Ill-posed Linear System Research Articles

Time varying discrete cosine transform based on shaping regularization and its application in seismic data analysis

Abstract The discrete cosine transform is a commonly used technique in the field of signal processing that employs cosine basis functions for signal analysis. Traditionally, the regression coefficients of the cosine basis functions are solely based on frequency information. This paper extends the regression coefficients associated with the cosine basis functions to take into account both frequency and time information, not just frequency information alone. This modification results in an ill-posed linear system, which requires regularization to prevent overfitting. To address this, the paper uses shaping regularization, a technique used to stabilize ill-posed problems. By doing so, the absolute values of these extended coefficients, now exhibiting variations in both frequency and time domains, are defined as the time-frequency distribution of that input signal. The numerical experiments conducted to validate this approach demonstrate that the proposed method yields a commendable time-frequency resolution. Consequently, it proves valuable for interpreting seismic data, showcasing its potential for applications in this field.

The regularization of spectral methods for hyperbolic Volterra integrodifferential equations with fractional power elliptic operator

Abstract In this study, a numerical approach is presented to solve the linear and nonlinear hyperbolic Volterra integrodifferential equations (HVIDEs). The regularization of a Legendre-collocation spectral method is applied for solving HVIDE of the second kind, with the time and space variables on the basis of Legendre-Gauss-Lobatto and Legendre-Gauss (LG) interpolation points, respectively. Concerning bounded domains, the provided HVIDE relation is transformed into three corresponding relations. Hence, a Legendre-collocation spectral approach is applied for solving this equation, and finally, ill-posed linear and nonlinear systems of algebraic equations are obtained; therefore different regularization methods are used to solve them. For an unbounded domain, a suitable mapping to convert the problem on a bounded domain is used and then apply the same proposed method for the bounded domain. For the two cases, the numerical results confirm the exponential convergence rate. The findings of this study are unprecedented for the regularization of the spectral method for the hyperbolic integrodifferential equation. The result in this work seems to be the first successful for the regularization of spectral method for the hyperbolic integrodifferential equation.

A numerical method for solving retrospective inverse problem of fractional parabolic equation

An effective numerical method for solving the inverse problem of time fractional parabolic equation is constructed in this paper. We use implicit finite difference method to discretize the problem and for the inverse problem we propose a conjugate gradient type regularization method to solve the discretized ill-posed linear systems. By comparing the different errors and the results with different perturbed data in several numerical experiments, our method is shown to solve the inverse problem, even with some noisy measurements, efficiently and stably.

A content-adaptive unstructured grid based regularized CT reconstruction method with a SART-type preconditioned fixed-point proximity algorithm

The goal of this study is to develop a new computed tomography (CT) image reconstruction method, aiming at improving the quality of the reconstructed images of existing methods while reducing computational costs. Existing CT reconstruction is modeled by pixel-based piecewise constant approximations of the integral equation that describes the CT projection data acquisition process. Using these approximations imposes a bottleneck model error and results in a discrete system of a large size. We propose to develop a content-adaptive unstructured grid (CAUG) based regularized CT reconstruction method to address these issues. Specifically, we design a CAUG of the image domain to sparsely represent the underlying image, and introduce a CAUG-based piecewise linear approximation of the integral equation by employing a collocation method. We further apply a regularization defined on the CAUG for the resulting ill-posed linear system, which may lead to a sparse linear representation for the underlying solution. The regularized CT reconstruction is formulated as a convex optimization problem, whose objective function consists of a weighted least square norm based fidelity term, a regularization term and a constraint term. Here, the corresponding weighted matrix is derived from the simultaneous algebraic reconstruction technique (SART). We then develop a SART-type preconditioned fixed-point proximity algorithm to solve the optimization problem. Convergence analysis is provided for the resulting iterative algorithm. Numerical experiments demonstrate the superiority of the proposed method over several existing methods in terms of both suppressing noise and reducing computational costs. These methods include the SART without regularization and with the quadratic regularization, the traditional total variation (TV) regularized reconstruction method and the TV superiorized conjugate gradient method on the pixel grid.

