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Small angle x-ray scattering (SAXS) is a widely recognized solution for measuring complex nanostructures. With the increasing demand for accurately assessing structural characteristics and optimizing manufacturing processes, uncertainty quantification in SAXS inverse problems has become a critical issue. However, traditional methods face challenges such as slow computation speed and inaccurate estimation of multidimensional parameters. To overcome these issues, we propose an uncertainty quantification approach suitable for SAXS measurement that approximates the posterior using Bayesian deep learning. The effectiveness and reliability of our method are illustrated by assessing structural parameters of synthetic 2D Si grating samples. The uncertainty quantification takes only about 2.3 s, thousands of times faster than the conventional Markov Chain Monte Carlo (MCMC) methods. Also, our method has superior repeatability for parameter measurement compared to the MCMC approaches. It provides the potential of efficient and reliable SAXS measurement in increasingly intricate semiconductor manufacturing. [ABSTRACT FROM AUTHOR]

Biomolecular condensates are viscoelastic materials defined by time-dependent, sequence-specific complex shear moduli. Here, we show that viscoelastic moduli can be computed directly using a generalization of the Rouse model that leverages information regarding intra- and inter-chain contacts, which we extract from equilibrium configurations of lattice-based Metropolis Monte Carlo (MMC) simulations of phase separation. The key ingredient of the generalized Rouse model is a graph Laplacian that we compute from equilibrium MMC simulations. We compute two flavors of graph Laplacians, one based on a single-chain graph that accounts only for intra-chain contacts, and the other referred to as a collective graph that accounts for inter-chain interactions. Calculations based on the single-chain graph systematically overestimate the storage and loss moduli, whereas calculations based on the collective graph reproduce the measured moduli with greater fidelity. However, in the long time, low-frequency domain, a mixture of the two graphs proves to be most accurate. In line with the theory of Rouse and contrary to recent assertions, we find that a continuous distribution of relaxation times exists in condensates. The single crossover frequency between dominantly elastic vs dominantly viscous behaviors does not imply a single relaxation time. Instead, it is influenced by the totality of the relaxation modes. Hence, our analysis affirms that viscoelastic fluid-like condensates are best described as generalized Maxwell fluids. Finally, we show that the complex shear moduli can be used to solve an inverse problem to obtain the relaxation time spectra that underlie the dynamics within condensates. This is of practical importance given advancements in passive and active microrheology measurements of condensate viscoelasticity. [ABSTRACT FROM AUTHOR]

Small angle x-ray scattering (SAXS) stands out as a promising solution in semiconductor metrology. The critical issue of SAXS metrology is to solve the SAXS inverse problem. With the increasing complexity of semiconductor devices, traditional strategies will face problems such as long iteration time and multiple solutions. To address these challenges, we develop a physical symmetry enhanced method to speed up the solution of the SAXS inverse problem for complex nanostructures. We incorporate the physical symmetry into a deep learning model, and a combined loss function is proposed to determine the correct structure in each step of training, which can continuously correct errors and make the model converge faster. The results show that the proposed method achieves high accuracy in determining the critical structural parameters of the complex profile gratings. Compared to traditional strategies, our method performs better in accuracy and does not require time-consuming iterations during reconstruction. The physical symmetry enhanced method provides a feasible way for achieving real-time reconstruction of complex profile nanostructures and is expected to promote the development of SAXS metrology. [ABSTRACT FROM AUTHOR]

Toward identifying efficient sound absorbers, we have formulated and analyzed the micro-perforated panels (MPPs) comprising cylindrical pores of arbitrary axial profiles for out-of-plane wave propagation. First, the forward problem was solved wherein an exact analytical expression for the absorption coefficient α was developed for these MPPs. The dependence of α was analyzed for various profiles comprising convex and concave-shaped sections including divergent and convergent linear and sinusoidal profiles, conic sections, Kilroy, and chirp shapes. The MPPs with pores having a diverging shape at the entrance were found to have higher sound absorption. The non-symmetric fluctuations in the profile led to fluctuations in the spectrum of α. Aiming to address the long-standing challenge of low-frequency sound absorption, we solved the inverse problem to identify the axial profile of the cylindrical pore for maximum sound absorption for frequencies up to 2500 Hz. Given the advances in additive manufacturing, the results of this comprehensive work help in designing MPPs comprising complex-shaped pores suitable for a particular spectral regime. [ABSTRACT FROM AUTHOR]

Previously, we introduced photomagnetic imaging (PMI) that synergistically utilizes laser light to slightly elevate the tissue temperature and magnetic resonance thermometry (MRT) to measure the induced temperature. The MRT temperature maps are then converted into absorption maps using a dedicated PMI image reconstruction algorithm. In the MRT maps, the presence of abnormalities such as tumors would create a notable high contrast due to their higher hemoglobin levels. In this study, we present a new artificial intelligence-based image reconstruction algorithm that improves the accuracy and spatial resolution of the recovered absorption maps while reducing the recovery time. Technically, a supervised machine learning approach was used to detect and delineate the boundary of tumors directly from the MRT maps based on their temperature contrast to the background. This information was further utilized as a soft functional a priori in the standard PMI algorithm to enhance the absorption recovery. Our new method was evaluated on a tissue-like phantom with two inclusions representing tumors. The reconstructed absorption map showed that the well-trained neural network not only increased the PMI spatial resolution but also improved the accuracy of the recovered absorption to as low as a 2% percentage error, reduced the artifacts by 15%, and accelerated the image reconstruction process approximately 9-fold.

