Methods For Inverse Problems Research Articles

Convergence of a series associated with the convexification method for coefficient inverse problems

Abstract This paper is concerned with the convergence of a series associated with a certain version of the convexification method. That version has been recently developed by the research group of the first author for solving coefficient inverse problems. The convexification method aims to construct a globally convex Tikhonov-like functional with a Carleman weight function in it. In the previous works, the construction of the strictly convex weighted Tikhonov-like functional assumes a truncated Fourier series (i.e. a finite series instead of an infinite one) for a function generated by the total wave field. In this paper, we prove a convergence property for this truncated Fourier series approximation. More precisely, we show that the residual of the approximate PDE obtained by using the truncated Fourier series tends to zero in L 2 {L^{2}} as the truncation index in the truncated Fourier series tends to infinity. The proof relies on a convergence result in the H 1 {H^{1}} -norm for a sequence of L 2 {L^{2}} -orthogonal projections on finite-dimensional subspaces spanned by elements of a special Fourier basis. However, due to the ill-posed nature of coefficient inverse problems, we cannot prove that the solution of that approximate PDE, which results from the minimization of that Tikhonov-like functional, converges to the correct solution.

Inverse integer optimization with an imperfect observation

This study develops and evaluates methods for inverse integer optimization problems with an imperfect observation where the unknown parameters are the cost coefficients. We propose a cutting plane algorithm for this problem and compare it to a heuristic which solves the inverse of the linear relaxation of the forward problem. We then propose a hybrid approach that initializes the cutting plane algorithm from the solution of the heuristic.

A Conjugate Gradient Method for Inverse Problems of Non-linear Coupled Diffusion Equations

In many fields there are many problems called inverse problems, which infer the reasons from the observations. The inverse problem of nonlinear diffusion equations plays a crucial role in the numerical simulation of reservoirs. This article constructs a conjugate gradient method to solve the inverse problem of a nonlinear diffusion equation within oil reservoir simulation. The numerical simulation is performed, and the results show the effectiveness of the method.

A Tutorial Introduction to Inverse Problems in Magnetic Resonance.

There has been a tremendous increase in applications of the inverse problem framework to parameter estimation in magnetic resonance. Attempting to capture both the basics of this formalism and modern developments would require an article of inordinate length. Therefore, in the following, we provide basic material as a practical introduction to the topic and an entree to the literature. First, we describe the formulation of linear and nonlinear inverse problems, with an emphasis on signal equations arising in magnetic resonance. We then describe the Fredholm equation of the first kind as a paradigm for these problems. This is followed by much more detailed considerations for determining solutions in the linear case, including central concepts such as condition number, regularization, and stability. Solution methods for nonlinear inverse problems are described next, followed by a treatment of their stability and regularization. Finally, we provide an introduction to compressed sensing, with signal reconstruction formulated as the solution to an inverse problem, making use of much of the previous material. Throughout, the emphasis is on outlines of the theory and on numerical examples, rather than on mathematical rigor and completeness.

Convexification and experimental data for a 3D inverse scattering problem with the moving point source

Inverse scattering problems of the reconstructions of physical properties of a medium from boundary measurements are substantially challenging ones. This work aims to verify the performance on experimentally collected data of a newly developed convexification method for a 3D coefficient inverse problem for the case of unknown objects buried in a sandbox. The measured backscatter data are generated by a point source moving along an interval of a straight line and the frequency is fixed. Using a special Fourier basis, the method of this work strongly relies on a new derivation of a boundary value problem for a system of coupled quasilinear elliptic equations. This problem, in turn, is solved via the minimization of a Tikhonov-like functional weighted by a Carleman weight function. Different from the continuous case, our weighted cost functional in the partial finite difference does not need the penalty term to gain the global convergence analysis. The numerical verification is performed using experimental data, which are raw backscatter data of the electric field. These data were collected using a microwave scattering facility at The University of North Carolina at Charlotte.

Convexification for an inverse parabolic problem

A convexification-based numerical method for a coefficient inverse problem for a parabolic PDE is presented. The key element of this method is the presence of the so-called Carleman weight function in the numerical scheme. Convergence analysis ensures the global convergence of this method, as opposed to the local convergence of the conventional least squares minimization techniques. Numerical results demonstrate a good performance.

