Nonlinear Inverse Problem Research Articles

Fast retrieval of XCO2 over east Asia based on Orbiting Carbon Observatory-2 (OCO-2) spectral measurements

Abstract. The increase in greenhouse gas concentrations, particularly CO2, has significant implications for global climate patterns and various aspects of human life. Spaceborne remote sensing satellites play a crucial role in high-resolution monitoring of atmospheric CO2. However, the next generation of greenhouse gas monitoring satellites is expected to face challenges, particularly in terms of computational efficiency in atmospheric CO2 retrieval and analysis. To address these challenges, this study focuses on improving the speed of retrieving the column-averaged dry-air mole fraction of carbon dioxide (XCO2) using spectral data from the Orbiting Carbon Observatory-2 (OCO-2) satellite while still maintaining retrieval accuracy. A novel approach based on neural network (NN) models is proposed to tackle the nonlinear inversion problems associated with XCO2 retrievals. The study employs a data-driven supervised learning method and explores two distinct training strategies. Firstly, training is conducted using experimental data obtained from the inversion of the operational optimization model, which is released as the OCO-2 satellite products. Secondly, training is performed using a simulated dataset generated by an accurate forward calculation model. The inversion performance and prediction performance of the machine learning model for XCO2 are compared, analyzed, and discussed for the observed region over east Asia. The results demonstrate that the model trained on simulated data accurately predicts XCO2 in the target area. Furthermore, when compared to OCO-2 satellite product data, the developed XCO2 retrieval model not only achieves rapid predictions (<1 ms) with good accuracy (1.8 ppm or approximately 0.45 %) but also effectively captures sudden increases in XCO2 plumes near industrial emission sources. The accuracy of the machine learning model retrieval results is validated against reliable data from Total Carbon Column Observing Network (TCCON) sites, demonstrating its ability to effectively capture CO2 seasonal variations and annual growth trends.

Error estimates for simplified Levenberg–Marquardt method for nonlinear ill-posed operator equations in Hilbert Spaces

Abstract In this paper, we consider the simplified Levenberg–Marquardt method for nonlinear ill-posed inverse problems in Hilbert spaces for obtaining stable approximations of solutions to the ill-posed nonlinear equations of the form F ⁢ ( u ) = y {F(u)=y} , where F : 𝒟 ⁢ ( F ) ⊂ 𝖴 → 𝖸 {F:\mathcal{D}(F)\subset\mathsf{U}\to\mathsf{Y}} is a nonlinear operator between Hilbert spaces 𝖴 {\mathsf{U}} and 𝖸 {\mathsf{Y}} . The method is defined as follows: u n + 1 δ = u n δ - ( T 0 ∗ ⁢ T 0 + α n ⁢ I ) - 1 ⁢ T 0 ∗ ⁢ ( F ⁢ ( u n δ ) - y δ ) , u_{n+1}^{\delta}=u_{n}^{\delta}-(T_{0}^{\ast}T_{0}+\alpha_{n}I)^{-1}T_{0}^{% \ast}(F(u_{n}^{\delta})-y^{\delta}), where T 0 = F ′ ⁢ ( u 0 ) {T_{0}=F^{\prime}(u_{0})} and T 0 ∗ = F ′ ⁢ ( u 0 ) ∗ {T_{0}^{\ast}=F^{\prime}(u_{0})^{\ast}} . Here F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} denotes the Frèchet derivative of F at an initial guess u 0 ∈ 𝒟 ⁢ ( F ) {u_{0}\in\mathcal{D}(F)} for the exact solution u † {u^{\dagger}} , F ′ ⁢ ( u 0 ) ∗ {F^{\prime}(u_{0})^{\ast}} is the adjoint of F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} and { α n } {\{\alpha_{n}\}} is an a priori chosen sequence of non-negative real numbers satisfying suitable properties. We use Morozov-type stopping rule to terminate the iterations. Under suitable non-linearity conditions on operator F, we show convergence of the method and also obtain a convergence rate result under a Hölder-type source condition on the element u 0 - u † {u_{0}-u^{\dagger}} . Furthermore, we derive convergence of the method for the case when no source conditions are used and the study concludes with numerical examples which validate the theoretical conclusions.

