Morozov's Discrepancy Principle Research Articles

Regularization of vertical derivatives of potential field data using Morozov's discrepancy principle

AbstractThe calculation of the vertical derivatives of potential field methods can be carried out in a stable manner by Tikhonov regularization, but this procedure requires the appropriate selection of a regularization parameter. For this purpose, we introduce a criterion based on Morozov's discrepancy principle that uses a preliminary approximation given by the vertical derivative of the smoothed data. The smoothing may be performed by a physical or a mathematically based low‐pass filter. The filtered data are computed only for estimating the regularization parameter; once it is found, we evaluate the regularized vertical derivative from the original data (not from the smoothed one) in the frequency domain. We verified from experiments with noise‐corrupted synthetic data, as well as gravity and magnetic field data, that the regularized vertical derivative has about the same smoothness as the one obtained from filtered data, but true anomalies are more easily distinguished from noise and the shapes of the anomalies are better preserved.

Active control of electromagnetic fields in layered media

This article presents a numerical strategy for actively manipulating electromagnetic (EM) fields in layered media. In particular, we develop a scheme to characterize an EM source that will generate some predetermined field patterns in prescribed disjoint exterior regions in layered media. The proposed question of specifying such an EM source is not an inverse source problem (ISP) since the existence of a solution is not guaranteed. Moreover, our problem allows for the possibility of prescribing different EM fields in mutually disjoint exterior regions. This question involves a linear inverse problem that requires solving a severely ill-posed optimization problem (i.e. suffering from possible non-existence or non-uniqueness of a solution). The forward operator is defined by expressing the EM fields as a function of the current at the source using the layered media Green's function (LMGF), accounting for the physical parameters of the layered media. This results to integral equations that are then discretized using the method of moments (MoM), yielding an ill-posed system of linear equations. Unlike in ISPs, stability with respect to data is not an issue here since no data is measured. Rather, stability with respect to input current approximation is important. To get such stable solutions, we applied two regularization methods, namely, the truncated singular value decomposition (TSVD) method and the Tikhonov regularization method with the Morozov Discrepancy Principle. We performed several numerical simulations to support the theoretical framework and analyzes, and to demonstrate the accuracy and feasibility of the proposed numerical algorithms.

Mixed Fractional-Order and High-Order Adaptive Image Denoising Algorithm Based on Weight Selection Function

In this paper, a mixed-order image denoising algorithm containing fractional-order and high-order regularization terms is proposed, which effectively suppresses the staircase effect generated by the TV model and its variants while better preserving the edges and details of the image. Adding different regularization penalties in different regions is fundamental to improving the denoising performance of the model. Therefore, a weight selection function is designed using the structure tensor to achieve a more effective selection of regularization terms in different regions. In each iteration, the regularization parameters are adaptively adjusted according to the Morozov discrepancy principle to promote the performance of the algorithm. Based on the primal–dual theory, the original algorithm is improved by using the predictor–corrector scheme to obtain a more accurate approximate solution while ensuring the convergence of the algorithm. The effectiveness of the proposed algorithm is demonstrated through simulation experiments.

Identifying an unknown coefficient in the fractional parabolic differential equation

In this study, we considered a fractional parabolic equation for identifying the unknown diffusion coefficient from the noisy measurement of the ultimate time solution. It is an inverse problem involving a nonlocal operator that is nonlinear and poorly formulated. By demonstrating the existence of this inverse problem singular solution with regard to the final observed data, we demonstrate the identifiability of this problem. The inverse problem is expressed as a regularized optimization problem that minimizes a cost function of the least-squares kind. We have covered various theoretical and practical difficulties pertaining to the problem under consideration. It has been demonstrated that a unique stable solution to the optimization problem exists. The Morozov discrepancy principle and the conjugate gradient approach are used to develop an iterative reconstruction procedure. The accuracy and efficiency of the suggested method are illustrated using a few numerical examples.

Fluid dynamics characterization of stirred-tank reactors via approximate Bayesian computational (ABC) for parameter estimation and model selection

The Approximate Bayesian Computing (ABC) was used in this work to perform the selection and calibration of models in reactor dynamics. Five phenomenologically based hydrodynamic models on dead volume, flow deviation and retro-mix anomalies were evaluated and classified into two groups. The inverse problems were solved from experimental data obtained by the technique of residence time distribution (RTD) in positive step, using the methylene blue tracer in a stirred-tank reactor. The experiments evaluated the influence of volumetric flow, effluent outlet position, agitation position and rotation on the hydrodynamic characterization of the reactor. The results revealed that the most influential parameter in the first group was the dead volume, while in the second group, the retro-mix anomaly was highly sensitive to rotation. The ABC algorithm applied in this work used the Morozov discrepancy principle to select the tolerance in the final population. Tolerances for other intermediate populations were obtained from the median distance of each particle accepted in the previous population. HIGHLIGHTS The application of the Bayesian approach in RTD problems was excellent for the hydrodynamic characterization of reactors. The ABC SMC algorithm proved to be efficient for the calibration and selection of hydrodynamic models of reactors. The most influential anomaly in the flow was the dead volume.

