This report extends our recent progress in tackling a challenging 3D inverse scattering problem governed by the Helmholtz equation. Our target application is to reconstruct dielectric constants, electric conductivities and shapes of front surfaces of objects buried very closely under the ground. These objects mimic explosives, like, e.g. antipersonnel land mines and improvised explosive devices. We

New method of scanning tomography of subsurface dielectric inhomogeneities by bistatic measurements of 2D distributions of the scattered signal with the variable source-receiver offset (base) is considered. This method is based on the solution of the corresponding inverse scattering problem – retrieval of 3D distribution of complex permittivity of inhomogeneities from the solution of 3D integral equation

In this paper, a non-linear parameter identification method for structures is presented, whereby the instantaneous power flow balance of the substructure of interest is enforced. The time-domain power flow into the non-linear substructure is balanced against the power transmitted to adjacent structures, damping and kinetic and strain energies. Enforcing this condition of matching the net power balance

Inverse estimation of the temperature/phase change-dependent refractive index of semitransparent media is studied in present work. The direct models (including the coupled radiation-conduction heat transfer and coupled radiation-phase change heat transfer exposed to the laser irradiation) are solved by the finite volume method. The inverse model adopts stochastic particle swarm optimization (SPSO)

An inverse source problem for an n-dimensional heat equation with a time-varying coefficient is investigated. The spatially dependent component of a source function is determined from a measurement at the final time. The inverse problem is regularized by a mollification method. Hölder-type stability estimates are proved. Error estimates of Hölder type are also proved for regularized solutions for both

Inverse problems are of importance in many fields of science and engineering. Electromagnetic inversion deals with estimating information contained in electromagnetic measurements. The inversion scheme needs to be designed properly to compensate for Gibbs oscillations effects in the solution, and hence give better validation for the estimated quantities. In this paper an inversion methodology based

The main mathematical results on the data processing in vibrational spectroscopy are presented. The approaches and algorithms proposed for molecular force field calculations have been constructed on a base of regularizing methods for solving nonlinear ill-posed problems and have been implemented in the software package SPECTRUM. These algorithms were used for constructing the regularized ab initio

We consider the problem of guaranteed estimation of unknown right-hand sides of the equations entering the statement of the exterior Neumann problems for the Helmholtz equation from indirect observations of their solutions. A method is developed for the determination of guaranteed a posteriori estimates of this right-hand sides which are compatible with measurement data. It is shown that such estimates

Support vector machine (SVM) is regarded as one of the most effective techniques for supervised learning, while the Gaussian kernel SVM is widely utilized due to its excellent performance capabilities. To ensure high performance of models, hyperparameters, i.e. kernel width and penalty factor must be determined appropriately. This paper studies the influence of hyperparameters on the Gaussian kernel

This paper proposes a new regularization technique using spectral graph wavelet for backward heat conduction problem (BHCP) on the graph. The method uses the fourth-order compact difference for an approximation of differential operators. Meanwhile, the error estimate between the exact and wavelet regularized solution is derived. Adaptive node arrangement is obtained using spectral graph wavelet. The

In this paper, we propose a novel reconstruction scheme for the low-frequency near-field electromagnetic imaging of high-contrast conductivity distributions inside shielded regions using the system of Maxwell's equations in 3D. In our novel scheme, we focus on estimating the shape characteristics of the electrical conductivity profile inside these regions from low-frequency electromagnetic data measured

In this paper, we consider the inverse problem of recovering an isotropic elastic tensor from the Neumann-to-Dirichlet map. To this end, we prove a Lipschitz stability estimate for Lamé parameters with certain regularity assumptions. In addition, we assume that the Lamé parameters belong to a known finite subspace with a priori known bounds and that they fulfil a monotonicity property. The proof relies

In previous work the author presented a novel method whereby relative values of the specific heat capacity and thermal conductivity of a composite rod may be accurately estimated by analyzing oscillations present within eigenvalue spectra. This work described how the solotone effect may be utilized to solve a parameter estimation inverse problem. In this paper, we show how the solotone inverse method

The knowledge of the variation of arc current is helpful in improving the breaking performance of a vacuum circuit breaker. As a novel non-intrusive method, arc magnetic performance testing technology has attracted attention. In view of the widely used regularization method of the standard form in arc current reconstructions cannot fully overcome the ill-posed nature of the Biot-Savart equation, the

In this paper we consider the identification of the initial condition for a distributed order diffusion equation. We first prove the unique existence and the regularity properties of the strong solution on the bounded temporal-spacial domain based on the eigenfunction expansions. The ill-posedness of the backward problem is interpreted by the compactness of the observation operator. Next the Laplace