Genetic Algorithm as the Solution of Non-Linear Inverse Heat Conduction Problems: A Novel Sequential Approach

Abstract Direct measurement of surface heat flux could be extremely challenging, if not impossible, in numerous applications. In such cases, the use of temperature measurement data from subsurface locations can facilitate the determination of surface heat flux and temperature through the solution of the inverse heat conduction problem (IHCP). Several different techniques have been developed over the years for solving IHCPs, with different levels of complexity and accuracy. The filter coefficient technique has proved to be a promising approach for solving IHCPs. Inspired by the filter coefficient approach, a novel method is presented in this paper for solving one-dimensional IHCPs in a domain with temperature-dependent material properties. A test case is developed in COMSOL Multiphysics where the front side of a slab is subject to known transient heat flux and the temperature distributions within the domain are calculated through numerical simulation. The IHCP solution in the form of filter coefficients is defined and a genetic algorithm (GA) is used for the calculation of filter matrix. The number of significant filter coefficients required to evaluate surface heat flux at each time-step is determined through trial and error and the optimal number is selected for evaluating the solution. The structure of the filter matrix is assessed and discussed. The resulting filter coefficients are then used to evaluate the surface heat flux for several different test cases and the performance of the proposed approach is assessed in detail. The results showed that the presented approach is robust and can result in finding optimal filter coefficients that can accurately estimate various types of surface heat flux profiles using temperature data from a limited number of time steps and in a near real-time fashion.

Nonlinear Inverse Problems for First Order Hyperbolic Equations

Nonlinear Inverse Problems for First Order Hyperbolic Equations

Chebyshev polynomial-based Fourier transformation and its use in low pass filter of gravity data

In this paper, we introduce the novel Chebyshev Polynomials Least-Squares Fourier Transformation (C-LSQ-FT) and its robust variant with the Iteratively Reweighted Least-Squares technique (C-IRLS-FT). These innovative techniques for Fourier transformation are predicated on the concept of inversion, and the C-LSQ-FT method establishes an overdetermined inverse problem within the realm of Fourier transformation. However, given the LSQ approach’s vulnerability to data outliers, we note the potential for considerable errors and potentially unrepresentative model estimations. To circumvent these shortcomings, we incorporate Steiner’s Most Frequent Value method into our framework, thereby providing a more reliable alternative. The fusion of the Iteratively Reweighted Least-Squares (IRLS) algorithm with Cauchy-Steiner weights enhances the robustness of our Fourier transformation process, culminating in the C-IRLS-FT method. We use Chebyshev polynomials as the basis functions in both methods, leading to the approximation of continuous Fourier spectra through a finite series of Chebyshev polynomials and their corresponding coefficients. The coefficients were obtained by solving an overdetermined non-linear inverse problem. We validated the performance of both the traditional Discrete Fourier Transform (DFT) and the newly developed C-IRLS-FT through numerical tests on synthetic datasets. The results distinctly exhibited the reduced sensitivity of the C-IRLS-FT method to outliers and dispersed noise, in comparison with the traditional DFT. We leveraged the newly proposed (C-IRLS-FT) technique in the application of low-pass filtering in the context of gravity data. The results corroborate the technique’s robustness and adaptability, making it a promising method for future applications in geophysical data processing.

[formula omitted] sparsity regularization for nonlinear ill-posed problems

In this paper, the α‖⋅‖ℓ1−β‖⋅‖ℓ2 sparsity regularization with parameters α≥β≥0 is studied for nonlinear ill-posed inverse problems. The well-posedness of the regularization is investigated. Compared to the case where α>β≥0, the results for the case α=β>0 are weaker due to the lack of coercivity and Radon-Riesz property of the regularization term. Under certain conditions on the nonlinearity of F, sparsity is shown for every minimizer of the α‖⋅‖ℓ1−β‖⋅‖ℓ2 regularized inverse problem. Moreover, for the case α>β≥0, convergence rates O(δ12) and O(δ) are proved for the regularized solution toward a sparse exact solution, under different yet commonly adopted conditions on the nonlinearity of F. The iterative soft thresholding algorithm is shown to be useful to solve the α‖⋅‖ℓ1−β‖⋅‖ℓ2 regularized problem for nonlinear ill-posed equations. Numerical results illustrate the efficiency of the proposed method.