Summary: In this article we deal with the problem of determining a non‐homogeneous N$$ N $$‐valued heat conductivity profile in a steady‐state heat conduction boundary‐value problem with mixed Dirichlet‐Neumann boundary conditions over a bounded domain in ℝn$$ {\mathbb{R}}^n $$, from the knowledge of the temperature field over the whole domain. In a previous work we developed a method based on a variational approach of the PDE leading to an optimality equation which is then projected into a finite dimensional space. Discretization of the optimality equation then yields a linear although severely ill‐posed equation which is then regularized via appropriate ad‐hoc penalizers based upon a‐priori information about the conductivities of all materials present. This process results in a generalized Tikhonov‐Phillips functional whose global minimizer yields our approximate solution to the inverse problem. In our previous work we showed that this approach yields quite satisfactory results in the cases of two different conductivities. We considered here an appropriate extension of that approach for the N$$ N $$ materials case and show a few numerical examples for the case N=3$$ N=3 $$ in which the method is able to produce very good reconstructions of the exact solution. [ABSTRACT FROM AUTHOR]

We consider the inverse conductivity problem with discontinuous conductivities. We show in a rigorous way, by a convergence analysis, that one can construct a completely discrete minimization problem whose solution is a good approximation of a solution to the inverse problem. The minimization problem contains a regularization term which is given by a total variation penalization and is characterized by a regularization parameter. The discretization involves at the same time the boundary measurements, by the use of the complete electrode model, the unknown conductivity and the solution to the direct problem. The electrodes are characterized by a parameter related to their size, which in turn controls the number of electrodes to be used. The discretization of the unknown and of the solution to the direct problem is characterized by another parameter related to the size of the mesh involved. In our analysis we show how to precisely choose the regularization, electrodes size and mesh size parameters with respect to the noise level in such a way that the solution to the discrete regularized problem is meaningful. In particular we obtain that the electrodes and mesh size parameters should decay polynomially with respect to the noise level. [ABSTRACT FROM AUTHOR]

Worldwide communication bandwidth growth has largely been driven by (1) multimedia demands, (2) multicommunication-point demands and (3) multicommunication-rate demands, and has increased dramatically over the last two decades due to e-commerce, internet communication and the explosion of cell-phone use, in particular for in-flight services, all of which necessitate broadband use and low latency. In order to accommodate this huge surge in demand, next generation "mega-constellations" of satellites are being proposed combining a mix of heterogeneous unit types in LEO, MEO and GEO orbital shells, in order to provide continuous lower-latency and high-bandwidth service which exploits a wide-range of frequencies for fast internet connections (broadband, which is not possible with single satellite-type orbital shell systems). Accordingly, in this work, we develop a computationally-efficient digital-twin framework for a constellation of satellites around an arbitrary planet ("Planet-X"). The rapid speed of these simulations enables the ability to explore satellite infrastructure parameter combinations, represented by a multicomponent satellite constellation design vector Λ = def (number of satellites, satellite orbital radii, satellite orbital speeds, satellite types), that can deliver desired communication signal or camera coverage on "Planet-X", while simultaneously incorporating satellite infrastructural resource constraints. In order to cast the objective mathematically, we set up the system design as an inverse problem to minimize a cost function via a Genetic Machine Learning Algorithm (G-MLA), which is well-suited for nonconvex optimization. Numerical examples are provided to illustrate the framework. [ABSTRACT FROM AUTHOR]

We address the discrete inverse conductance problem for well-connected spider networks, that is, to recover the conductance function on a well-connected spider network from the Dirichlet-to-Neumann map. It is well-known that this inverse problem is exponentially ill-posed, requiring the implementation of a regularization strategy for numerical solutions. Our focus lies in exploring whether prior knowledge of the conductance being piecewise constant within a partition of the edge set comprising a few subsets enables stable conductance recovery. To achieve this, we propose formulating the problem as a polynomial optimization problem, incorporating a regularization term that accounts for the piecewise constant hypothesis. We show several experimental examples in which the stable conductance recovery under the aforementioned hypothesis is feasible. [ABSTRACT FROM AUTHOR]

This article develops the theory of the conformal mapping of the surface of constant height above the ellipsoid (ChS), which has implications for the design of low-distortion projections. This paper gives a development of the Mercator, polar stereographic, Lambert conformal conic (LCC) and transverse Mercator (TM) projections based on the ChS, a better Earth model than the reference ellipsoid (RE) for positions at height. This article provides new formulas for the LCC and TM inverse scale problems – when given a value for the local scale function, where does it occur? These formulas apply to both traditional mapping on the RE and novel mapping on the ChS. [ABSTRACT FROM AUTHOR]

Starting from the seminal works of Toupin, Mindlin and Germain, a wide class of generalized elastic models have been proposed via the principle of virtual work, by postulating expressions of the elastic energy enriched by additional kinematic descriptors or by higher gradients of the placement. More recently, such models have been adopted to describe phenomena which are not consistent with the Cauchy-Born continuum, namely the size dependence of apparent elastic moduli observed for micro and nano-objects, wave dispersion, optical modes and band gaps in the dynamics of heterogeneous media. For those structures the mechanical response is affected by surface effects which are predominant with respect to the bulk, and the scale of the external actions interferes with the characteristic size of the heterogeneities. Generalized continua are very often referred to as media with microstructure although a rigorous deduction is lacking between the specific microstructural features and the constitutive equations. While in the forward modelling predictions of the observations are provided, the actual observations at multiple scales can be used inversely to integrate some lack of information about the model. In this review paper, generalized continua are investigated from the standpoint of inverse problems, focusing onto three topics, tightly connected and located at the border between multiscale modelling and the experimental assessment, namely: (i) parameter identification of generalized elastic models, including asymptotic methods and homogenization strategies; (ii) design of non-conventional tests, possibly integrated with full field measurements and advanced modelling; (iii) the synthesis of meta-materials, namely the identification of the microstructures which fit a target behaviour at the macroscale. The scientific literature on generalized elastic media, with the focus on the higher gradient models, is fathomed in search of questions and methods which are typical of inverse problems theory and issues related to parameter estimation, providing hints and perspectives for future research. [ABSTRACT FROM AUTHOR]