The Lagrangian and Hamiltonian for the Two-Dimensional Mathews-Lakshmanan Oscillator

The purpose of this paper is to illustrate the theory and methods of analytical mechanics that can be effectively applied to the research of some nonlinear nonconservative systems through the case study of two-dimensionally coupled Mathews-Lakshmanan oscillator (abbreviated as M-L oscillator). (1) According to the inverse problem method of Lagrangian mechanics, the Lagrangian and Hamiltonian function in the form of rectangular coordinates of the two-dimensional M-L oscillator is directly constructed from an integral of the two-dimensional M-L oscillators. (2) The Lagrange and Hamiltonian function in the form of polar coordinate was rewritten by using coordinate transformation. (3) By introducing the vector form variables, the two-dimensional M-L oscillator motion differential equation, the first integral, and the Lagrange function are written. Therefore, the two-dimensional M-L oscillator is directly extended to the three-dimensional case, and it is proved that the three-dimensional M-L oscillator can be reduced to the two-dimensional case. (4) The two direct integration methods were provided to solve the two-dimensional M-L oscillator by using polar coordinate Lagrangian and pointed out that the one-dimensional M-L oscillator is a special case of the two-dimensional M-L oscillator.

On the convergence of the Laplace approximation and noise-level-robustness of Laplace-based Monte Carlo methods for Bayesian inverse problems

The Bayesian approach to inverse problems provides a rigorous framework for the incorporation and quantification of uncertainties in measurements, parameters and models. We are interested in designing numerical methods which are robust w.r.t. the size of the observational noise, i.e., methods which behave well in case of concentrated posterior measures. The concentration of the posterior is a highly desirable situation in practice, since it relates to informative or large data. However, it can pose a computational challenge for numerical methods based on the prior measure. We propose to employ the Laplace approximation of the posterior as the base measure for numerical integration in this context. The Laplace approximation is a Gaussian measure centered at the maximum a-posteriori estimate and with covariance matrix depending on the logposterior density. We discuss convergence results of the Laplace approximation in terms of the Hellinger distance and analyze the efficiency of Monte Carlo methods based on it. In particular, we show that Laplace-based importance sampling and Laplace-based quasi-Monte-Carlo methods are robust w.r.t. the concentration of the posterior for large classes of posterior distributions and integrands whereas prior-based importance sampling and plain quasi-Monte Carlo are not. Numerical experiments are presented to illustrate the theoretical findings.

Inverse Problem Method to Optimize Cascade Heat Exchange Network in Central Heating System

In central heating systems, heat is often transferred from heat source to users by the heat network where several heat exchangers are installed at the heat source, substations, and terminals. In this article, the mathematical optimization model of the cascade heat exchange network with three-stage heat exchangers in a series is established. The aim is to maximize the cold fluid temperature for a given hot fluid temperature and overall heating capacity, the optimal heat exchange area distribution, and the medium fluid flow rates are determined through an inverse problem and variation method. The results show that the heat exchange areas should be distributed equally for each heat exchanger. It also indicates that in order to improve the thermal performance of the whole system, more heat exchange areas should be allocated to the heat exchanger where flow rate difference between two fluids is relatively small.

Learning performance in inverse Ising problems with sparse teacher couplings

We investigate the learning performance of the pseudolikelihood maximization method for inverse Ising problems. In the teacher–student scenario under the assumption that the teacher’s couplings are sparse and the student does not know the graphical structure, the learning curve and order parameters are assessed in the typical case using the replica and cavity methods from statistical mechanics. Our formulation is also applicable to a certain class of cost functions having locality; the standard likelihood does not belong to that class. The derived analytical formulas indicate that the perfect inference of the presence/absence of the teacher’s couplings is possible in the thermodynamic limit taking the number of spins N as infinity while keeping the dataset size M proportional to N, as long as α = M/N > 2. Meanwhile, the formulas also show that the estimated coupling values corresponding to the truly existing ones in the teacher tend to be overestimated in the absolute value, manifesting the presence of estimation bias. These results are considered to be exact in the thermodynamic limit on locally tree-like networks, such as the regular random or Erdős–Rényi graphs. Numerical simulation results fully support the theoretical predictions. Additional biases in the estimators on loopy graphs are also discussed.

Problem of Determination of Inhomogeneity Parameters of Dielectric Body by Measurement of the Near Field

The article is devoted to the development of a method for solving the inverse problem. We considered the problem of propagation of electromagnetic wave inside the body located in free space. The method of reconstruction of the parameters inhomogeneity from a known field is applied. The numerical results of the inhomogeneity bodies is presented.