A replica exchange preconditioned Crank-Nicolson Langevin dynamic MCMC method with multi-variance strategy for Bayesian inverse problems

The preconditioned Crank-Nicolson Markov Chain Monte Carlo (pCN-MCMC) method is an efficient approach for large-scale Bayesian inference problems. It is constructed by discretizing a stochastic partial differential equation (SPDE). When the gradient information is coupled in this SPDE, the corresponding SPDE is also referred to as the Langevin diffusion. In this work, we present the replica exchange preconditioned Crank-Nicolson Langevin dynamic (repCNLD) Monte Carlo method to speed up the convergence of the pCN-MCMC and tackle the challenge of multimodal distribution. We use the Crank-Nicolson scheme to discretize the repLD and set the preconditioned matrix as the covariance matrix of the Gaussian prior. The discretization error is analyzed. Moreover, we use the multi-variance strategy (m-repCNLD) to improve computational efficiency. Perturbations are introduced into the scheme due to the likelihood estimation error, leading to the biased swapping rate. To correct the bias, we provide an unbiased estimator derived from some assumptions on the forward solvers. The accuracy and efficiency of the proposed method are demonstrated by sampling Gaussian mixture distribution and solving PDE-based nonlinear inverse problems numerically.

Ensemble variational Bayesian approximation for the inversion and uncertainty quantification of Darcy flows in heterogeneous porous media with random parameters

The inversion and uncertainty quantification of physical parameters are important for many science and engineering problems. For nonlinear parameter inversion problems of flows in hydrocarbon reservoirs, estimating the true posterior probability distribution function for the random parameter is difficult when the dimension of the parameter is relatively high. Inversion methods such as gradient-based or heuristic optimization methods searching for the maximum a posterior are generally point-estimate, while uncertainty quantification using MCMC is computationally expensive. In the current study, a new ensemble variational Bayesian approximation method is developed to estimate the posterior probability density using an ensemble of realizations regarded as a trial sample distribution. The objective is to optimize the trial sample distribution such that the evidence lower bound is maximized indicating that the distance between the trial and true posterior distribution is minimized. A subspace is built using principle component analysis based on finite samples to obtain a diagonal positive-definite covariance matrix, and ensemble-based data assimilation is used to optimize the trial distribution efficiently. Heterogeneous 2D and 3D test cases of transient single- and two-phase Darcy flows are presented for validation.

M. M. Lavrentiev-type systems and reconstructing parameters of viscoelastic media

Abstract We consider a nonlinear coefficient inverse problem of reconstructing the density and the memory matrix of a viscoelastic medium by probing the medium with a family of wave fields excited by moment tensor point sources. A spatially non-overdetermined formulation is investigated, in which the manifolds of point sources and detectors do not coincide and have a total dimension equal to three. The requirements for these manifolds are established to ensure the unique solvability of the studied inverse problem. The results are achieved by reducing the problem to a chain of connected systems of linear integral equations of the M. M. Lavrentiev type.

Derivative-Free Iterative One-Step Reconstruction for Multispectral CT.

Image reconstruction in multispectral computed tomography (MSCT) requires solving a challenging nonlinear inverse problem, commonly tackled via iterative optimization algorithms. Existing methods necessitate computing the derivative of the forward map and potentially its regularized inverse. In this work, we present a simple yet highly effective algorithm for MSCT image reconstruction, utilizing iterative update mechanisms that leverage the full forward model in the forward step and a derivative-free adjoint problem. Our approach demonstrates both fast convergence and superior performance compared to existing algorithms, making it an interesting candidate for future work. We also discuss further generalizations of our method and its combination with additional regularization and other data discrepancy terms.

Inverse moving source problems for parabolic equations

This paper is concerned with the inverse moving source problems for parabolic equations. Given the temporal function, we prove the uniqueness of the nonlinear inverse problem of determining the orbit function by final data measured in a bounded domain. On the other hand, given the orbit function we also show that the profile function can be uniquely determined by final data measured in a bounded domain away from the domain enclosing the moving orbit. The proofs adopt the Fourier approach and results from complex analysis.

