Types Of Inverse Problems Research Articles

Using machine learning techniques in inverse problems of acoustical oceanography

AbstractThe goal of the work presented here is to study a novel approach for inverting acoustic signals recorded in the marine environment for the estimation of environmental parameters of the water column and/or the seabed. The proposed approach is based on signal feature extraction using a discrete wavelet packet transform, applied to the measured signal, and hidden Markov models that exploit the sequential patterns of the signals. The signal feature is thereafter used in the framework of a mixture density network, which, after training with sets of simulated signals calculated within a predefined search space, provides conditional posterior distributions of the recoverable parameters. The technique is tested with two test cases corresponding to different types of inverse problems. The first case corresponds to a simple problem of geoacoustic inversion, while the second is referred to a, rather unusual, still interesting problem of recovering the shape of a seamount using long‐range acoustic data. Both test cases are based on simulated experiments. The inversion results obtained using the proposed scheme are compared with inversion results using statistical features of the acoustic signal, which is another inversion approach well documented in the literature and is also based on the wavelet packet transform of the measured signal.

On solvability of some inverse problems for a nonlocal fourth-order parabolic equation with multiple involution

<abstract><p>In this paper, the solvability of some inverse problems for a nonlocal analogue of a fourth-order parabolic equation was studied. For this purpose, a nonlocal analogue of the biharmonic operator was introduced. When defining this operator, transformations of the involution type were used. In a parallelepiped, the eigenfunctions and eigenvalues of the Dirichlet type problem for a nonlocal biharmonic operator were studied. The eigenfunctions and eigenvalues for this problem were constructed explicitly and the completeness of the system of eigenfunctions was proved. Two types of inverse problems on finding a solution to the equation and its righthand side were studied. In the two problems, both of the righthand terms depending on the spatial variable and the temporal variable were obtained by using the Fourier variable separation method or reducing it to an integral equation. The theorems for the existence and uniqueness of the solution were proved.</p></abstract>

A Haar wavelets based approximation for nonlinear inverse problems influenced by unknown heat source

In this discussion, a new numerical algorithm focused on the Haar wavelet is used to solve linear and nonlinear inverse problems with unknown heat source. The heat source is dependent on time and space variables. These types of inverse problems are ill‐posed and are challenging to solve accurately. The linearization technique converted the nonlinear problem into simple nonhomogeneous partial differential equation. In this Haar wavelet collocation method (HWCM), the time part is discretized by using finite difference approximation, and space variables are handled by Haar series approximation. The main contribution of the proposed method is transforming this ill‐posed problem into well‐conditioned algebraic equation with the help of Haar functions, and hence, there is no need to implement any sort of regularization technique. The results of numerical method are efficient and stable for this ill‐posed problems containing different noisy levels. We have utilized the proposed method on several numerical examples and have valuable efficiency and accuracy.

GenES-MDA: A generic open-source software package to solve inverse problems via the Ensemble Smoother with Multiple Data Assimilation

Ensemble Kalman filter methods have been successfully applied for data assimilation and parameter estimation through inverse modeling in various scientific fields. We have developed a new generic software package for the solution of inverse problems implementing the Ensemble Smoother with Multiple Data Assimilation (genES-MDA). It is an open-source, platform-independent Python-based program. Its aim is to facilitate the management and configuration of the ES-MDA through several programming tools that help in the preparation of the different steps of ES-MDA. genES-MDA has a flexible workflow that can be easily adapted for the implementation of different variants of the ensemble Kalman filter and for the solution of generic inverse problems. This paper presents a description of the package and some application examples. genES-MDA has been tested in three synthetic case studies: the solution of the reverse flow routing for the estimation of the inflow hydrograph to a river reach using observed water levels and a calibrated forward model of the river system, the identification of a hydraulic conductivity field using piezometric observations and a known forward flow model, and the estimation of the release history of a contaminant spill in an aquifer from measured concentration data and a known flow and transport model. The results of all these tests have demonstrated the flexibility of genES-MDA and its capabilities to efficiently solve different types of inverse problems.