Numerical method for solving inverse source problem for Poisson equation

We consider an inverse problem for the Poisson equation [Formula: see text] in the square [Formula: see text] which consists of determining the source [Formula: see text] from boundary measurements. Such problem is ill-posed. We restrict ourselves to a class of functions [Formula: see text]. To illustrate our method, we first assume that [Formula: see text] and [Formula: see text] are known functions with partial data at the boundary. For the reconstruction, we consider approximations by the Fourier series, therefore we obtain an ill-posed linear system which requires a regularization strategy. In the general case, we propose an iterative algorithm based on the full data at the boundary. Finally, some numerical results are presented to show the effectiveness of the proposed reconstruction algorithms.

Regularized iterative method for ill-posed linear systems based on matrix splitting

In this paper, the concept of matrix splitting is introduced to solve a large sparse ill-posed linear system via Tikhonov?s regularization. In the regularization process, we convert the ill-posed system to a well-posed system. The convergence of such a well-posed system is discussed by using different types of matrix splittings. Comparison analysis of both systems are studied by operating certain types of weak splittings. Further, we have extended the double splitting of [Song J. and Song Y, Calcolo 48(3), 245-260, 2011] to double weak splitting of type II for nonsingular symmetric matrices. In addition to that, some more comparison results are presented with the help of such weak double splittings of type I and type II.

UAV Tomographic Synthetic Aperture Radar for Landmine Detection

The development of the Unmanned Aerial Vehicle (UAV) and communication systems contributed to the availability of more applications using UAVs in military and civilians purposes. Anti-personnel landmines deployed by militia groups in conflict zones are a life threat for civilians and need cautious handling while removing. The UAV Tomographic Synthetic Aperture Radar (TSAR) can reconstruct three-dimension images of the investigation domain to prescreen nonmetallic landmines. A nonmetallic landmine cannot be detected using conventional ground penetrating radars when the scattering field is undetected due to the dielectric permittivity. In this paper, imaging the underground for detecting landmine using TSAR is proposed. The TSAR has the capability of prosing the data in discrete mode regardless of the altitude of UAV’s radar. A landmine is always buried less than a feet depth. L-band frequency is used to provide high resolution and to penetrate deep in dry soil. More than one UAVs are used to multistatic scan the investigation space. The geometric diversity of multistatic distribution of the sensors will provide more information about the buried nonmetallic landmines, certain features, and their location. The data collected from the sensors will align with the geolocation data obtained from the UAV’s system for processing. Dynamic flying can be used to predict the electromagnetic response of the scattering field to create a dynamic matching filter using the Green’s function under first-order Born approximation. The occurring air-soil interference has been removed as an unwanted reflection from the ground while keeping the signal coming from underground. Using the Born approximation assumption created an ill-posed linear system solved by the Conjugate Gradient algorithm. Simulation results are presented to validate the method.

Proximal splitting methods for fluorescence molecular tomography

The inverse problems in fluorescence molecular tomography involve finding stable and meaningful solutions to under-determined and ill-posed linear systems of equations. A promising approach consists of minimizing an objective function, which includes a quadratic data-fidelity term combined with a nonquadratic and nonsmooth convex regularizer. Choosing l1-norm as an example of this regularizer, our paper proposes three proximal splitting algorithms for the regularized output least-squares formulation of these problems. We evaluate costs of parameter initialization and reconstruction speeds of the three algorithms that split the original problem in different ways, including two types of techniques based on the forward-backward method and the primal-dual method, respectively. Extensive numerical experiments show that in a wide range of test cases (covering different distances between fluorescent targets, noise levels and numbers of sources), all of the three algorithms perform well and produce very nearly the same reconstructed images, with the primal-dual-type algorithm being significantly faster (in terms of CPU time taken by both initialization and reconstruction) than two variants of the forward-backward-type one, as well as two algorithms with publicly available implementations.

Direct recovery of wave-penetrable scatterers in a stratified ocean waveguide

We shall derive a direct sampling method for the recovery of wave-penetrable inhomogeneous scatterers in a three-dimensional stratified ocean waveguide. The proposed method is simple, straightforward and computationally efficient since no matrix inversions, optimizations or solutions of ill-posed linear systems are involved. Moreover, it is highly tolerant to noise and is applicable with a few scattered fields corresponding to only two or four incidences. In addition, a priori knowledge of either the number of disconnected components or the physical features of the unknown obstacles is not required by the direct sampling method. A mathematical derivation is provided for its validation and several extensive numerical experiments are shown to evaluate the effectiveness and feasibility of the newly proposed method.