Reflectivity inversion is a key step in reservoir prediction. Conventional sparse‐spike deconvolution assumes that the reflectivity (reflection coefficient series) is sparse and solves for the reflection coefficients by an L1‐norm inversion process. Spectral inversion is an alternative to sparse‐spike deconvolution, which is based on the odd–even decomposition algorithm and can accurately identify thin layers and reduce the wavelet tuning effect without using constraints from logging data, from horizon interpretations or from an initial model of the reflectivity. In seismic processing, an error exists in wavelet extraction because of complex geological structures, resulting in the low accuracy of deconvolution and inversion. Blind deconvolution is an effective method for solving the problem mentioned above, which comprises seismic wavelet and reflectivity sequence, assuming that the wavelets that affect some subsets of the seismic data are approximately the same. Therefore, we combined blind deconvolution with spectral inversion to propose blind spectral inversion. Given an initial wavelet, we can calculate the reflectivity based on spectral inversion and update the wavelet for the next iteration. During the update processing, we add the smoothness of the wavelet amplitude spectrum as a regularization term, thus reducing the wavelet oscillation in the time domain, increasing the similarity between inverted and initial wavelets, and improving the stability of the solution. The blind spectral inversion method inherits the wavelet robustness of blind deconvolution and high resolution of spectral inversion, which is suitable for reflectivity inversion. Applications to synthetic and field seismic datasets demonstrate that the blind spectral inversion method can accurately calculate the reflectivity even when there is an error in wavelet extraction. [ABSTRACT FROM AUTHOR]

Time‐lapse images carry out important information about dynamic changes in Earth's interior, which can be inferred using different full waveform inversion schemes. The estimation process is performed by manipulating more than one seismic dataset, associated with the baseline and monitors surveys. The time‐lapse variations can be so minute and localized that quantifying the uncertainties becomes fundamental to assessing the reliability of the results. The Bayesian formulation of the full waveform inversion problem naturally provides confidence levels in the solution, but evaluating the uncertainty of time‐lapse seismic inversion remains a challenge due to the ill‐posedness and high dimensionality of the problem. The Hamiltonian Monte Carlo can effectively sample over high‐dimensional distributions with affordable computational efforts. In this context, we explore the sequential approach in a Bayesian fashion for time‐lapse full waveform inversion using the Hamiltonian Monte Carlo method. The idea relies on integrating the baseline survey information as prior knowledge to the monitor estimation. We compare this methodology with a parallel scheme in perfect and a simple perturbed acquisition geometry scenario considering the Marmousi and a typical Brazilian pre‐salt velocity model. We also investigate the correlation effect between baseline and monitor samples on the propagated uncertainties. The results show that samples between different surveys are weakly correlated in the sequential case, while the parallel strategy provides time‐lapse images with lower dispersion. Our findings demonstrate that both methodologies are robust in providing uncertainties even in non‐repeatable scenarios. [ABSTRACT FROM AUTHOR]

Surface geometry inversion of geophysical data has recently been introduced as an effective approach for generating surface‐based geological models. The models obtained through surface geometry inversion clearly delineate the contacts between distinct rock units, making them easily interpretable by geologists. Surface geometry inversion has shown promising preliminary results in other works, but the practical application of surface geometry inversion on real geophysical data has not been thoroughly investigated. To move towards a better understanding of the practicalities involved, we applied surface geometry inversion to a real magnetic dataset acquired over two kimberlite pipes located in north‐central Botswana. The objective was to assess the effectiveness and limitations of the surface geometry inversion approach in accurately characterizing the subsurface geometry and identifying the boundaries of the kimberlite pipes. We first perform an anomaly separation approach to isolate the magnetic anomalies associated with the kimberlite pipes. A surface geometry inversion algorithm was applied to the original and separated datasets using various initial models and other control parameters. Several tests were performed to investigate the effects that data processing, initial models, and other parameter choices have on the surface geometry inversion results. We successfully recover the geometry, extension and dip of the two kimberlite pipes. We discuss the results of our various tests and provide advice for practitioners interested in applying surface geometry inversion methods to their data. Our work indicates that surface geometry inversion can be used as a complementary approach to voxel inversion, and we propose an iterative surface geometry inversion algorithm as a possible alternative approach to voxel inversion for simple geological scenarios. This work provides valuable insights into the appropriate application of surface geometry inversion on real geophysical datasets. [ABSTRACT FROM AUTHOR]

The Wiener–Hopf factorisation of a Lévy or Markov additive process describes the way that it attains new extrema in terms of a pair of so-called ladder height processes. Vigon's theory of friendship for Lévy processes addresses the inverse problem: when does a process exist which has certain prescribed ladder height processes? We give a complete answer to this problem for Markov additive processes, provide simpler sufficient conditions for constructing processes using friendship, and address in part the question of the uniqueness of the Wiener–Hopf factorisation for Markov additive processes. [ABSTRACT FROM AUTHOR]

We discuss the identification of a time‐dependent potential in a time‐fractional diffusion model from a boundary measurement taken at a single point. Theoretically, we establish a conditional Lipschitz stability for this inverse problem. Numerically, we develop an easily implementable iterative algorithm to recover the unknown coefficient, and also derive rigorous error bounds for the discrete reconstruction. These results are attained by leveraging the (discrete) solution theory of direct problems, and applying error estimates that are optimal with respect to problem data regularity. Numerical simulations are provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

