Table of contents

This paper introduces a computational framework to reconstruct and forecast a partially observed state that evolves according to an unknown or expensive-to-simulate dynamical system. Our reduced-order autodifferentiable ensemble Kalman filters (ROAD-EnKFs) learn a latent low-dimensional surrogate model for the dynamics and a decoder that maps from the latent space to the state space. The learned dynamics and decoder are then used within an EnKF to reconstruct and forecast the state. Numerical experiments show that if the state dynamics exhibit a hidden low-dimensional structure, ROAD-EnKFs achieve higher accuracy at lower computational cost compared to existing methods. If such structure is not expressed in the latent state dynamics, ROAD-EnKFs achieve similar accuracy at lower cost, making them a promising approach for surrogate state reconstruction and forecasting.

Terahertz time-domain spectroscopy (THz-TDS) system is a powerful tool in material spectral analysis and non-destructive testing (NDT). However, the data acquisition is time-consuming due to the point-by-point scanning process, which leads to limitations in THz imaging applications. In this paper, a robust adaptive-thresholding primal-dual sparse recovery (APSR) method is proposed to reconstruct high-resolution THz images and reduce the acquisition time. A sparsity averaging dictionary, which is a concatenate of multiple bases, is applied to improve the robustness and generalization. A reweighted $\ell_1$ scheme is also adopted to enhance the sparse solution. As a crucial consideration in the sparse recovery method, we propose a robust adaptive threshold estimator based on the median absolute deviation in the sense that the final threshold determines the reconstruction quality and the convergence rate. Meanwhile, together with the Nesterov acceleration technique, a large threshold can be applied at the beginning of the iteration to help accelerate the convergence. Numerical experiments demonstrate that the APSR method can reconstruct THz images with robustness even with low SNR (of 30 dB) and low sampling rate (of 30%) compared to conventional methods. In the real THz-TDS system, our proposed sparse imaging method can successfully recover a composite object with much fewer THz spectra and reduces the acquisition time to 30% at the cost of a relative error of 2.5%, demonstrating the efficiency of our proposed method and the penetrating ability of THz technology in the NDT applications.

A coefficient inverse problem for the radiative transport equation is considered. The globally convergent numerical method, the so-called convexification, is developed. For the first time, the viscosity solution is considered for a boundary value problem for the resulting system of two coupled partial differential equations. A Lipschitz stability estimate is proved for this boundary value problem using a Carleman estimate for the Laplace operator. Next, the global convergence analysis is provided via that Carleman estimate. Results of numerical experiments demonstrate a high computational efficiency of this approach.

In a task where many similar inverse problems must be solved, evaluating costly simulations is impractical. Therefore, replacing the model y with a surrogate model y_s that can be evaluated quickly leads to a significant speedup. The approximation quality of the surrogate model depends strongly on the number, position, and accuracy of the sample points. With an additional finite computational budget, this leads to a problem of (computer) experimental design. In contrast to the selection of sample points, the trade-off between accuracy and effort has hardly been studied systematically. We therefore propose an adaptive algorithm to find an optimal design in terms of position and accuracy. Pursuing a sequential design by incrementally appending the computational budget leads to a convex and constrained optimization problem. As a surrogate, we construct a Gaussian process regression model. We measure the global approximation error in terms of its impact on the accuracy of the identified parameter and aim for a uniform absolute tolerance, assuming that y_s is computed by finite element calculations. A priori error estimates and a coarse estimate of computational effort relate the expected improvement of the surrogate model error to computational effort, resulting in the most efficient combination of sample point and evaluation tolerance. We also allow for improving the accuracy of already existing sample points by continuing previously truncated finite element solution procedures.

In this research work, we investigate the Cauchy problem for the Helmholtz equation. Considering the completion data problem in a bounded cylindrical domain with Neumann and Dirichlet conditions on a part of the boundary. An immediate approximation of missing boundary data is obtained using a method that factorizes the boundary value problem. This factorization uses the Neumann to Dirichlet or Dirichlet to Neumann operators that satisfy the Riccati equation. Some singularities appear in the solution of the Riccati equation for a particular length of the waveguide of the Helmholtz equation. We elaborate a new numerical method called 'adaptive anadromic regularization method' that can solve these operators beyond the singularity. In addition, we introduce a scaling matrix technique to the linear matrix equations associated with the Riccati equations to generate normalized solutions. Our numerical procedure not only approximates missing boundary data, but also provides an error estimate that allows efficient time stepping. Numerical tests proved to confirm the theory even in the presence of high noise levels.