Identification of an unknown spatial source function in a multidimensional hyperbolic partial differential equation with interior degeneracy

A reconstruction of an unknown source function is considered for hyperbolic partial differential equations with interior degeneracy. We identify the spatial element of the source term of a degenerate wave equation using the final observation data. The existence and uniqueness of the direct problem with interior degeneracy within the spatial domain are stated and proved. The inverse problem can be formulated as a nonlinear optimization problem and the unknown source term can be characterized as the solution to a minimization problem. The Tikhonov regularization technique is employed to accomplish the inclusion of noise in the input data, based on the insertion of the regularization term into the cost functional. The conjugate gradient algorithm in conjunction with Morozov's discrepancy principle as a stopping criterion is then utilized to develop an iterative reconstruction procedure. Finally, some numerical simulation results are provided to show the performance of the proposed scheme in one and two dimensions.

Approximation of the Tikhonov regularization parameter through Aitken's extrapolation

In the present work, we study the determination of the regularization parameter and the computation of the regularized solution in Tikhonov regularization, by the Aitken's extrapolation method. In particular, this convergence acceleration method is adjusted for the approximation of quadratic forms that appear in regularization methods, such as the generalized cross-validation method, the quasi-optimality criterion, the Gfrerer/Raus method and the Morozov's discrepancy principle. We present several numerical examples to illustrate the effectiveness of the derived estimates for approximating the regularization parameter for several linear discrete ill-posed problems and we compare the described method with further existing methods, for the determination of the regularized solution.

Sensitivity analysis for active electromagnetic field manipulation in free space

ABSTRACT This paper presents a detailed sensitivity analysis of the active manipulation scheme for electromagnetic (EM) fields in free space. The active EM fields control strategy is designed to construct surface current sources (electric and/or magnetic) that can manipulate the EM fields in prescribed exterior regions. The active EM field control is formulated as an inverse source problem. We follow the numerical strategies in our previous works, which employ the Debye potential representation and integral equation representation in the forward modelling. We consider two regularization approaches to the inverse problem to approximate a current source, namely the truncated singular value decomposition (TSVD) and the Tikhonov regularization with the Morozov discrepancy principle. Moreover, we discuss the sensitivity of the active scheme (concerning power budget, control accuracy, and quality factor) as a function of the frequency, the distance between the control region and the source, the mutual distance between the control regions, and the size of the control region. The numerical simulations demonstrate some challenges and limitations of the active EM field control scheme.

Determination of the initial density in nonlocal diffusion from final time measurements

<p style='text-indent:20px;'>This paper is concerned with an inverse problem related to a fractional parabolic equation. We aim to reconstruct an unknown initial condition from noise measurement of the final time solution. It is a typical nonlinear and ill-posed inverse problem related to a nonlocal operator. The considered problem is motivated by a probabilistic framework when the initial condition represents the initial probability distribution of the position of a particle. We show the identifiability of this inverse problem by proving the existence of its unique solution with respect to the final observed data. The inverse problem is formulated as a regularized optimization one minimizing a least-squares type cost functional. In this work, we have discussed some theoretical and practical issues related to the considered problem. The existence, uniqueness, and stability of the optimization problem solution have been proved. The conjugate gradient method combined with Morozov's discrepancy principle are exploited for building an iterative reconstruction process. Some numerical examples are carried out showing the accuracy and efficiency of the proposed method.</p>

A New Hybrid Method for Solving Inverse Heat Conduction Problems

Solving the inverse problems, especially in the field of heat transfer, is one of the challenges of engineering due to its importance in industrial applications. It is well-known that inverse heat conduction problems (IHCPs) are severely ill-posed, which means that small disturbances in the input may cause extremely large errors in the solution. This paper introduces an accurate method for solving inverse problems by combining Tikhonov's regularization and the genetic algorithm. Finding the regularization parameter as the decisive parameter is modelled by this method, a few sample problems were solved to investigate the efficiency and accuracy of the proposed method. A linear sum of fundamental solutions with unknown constant coefficients assumed as an approximated solution to the sample IHCP problem and collocation method is used to minimize residues in the collocation points. In this contribution, we use Morozov's discrepancy principle and Quasi-Optimality criterion for defining the objective function, which must be minimized to yield the value of the optimum regularization parameter.