New exact solutions for unidirectional unsteady flows of incompressible viscous fluids with linear dependence of viscosity on the pressure between two infinite horizontal parallel plates are established when the lower plate is moving in its plane with an arbitrary time-dependent velocity. In addition to being useful solutions to idealizations of technologically relevant problems such exact solutions

Seismic inversion is an effective way to get stress distribution of rock in coal mine, but its usage is still limited by multi-solution problem in practice. As seismometers can only be deployed in limited areas in underground coal mine, it is impossible to solve multi-solution problem by adding more seismometers, thus, we propose a new seismic inversion method which introduces stress value changes

We developed a method of inverse problem solving in semiconductor photoacoustics based on neural networks application. Simple structured neural networks, trained on a large set of data obtained by the well – known theoretical models in the 20 Hz–20 kHz modulation frequency range, are applied to determine thermal diffusivity, coefficient of linear expansion and thickness of n – type silicon samples

In this paper we solve numerically some geometric inverse problems for linear and semilinear elliptic equations with the method of fundamental solutions (MFS). More precisely, we focus our work on the numerical reconstruction of the unknown domain where the equation holds. We first consider two dimensional problem and we present several numerical experiments with different type of geo metries to be

The possibilities of using asymptotic analysis for solving the inverse problem of restoring the parameters of the source of nitrogen oxide industrial emissions into the atmosphere are demonstrated. The asymptotic analysis allows to reduce the subproblem for a three-dimensional singularly perturbed equation of the reaction–diffusion–advection type to a much simpler problem for numerical solution. This

In this paper, the acoustic full waveform inversion based on the wavelet method is proposed. It allows to determine density and velocity parameters simultaneously in the time domain. The forward problem is solved based on the wavelet method. The numerical schemes for modelling are constructed in detail. The stability condition is derived for the first time. The inversion is an optimization process

In this work, the adjoint to the transport operator is used to estimate the spatial distribution of an energy dependent neutral particles source in a homogeneous one-dimensional medium, from measurements of internal detectors. An analytical discrete ordinates formulation, the ADO method, is applied to derive a spatially explicit solution for the adjoint flux. A Bayesian approach is considered to relate

An approach is developed to read and simulate experimental data using inverse problems. The idea is founded on a locally sequential refinement of the reconstruction. The regularization that occurs under a scheme of separate matching with observations is the basis for a numerical solution. The simultaneous identification of several unknowns using a set of mathematical models and functional representations

For solving the inverse Cauchy problem of a three-dimensional (3D) non-homogeneous sideways heat equation in a cuboid, we first transform it into a partial homogeneous boundary value problem, using a time-dependent two-dimensional (2D) homogenization technique. After a time step-by-step approach of the inverse Cauchy problem, it becomes an inverse Cauchy problem of the 3D modified Helmholtz equation

The studies on inverse problems exist extensively in aerospace, mechanical, identification, detection, scanning imaging and other fields. Its ill-posed characteristics often lead to large oscillations in the solution of the inverse problem. In this study, the truncated generalized singular value decomposition (TGSVD) method is introduced to identify two kinds of moving forces, single and multi-axial

An inverse source problem related to the Poisson equation is the main concern of this work. Specifically, we deal with the reconstruction of a mass distribution in a geometrical domain from a partial boundary measurement of the associated potential. The considered problem is motivated by various applications such as the identification of geological anomalies underneath the Earth's surface. The proposed

In this paper, we deal with a nonlinear inverse problem for recovering a time-dependent potential term in a time fractional diffusion equation from an additional measurement in the form of integral over the space domain. By using the fixed point theorem, the existence, uniqueness, regularity and stability of the direct problem are proved. The uniqueness of the inverse problem is proved by the property

Inverse nodal problems for the Sturm–Liouville operator are to reconstruct this operator from the given nodal points(zeros) of its eigenfunctions. In this paper, the potential q ( x ) up to its mean value on the whole interval is uniquely determined by two adjacent twin-dense nodal subsets on arbitrarily-half subintervals corresponding to eigenvalues with various boundary conditions and an example

The parameterized generalized inverse eigenvalue problems containing multiplicative and additive inverse eigenvalue problems appear in vibrating systems design, structural design, and inverse Sturm–Liouville problems. In this article, by using the Cholesky factorization and the Jacobi method, we propose two efficient algorithms based on Newton's method and the L-BFGS-B method for solving these problems

