnetic drug targeting or magnetic hyperthermia therapy. Hence, it is desirable to design MRXI setups such that the reconstruction accuracy is maximized for a given set of design parameters. Several attempts exist in literature that focus on the improvement of MRXI and other related linear inverse problems with respect to various figures of merit. However, to date it remains unclear, which approach leads to the largest benefit for the reconstruction accuracy. Thus, the aim of this study is to compare the different figures of merit, thereby determining the most reliable and effective optimization approach for magnetorelaxometry setups. Approach. In the present simulation study, we translate these figures of merit to various cost functions, allowing us to optimize the electromagnetic coil positions and radii of two distinct MRXI setups with an adapted tabu search algorithm. Multiple artificial MNP phantoms are reconstructed employing the optimized setups and the resulting imaging qualities are subsequently compared. Main results. The extensive amount of generated synthetic data unprecedented in previous MRXI studies identifies the condition number as the most reliable indicator for good imaging results. This is the case for both the qualitative as well as the quantitative reconstruction accuracies. Significance. The results of this study show that optimized coil configurations increase the reconstruction quality compared to the state-of-the-art. The insights obtained here can also be extended to other design parameters of MRXI setups, thus enabling more reliable reconstructions of MNP ensembles which will ultimately render the aforementioned treatment methods safer and more efficient.

Inverse characterization of composites using guided waves and convolutional neural networks with dual-branch feature fusion

In this work, ultrasonic guided waves and a dual-branch version of convolutional neural networks are used to solve two different but related inverse problems, i.e., finding layup sequence type and identifying material properties. In the forward problem, polar group velocity representations are obtained for two fundamental Lamb wave modes using the stiffness matrix method. For the inverse problems, a supervised classification-based network is implemented to classify the polar representations into different layup sequence types (inverse problem − 1) and a regression-based network is utilized to identify the material properties (inverse problem − 2).

Joining Forces for Reconstruction Inverse Problems

A spectral inverse problem concerns the reconstruction of parameters of a parent graph from prescribed spectral data of subgraphs. Also referred to as the P–NP Isomorphism Problem, Reconstruction or Exact Graph Matching, the aim is to seek sets of parameters to determine a graph uniquely. Other related inverse problems, including the Polynomial Reconstruction Problem (PRP), involve the recovery of graph invariants. The PRP seeks to extract the spectrum of a graph from the deck of cards each showing the spectrum of a vertex-deleted subgraph. We show how various algebraic methods join forces to reconstruct a graph or its invariants from a minimal set of restricted eigenvalue-eigenvector information of the parent graph or its subgraphs. We show how functions of the entries of eigenvectors of the adjacency matrix A of a graph can be retrieved from the spectrum of eigenvalues of A. We establish that there are two subclasses of disconnected graphs with each card of the deck showing a common eigenvalue. These could occur as possible counter examples to the positive solution of the PRP.

Deep denoising for multi-dimensional synchrotron X-ray tomography without high-quality reference data

Synchrotron X-ray tomography enables the examination of the internal structure of materials at submicron spatial resolution and subsecond temporal resolution. Unavoidable experimental constraints can impose dose and time limits on the measurements, introducing noise in the reconstructed images. Convolutional neural networks (CNNs) have emerged as a powerful tool to remove noise from reconstructed images. However, their training typically requires collecting a dataset of paired noisy and high-quality measurements, which is a major obstacle to their use in practice. To circumvent this problem, methods for CNN-based denoising have recently been proposed that require no separate training data beyond the already available noisy reconstructions. Among these, the Noise2Inverse method is specifically designed for tomography and related inverse problems. To date, applications of Noise2Inverse have only taken into account 2D spatial information. In this paper, we expand the application of Noise2Inverse in space, time, and spectrum-like domains. This development enhances applications to static and dynamic micro-tomography as well as X-ray diffraction tomography. Results on real-world datasets establish that Noise2Inverse is capable of accurate denoising and enables a substantial reduction in acquisition time while maintaining image quality.

Stability properties of a crack inverse problem in half space

We show in this paper a Lipschitz stability result for a crack inverse problem in half space. The direct problem is a Laplace equation with zero Neumann condition on the top boundary. The forcing term is a discontinuity across the crack. This formulation can be related to geological faults in elastic media or to irrotational incompressible flows in a half space minus an inner wall. The direct problem is well posed in an appropriate functional space. We study the related inverse problem where the jump across the crack is unknown, and more importantly, the geometry and the location of the crack are unknown. The data for the inverse problem are of Dirichlet type over a portion of the top boundary. We prove that this inverse problem is uniquely solvable under some assumptions on the geometry of the crack. The highlight of this paper is showing a stability result for this inverse problem. Assuming that the crack is planar, we show that reconstructing the plane containing the crack is Lipschitz stable despite the fact that the forcing term for the underlying PDE is unknown. This uniform stability result holds under the assumption that the forcing term is bounded above and the Dirichlet data are bounded below away from zero in appropriate norms.