Feasibility analysis for active near/far field acoustic pattern synthesis in free space and shallow water environments

In this paper, a detailed sensitivity and feasibility analysis of the active manipulation scheme for scalar Helmholtz fields proposed in our previous works, in both free space and constant-depth homogeneous ocean environments, is presented. We apply the method of moments (MoM) together with Tikhonov regularization with the Morozov discrepancy principle to investigate the effects of varying the problem parameters to the accuracy and feasibility of the proposed active field control strategy. We discuss the feasibility of the active scheme (with respect to power budget, control accuracy and process error) as a function of the frequency, the distance between the control region and the source, the mutual distance between the control regions, and the size of the control region. Process error is considered as well to investigate the possibility of an accurate active control in the presence of manufacturing or feeding noise. The numerical simulations show the accuracy of the active field control scheme and indicate some challenges and limitations for its physical implementation.

Parameter Selection Criteria for Tomo-SAR Focusing

The synthetic aperture radar (SAR) tomography (TomoSAR) inverse problem is commonly tackled in the context of the direction-of-arrival estimation theory. The latter allows achieving super resolution, along with ambiguity levels reduction, thanks to the use of parametric focusing methods, as multiple signal classification (MUSIC) and statistical regularization techniques, like the maximum-likelihood-inspired adaptive robust iterative approach (MARIA). Nevertheless, in order to correctly suit the considered signal model, MUSIC and most regularization approaches require an appropriate setting of the involved parameters. In both cases, the accuracy of the retrieved solutions depends on the right selection of the assigned values. Thus, with the aim of dealing with such an issue, this article addresses several parameter selection strategies, adapted specifically to the TomoSAR scenario. Parametric techniques as MUSIC solve the TomoSAR problem in a different manner as the regularization methods do, hence, each approach demands different methodologies for the proper estimation of their parameters. Consequently, we refer to the Kullback-Leibler information criterion for the model order selection of parametric techniques as MUSIC, whereas we rather explore the Morozov's discrepancy principle, the L-Curve, the Stein's unbiased risk estimate, and the generalized cross-validation to choose the regularization parameters. After the incorporation of these criteria to MUSIC and MARIA, respectively, their capabilities are first analyzed through simulations, and later on, utilizing real data acquired from an urban area.

Optimal combinations of Tikhonov regularization orders for IHCPs

In achieving a regularized solution of inverse heat conduction problems (IHCPs), Tikhonov Regularization works based on adding either a zeroth-, a first-, or a second-order term to the sum of squared errors function. In other words, considering combinations of these terms are not often considered. This work investigates employing standard optimization techniques in order to obtain optimal regularization parameters when combinations of these three regularization terms are used. Five different heat pulses are used as test cases: step, triangular, quadratic, quartic, and half-sine. The criterion used to find the optimal value for the regularization parameters is sum of the squares of deviations between the estimated heat flux and the exact heat flux pulse. A hybrid method which utilizes both the Genetic Algorithm and the Pattern Search is used for the optimization through functions defined in MATLAB software. Moreover, a general case containing all five heat flux test cases is considered in finding the optimal combination of three versions of Tikhonov method. All separate and general optimal combined models are used in estimating five pulse heat flux functions, and for each case, the RMS errors are calculated to give an insight toward the combined Tikhonov regularization technique. The proposed approach is also used to recover the surface heat flux using measured temperature data from an experiment. Some discussion about the Morozov discrepancy principle for this application is considered at the end.

Chebyshev pseudospectral method in the reconstruction of orthotropic conductivity

In this paper, we present a method to reconstruct the spatially varying conductivity tensor in isotropic and orthotropic materials, involved in a two-dimensional transient anisotropic model with Robin boundary conditions. For the reconstruction, the partial differential equation is solved by a semi-discrete method that combines a pseudospectral collocation method for spatial variables and Crank–Nicolson for time. The conductivity tensor is reconstructed through a non-linear least-squares problem solved by Levenberg–Marquardt method (LMM), along with Morozov's discrepancy principle as stopping rule to cope with noise in the data. Unlike classic LMM implementations that mitigate poor conditioning in calculating iterates using nonsingular diagonal scaling matrices, in this paper, singular regularization matrices are used. The impact of such a modification is illustrated with numerical experiments using discrete differential operators as scaling matrices. Numerical results show that accurate conductivity values can be obtained using a fairly small number of discretization points at a very low computational cost.

Regularized minimal residual method for permittivity reconstruction in microwave imaging

AbstractIn this paper, a regularized reconstruction based on the minimal residual method is proposed for microwave imaging applications. The method provides optimum regularization parameter to estimate the distribution of permittivity values of unknown scatterers under test. Initially, the method is applied to Born approximated linear model for weak scatterers. The performance of this approach is compared with the commonly adopted Morozov's discrepancy principle used in conjunction with the Tikhonov regularization. The effectiveness of the method is assisted by simulating the various numerical examples of synthetic and experimental data. The results of numerical simulations validate that the proposed method is highly effective. Thereafter, a non‐linear inverse problem based on the inexact Newton method is examined for the estimation of strong scatterers. Here also, the proposed method is found to be helpful in terms of improving the image accuracy.