Given a graph G , consider the family of real symmetric matrices with the property that the pattern of their nonzero off-diagonal entries corresponds to the edges of G. For the past 30 years a central problem has been to determine which spectra are realizable in this matrix class. Using combinatorial methods, we identify a family of graphs and multiplicity lists whose realizable spectra are highly restricted. In particular, we construct trees with multiplicity lists that require a unique spectrum, up to shifting and scaling. This represents the most extreme possible failure of spectral arbitrariness for a multiplicity list, and greatly extends all previously known instances of this phenomenon, in which only single linear constraints on the eigenvalues were observed. [ABSTRACT FROM AUTHOR]

We explore the possibility for using boundary measurements to recover a sparse source term f(x) in the potential equation. Employing weighted sparsity regularization and standard results for subgradients, we derive simple-to-check criteria which assure that a number of sinks (f(x)<0) and sources (f(x)>0) can be identified. Furthermore, we present two cases for which these criteria always are fulfilled: (a) well-separated sources and sinks, and (b) many sources or sinks located at the boundary plus one interior source/sink. Our approach is such that the linearity of the associated forward operator is preserved in the discrete formulation. The theory is therefore conveniently developed in terms of Euclidean spaces, and it can be applied to a wide range of problems. In particular, it can be applied to both isotropic and anisotropic cases. We present a series of numerical experiments. This work is motivated by the observation that standard methods typically suggest that internal sinks and sources are located close to the boundary. [ABSTRACT FROM AUTHOR]

This paper deals with the effect of fractional order on the fractional Euler-Bernoulli beams model. This model is based on the elastic curve that can be approximated using tangent lines and the osculating circle. We introduce a general form of the fractional Green's function for the highest-order term 1 < α ≤ 2 , which gives a unique solution for the geometrical fractional bending beams with an inhomogeneous fractional differential equation. The analytical closed-form responses were obtained by inverse problem technique and integration over Green's function kernel. The theory and properties of fractional Green's function method are described by generalizing classical theory to the initial and two-point fractional boundary value problems. Static analysis of the numerical examples, such as determinate and indeterminate beams with typical loads and beam property boundary conditions, are presented and discussed. The results were verified using the direct integration method. The fractional conventional ratio indicates the beam's deformation deviation from its classical state. The analysis results reveal that changing the fractional parameter significantly impacts the deflection behavior of flexural beams due to the beam's characteristics and the fractional order. The behavior of the fractional beam is identical to the classical state when the fractional parameter is an integer number. [ABSTRACT FROM AUTHOR]

Aim. Development of method for prediction by energy-saturated extended technical objects magnetic silencing based on magnetostatics geometric inverse problems solution and magnetic field spatial spheroidal harmonics calculated in prolate spheroidal coordinate system taking into account of technical objects magnetic characteristics uncertainties. Methodology. Spatial prolate spheroidal harmonics of extended technical objects magnetic field model calculated as magnetostatics geometric inverse problems solution in the form of nonlinear minimax optimization problem based on near field measurements for prediction far extended technical objects magnetic field magnitude. Nonlinear objective function calculated as the weighted sum of squared residuals between the measured and predicted magnetic field COMSOL Multiphysics software package used. Nonlinear minimax optimization problems solutions calculated based on particle swarm nonlinear optimization algorithms. Results. Results of prediction extended technical objects far magnetic field magnitude based on extended technical objects spatial prolate spheroidal harmonics of the magnetic field model in the prolate spheroidal coordinate system using near field measurements with consideration of extended technical objects magnetic characteristics uncertainty. Originality. The method for prediction by extended technical objects magnetic cleanliness based on spatial prolate spheroidal harmonics of the magnetic field model in the prolate spheroidal coordinate system with consideration of magnetic characteristics uncertainty is developed. Practical value. The important practical problem of prediction extended technical objects magnetic silencing based on the spatial prolate spheroidal harmonics of the magnetic field model in the prolate spheroidal coordinate system with consideration of extended technical objects magnetic characteristics uncertainty solved. [ABSTRACT FROM AUTHOR]

In this paper we analyze and solve eigenvalue programs, which consist of the task of minimizing a function subject to constraints on the "eigenvalues" of the decision variable. Here, by making use of the FTvN systems framework introduced by Gowda, we interpret "eigenvalues" in a broad fashion going beyond the usual eigenvalues of matrices. This allows us to shed new light on classical problems such as inverse eigenvalue problems and also leads to new applications. In particular, after analyzing and developing a simple projected gradient algorithm for general eigenvalue programs, we show that eigenvalue programs can be used to express what we call vanishing quadratic constraints. A vanishing quadratic constraint requires that a given system of convex quadratic inequalities be satisfied and at least a certain number of those inequalities must be tight. As a particular case, this includes the problem of finding a point x in the intersection of m ellipsoids in such a way that x is also in the boundary of at least ℓ of the ellipsoids, for some fixed ℓ > 0 . At the end, we also present some numerical experiments. [ABSTRACT FROM AUTHOR]

Summary: In modelling two‐phase flows, accurate representation of interfaces is crucial. A class of methods for interface reconstruction are based on isosurface extraction, which involves a non‐iterative, interpolation based approach. These approaches have been shown to be faster by an order of magnitude than the conventional PLIC schemes. In this work, we present a new isosurface extraction based interface reconstruction scheme based on the Marching Cubes algorithm (MC), which is commonly used in computer graphics for visualizing isosurfaces. The MC algorithm apriori lists and categorizes all possible interface configurations in a single grid cell into a Look Up Table (LUT), which makes this approach fast and robust. We also show that for certain interface configurations, the inverse problem of obtaining the isovalue from the cell volume fraction is not surjective, and a special treatment is required while handling these cases. We then demonstrate the capabilities of the method through benchmark cases for 2D and 3D static/dynamic interface reconstruction. [ABSTRACT FROM AUTHOR]