A Generative Variational Model for Inverse Problems in Imaging

This paper is concerned with the development, analysis and numerical realization of a novel variational model for the regularization of inverse problems in imaging. The proposed model is inspired by the architecture of generative convolutional neural networks; it aims to generate the unknown from variables in a latent space via multi-layer convolutions and non-linear penalties, and penalizes an associated cost. In contrast to conventional neural-network-based approaches, however, the convolution kernels are learned directly from the measured data such that no training is required. The present work provides a mathematical analysis of the proposed model in a function space setting, including proofs for regularity and existence/stability of solutions, and convergence for vanishing noise. Moreover, in a discretized setting, a numerical algorithm for solving various types of inverse problems with the proposed model is derived. Numerical results are provided for applications in inpainting, denoising, deblurring under noise, super-resolution and JPEG decompression with multiple test images.

On some inverse problem for bi-parabolic equation with observed data in L^{p} spaces

The bi-parabolic equation has many practical significance in the field of heat transfer. The objective of the paper is to provide a regularized problem for bi-parabolic equation when the observed data are obtained in \(L^p\). We are interested in looking at three types of inverse problems. Regularization results in the \(L^2\) space appears in many related papers, but the survey results are rare in \(L^p\), \(p \neq 2\). The first problem related to the inverse source problem when the source function has split form. For this problem, we introduce the error between the Fourier regularized solution and the exact solution in \(L^p\) spaces. For the inverse initial problem for both linear and nonlinear cases, we applied the Fourier series truncation method. Under the terminal input data observed in \(L^p\), we obtain the approximated solution also in the space \(L^p\). Under some reasonable smoothness assumptions of the exact solution, the error between the the regularized solution and the exact solution are derived in the space \(L^p\). This paper seems to generalize to previous results for bi-parabolic equation on this direction.

Adversarial Training for Solving Inverse Problems in Image Processing.

Inverse problems are a group of important mathematical problems that aim at estimating source data x and operation parameters z from inadequate observations y . In the image processing field, most recent deep learning-based methods simply deal with such problems under a pixel-wise regression framework (from y to x ) while ignoring the physics behind. In this paper, we re-examine these problems under a different viewpoint and propose a novel framework for solving certain types of inverse problems in image processing. Instead of predicting x directly from y , we train a deep neural network to estimate the degradation parameters z under an adversarial training paradigm. We show that if the degradation behind satisfies some certain assumptions, the solution to the problem can be improved by introducing additional adversarial constraints to the parameter space and the training may not even require pair-wise supervision. In our experiment, we apply our method to a variety of real-world problems, including image denoising, image deraining, image shadow removal, non-uniform illumination correction, and underdetermined blind source separation of images or speech signals. The results on multiple tasks demonstrate the effectiveness of our method.

Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker--Planck Equation and Physics-Informed Neural Networks

The Fokker-Planck (FP) equation governing the evolution of the probability density function (PDF) is applicable to many disciplines but it requires specification of the coefficients for each case, which can be functions of space-time and not just constants, hence requiring the development of a data-driven modeling approach. When the data available is directly on the PDF, then there exist methods for inverse problems that can be employed to infer the coefficients and thus determine the FP equation and subsequently obtain its solution. Herein, we address a more realistic scenario, where only sparse data are given on the particles' positions at a few time instants, which are not sufficient to accurately construct directly the PDF even at those times from existing methods, e.g., kernel estimation algorithms. To this end, we develop a general framework based on physics-informed neural networks (PINNs) that introduces a new loss function using the Kullback-Leibler divergence to connect the stochastic samples with the FP equation, to simultaneously learn the equation and infer the multi-dimensional PDF at all times. In particular, we consider two types of inverse problems, type I where the FP equation is known but the initial PDF is unknown, and type II in which, in addition to unknown initial PDF, the drift and diffusion terms are also unknown. In both cases, we investigate problems with either Brownian or Levy noise or a combination of both. We demonstrate the new PINN framework in detail in the one-dimensional case (1D) but we also provide results for up to 5D demonstrating that we can infer both the FP equation and} dynamics simultaneously at all times with high accuracy using only very few discrete observations of the particles.

Carleman estimates for the structurally damped plate equations with Robin boundary conditions and applications

In this paper, we consider Carleman estimates for the damped fourth-order plate operators in a bounded smooth domain with Robin boundary conditions. Because the appearance of such kind of structural damping, the speed of propagation for solutions to the plate equation is infinite and the corresponding properties of the solution similar to heat equations, and is significantly different from that of the usual plate equations without damping. As applications, we consider two types of inverse problems in determining source terms for the plate equation with structural damping. The time-dependent measurements can be restricted to an arbitrary small sub-domain or arbitrary sub-boundary, respectively. Under some assumptions on the regularity of the solutions and coefficients, we prove the global stability results for these inverse problems.