Deep unrolling networks with recurrent momentum acceleration for nonlinear inverse problems

Combining the strengths of model-based iterative algorithms and data-driven deep learning solutions, deep unrolling networks (DuNets) have become a popular tool to solve inverse imaging problems. Although DuNets have been successfully applied to many linear inverse problems, their performance tends to be impaired by nonlinear problems. Inspired by momentum acceleration techniques that are often used in optimization algorithms, we propose a recurrent momentum acceleration (RMA) framework that uses a long short-term memory recurrent neural network (LSTM-RNN) to simulate the momentum acceleration process. The RMA module leverages the ability of the LSTM-RNN to learn and retain knowledge from the previous gradients. We apply RMA to two popular DuNets—the learned proximal gradient descent (LPGD) and the learned primal-dual (LPD) methods, resulting in LPGD-RMA and LPD-RMA, respectively. We provide experimental results on two nonlinear inverse problems: a nonlinear deconvolution problem, and an electrical impedance tomography problem with limited boundary measurements. In the first experiment we have observed that the improvement due to RMA largely increases with respect to the nonlinearity of the problem. The results of the second example further demonstrate that the RMA schemes can significantly improve the performance of DuNets in strongly ill-posed problems.

Adaptive neural network surrogate model for solving the nonlinear elastic inverse problem via Bayesian inference

Abstract In this paper, we consider a Bayesian method for nonlinear elastic inverse problems. As a working model, we are interested in the inverse problem of restoring elastic properties from measured tissue displacement. In order to reduce the computational cost, we will use the following multi-fidelity model approach. First, we construct a surrogate low-fidelity DNNs-based model in the prior distribution, then use a certain number of simulations of high fidelity model associated with an adaptive strategy online to update the low-fidelity model locally. Numerical examples show that the proposed method can solve nonlinear elastic inverse problems efficiently and accurately.

A model-based MR parameter mapping network robust to substantial variations in acquisition settings

Deep learning methods show great potential for the efficient and precise estimation of quantitative parameter maps from multiple magnetic resonance (MR) images. Current deep learning-based MR parameter mapping (MPM) methods are mostly trained and tested using data with specific acquisition settings. However, scan protocols usually vary with centers, scanners, and studies in practice. Thus, deep learning methods applicable to MPM with varying acquisition settings are highly required but still rarely investigated. In this work, we develop a model-based deep network termed MMPM-Net for robust MPM with varying acquisition settings. A deep learning-based denoiser is introduced to construct the regularization term in the nonlinear inversion problem of MPM. The alternating direction method of multipliers is used to solve the optimization problem and then unrolled to construct MMPM-Net. The variation in acquisition parameters can be addressed by the data fidelity component in MMPM-Net. Extensive experiments are performed on R2 mapping and R1 mapping datasets with substantial variations in acquisition settings, and the results demonstrate that the proposed MMPM-Net method outperforms other state-of-the-art MR parameter mapping methods both qualitatively and quantitatively.

Uniqueness of Nonlinear Inverse Problem for Sturm–Liouville Operator with Multiple Delays

The inverse problem concerns how to reconstruct the operator from given spectral data. The main goal of this paper is to address nonlinear inverse Sturm–Liouville problem with multiple delays. By using a new technique and method: zero function extension, we establish the uniqueness result and practical method for recovering the nonlinear inverse problem from two spectra.

