Abstract This work develops a machine learned structural design model for continuous beam systems from the inverse problem perspective. After demarcating between forward, optimisation and inverse machine learned operators, the investigation proposes a novel methodology based on the recently developed influence zone concept which represents a fundamental shift in approach compared to traditional structural design methods. The aim of this approach is to conceptualise a non-iterative structural design model that predicts cross-section requirements for continuous beam systems of arbitrary system size. After generating a dataset of known solutions, an appropriate neural network architecture is identified, trained, and tested against unseen data. The results show a mean absolute percentage testing error of 1.6% for cross-section property predictions, along with a good ability of the neural network to generalise well to structural systems of variable size. The CBeamXP dataset generated in this work and an associated python-based neural network training script are available at an open-source data repository to allow for the reproducibility of results and to encourage further investigations.

The objective of this work is to quantify the reconstruction error in sparse inverse problems with measures and stochastic noise, motivated by optimal sensor placement. To be useful in this context, the error quantities must be explicit in the sensor configuration and robust with respect to the source, yet relatively easy to compute in practice, compared to a direct evaluation of the error by a large number of samples. In particular, we consider the identification of a measure consisting of an unknown linear combination of point sources from a finite number of measurements contaminated by Gaussian noise. The statistical framework for recovery relies on two main ingredients: first, a convex but non-smooth variational Tikhonov point estimator over the space of Radon measures and, second, a suitable mean-squared error based on its Hellinger–Kantorovich distance to the ground truth. To quantify the error, we employ a non-degenerate source condition as well as careful linearization arguments to derive a computable upper bound. This leads to asymptotically sharp error estimates in expectation that are explicit in the sensor configuration. Thus they can be used to estimate the expected reconstruction error for a given sensor configuration and guide the placement of sensors in sparse inverse problems.

We establish sharp stability estimates of logarithmic type in determining an impedance obstacle in R2 . The obstacle is the polygonal shape and the surface impedance parameter is non-zero constant. We establish the stability results using a single far-field pattern, constituting a longstanding problem in the inverse scattering theory. This is the first stability result in the literature in determining an impedance obstacle by a single far-field measurement. The stability in simultaneously determining the obstacle and the boundary impedance is established in terms of the classical Hausdorff distance. Several technical novelties and developments in the mathematical strategy developed for establishing the aforementioned stability results exist. First, the stability analysis is conducted around a corner point in a micro-local manner. Second, our stability estimates establish explicit relationships between the obstacle’s geometric configurations and the wave field’s vanishing order at the corner point. Third, we develop novel error propagation techniques to tackle singularities of the wave field at a corner with the impedance boundary condition.

This paper presents a multiple material-phase level-set approach for acoustic full-waveform inversion in the time domain. By using a single level set (LS) function, several level values are used to define virtual boundaries between material phases with different (and known) wave propagation velocities. The aim of the proposed approach is to provide a suitable framework to identify multiple/nested inclusions or a finite number of almost homogeneous sedimentary layers with sharp interfaces between them. The use of a single LS function provides a significant reduction in the number of variables to be identified, when compared with the usual multi-material phase approaches defined by multiple functions, especially for problems with a high number of degrees of freedom. Numerical experiments show satisfactory results in identifying simultaneously different interfaces. Cases with and without inverse crime are evaluated, showing that the approach is reasonably robust in dealing with such a condition.

We consider an inverse problem of reconstructing a degeneracy point in the diffusion coefficient in a one-dimensional parabolic equation by measuring the normal derivative on one side of the domain boundary. We analyze the sensitivity of the inverse problem to the initial data. We give sufficient conditions on the initial data for uniqueness and stability for the one-point measurement and show some examples of positive and negative results. On the other hand, we present more general uniqueness results, also for the identification of an initial data by measurements distributed over time. The proofs are based on an explicit form of the solution by means of Bessel functions of the first type. Finally, the theoretical results are supported by numerical experiments.

We study inverse source problems associated to semilinear elliptic equations of the form Δux+ax,u=Fx on a bounded domain Ω⊂Rn , n⩾2 . We show that it is possible to use nonlinearity to recover both the source F and the nonlinearity a(x,u) simultaneously and uniquely for a class of nonlinearities. This is in contrast to inverse source problems for linear equations, which always have a natural (gauge) symmetry that obstructs the unique recovery of the source. The class of nonlinearities for which we can uniquely recover the source and nonlinearity, includes a class of polynomials, which we characterize, and exponential nonlinearities.For general nonlinearities a(x,u) , we recover the source F(x) and the Taylor coefficients ∂uka(x,u) up to a gauge symmetry. We recover general polynomial nonlinearities up to the gauge symmetry. Our results also generalize results of Feizmohammadi and Oksanen (2020 J. Differ. Equ. 269 4683–719), Lassas et al (2020 Rev. Mat. Iberoam. 37 1553–80) by removing the assumption that u≡0 is a solution. To prove our results, we consider linearizations around possibly large solutions.Our results can lead to new practical applications, because we show that many practical models do not have the obstruction for unique recovery that has restricted the applicability of inverse source problems for linear models.

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.

In passive imaging, one attempts to reconstruct some coefficients in a wave equation from correlations of observed randomly excited solutions to this wave equation. Many methods proposed for this class of inverse problem so far are only qualitative, e.g. trying to identify the support of a perturbation. Major challenges are the increase in dimensionality when computing correlations from primary data in a preprocessing step, and often very poor pointwise signal-to-noise ratios. In this paper, we propose an approach that addresses both of these challenges: it works only on the primary data while implicitly using the full information contained in the correlation data, and it provides quantitative estimates and convergence by iteration. Our work is motivated by helioseismic holography, a well-established imaging method to map heterogenities and flows in the solar interior. We show that the back-propagation used in classical helioseismic holography can be interpreted as the adjoint of the Fréchet derivative of the operator which maps the properties of the solar interior to the correlation data on the solar surface. The theoretical and numerical framework for passive imaging problems developed in this paper extends helioseismic holography to nonlinear problems and allows for quantitative reconstructions. We present a proof of concept in uniform media.

The main contribution of this paper is the derivation of non-asymptotic convergence rates for Nyström kernel canonical correlation analysis (CCA) in a setting of statistical learning. Our theoretical results reveal that, under certain conditions, Nyström kernel CCA can achieve a convergence rate comparable to that of the standard kernel CCA, while offering significant computational savings. This finding has important implications for the practical application of kernel CCA, particularly in scenarios where computational efficiency is crucial. Numerical experiments are provided to demonstrate the effectiveness of Nyström kernel CCA.

Inverse problems, which are related to Maxwell’s equations, in the presence of nonlinear materials is a quite new topic in the literature. The lack of contributions in this area can be ascribed to the significant challenges that such problems pose. Retrieving the spatial behavior of some unknown physical property, from boundary measurements, is a nonlinear and highly ill-posed problem even in the presence of linear materials. Furthermore, this complexity grows exponentially in the presence of nonlinear materials. In the tomography of linear materials, the Monotonicity Principle (MP) is the foundation of a class of non-iterative algorithms able to guarantee excellent performances and compatibility with real-time applications. Recently, the MP has been extended to nonlinear materials under very general assumptions. Starting from the theoretical background for this extension, we develop a first real-time inversion method for the inverse obstacle problem in the presence of nonlinear materials. The proposed method is intendend for all problems governed by the quasilinear Laplace equation, i.e. static problems involving nonlinear materials. In this paper, we provide some preliminary results which give the foundation of our method and some extended numerical examples.