The inverse solution of the coupled nonlinear reaction–diffusion equations by the Haar wavelets

ABSTRACTIn this paper, a numerical method is proposed for the numerical solution of the coupled nonlinear reaction–diffusion equations with suitable initial and boundary conditions by using the Haar wavelet method to determine the unknown boundary conditions. More precisely, we apply the Haar wavelet method for discretizing the space derivatives and then use a quasilinearization technique to linearize the nonlinear term in the equations. This process generates an ill-posed linear system of equations. Hence, to regularize the resultant ill-posed linear system of equations, we employ the Tikhonov regularization method to obtain a stable numerical approximation to the solution. We also prove the convergence of order one (i.e. ) and discuss the error estimation and stability computation for the proposed method. Finally, we report some numerical results which in compared with the finite difference method and the radial basis function method show the efficiency and capability of the proposed method.

A harmonic polynomial method with a regularization strategy for the boundary value problems of Laplace’s equation

This paper concerns a boundary value problem of Laplace’s equation, which is solved by determining the unknown coefficients in the expansion of harmonic polynomials. A regularization method is proposed to tackle the resulting ill-posed linear system. The stability and convergence results are provided and a validating numerical experiment is presented.

Detecting buried wave-penetrable scatterers in a two-layered medium

In a two-layered medium, we prove that a buried inhomogeneous scatterer is uniquely determined from the wave field data measured in the upper half-space with respect to many incident point sources. Moreover, we extend the multilevel sampling method in Liu and Zou (2013) to numerically reconstruct the buried scatterer applying only few incident fields and partial scattered data. The extended recovery scheme only involves matrix–vector operations and does not need to solve any large-scale ill-posed linear systems or any optimization process. It is feasible to deal with the scatterers of different features and easy to implement, highly tolerant to noise and computationally quite cheap. We can regard it as an effective yet simple computational method to provide a reliable initial guess for the implementation in existing more accurate and refined optimization-type reconstruction algorithms.

A Novel Fast Tensor-Based Preconditioner for Image Restoration.

Image restoration is one of the main parts of image processing. Mathematically, this problem can be modeled as a large-scale structured ill-posed linear system. Ill-posedness of this problem results in low-convergence rate of iterative solvers. For speeding up the convergence, preconditioning usually is used. Despite the existing preconditioners for image restoration, which are constructed based on approximations of the blurring matrix, in this paper, we propose a novel preconditioner with a different viewpoint. Here, we show that image restoration problem can be modeled as a tensor contractive linear equation. This modeling enables us to propose a new preconditioner based on an approximation of the blurring tensor operator. Due to the particular structure of the blurring tensor for zero boundaries, we show that the truncated higher order singular value decomposition of the blurring tensor is obtained very fast and so could be used as a preconditioner. Experimental results confirm the efficiency of this new preconditioner in image restoration and its outperformance in comparison with the other well-known preconditioners.

Numerical solution of two-dimensional inverse force function in the wave equation with nonlocal boundary conditions

In this paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the inverse time-dependent force function in the wave equation on regular and irregular domains. The SMRPI is developed for identifying the force function which satisfies in the wave equation subject to the integral overspecification over a portion of the spatial domain or to the overspecification at a point in the spatial domain. This method is based on erudite combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of SMRPI. Since the problem is known to be ill-posed, Thikhonov regularization strategy is employed to solve effectively the discrete ill-posed resultant linear system. Three numerical examples are tested to show that numerical results are accurate for exact data and stable with noisy data.

Cubic B-spline method for the solution of an inverse parabolic system

In this paper, a numerical method is proposed for the numerical solution of a linear system with suitable initial and boundary conditions using the cubic B-spline collocation scheme to determine the unknown boundary condition. We apply the cubic B-spline for the spatial variable and the derivatives which, generate an ill-posed linear system of equations. In this regard, to overcome, to this drawback, we employ the Tikhonov regularization method for solving the resultant linear system. It is proved that the proposed method has the order of convergence . Also, the conditional stability using the Von-Neumann method is established under suitable assumptions. Finally, some numerical results are reported to show the efficiency and robustness of the proposed approach for solving the inverse problems.

Forward-backward pursuit algorithm for Cerenkov luminescence tomography.