This work is focused on the detection of interlayer damages in multi-layer composite materials, which use the inverse analysis approach based on the reduced-order models (ROMs). Direct problems including heat conduction through the solid and heat radiation to the space are solved by finite element method (FEM), and the solutions of transient nonlinear analysis are collected to build the snapshot data

Imaging defects in austenitic welds presents a significant challenge for the ultrasonic non-destructive testing community. Due to the heating process during their manufacture, a dendritic structure develops, exhibiting large grains with locally anisotropic properties which cause the ultrasonic waves to scatter and refract. When basic imaging algorithms, which typically make constant wave speed assumptions

It is well known that the problem of numerical differentiation is an ill-posed problem and one requires regularization methods to approximate the solution. The commonly practiced regularization methods are (external) parameter-based like Tikhonov regularization, which has certain inherent difficulties associated with them. In such scenarios, iterative regularization methods serve as an attractive alternative

In this paper we study inverse problems of electric conductivity that arise in the design of spherical shielding or cloaking shells and other functional devices used to control static electric fields. The shells are considered consisting of a finite number of layers filled with homogeneous isotropic or anisotropic medium. The inverse problems under study are reduced to control problems with the layers

The goal of this study is to develop a tool that will evaluate the moisture content gradient in the depth of reinforced concretes in the context of building safety. The monitoring of ageing phenomena, such as the corrosion of steel reinforcements, depends on the amount of water and on its gradient as they are directly related to the dielectric values of the concrete. In this paper, to obtain these

The paper deals with the identification from some pumping tests of unknown storativity and transmissivity functions defining a 2 D confined aquifer. We introduce an appropriate change of variables that transforms the groundwater equation into a diffusion-reaction one wherein the diffusion term is defined by the fraction transmissivity/storativity and the reaction term yields the right hand side of

A method for the path synthesis of a planar five-bar mechanism based on wavelet analysis theory is proposed in this paper. Wavelet transform analysis is used to extract wavelet characteristic parameters(WCPs) from the basic dimensional types(BDTs) database. Combined the numerical atlas method with multidimensional search tree, a four dimensions search tree with a depth of five is obtained, in which

This paper is devoted to investigating a two-dimensional inverse anomalous diffusion problem. The missing solely time-dependent Dirichlet boundary condition is recovered by imposing an additional integral measurement over the domain. An efficient computational technique based on a combination of a time integration scheme and local meshless Petrov–Galerkin method is implemented to solve the governing

(2020). Celebration of 90th birthday of James Vere Beck. Inverse Problems in Science and Engineering: Vol. 28, No. 5, pp. 599-600.

In this paper, the bi-Helmholtz equation with Cauchy conditions is nominated in a n-dimensional strip domain. It is shown that this Cauchy problem may be ill-posed in the sense of Hadamard. In order to overcome the ill-posedness, a suitable regularization method has to be applied to the Cauchy problem. Hence, two well-known wavelet and Fourier regularization methods are used for solving this ill-posed

We discuss inverse problems to finding the time-dependent coefficient for the multidimensional Cauchy problems for both strictly hyperbolic equations and polyharmonic heat equations. We also extend our techniques to the general inverse Cauchy problems for evolution equations.

Electrical impedance tomography (EIT) is a boundary measurement inverse technique targeting reconstruction of the conductivity distribution of the interior of a physical body based on boundary measurement data. Typically, the measured data are uncertain because of various error sources; thus, there are many uncertainties in the reconstructed image. This study attempts to quantify the effects of these

In this paper, the dynamic reliability and global sensitivity analysis method of double random vibration system is proposed, considering structure uncertainty and excitation uncertainty of complex mechanical and structural systems under random load. Based on the Miner criterion of fatigue failure of structures, the dynamic responses of structures under the condition of double random vibration are analyzed

In this paper, we consider the two-dimensional Fredholm integral equation of the first kind. The kernel of this equation models the scattering of a laser beam by spheroids having the same fixed orientation. The unknown function under the integral describes the distribution of spheroids along two semi-axes. The input data are the diffraction pattern corresponding to the scattering of the laser beam

In this paper, we combine the maximum entropy principle with binomial tree to construct a non-recombining entropy binomial tree pricing model under the volatility that is a function of time, and give the rate of convergence. The model may yield an unbiased and objective probability. In addition, we research the calibration problem of volatility with the entropy binomial tree, and adopt an exterior

We consider the problem of computing the initial condition for a general parabolic equation from the Cauchy lateral data. The stability of this problem is well-known to be logarithmic. In this paper, we introduce an approximate model, as a coupled linear system of elliptic partial differential equations. Solution to this model is the vector of Fourier coefficients of the solutions to the parabolic