This article deals with linear time-fractional diffusion equations with time-dependent singular source term. Whether the order of the time-fractional derivative is multi-term, distributed or space-dependent, we prove that the system admits a unique weak solution enjoying a Duhamel representation, provided that the time-dependence of the source term is a distribution. As an application, the square integrable space-dependent part and the distributional time-dependent part of the source term of a multi-term time-fractional diffusion equation are simultaneously recovered by partial internal observation of the solution.

We consider scattering for the discrete Schrödinger operator on the square lattice $\mathbb{Z}^d$, $d \unicode{x2A7E} 1$, with compactly supported potential. We give formulas for finding the phased scattering amplitude from phaseless near-field scattering data.

A nonlinear optimization method is proposed for the solution of inverse medium problems with spatially varying properties. To avoid the prohibitively large number of unknown control variables resulting from standard grid-based representations, the misfit is instead minimized in a small subspace spanned by the first few eigenfunctions of a judicious elliptic operator, which itself depends on the previous iteration. By repeatedly adapting both the dimension and the basis of the search space, regularization is inherently incorporated at each iteration without the need for extra Tikhonov penalization. Convergence is proved under an angle condition, which is included into the resulting Adaptive Spectral Inversion (ASI) algorithm. The ASI approach compares favorably to standard grid-based inversion using L²-Tikhonov regularization when applied to an elliptic inverse problem. The improved accuracy resulting from the newly included angle condition is further demonstrated via numerical experiments from time-dependent inverse scattering problems.

We study stability properties of the expected utility function in Bayesian optimal experimental design. We provide a framework for this problem in a non-parametric setting and prove a convergence rate of the expected utility with respect to a likelihood perturbation. This rate is uniform over the design space and its sharpness in the general setting is demonstrated by proving a lower bound in a special case. To make the problem more concrete we proceed by considering non-linear Bayesian inverse problems with Gaussian likelihood and prove that the assumptions set out for the general case are satisfied and regain the stability of the expected utility with respect to perturbations to the observation map. Theoretical convergence rates are demonstrated numerically in three different examples.

In this paper, we study the partial data inverse boundary value problem for the Schrödinger operator at a high frequency $k\unicode{x2A7E} 1$ in a bounded domain with smooth boundary in $\mathbb{R}^n$, $n\unicode{x2A7E}3$. Assuming that the potential is known in a neighborhood of the boundary, we obtain the logarithmic stability when both Dirichlet data and Neumann data are taken on arbitrary open subsets of the boundary where the two sets can be disjointed. Our results also show that the logarithmic stability can be improved to the one of Hölder type in the high frequency regime. To achieve those goals, we used a method by combining the CGO solution, Runge approximation and Carleman estimate.

This article shows that a large class of posterior measures that are absolutely continuous with respect to a Gaussian prior have strong maximum a posteriori estimators in the sense of Dashti et al (2013 Inverse Problems29 095017). This result holds in any separable Banach space and applies in particular to nonparametric Bayesian inverse problems with additive noise. When applied to Bayesian inverse problems, this significantly extends existing results on maximum a posteriori estimators by relaxing the conditions on the log-likelihood and on the space in which the inverse problem is set.

The present paper deals with the data-driven design of regularizers in the form of artificial neural networks, for solving certain inverse problems formulated as optimal control problems. These regularizers aim at improving accuracy, wellposedness or compensating uncertainties for a given class of optimal control problems (inner-problems). Parameterized as neural networks, their weights are chosen in order to reduce a misfit between data and observations of the state solution of the inner- optimal control problems. Learning these weights constitutes the outer-problem. Based on necessary first-order optimality conditions for the inner-problems, a relaxation approach is proposed in order to implement efficient solving of these inner-problems, namely the forward operator of the outer-problem. Optimality conditions are derived for the latter, and are implemented in numerical illustrations dealing with the inverse conductivity problem. The numerical tests show the feasibility of the relaxation approach, first for rediscovering standard L²-regularizers, and next for designing regularizers that compensate unknown noise on the observed state of the inner-problem.

In this work, we ask and answer the question: when is the viscosity of a fluid uniquely determined from spatially sparse measurements of its velocity field? We pose the question mathematically as an optimization problem using the determining map (the mapping of data to an approximation made via a nudging algorithm) to define a loss functional, the minimization of which solves the inverse problem of identifying the true viscosity given the measurement data. We give explicit a priori conditions for the well-posedness of this inverse problem. In addition, we show that smallness of the loss functional implies proximity to the true viscosity. We then present an algorithm for solving the inverse problem and provide a priori verifiable conditions that ensure its convergence.