A dual approach to Kohn–Vogelius regularization applied to data completion problem

This paper focuses on the data completion problem, which is well known to be an ill-posed inverse problem. We propose a dual regularization strategy without regularization parameter, based on the minimization of a functional which, instead of acting on the space of solutions, acts on the space of data. We prove the well-posedness of the minimization problem and the convergence of our regularized solution to the exact solution when the amount of noise on the data goes to 0. Moreover we prove that the regularized solution satisfies the well-known Morozov discrepancy principle. Our regularization strategy is closely related to the usual Kohn–Vogelius minimization strategy. In particular, we show that it allows not only to stably obtain a good reconstruction of the missing data of the Cauchy problem but also to determine the unique parameter of regularization for the Kohn–Vogelius strategy that satisfies the Morozov discrepancy principle. Finally, we present numerical results, in two and three dimensions, to highlight the efficiency of the proposed method.

Spectral representation of lattice gluon and ghost propagators at zero temperature

We consider the analytic continuation of Euclidean propagator data obtained from 4D simulations to Minkowski space. In order to perform this continuation, the common approach is to first extract the Källén-Lehmann spectral density of the field. Once this is known, it can be extended to Minkowski space to yield the Minkowski propagator. However, obtaining the Källén-Lehmann spectral density from propagator data is a well known ill-posed numerical problem. To regularise this problem we implement an appropriate version of Tikhonov regularisation supplemented with the Morozov discrepancy principle. We will then apply this to various toy model data to demonstrate the conditions of validity for this method, and finally to zero temperature gluon and ghost lattice QCD data. We carefully explain how to deal with the IR singularity of the massless ghost propagator. We also uncover the numerically different performance when using two—mathematically equivalent—versions of the Källén-Lehmann spectral integral.

A Numerical Method Based on the Jacobi Polynomials to Reconstruct an Unknown Source Term in a Time Fractional Diffusion-wave Equation

In this paper, we consider an inverse problem of identifying an unknown time dependent source function in a time-fractional diffusion-wave equation. First, some basic properties of the shifted Jacobi polynomials (SJPs) are presented. Then, the analytical solution of the direct problem is given and used to obtain an approximation of the unknown source function in a series of SJPs. Due to ill-posedness of this inverse problem, the Tikhonov regularization method with Morozov's discrepancy principle criterion is applied to find a stable solution. After that, an error bound is obtained for the approximation of the unknown source function. Finally, some numerical examples are provided to show effectiveness and robustness of the proposed algorithm.

Tikhonov-type regularization and the finite element method applied to point source estimation in the atmosphere

This work is concerned with the source identification problem in air pollution modeling. The forward problem is described by an advection-diffusion equation with physically realistic coefficients. It is solved by the combination of adaptive meshes and a stabilized finite element method. The source is estimated by Tikhonov-type regularization with a composite misfit function and an entropic penalty term. The composite misfit is given by the convex combination of ℓ1 and ℓ2 norms of the discrepancy between observed and predicted concentrations. Morozov's discrepancy principle is applied to simultaneously choose the regularization and the misfit combination parameters. Numerical experiments using Copenhagen field data is used to validate numerically the proposed methodology.

New methods for numerical estimation of convective heat transfer coefficient in circular ducts

Estimating the internal heat transfer coefficient in ducts is becoming increasingly important in many engineering applications. Unlike traditional estimation methods which seek to recover the unknown coefficient by using optimization tools such as iterative methods or linear system solvers, new solution methods based on forward/inverse analytical solutions have gained noticeable interest. Two recently proposed non traditional methods are an explicit Fourier analysis-based (EFA) reconstruction method of Bazán and Bedin and a filtered reciprocity functional (FRF) approach of Mocerino et al. Both methods provide approximate solutions that are expressed as a sum of weighted harmonics where the number of terms used in the reconstruction depends on the noise level in the data and is selected using the Morozov's discrepancy principle (DP). In the present work, these methods are analyzed to identify similarities and differences as well as to introduce improvements. In particular, limitations of the FRF method in determining the truncation parameter are discussed and circumvented. This gives rise to new methods for heat flux estimation that are easy to implement and extremely inexpensive when compared to existing techniques. To illustrate the effectiveness of the new methods we present numerical results on several test problems including an analytical model obtained by synthesizing experimental data reported by Bozzoli et al. and several benchmark problems reported by Mocerino et al.