In this paper, we construct the first provably-secure isogeny-based (partially) blind signature scheme. While at a high level the scheme resembles the Schnorr blind signature, our work does not directly follow from that construction, since isogenies do not offer as rich an algebraic structure. Specifically, our protocol does not fit into the linear identification protocol abstraction introduced by Hauck, Kiltz, and Loss (EUROCYRPT'19), which was used to generically construct Schnorr-like blind signatures based on modules such as classical groups and lattices. Consequently, our scheme is provably secure in the random oracle model (ROM) against poly-logarithmically-many concurrent sessions assuming the subexponential hardness of the group action inverse problem. In more detail, our blind signature exploits the quadratic twist of an elliptic curve in an essential way to endow isogenies with a strictly richer structure than abstract group actions (but still more restrictive than modules). The basic scheme has public key size 128 B and signature size 8 KB under the CSIDH-512 parameter sets—these are the smallest among all provably secure post-quantum secure blind signatures. Relying on a new ring variant of the group action inverse problem (rGAIP ), we can halve the signature size to 4 KB while increasing the public key size to 512 B. We provide preliminary cryptanalysis of rGAIP and show that for certain parameter settings, it is essentially as secure as the standard GAIP . Finally, we show a novel way to turn our blind signature into a partially blind signature, where we deviate from prior methods since they require hashing into the set of public keys while hiding the corresponding secret key—constructing such a hash function in the isogeny setting remains an open problem. [ABSTRACT FROM AUTHOR]

In this work, we discuss the use of a recently introduced machine learning (ML) technique known as Fourier neural operators (FNO) as an efficient alternative to the traditional solution of the time‐dependent Schrödinger equation (TDSE). FNOs are ML models which are employed in the approximated solution of partial differential equations. For a wavepacket propagating in an anharmonic potential and for a tunneling system, we show that the FNO approach can accurately and faithfully model wavepacket propagation via the density. Additionally, we demonstrate that FNOs can be a suitable replacement for traditional TDSE solvers in cases where the results of the quantum dynamical simulation are required repeatedly such as in the case of parameter optimization problems (e.g., control). The speed‐up from the FNO method allows for its combination with the Markov‐chain Monte Carlo approach in applications that involve solving inverse problems such as optimal and coherent laser control of the outcome of dynamical processes. [ABSTRACT FROM AUTHOR]

If an unknown finite set C ⊂ Z 2 is cut by lines parallel to given directions, then one may count the number of points of C that are intercepted by each line, that is, the projections of C in the given directions. The inverse problem consists in reconstructing the set C , interpreted as a binary image, from the knowledge of its projections. In general, this challenging combinatorial problem, also related to the tomographic reconstruction of an unknown homogeneous object by means of X-rays, is ill-posed, meaning that different binary images exist that match the available projections. Therefore, as a preliminary step, one can try to find conditions to be imposed on the considered directions in order to limit the number of allowed solutions. In this paper we address the above problems for sets C contained in a finite assigned lattice grid, and generalize some results known in literature. First, we describe special sets of lattice directions, called simple cycles, and focus on some of their properties. Then we prove that uniqueness of reconstruction for binary images is guaranteed if and only if the line sums are computed along suitable simple cycles having even cardinality. As a second item, we prove that the unique binary solution can be explicitly reconstructed from a real-valued solution having minimal Euclidean norm. This leads to an explicit reconstruction algorithm, tested on four different phantoms and compared with previous results, which points out a significant improvement of the corresponding performance. [ABSTRACT FROM AUTHOR]

ABSTRACT In this article, we investigate the numerical solutions of the inverse eigenvalue problem (IEP). Motivated by the techniques of Aishima in 2018, we propose here an improved two‐step method for the IEP. Compared with other existing methods, the proposed method has higher convergence order and/or requires less operations. Under the assumption that the relative generalized Jacobian matrices at a solution are nonsingular, the proposed method is proved to be convergent with cubic root‐convergence rate. Numerical examples demonstrate the effectiveness of the improved method. [ABSTRACT FROM AUTHOR]

In this article, the inverse problems for the wave equation in a medium in which multiple types of cavities and inclusion exist in a mixture are considered. From the point of view of the indicator function of the enclosure method, there are two types of heterogeneous parts: “minus group” and “plus group.” For example, cavities with the Dirichlet boundary condition belong to the minus group, while inclusions with smaller propagation velocity belong to the plus group. The heterogeneous part of the minus group gives a negative sign to the indicator function, and the heterogeneous part of the plus group gives a positive sign. In general, the presence of many types of heterogeneous parts causes cancelation of the sign of the indicator function. Such cases are referred to as “mixed cases.” Here, we consider the case that the shortest length obtained from the indicator function is attained only by heterogeneous parts of the same group. This case is called the “mixed but separated case,” and it is shown that the method of elliptic estimates developed by Ikehata works well. We also show that the case of a two‐layered background medium with a flat layer can be considered in the same way as the case of a homogeneous background medium. [ABSTRACT FROM AUTHOR]

The study of the ultrasonic acoustic response of materials is of increasing interest for the characterisation of high-frequency applications. In this work, some elastic properties (acoustic velocity and acoustic attenuation) and some physical properties (thickness and density), together with the phase shift of the transducer, are estimated via one method. Scanning acoustic microscopy was performed via an innovative and different method using a high-frequency (50 MHz) planar wave transducer. In addition, the Debye series model (DSM) was used to model the acoustic response of the bilayer structure of the glass substrate and the resin‒epoxy sample. The inverse problem of estimating the acoustic parameters that results in a DSM reflected echo that matches the experimental signal is solved by proposing an innovative step-by-step curve fitting algorithm. The algorithm for solving the inverse problem is then optimised via relative mean square error (RMSE) curves, resulting in a robust method for assessing the uniqueness of the multiparameter solution. [ABSTRACT FROM AUTHOR]