The multiscale hierarchical decomposition method (MHDM) was introduced in Tadmor et al (2004 Multiscale Model. Simul.2 554–79; 2008 Commun. Math. Sci.6 281–307) as an iterative method for total variation (TV) regularization, with the aim of recovering details at various scales from images corrupted by additive or multiplicative noise. Given its success beyond image restoration, we extend the MHDM iterates in order to solve larger classes of linear ill-posed problems in Banach spaces. Thus, we define the MHDM for more general convex or even nonconvex penalties, and provide convergence results for the data fidelity term. We also propose a flexible version of the method using adaptive convex functionals for regularization, and show an interesting multiscale decomposition of the data. This decomposition result is highlighted for the Bregman iteration method that can be expressed as an adaptive MHDM. Furthermore, we state necessary and sufficient conditions when the MHDM iteration agrees with the variational Tikhonov regularization, which is the case, for instance, for one-dimensional TV denoising. Finally, we investigate several particular instances and perform numerical experiments that point out the robust behavior of the MHDM.

The ensemble Kalman filter (EnKF) is a Monte Carlo approximation of the Kalman filter for high dimensional linear Gaussian state space models. EnKF methods have also been developed for parameter inference of static Bayesian models with a Gaussian likelihood, in a way that is analogous to likelihood tempering sequential Monte Carlo (SMC). These methods are commonly referred to as ensemble Kalman inversion (EKI). Unlike SMC, the inference from EKI is asymptotically biased if the likelihood is non-linear and/or non-Gaussian and if the priors are non-Gaussian. However, it is significantly faster to run. Currently, a large limitation of EKI methods is that the covariance of the measurement error is assumed to be fully known. We develop a new method, which we call component-wise iterative EKI (CW-IEKI), that allows elements of the covariance matrix to be inferred alongside the model parameters at negligible extra cost. This novel method is compared to SMC on a linear Gaussian example as well as four examples with non-linear dynamics (i.e. non-linear function of the model parameters). The non-linear examples include a set of population models applied to synthetic data, a model of nitrogen mineralisation in soil that is based on the Agricultural Production Systems Simulator, a model predicting seagrass decline due to stress from water temperature and light, and a model predicting coral calcification rates. On our examples, we find that CW-IEKI has relatively similar predictive performance to SMC, albeit with greater uncertainty, and it has a significantly faster run time.

We consider the Born and inverse Born series for scalar waves with a cubic nonlinearity of Kerr type. We find a recursive formula for the operators in the Born series and prove their boundedness. This result gives conditions which guarantee convergence of the Born series, and subsequently yields conditions which guarantee convergence of the inverse Born series. We also use fixed point theory to give alternate explicit conditions for convergence of the Born series. We illustrate our results with numerical experiments.

We present the first numerical study of multipoint formulas for finding leading coefficients in asymptotic expansions arising in potential and scattering theories. In particular, we implement different formulas for finding the Fourier transform of potential from the scattering amplitude at several high energies. We show that the aforementioned approach can be used for essential numerical improvements of classical results including the slowly convergent Born–Faddeev formula for inverse scattering at high energies. The approach of multipoint formulas can be also used for recovering the x-ray transform of potential from boundary values of the scattering wave functions at several high energies. Determination of total charge (electric or gravitational) from several exterior measurements is also considered. In addition, we show that the aforementioned multipoint formulas admit an efficient regularization for the case of random noise. In particular, we proceed from theoretical works (Novikov 2020 Inverse Problems36 095001; 2021 Russ. Math. Surv.76 723–5).

In many applications it is desirable to inverse-calculate the distributed loading on a structure using a limited number of sensors. Yet, the calculated loads can be extremely sensitive to the placement of these sensors. In the case of predicting point loading applied at a known location, best results are typically achieved when one sensor is collocated with the force. However, the extension of this rule to distributed loading remains uncertain, and even simple sensor system design scenarios often require the designer to directly optimize the sensor placements using a numerical model. In an effort to provide designers with guidance, we identify optimal sensor configurations for predicting static distributed loads on beams with classical boundary conditions. An influence coefficient method, wherein the strain is related linearly to the static load, is used to estimate the applied forces. The loading distribution on the structure is assumed to be either a piece-wise linearly-distributed load or a uniformly-distributed load, allowing for distributed loads to be estimated using the magnitudes of a small number of control points. Given the simplicity of the beam structure, the equations of the influence coefficient method are derived analytically, which allows for the sensor placement to be specified using continuous optimization methods. The condition number of the influence coefficient matrix is used as a surrogate for error during optimization. 'Rules of thumb' for sensor placement are presented based on the optimization results. Results show that the optimal and rule-of-thumb sensor configurations are more resistant to input noise than naïve configurations, with the rule-of-thumb configurations yielding similar force predictions relative to the optimal configurations. We expect the rules of thumb to be useful guidelines for engineers designing tests on beam-like structures such as aircraft wings or marine propellers where the inverse calculation of distributed loads is of interest.