Cerenkov luminescence tomography (CLT) is a powerful imaging technique that allows dynamically and three-dimensionally resolving the metabolic process of radiopharmaceuticals. It uses optical method to detect radiopharmaceuticals with low cost and high sensitivity. However, because of the strong absorption and scatter of biological tissues, the reconstruction of CLT is always converted to an ill-posed linear system which is hard to solve. An accurate and fast reconstruct algorithm becomes a current issue. The traditional reconstruction algorithm based on l2 norm regularization is too smooth and with low accuracy. Some novel sparse reconstruction algorithm has satisfying accuracy and convergence rate, but lose its accuracy for multi-source situation. In this work, a novel CLT method based on forward-backward greedy algorithm is proposed to solve the ill-posed problem. Digital simulations and in vivo experiment were conducted to test the algorithm. The reconstruct results were compared with traditional orthogonal matching pursuit (OMP) algorithm and Tikhonov algorithm. Both the Digital simulations and in vivo experiment show that this approach can reconstruct the distribution of radiopharmaceuticals effectively and accurately.

Numerical solutions of 3D Cauchy problems of elliptic operators in cylindrical domain using local weak equations and radial basis functions

ABSTRACTThis paper is concerned with the numerical solutions of 3D Cauchy problems of elliptic differential operators in the cylindrical domain. We assume that the measurements are only available on the outer boundary while the interior boundary is inaccessible and the solution should be obtained from the measurements from the outer layer. The proposed discretization approach uses the local weak equations and radial basis functions. Since the Cauchy problem is known to be ill-posed, the Thikhonov regularization strategy is employed to solve effectively the discrete ill-posed resultant linear system of equations. Numerical results of a different kind of test problems reveal that the method is very effective.

Reconstruction of inaccessible boundary value in a sideways parabolic problem with variable coefficients—Forward collocation with finite integration method

We investigate a sideways problem of reconstructing an inaccessible boundary value for parabolic equation with variable coefficients. Formulating the sideways problem into a sequence of well-posed direct problems (DP) and a system of Ordinary Differential Equations (ODE), we combine the recently developed finite integration method (FIM) with radial basis functions (RBF) to iteratively obtain the solution of each DP by solving an ill-posed linear system. The use of numerical integration instead of finite quotient formula in FIM completely avoids the well known roundoff-discretization errors problem in finite difference method and the use of RBF as forward collocation method (FCM) gives a truly meshless computational scheme. For tackling the ill-posedness of the sideways problem, we adapt the traditional Tikhonov regularization technique to obtain stable solution to the system of ODEs. Convergence analysis is then derived and error estimate shows that the error tends to zero when perturbation δ→0. We can then obtain highly accurate and stable solution under some assumptions. Numerical results validate the feasibility and effectiveness of the proposed numerical algorithms.

Sparsity-based signal processing for noise radar imaging

Noise radar systems transmitting incoherent signal sequences have been proposed as powerful candidates for implementing compressively sampled detection and imaging systems. This paper presents an analysis of compressively sampled noise radar systems by formulating ultrawideband (UWB) compressive noise radar imaging as a problem of inverting ill-posed linear systems with circulant system matrices. The nonlinear nature of compressive signal recovery presents challenges in characterizing the performance of radar imaging systems. The suitability of noise waveforms for compressive radar is demonstrated using phase transition diagrams and transform point spread functions (TPSFs). The numerical simulations are designed to provide a compelling validation of the system. Nonidealities occurring in practical compressive noise radar systems are addressed by studying the properties of the transmit waveform. The results suggest that waveforms and system matrices that arise in practical noise radar systems are suitable for compressive signal recovery. Field imaging experiments on various target scenarios using a UWB millimeter wave noise radar validate our analytical results and the theoretical guarantees of compressive sensing.

Proposed and Method Presentation of Bridge Model Updating

This article briefly reviews the two methods of finite element model (FEM) updating, such as direct matrix methods and the sensitivity-based model updating methods. In addition, the problem in bridge structure model updating often needs to solve large-scale ill-posed linear systems. Therefore, two regularization methods of Tikhonov and TSVD were introduced. Meanwhile, for these systems, it is proposed that the application of the two kinds of regularization method to solve the problem which the test data contaminated by noise may rarely lead to a physically meaningful updated model.