Modeling contact between deformable solids is a fundamental problem in computer animation, mechanical design, and robotics. Existing methods based on C0‐discretizations—piece‐wise linear or polynomial surfaces—suffer from discontinuities and irregularities in tangential contact forces, which can significantly affect simulation outcomes and even prevent convergence. In this work, we show that these limitations can be overcome with a smooth surface representation based on Implicit Moving Least Squares (IMLS). In particular, we propose a self collision detection scheme tailored to IMLS surfaces that enables robust and efficient handling of challenging self contacts. Through a series of test cases, we show that our approach offers advantages over existing methods in terms of accuracy and robustness for both forward and inverse problems. [ABSTRACT FROM AUTHOR]

The reconstruction of MR images has always been a challenging inverse problem in medical imaging. Acceleration of MR scanning is of great importance for clinical research and cutting-edge applications. One of the primary efforts to achieve this is using compressed sensing (CS) theory. The CS aims to reconstruct MR images using a small number of sampled data in k-space. The CS-MRI techniques face challenges, including the potential loss of fine structure and increased computational complexity. We introduce a novel framework based on a regularized sparse recovery problem and a sharpening step to improve the CS-MRI approaches regarding fine structure loss under high acceleration factors. This problem is solved via the Half Quadratic Splitting (HQS) approach. The inverse problem for reconstructing MR images is converted into two distinct sub-problems, each of which can be solved separately. One key feature of the proposed approach is the replacement of one sub-problem with a denoiser. This regularization assists the optimization of the Smoothed (SL0) norm in escaping local minimums and enhances its precision. The proposed method consists of smoothing, feature modification, and Smoothed cost function optimization. The proposed approach improves the SL0 algorithm for MRI reconstruction without complicating it. The convergence of the proposed approach is illustrated analytically. The experimental results show an acceptable performance of the proposed method compared to the network-based approaches. [ABSTRACT FROM AUTHOR]

For an inverse coefficient problem of determining a state‐varying factor in the corresponding Hamiltonian for a mean field game system, we prove the global Lipschitz stability by spatial data of one component and interior data in an arbitrarily chosen subdomain over a time interval. The proof is based on Carleman estimates with different norms. [ABSTRACT FROM AUTHOR]

We consider an inverse problem of determining the coefficients of a fractional p -Laplace equation in the exterior domain. Assuming suitable local regularity of the coefficients in the exterior domain, we offer an explicit reconstruction formula in the region where the exterior measurements are performed. This formula is then used to establish a global uniqueness result for real-analytic coefficients. In addition, we also derive a stability estimate for the unique determination of the coefficients in the exterior measurement set. [ABSTRACT FROM AUTHOR]

In this article, the mathematical study of the problem of identifying the space-dependent source term, in transport processes given by an n -dimensional linear parabolic equation, from space-dependent noisy measurements taken at an arbitrary fixed time is conducted. The problem is analytically solved using Fourier techniques, and it is shown that this solution is not stable. Three families of uniparametric regularization operators are proposed to address the instability of the solution. Each of them is designed to compensate for the factor that causes the instability of the inverse operator. Additionally, a selection rule for the regularization parameter is included, and an error bound for each estimation of Hölder type is obtained. Numerical examples of different characteristics are presented to demonstrate the benefits of the proposed strategies. [ABSTRACT FROM AUTHOR]

The inverse eigenvalue problem studies the possible spectra among matrices whose off-diagonal entries have their zero-nonzero patterns described by the adjacency of a graph G. In this paper, we refer to the i -nullity pair of a matrix A as (null (A) , null (A (i)) , where A (i) is the matrix obtained from A by removing the i -th row and column. The inverse i -nullity pair problem is considered for complete graphs, cycles, and trees. The strong nullity interlacing property is introduced, and the corresponding supergraph lemma and decontraction lemma are developed as new tools for constructing matrices with a given nullity pair. [ABSTRACT FROM AUTHOR]

Given a pair of real symmetric matrices A , B ∈ R n × n with nonzero patterns determined by the edges of any pair of chosen graphs on n vertices, we consider an inverse eigenvalue problem for the structured matrix C = [ A B I O ] ∈ R 2 n × 2 n. We conjecture that C can attain any spectrum that is closed under conjugation. We use a structured Jacobian method to prove this conjecture for A and B of orders at most 4 or when the graph of A has a Hamilton path, and prove a weaker version of this conjecture for any pair of graphs with a restriction on the multiplicities of eigenvalues of C. [ABSTRACT FROM AUTHOR]

Physics informed neural network (PINN) demonstrates powerful capabilities in solving forward and inverse problems of nonlinear partial differential equations (NLPDEs) through combining data-driven and physical constraints. In this paper, two PINN methods that adopt tanh and sine as activation functions, respectively, are used to study data-driven solutions and parameter estimations of a family of high order KdV equations. Compared to the standard PINN with the tanh activation function, the PINN framework using the sine activation function can effectively learn the single soliton solution, double soliton solution, periodic traveling wave solution, and kink solution of the proposed equations with higher precision. The PINN framework using the sine activation function shows better performance in parameter estimation. In addition, the experiments show that the complexity of the equation influences the accuracy and efficiency of the PINN method. The outcomes of this study are poised to enhance the application of deep learning techniques in solving solutions and modeling of higher-order NLPDEs. [ABSTRACT FROM AUTHOR]