Inverse estimation of number and location of discrete heaters in radiant furnaces using artificial neural networks and genetic algorithm

The inverse problem of the determination of optimum number and location of discrete heaters in a three dimensional radiant heating furnace is solved by developing an artificial neural network based model to replace the radiative transfer equation solver and coupling it with genetic algorithm. The furnace medium is considered to be participating with a mixture of H2O and CO2. For the purpose of estimation of the best configuration of the heaters to achieve a desired uniform heat flux at the design surface, the inverse problem has been treated as an optimization problem. A generalized radiative transfer code based on Finite Volume Method and spectral line based weighted sum of gray gases method has been developed and validated with the available benchmarks. This code serves as the basis to develop the neural network based model for the flux estimation at design surface. The ANN function has been tested upon 5 random test cases. The ANN based replacement of RTE for this problem gives an error less than 4% for the local heat flux values at the order of 103 times less computation time. A design case is then considered where the desired flux levels should be uniform on the design surface in a very tight tolerance. Results obtained show that this novel method of ANN coupled with GA proves to be a fast, robust and efficient tool for solving such types of inverse problems.

Conjugate gradient based acceleration for inverse problems

The conjugate gradient method is a widely used algorithm for the numerical solution of a system of linear equations. It is particularly attractive because it allows one to take advantage of sparse matrices and produces (in case of infinite precision arithmetic) the exact solution after a finite number of iterations. It is thus well suited for many types of inverse problems. On the other hand, the method requires the computation of the gradient. Here difficulty can arise, since the functional of interest to the given inverse problem may not be differentiable. In this paper, we review two approaches to deal with this situation: iteratively reweighted least squares and convolution smoothing. We apply the methods to a more generalized, two parameter penalty functional. We show advantages of the proposed algorithms using examples from a geotomographical application and for synthetically constructed multi-scale reconstruction and regularization parameter estimation.

Fundamental characteristics of solids from low-temperature heat capacity

A new approach to obtaining fundamental characteristics of solids is provided. The key element of this approach is related to solving the inverse problem of how to reconstruct phonon density of states g(ω) from low-temperature heat capacity. The original method for numerical solution of this problem allows to calculate the g(ω) dependence with correct description of its shape and correct proportion of the number of vibrational modes in different frequency intervals. This allows to accurately calculate the moments of the g(ω) and related characteristic temperatures which, being constants, unambiguously describe individual substance. Also this allows to calculate a zero-point energy of crystal with high accuracy. This, in turn, opens up the possibility for calculating the full internal energy of a solid. The knowledge of the g(ω) allows calculating the isochoric thermodynamic functions in the entire region of existence of solid phase. Special attention is paid to the estimation of the accuracy of the characteristics obtained. The capacities of the approach are demonstrated on a model object and a real-world object, the latter is represented by a single-crystal lithium molybdate. The described approach is universal and opens up new possibilities for studying solids. The proposed computer algorithm can be used to numerical solve other types of inverse problems that include the integrated form.

Inverse problems of anomalous diffusion theory: An artificial neural network approach

The results are presented of computer simulation of the operation of a three-layer perceptron trained for solving inverse problems of anomalous diffusion theory. Several types of inverse problems are considered, including the problem of determining the Hurst exponent of a selfsimilar medium.

ИСПОЛЬЗОВАНИЕ МЕТОДОВ ТЕОРИИ ОПТИМАЛЬНОГО УПРАВЛЕНИЯ ПРИ РЕШЕНИИ ОБРАТНЫХ ЗАДАЧ ОЦЕНКИ РЕЗУЛЬТАТОВ ИСПЫТАНИЙ ТЕХНИЧЕСКИХ СИСТЕМ

The article draws attention to the identity of the methods for solving problems of identification arising from the evaluation of test results of complex dynamic systems, the classical problems of the theory of optimal control. Shows how to use the latest techniques can be very effective for solving certain types of inverse problems both during active and passive experiment. The authors believe that there are many inverse problems, where it will be possible using effective methods of optimal control.