Определение теплофизических свойств керамики с добавкой пиролизного остатка твердых отходов

A promising environmentally friendly method for processing municipal solid waste is pyrolysis. Pyrolysis allows obtaining a combination of solid, liquid and gaseous products in various proportions due to changes in process parameters. A promising area for the use of solid pyrolysis products is the production of various materials for industrial, civil and road construction. There are some known methods for producing cement, concrete and foam glass with improved properties on pyrolysis basis. In modern building technologies, ceramic materials are widely used. Their production requires various expensive additives such as quartz sand, dolomite, coal, coal mining rocks, etc. Thus, it seems relevant to use the residue of the pyrolysis of municipal solid waste as a free of charge structure-forming additive for the production of ceramic material. To determine the scope of the material application and to perform subsequent thermotechnical calculations, the data on its thermophysical characteristics is required. To determine the thermophysical characteristics of the ceramic material, the experimental method of the cylindrical layer and the calculation-experimental method of inverse heat conduction problem are used. The experimental studies of thermophysical properties of the new ceramic material have allowed obtaining the dependence of the average specific heat capacity and thermal diffusivity of the ceramic material on temperatures in the range of 30–90 оС, and the coefficient of thermal conductivity on the temperatures in the range of 30–500 оС. These characteristics enable to perform calculations of thermal processes that occur in the material during its industrial use. The coefficient of thermal conductivity of the material varies from 0,3 to 0,65 W/m×K, thermal diffusivity varies within (1,8–2,3)×10–7 m2/s, and the average specific heat capacity changes from 780 to 1300 J/kg×K, which is comparable with the thermophysical characteristics of known ceramic materials. It is advisable to use the pyrolysis residue of municipal solid waste as a free raw material instead of expensive additives to obtain ceramic material. It is recommended to use the new ceramic material in the production technology of building products for various purposes.

Three-dimensional magnetotelluric inversion using L-BFGS

The gradient-based optimization methods are preferable for the large-scale three-dimensional (3D) magnetotelluric (MT) inverse problem. Compared with the popular nonlinear conjugate gradient (NLCG) method, however, the limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) method is less adopted. This paper aims to implement a L-BFGS-based inversion algorithm for the 3D MT problem. And we develop our code on top of the ModEM package, which is highly extensible and popular among the MT community. To accelerate the convergence speed, the preconditioning technique by the affine linear transformation of the original model parameters is used. Two modifications of the conventional L-BFGS algorithm are also made to get a comparable convergence rate with the NLCG method. The impacts of the preconditioner parameters, the regularization parameters, the starting model, etc., on the inversion are evaluated by synthetic examples for both L-BFGS and NLCG methods. And the real MT Kayabe dataset is also inverted by the inversion algorithms. The synthetic tests show that through our L-BFGS inversion algorithm the similar resistivity models can be obtained with that from the NLCG method. For the real data inversion, the L-BFGS method performs more efficiently and reasonable results could be obtained by less iterations of the inversion process than the NLCG method. Thus, we suggest the common usage of the L-BFGS method for the 3D MT inverse problem.

Идентификация режима работы пропеллера беспилотного летательного аппарата по акустическому сигналу

The paper deals with a special case of identifying the operation mode of an unmanned aircraft (UAV) engine by an acoustic signal (provided that a known type of propellers is used), the main frequency extraction from the signal, and a numerical experiment. A full-scale experiment was performed with the recording of the acoustic signal of a real quadrocopter. The signal is analyzed by the method of frequency filters and Fourier transform. A numerical experiment was performed on a scanned model of a UAV propeller, rotating at a frequency obtained from an acoustic signal. The experiment involved a hybrid finite-volume method on the mesh of the finite element method of the ANSYS CFX simulation environment. The correspondence of the required lifting force with the permissible error is revealed, and as a result, the ability to identify the engine operating mode by an acoustic signal is shown. The field measurements of the acoustic signal are compared with the values recovered from a numerical experiment. A significant amount of data has been obtained suitable for use in the inverse problem and semi-analytic methods.

A novel homogenization function method for inverse source problem of nonlinear time-fractional wave equation

A novel numerical technique is developed in this paper to accurately and efficiently resolve the inverse source problem of the nonlinear time-fractional wave equation. Based on all given conditions, the homogenization function of nonlinear time-fractional wave equation can be derived, and then a family of homogenization functions is obtained. Furthermore, a numerical model is established by the superposition of homogenization functions and used for tackling inverse source problem. The proposed method is free of mesh generation, numerical integration, iteration, regularization and fundamental solutions, and it is easy to program and implement on the existing software. Three numerical experiments demonstrate the accuracy and convergence of the proposed strategy for the inverse source problem even with high noise imposed on the boundary conditions.

Multiplicatively Regularized Iterative Updated Background Inversion Method for Inverse Scattering Problems

A multiplicatively regularized iterative updated background method (MR-IUBM) based on difference integral equations is proposed to solve the electromagnetic inverse scattering problems (ISPs) involving both inhomogeneous and homogeneous background media. This aims to improve the ability for solving highly nonlinear ISPs with source-typed inversion solver by consistently updating the inhomogeneous background media from the updated unknown scatterers via the use of the Green's function with homogeneous medium. In virtue of MR into the cost function, the stability and the capability of tackling highly nonlinear ISPs can be further greatly improved. Synthetic and experimental results are presented to illustrate the efficiency of this method.