Recently, Migórski‐Dudek  investigated a steady Oseen flow for a generalized Newtonian incompressible fluid with unilateral and frictional‐type boundary conditions. They established an existence theorem for the steady Oseen model when the divergence‐free convection field b$\mathbf {b}$ and acting volume force f$\mathbf {f}$ are known. However, prior knowledge of b$\mathbf {b}$ and f$\mathbf {f}$ is often impossible in practical engineering applications. This leads to the question of whether the divergence‐free convection field b$\mathbf {b}$ and acting volume force f$\mathbf {f}$ can be determined. The main objective of this paper is to provide a positive response to this challenging and intriguing question. Specifically, we aim to formulate an inverse problem for identifying b$\mathbf {b}$ and f$\mathbf {f}$, characterized by a nonlinear and nonsmooth regularized optimization problem. Initially, we introduce a variational selection mapping for the steady Oseen model. We then establish the boundedness and generalized continuity of this variational selection. Finally, by leveraging optimization theory, convex analysis, and nonsmooth analysis, we develop an abstract regularization framework for the considered inverse problem and establish the existence of a solution.  [ABSTRACT FROM AUTHOR]

The initial-boundary and the inverse coefficient problems for the semilinear hyperbolic equation with strong damping are considered in this study. The conditions for the existence and uniqueness of solutions in Sobolev spaces to these problems have been established. The inverse problem involves determining the unknown time-dependent parameter in the right-hand side function of the equation using an additional integral type overdetermination condition. [ABSTRACT FROM AUTHOR]

This research introduces an accelerated training approach for Vanilla Physics-Informed Neural Networks (PINNs) that addresses three factors affecting the loss function: the initial weight state of the neural network, the ratio of domain to boundary points, and the loss weighting factor. The proposed method involves two phases. In the initial phase, a unique loss function is created using a subset of boundary conditions and partial differential equation terms. Furthermore, we introduce preprocessing procedures that aim to decrease the variance during initialization and choose domain points according to the initial weight state of various neural networks. The second phase resembles Vanilla-PINN training, but a portion of the random weights are substituted with weights from the first phase. This implies that the neural network's structure is designed to prioritize the boundary conditions, subsequently affecting the overall convergence. The study evaluates the method using three benchmarks: two-dimensional flow over a cylinder, an inverse problem of inlet velocity determination, and the Burger equation. Incorporating weights generated in the first training phase neutralizes imbalance effects. Notably, the proposed approach outperforms Vanilla-PINN in terms of speed, convergence likelihood and eliminates the need for hyperparameter tuning to balance the loss function. [ABSTRACT FROM AUTHOR]

We study the inverse problem for a differential equation of 2b$$ 2b $$‐order with the Caputo fractional derivative over time and Schwartz‐type distribution in its right‐hand side. The generalized solution of the Cauchy problem for such an equation, space‐dependent part of a source, and a time‐dependent reaction coefficient in the equation are unknown. We find sufficient conditions for unique local in time solvability of the inverse problem under time‐ and space‐integral overdetermination conditions. [ABSTRACT FROM AUTHOR]

Least‐squares reverse time migration is an increasingly popular technique for subsurface imaging, especially in the presence of complex geological structures. However, elastic least‐squares reverse time migration algorithms face significant practical and numerical challenges when migrating multi‐component seismic data acquired from irregular topography. Many associated issues can be avoided by abandoning the Cartesian coordinate system and migrating the data to a generalized topographic coordinate system conformal to surface topology. We introduce a generalized anisotropic elastic least‐squares reverse time migration methodology that uses the numerical solutions of tensorial elastodynamics for propagating wavefields in computational domains influenced by free‐surface topography. We define a coordinate mapping assuming unstretched vertically translated meshes that transform an irregular physical domain to a regular computational domain on which calculating numerical elastodynamics solutions is straightforward. This allows us to obtain numerical solutions of forward and adjoint elastodynamics and generate subsurface images directly in topographic coordinates using a tensorial energy‐norm imaging condition. Numerical examples demonstrate that the proposed generalized elastic least‐squares reverse time migration algorithm is suitable for generating high‐quality images with reduced artefacts and better balanced reflectivity that can accurately explain observed data acquired from topography in a medium characterized by arbitrary heterogeneity and anisotropy. Finally, the computational cost of our method is comparable to that of an equivalent Cartesian elastic least‐squares reverse time migration numerical implementation. [ABSTRACT FROM AUTHOR]

AbstractWe explore the possibility for using boundary data to identify sources in elliptic PDEs. Even though the associated forward operator has a large null space, it turns out that box constraints, combined with weighted sparsity regularization, can enable rather accurate recovery of sources with constant magnitude/strength. In addition, for sources with varying strength, the support of the inverse solution will be a subset of the support of the true source. We present both an analysis of the problem and a series of numerical experiments. Our work only addresses discretized problems. The reason for introducing the weighting procedure is that standard (unweighted) sparsity regularization fails to provide adequate results for the source identification task considered in this paper. This investigation is also motivated by applications, e.g. recovering mass distributions from measurements of gravitational fields and inverse scattering. We develop the methodology and the analysis in terms of Euclidean spaces, and our results can therefore be applied to many problems. For example, the results are equally applicable to models involving the screened Poisson equation as to models using the Helmholtz equation, with both large and small wave numbers. [ABSTRACT FROM AUTHOR]

We investigate models for nonlinear ultrasound propagation in soft biological tissue based on the one that serves as the core for the software package k‐Wave. The systems are solved for the acoustic particle velocity, mass density, and acoustic pressure and involve a fractional absorption operator. We first consider a system that incorporates additional viscosity in the equation for momentum conservation. By constructing a Galerkin approximation procedure, we prove the local existence of its solutions. In view of inverse problems arising from imaging tasks, the theory allows for the variable background mass density, speed of sound, and the nonlinearity parameter in the systems. Second, under stronger conditions on the data, we take the vanishing viscosity limit of the problem, thereby rigorously establishing the existence of solutions for the limiting system as well. [ABSTRACT FROM AUTHOR]