Learned inverse problem solvers exhibit remarkable performance in applications like image reconstruction tasks. These data-driven reconstruction methods often follow a two-step procedure. First, one trains the often neural network-based reconstruction scheme via a dataset. Second, one applies the scheme to new measurements to obtain reconstructions. We follow these steps but parameterize the reconstruction scheme with invertible residual networks (iResNets). We demonstrate that the invertibility enables investigating the influence of the training and architecture choices on the resulting reconstruction scheme. For example, assuming local approximation properties of the network, we show that these schemes become convergent regularizations. In addition, the investigations reveal a formal link to the linear regularization theory of linear inverse problems and provide a nonlinear spectral regularization for particular architecture classes. On the numerical side, we investigate the local approximation property of selected trained architectures and present a series of experiments on the MNIST dataset that underpin and extend our theoretical findings.

We consider the problem of solving linear least squares problems in a framework where only evaluations of the linear map are possible. We derive randomized methods that do not need any other matrix operations than forward evaluations, especially no evaluation of the adjoint map is needed. Our method is motivated by the simple observation that one can get an unbiased estimate of the application of the adjoint. We show convergence of the method and then derive a more efficient method that uses an exact linesearch. This method, called random descent, resembles known methods in other context and has the randomized coordinate descent method as special case. We provide convergence analysis of the random descent method emphasizing the dependence on the underlying distribution of the random vectors. Furthermore we investigate the applicability of the method in the context of ill-posed inverse problems and show that the method can have beneficial properties when the unknown solution is rough. We illustrate the theoretical findings in numerical examples. One particular result is that the random descent method actually outperforms established transposed-free methods (TFQMR and CGS) in examples.

The aim of magnetorelaxometry imaging is to determine the distribution of magnetic nanoparticles inside a subject by measuring the relaxation of the superposition magnetic field generated by the nanoparticles after they have first been aligned using an external activation magnetic field that has subsequently been switched off. This work applies techniques of Bayesian optimal experimental design to (sequentially) selecting the positions for the activation coil in order to increase the value of data and enable more accurate reconstructions in a simplified measurement setup. Both Gaussian and total variation (TV) prior models are considered for the distribution of the nanoparticles. The former allows simultaneous offline computation of optimized designs for multiple consecutive activations, while the latter introduces adaptability into the algorithm by using previously measured data in choosing the position of the next activation. The TV prior has a desirable edge-enhancing characteristic, but with the downside that the computationally attractive Gaussian form of the posterior density is lost. To overcome this challenge, the lagged diffusivity iteration is used to provide an approximate Gaussian posterior model and allow the use of the standard Bayesian A- and D-optimality criteria for the TV prior as well. Two-dimensional numerical experiments are performed on a few sample targets, with the conclusion that the optimized activation positions lead, in general, to better reconstructions than symmetric reference setups when the target distribution or region of interest are nonsymmetric in shape.

Owing to the harsh environment, the support conditions of wind turbines inevitably degrade/change over their lifetime, however, the evolution mechanism is not yet well understood. Although the damping parameters are sensitive to structural support and connection conditions, they are difficult to measure and quantify, which is a challenging inverse problem. This study aims to develop an approach to obtain a statistical time-domain damping parameter (STDP) based on operational vibration signals, and to utilize the parameter to identify support conditions of wind turbines. The proposed approach transforms operational vibration signals to free vibration signals by using the random decrement technique and then performs nonparametric statistical analysis to quantify the statistically significant changes in the damping characteristics of a structure. The effectiveness of the STDP method is verified by two challenging cases of bolted connection damage and soil-structure interaction condition changes. The regression analysis demonstrates the ability of the STDP method for the identification of structural overall damping. In contrast with classic modal analysis methods, the proposed method provides a monotonic relationship between the STDP and support conditions, which is significant for structural condition identification.

In the recent manuscript (Ikehata 2022 Inverse Problems38 075009), the author made some comments on our previous work (Sini and Yoshida 2012 Inverse Problems28 055013) saying that few arguments in the proof of lemma 3.5 might be incomplete for $C^{0, 1}$-regular domains. In this note, we reply to his comments by showing that, for C¹-regular domains, those arguments are correct.

What should you do when you re-read the paper you intended to cite and found an error in the proof? In section 4 of Ikehata (2022 Inverse Problems38 075009) the author pointed out that the proof of lemma 3.5 in Sini and Yoshida (2012 Inverse Problems28 055013; 2013 Inverse Problems29 039501) does not work for a Lipschitz obstacle case, by applying their approach to an essentially same lemma (lemma 3.2 in Ikehata (2022 Inverse Problems38 075009)). For this, Sini (2023 Inverse Problems accepted) gave a comment. This is a reply to their comment.