A General Result on the Mean Integrated Squared Error of the Hard Thresholding Wavelet Estimator underα-Mixing Dependence

We consider the estimation of an unknown functionffor weakly dependent data (α-mixing) in a general setting. Our contribution is theoretical: we prove that a hard thresholding wavelet estimator attains a sharp rate of convergence under the mean integrated squared error (MISE) over Besov balls without imposing too restrictive assumptions on the model. Applications are given for two types of inverse problems: the deconvolution density estimation and the density estimation in a GARCH-type model, both improve existing results in this dependent context. Another application concerns the regression model with random design.

A geometric approach to joint inversion with applications to contaminant source zone characterization

This paper presents a new joint inversion approach to shape-based inverse problems. Given two sets of data from distinct physical models, the main objective is to obtain a unified characterization of inclusions within the spatial domain of the physical properties to be reconstructed. Although our proposed method generally applies to many types of inverse problems, the main motivation here is to characterize subsurface contaminant source zones by processing down-gradient hydrological data and cross-gradient electrical resistance tomography observations. Inspired by Newton's method for multi-objective optimization, we present an iterative inversion scheme in which descent steps are chosen to simultaneously reduce both data-model misfit terms. Such an approach, however, requires solving a non-smooth convex problem at every iteration, which is computationally expensive for a pixel-based inversion over the whole domain. Instead, we employ a parametric level set technique that substantially reduces the number of underlying parameters, making the inversion computationally tractable. The performance of the technique is examined and discussed through the reconstruction of source zone architectures that are representative of dense non-aqueous phase liquid (DNAPL) contaminant release in a statistically homogenous sandy aquifer. In these examples, the geometric configuration of the DNAPL mass is considered along with additional information about its spatial variability within the contaminated zone, such as the identification of low and high saturation regions. Comparison of the reconstructions with the true DNAPL architectures highlights the superior performance of the model-based technique and joint inversion scheme.

Computational modeling of monoenergetic neutral particle inverse transport problems in slab geometry

Abstract An analytical numerical method applied to three different types of monoenergetic neutral particle inverse transport problems in the discrete ordinates (SN) formulation is presented: (a) boundary condition estimation; (b) interior source estimation; and (c) effective slab length estimation. These three types of inverse problems governed by the linear integrodifferential transport equation in SN formulation are related respectively to medical physics; nuclear waste storage; and non-destructive testing in industry. Numerical results and a brief discussion are given to conclude the paper.

A Survey on Inverse Problems for Applied Sciences

The aim of this paper is to introduce inversion-based engineering applications and to investigate some of the important ones from mathematical point of view. To do this we employ acoustic, electromagnetic, and elastic waves for presenting different types of inverse problems. More specifically, we first study location, shape, and boundary parameter reconstruction algorithms for the inaccessible targets in acoustics. The inverse problems for the time-dependent differential equations of isotropic and anisotropic elasticity are reviewed in the following section of the paper. These problems were the objects of the study by many authors in the last several decades. The physical interpretations for almost all of these problems are given, and the geophysical applications for some of them are described. In our last section, an introduction with many links into the literature is given for modern algorithms which combine techniques from classical inverse problems with stochastic tools into ensemble methods both for data assimilation as well as for forecasting.

Comparative analysis of inverse coefficient problems for parabolic equations. Part I: adjoint problem approach

This article presents a mathematical and computational analysis of the adjoint problem approach for parabolic inverse coefficient (or inverse heat conduction) problems based on boundary measured data. In Part I the mathematical analysis is given for three classes of typical inverse coefficient problems with various Neumann or/and Dirichlet types of measured output data. Although all three types of considered inverse coefficient problems are severely ill-posed, comparative numerical analysis show that the ill-posedness depends also on where the Neumann and Dirichlet conditions are given: in the direct problem or as an output data. For all these types of inverse problems the integral identities relating solutions of direct problems and appropriate adjoint problems solutions are derived. These integral identities permit proof of monotonicity, Lipschitz continuity, and hence invertibility of the corresponding input–output mappings. Based on these results solvability of all three types of inverse coefficient problems are proved. The degree of ill-posedness of inverse problems is demonstrated on numerical test examples.

Entropy description of measured information in mathematical and physical inverse problems

There are two types of inverse problems: Optimization designation and parameter identification. Before the parameter identification of mathematical and physical inverse problems, it is necessary to determine the number and position of measurement points in analysis and evaluation of a large number of measured data. In this paper, a mathematical methodology is proposed to describe the influence of the number and position of measurement points on the reconstruction precision. Information entropy and Bayesian theory are used in the description. Finally, a numerical experiment shows that the methodology is effective.