Inverse problems are commonly used in many fields as they enable the estimation of parameters that cannot be experimentally measured. However, the complex nature of inverse problems requires a strong background in data and signal processing. Moreover, ill-posed problems yield parameters that have a strong linear dependence on the problem. The ill-posed nature of these problems lead to many errors in numerical computations that can make parameter identification nearly impossible. In this paper, a new data processing tool is proposed to maximize the sensitivity of the model to the parameters of interest, while reducing the correlation between them. The effectiveness of the toll is demonstrated through a given inverse problem example using Periodically Pulsed Photothermal Radiometry (PPTR). [ABSTRACT FROM AUTHOR]

This study focuses on the optimization process of crash box design. The design optimization process is resource‐intensive and requires multiple dynamic simulations. Numerous design parameters can be varied to satisfy the crashworthiness objectives. Therefore an intelligent and robust machine learning (ML) framework has been developed. This framework can be employed to assist in the optimization of various crashworthiness components with different crashworthiness objectives. Here, a reinforcement learning‐based (RL) machine learning framework is developed. It consists of a finite element method (FEM) surrogate and RL environment. The FEM surrogate is trained using data from FEM simulations as well as synthetic data generated by a Generative Adversarial Network (GAN). An inverse problem is solved for optimizing the geometrical parameters of the boundary value problem while the RL is receiving input from the FEM surrogate. RL has proven to be accurate in many fields due to its ability to explore and exploit the learned dynamics. Conventional algorithms require numerous iterations and tuned functions to achieve appropriate results and need to be initialized after a slight change in the problem. Due to an optimization of input parameters in an inverse problem, RL is chosen for the present investigation. For the simulation data needed, the crash box is designed with linear first‐order accurate four nodal shell elements. Also, bilinear elastoplastic material along with geometrical nonlinearity is used to simulate the dynamic deformation process. The RL agents learn to vary the crash box design parameters and search for the optimal parameters. The optimal parameters are based on the user‐defined crashworthiness objectives. [ABSTRACT FROM AUTHOR]

In this paper, we prove an inverse resonance theorem for the half-solid with vanishing stresses on the surface via Weyl-Titchmarsh function. Using a semi-classical approach, it is possible to simplify this three-dimensional problem of the elastic wave equation for the half-solid as a Schrödinger equation with Robin boundary conditions on the half-line. The goal of the paper is to establish a method to recover the potential from eigenvalues and resonances through the Weyl-Titchmarsh function for non-self-adjoint problems and to establish a one-toone and onto map between suitable function spaces. Moreover, we produce an algorithm in order to retrieve the shear modulus from the eigenvalues and resonances, via the spectral data. [ABSTRACT FROM AUTHOR]

Objective: This study provided a quantitative prediction of best-corrected visual acuity (BCVA) for cataract patients using the inverse problem algorithm (IPA) technique earlier proposed by the authors. Methods: To this end, seven risk factors (age, BMI, MAP, IOP, HbA1c, LDL-C, and gender) were linked by a semi-empirical formula by normalizing each factor into a dimensionless range of −1.0 to +1.0. The adopted inverse problem algorithm (IPA) technique was run via a self-developed program in STATISTICA 7.0, featuring a 29-term nonlinear equation considering seven risk factors, cross-interaction between various pairs of factors, and one constant term [7 + (7 × 6)/2 + 1 = 29]. The IPA neglected quadratic, triple, or quadruple factors′ cross-interactions. This study used a dataset of 632 cataract patients to attain a reliable BCVA prediction with a variance of 0.929. A verification dataset of 160 patients with similar symptoms was used to verify this approach′s feasibility, reaching a good correlation with R2 = 0.909. Results: The verification group′s derived average AT (agreement) (9.12 ± 27.00%) indicated a slight deviation between the theoretical prediction and practical BCVA. The significant factors were age, body mass index (BMI), and intraocular pressure (IOP), whereas mean arterial pressure (MAP), hemoglobin A1c (HbA1c), low-density-lipoprotein cholesterol (LDL-C), and gender insignificantly contributed to BCVA. Conclusions: The proposed approach is instrumental in AI-assisted clinical diagnosis, yielding robust BCVA predictions for individual cataract patients based on their biological indices before the ophthalmological examination procedure. [ABSTRACT FROM AUTHOR]

For over 35 years of research, the debate about the systematic compositionality of neural networks remains unchanged, arguing that existing artificial neural networks are inadequate cognitive models. Recent advancements in deep learning have significantly shaped the landscape of popular domains, however, the systematic combination of previously trained neural networks remains an open challenge. This study presents how to dynamically synthesize a neural network for the design of broadband electromagnetic metasurfaces. The underlying mechanism relies on an assembly network to adaptively integrate pre‐trained inherited networks in a transparent manner that corresponds to the metasurface assembly in physical space. This framework is poised to curtail data requirements and augment network flexibility, promising heightened practical utility in complex composition‐based tasks. Importantly, the intricate coupling effects between different metasurface segments are accurately captured. The approach for two broadband metasurface inverse design problems is exemplified, reaching accuracies of 96.7% and 95.5%. Along the way, the importance of suitably formatting the spectral data is highlighted to capture sharp spectral features. This study marks a significant leap forward in inheriting pre‐existing knowledge in neural‐network‐based inverse design, improving its adaptability for applications involving dynamically evolving tasks. [ABSTRACT FROM AUTHOR]