On mathematical problems of two-coefficient inverse problems of ultrasonic tomography

This paper proves the theorem of uniqueness for the solution of a coefficient inverse problem for the wave equation in (with two unknown coefficients: speed of sound and absorption. The original nonlinear coefficient inverse problem is reduced to an equivalent system of two uniquely solvable linear integral equations of the first kind with respect to the sound speed and absorption coefficients. Estimates are made, substantiating the multistage method for two unknown coefficients. These estimates show that given sufficiently low frequencies and small inhomogeneities, the residual functional for the nonlinear inverse problem approaches a convex one. This solution method for nonlinear coefficient inverse problems is not linked to the limit approach as frequency tends to zero, but assumes solving the inverse problem using sufficiently low, but not zero, frequencies at the first stage. For small inhomogeneities that are typical, for instance, for medical tasks, carrying out real experiments at such frequencies does not present major difficulties. The capabilities of the method are demonstrated on a model inverse problem with unknown sound speed and absorption coefficients. The method effectively solves the nonlinear problem with parameter values typical for tomographic diagnostics of soft tissues in medicine. A resolution of approximately 2 mm was achieved using an average sounding pulse wavelength of 5 mm.

Generalized variational framework with minimax optimization for parametric blind deconvolution

Blind deconvolution (BD), which aims to separate unknown convolved signals, is a fundamental problem in signal processing. Due to the ill-posedness and underdetermination of the convolution system, it is a challenging nonlinear inverse problem. This paper is devoted to the algorithmic studies of parametric BD, which is typically applied to recover images from ad hoc optical modalities. We propose a generalized variational framework for parametric BD with various priors and potential functions. By using the conjugate theory in convex analysis, the framework can be cast into a nonlinear saddle point problem. We employ the recent advances in minimax optimization to solve the parametric BD by the nonlinear primal-dual hybrid gradient method, with all subproblems admitting closed-form solutions. Numerical simulations on synthetic and real datasets demonstrate the compelling performance of the minimax optimization approach for solving parametric BD.

Structural Tectonic Scheme Creation Based on Seismic-Gravity Models and Isostasy Usage: Ural Case

Process of Earth’s density models creation leads to the solution of direct and inverse gravimetry problems. The inverse problem of gravimetry is a classic example of an ill-posed problem: in the common statement, its solution is not unique and unstably depends on input data. Therefore, it is necessary to determine solutions belonging substantial sets of correctness, choosing reasonable models of an initial approximation. In this paper the application of complex interpretation methods of seismic and gravitational data for the creation of three-dimensional models of crust and the upper mantle are presented. Original algorithms and programs were developed for implementation of these methods. They contain solution of non-linear (structural) inverse problem and the solution of the linear three-dimensional inverse problem taking note of the side sources. Coefficients of the “density-velocity” correlation formulas for a number of geo-traverses were defined. Also, we suggest a technique of tectonic maps construction, which is based on the lithostatic pressure calculation. Its idea can be applied to both two- and three-dimensional cases. In the 2D case we show the way to split the mantle to blocks with vertical boundaries. If lithostatic compensation hypothesis is adopted, the method also allows one to calculate density value for each block. Such separation of the mantle helps to diminish discrepancy between model and observed fields. In 3D case we suggest a method, which can be used to construct tectonic structure maps with information about approximate depth and height of each tectonic block.

A Bayesian approach for consistent reconstruction of inclusions

This paper considers a Bayesian approach for inclusion detection in nonlinear inverse problems using two known and popular push-forward prior distributions: the star-shaped and level set prior distributions. We analyze the convergence of the corresponding posterior distributions in a small measurement noise limit. The methodology is general; it works for priors arising from any Hölder continuous transformation of Gaussian random fields and is applicable to a range of inverse problems. The level set and star-shaped prior distributions are examples of push-forward priors under Hölder continuous transformations that take advantage of the structure of inclusion detection problems. We show that the corresponding posterior mean converges to the ground truth in a proper probabilistic sense. Numerical tests on a two-dimensional quantitative photoacoustic tomography problem showcase the approach. The results highlight the convergence properties of the posterior distributions and the ability of the methodology to detect inclusions with sufficiently regular boundaries.

Application of the Two-sided difference quotient in the solution of nonlinear Ill-posed inverse self-adjoint elliptic problem

When we use a discretization by finite differences, to solve differential equations we find problems at the border of the domain of the solution. If the solution is also immersed in a ill-posed inverse problem; we can find very bad solutions. In this paper we apply a discretization of two - sided difference quotients method to solve Ill-posed inverse self-adjoint elliptic problem [Kirsch(2011)]. Some numerical examples showing the effectiveness of this method and we will use mollification techniques to smooth the solutions.