Ultrasound Tomography (UT) is primarily used for the detection of malignant tissue in the human breast. However, the reconstruction algorithms used for UT require large computational time and are based upon solving a nonlinear, ill-posed inverse problem. We constructed and solved the inverse scattering problem from UT using the Distorted Born Iterative method. Since this problem is ill-posed, this

In this paper, we have developed the spectral theory for a conformable fractional Sturm-Liouville problem Lα(q(x),h,H) with boundary conditions which include conformable fractional derivatives of order α, 0<α⩽1, and prove a completeness theorem and an expansion theorem. We obtain the canonical infinite product constructed by the zero set of the characteristic function of Lα. Also, we calculate the

The Dirac system with an integral delay in the form of convolution is considered. We suppose that the convolution kernel, having four independent components, is known a priori on a part of the interval. A partial inverse problem is studied, which consists in recovering the kernel on the remaining part of the interval from some fractional parts of two spectra. Uniqueness theorems are proved, and a constructive

This study newly demonstrates a compressed sensing (CS)-based reconstruction scheme for computed tomography (CT) with a translational trajectory. CT with a translational trajectory has many potential applications in the field of non-destructive testing (NDT) and medicine. For example, in the X-ray inspection of a large civil engineering structure, the movements of the source and detector are strictly

This paper deals with the reconstruction of small-amplitude perturbations in the electric properties (permittivity and conductivity) of a medium from boundary measurements of the electric field at a fixed frequency. The underlying model is the three-dimensional time-harmonic Maxwell equations in the electric field. Sensitivity analysis with respect to the parameters is performed, and explicit relations

An inverse microtomography problem is under consideration in a class of functions with bounded V H variation. An algorithm for solving this problem is proposed based on Tikhonov's regularization with a special regularizer. The algorithm ensures piecewise uniform convergence of approximate solutions to exact solution of the inverse problem. In addition, the question of a-posteriori error estimate of

An inverse spectral problem is studied for the matrix Sturm–Liouville operator on a finite interval with the general self-adjoint boundary condition. We obtain a constructive solution based on the method of spectral mappings for the considered inverse problem. The nonlinear inverse problem is reduced to a linear equation in a special Banach space of infinite matrix sequences. In addition, we apply

The monotonicity method for the inverse acoustic scattering problem is to understand the inclusion relation between an unknown object and artificial one by comparing the far field operator with the artificial operator. This paper introduces the development of this method to the inverse crack scattering problem. Our aim is to give the following two indicators: One (Theorem 1.1) is to determine whether

The AISI 304 stainless steel has largely been used in industry owing to its good formability characteristics. Therefore, the use of proper material parameters is essential for a successful numerical simulation of forming operations. Within this framework, this work addresses a parameter identification strategy using strategies based on multiple mechanical tests – a multi-test optimization scheme. The

This paper concerns a one-phase inverse Stefan problem in one-dimensional space. The problem is ill-posed in the sense that the solution does not depend continuously on the data. We also consider noisy data for this problem. As such, we first regularize the proposed problem by the discrete mollification method. We apply the integration matrix using Bernstein basis polynomials for the discrete mollification

Tikhonov regularization is an usual method to solve an ill-posed problem, recommended when the input data are contaminated with noise. However, in some cases, the use of this technique is not sufficient to provide good solutions. In this work, an improvement of the Tikhonov regularization method was proposed and tested in the inverse black body radiation problem. The method proposed consisted in including

Several alternatives have been proposed to model the radiation profile emitted by flames. Among them there is the weighted-multi-point-source-model (WMP), which is able to provide good predictions for medium and far distances. Despite being a simple model, the determination of its parameters can be difficult, which creates the opportunity to apply optimisation methods in order to find the best answer

The unloaded configuration of a body without residual stresses is generally used as the starting point to compute the displacement field but may be unknown for some applications. In this paper, we propose a novel inverse formulation to identify the unloaded configuration of a deformed hyperelastic body using finite element based discretization schemes. The inverse problem consists of finding a domain

An algorithm for the reconstruction of a symmetric matrix from the eigenvalues of finite dimensional perturbations is given. The sufficient condition of determining the finite symmetric matrices is derived.

The scalar problem of reconstruction of an unknown refractive index of an inhomogeneous solid is considered. The original boundary value problem for the Helmholtz equation with an unknown refractive index is reduced to the source-type integral equation. The solution to the inverse problem is obtained in two steps. First, the integral equation of the first kind is solved in the inhomogeneity domain