Regularized ensemble Kalman methods for inverse problems

Inverse problems are common and important in many applications in computational physics but are inherently ill-posed with many possible model parameters resulting in satisfactory results in the observation space. When solving the inverse problem with adjoint-based optimization, the problem can be regularized by adding additional constraints in the cost function. However, similar regularizations have not been used in ensemble-based methods, where the same optimization is done implicitly through the analysis step rather than through explicit minimization of the cost function. Ensemble-based methods, and in particular ensemble Kalman methods, have gained popularity in practice where physics models typically do not have readily available adjoint capabilities. While the model outputs can be improved by incorporating observations using these methods, the lack of regularization means the inference of the model parameters remains ill-posed. Here we propose a regularized ensemble Kalman method capable of enforcing regularization constraints. Specifically, we derive a modified analysis scheme that implicitly minimizes a cost function with generalized constraints. We demonstrate the method's ability to regularize the inverse problem with three cases of increasing complexity, starting with inferring scalar model parameters. As a final case, we utilize the proposed method to infer the closure field in the Reynolds-averaged Navier–Stokes equations, a problem of significant importance in fluid dynamics and many engineering applications.

Numerical Modeling of Heat and Mass Transport with Inner Heat Exchange in Unsaturated Porous Media

We are focused to the numerical modelling of heat, contaminant and water transport in unsaturated porous media in 3D. The heat exchange between water and porous media matrix is taken into the account. The determination of heat energy transmission coefficient and matrix heat conductivity is solved by means of inverse problem methods. The mathematical model represents the conservation of heat, contaminant and water mass balance. It is expressed by coupled non-linear system of parabolic-elliptic equations. Mathematical model for water transport in unsaturated porous media is represented by Richard's type equation. Heat transport by water includes water flux, molecular diffusion and dispersion. A successful experiment scenario is suggested to determine the required parameters including heat transmission and matrix heat conductivity coefficients. Additionally we investigate contaminant transport with heat transmission and contaminant adsorption. The obtained experiments support our method suitable for solution of direct and inverse problems. This problem we have discussed previously in 1D model, but preferential streamlines in 1D thin tubes shadow accurate results in determination of required parameters. In our presented setting we consider a cylindrical sample which is suitable in laboratory experiments for inverse problems.

Sampled limited memory methods for massive linear inverse problems

In many modern imaging applications the desire to reconstruct high resolution images, coupled with the abundance of data from acquisition using ultra-fast detectors, have led to new challenges in image reconstruction. A main challenge is that the resulting linear inverse problems are massive. The size of the forward model matrix exceeds the storage capabilities of computer memory, or the observational dataset is enormous and not available all at once. Row-action methods that iterate over samples of rows can be used to approximate the solution while avoiding memory and data availability constraints. However, their overall convergence can be slow. In this paper, we introduce a sampled limited memory row-action method for linear least squares problems, where an approximation of the global curvature of the underlying least squares problem is used to speed up the initial convergence and to improve the accuracy of iterates. We show that this limited memory method is a generalization of the damped block Kaczmarz method, and we prove linear convergence of the expectation of the iterates and of the error norm up to a convergence horizon. Numerical experiments demonstrate the benefits of these sampled limited memory row-action methods for massive 2D and 3D inverse problems in tomography applications.

Tsallis Entropy, Likelihood, and the Robust Seismic Inversion.

The nonextensive statistical mechanics proposed by Tsallis have been successfully used to model and analyze many complex phenomena. Here, we study the role of the generalized Tsallis statistics on the inverse problem theory. Most inverse problems are formulated as an optimisation problem that aims to estimate the physical parameters of a system from indirect and partial observations. In the conventional approach, the misfit function that is to be minimized is based on the least-squares distance between the observed data and the modelled data (residuals or errors), in which the residuals are assumed to follow a Gaussian distribution. However, in many real situations, the error is typically non-Gaussian, and therefore this technique tends to fail. This problem has motivated us to study misfit functions based on non-Gaussian statistics. In this work, we derive a misfit function based on the q-Gaussian distribution associated with the maximum entropy principle in the Tsallis formalism. We tested our method in a typical geophysical data inverse problem, called post-stack inversion (PSI), in which the physical parameters to be estimated are the Earth’s reflectivity. Our results show that the PSI based on Tsallis statistics outperforms the conventional PSI, especially in the non-Gaussian noisy-data case.