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In this article, the mathematical study of the problem of identifying the space-dependent source term, in transport processes given by an n -dimensional linear parabolic equation, from space-dependent noisy measurements taken at an arbitrary fixed time is conducted. The problem is analytically solved using Fourier techniques, and it is shown that this solution is not stable. Three families of uniparametric regularization operators are proposed to address the instability of the solution. Each of them is designed to compensate for the factor that causes the instability of the inverse operator. Additionally, a selection rule for the regularization parameter is included, and an error bound for each estimation of Hölder type is obtained. Numerical examples of different characteristics are presented to demonstrate the benefits of the proposed strategies. [ABSTRACT FROM AUTHOR]

Doppler positioning, as an early form of positioning, has regained significant research interest in the context of Low Earth Orbit (LEO) satellites.Given the LEO mega-constellation scenario, the objective function of Doppler positioning manifests significant nonlinearity, leading to ill-conditioning challenges for prevalent algorithms like iterative least squares (LS) estimation, especially in cases where inappropriate initial values are selected. In this study, we investigate the causes of ill-posed problems from two perspectives. Firstly, we analyze the linearization errors of the Doppler observation equations in relation to satellite orbital altitude and initial value errors, revealing instances where traditional algorithms may fail to converge. Secondly, from an optimization theory perspective, we demonstrate the occurrence of convergence to locally non-unique solutions for Doppler positioning. Subsequently, to address these ill-conditioning issues, we introduce Tikhonov regularization terms in the objective function to constrain algorithm divergence, with a fitted model for the regularization coefficient. Finally, we conduct comprehensive simulation experiments in both dynamic and static scenarios to validate the performance of the proposed algorithm. On the one hand, when the initial values are set to 0, our algorithm achieves high-precision positioning, whereas the iterative LS fails to converge. On the other hand, in certain simulation scenarios, the iterative LS converges to locally non-unique solutions, resulting in positioning errors exceeding 50 km in the north and east directions, several hundred kilometers in the vertical direction, and velocity errors surpassing 120 m/s. In contrast, our algorithm demonstrates typical errors of a position error of 6.8462 m, velocity error of 0.0137 m/s, and clock drift error of 8.3746 × 10 − 6 s/s. This work provides an effective solution to the sensitivity issue of initial points in Doppler positioning and can serve as a reference for the algorithm design of Doppler positioning receivers with LEO mega-constellations. [ABSTRACT FROM AUTHOR]

This study aims to explore two classes of ill‐posed problems governed by a nonclassical fractional heat equation with an involution perturbation. To achieve a stable solution, we introduce a modified pseudo‐parabolic regularization method that involves a correction term through a mixed fractional derivation operator, resulting in a sequence of well‐posed problems that depend on a regularization parameter ε$$ \varepsilon $$. We rigorously demonstrate that the resulting approximate problems are well‐posed and present convergence results for this regularization approach. [ABSTRACT FROM AUTHOR]

Tau Toolbox is a mathematical library for the solution of integro-differential problems, based on the spectral Lanczos' Tau method. Over the past few years, a class within the library, called polynomial, has been developed for approximating functions by classical orthogonal polynomials and it is intended to be an easy-to-use yet efficient object-oriented framework. In this work we discuss how this class has been designed to solve linear ill-posed problems and we provide a description of the available methods, Tikhonov regularization and truncated singular value expansion. For the solution of the Fredholm integral equation of the first kind, which is built from a low-rank approximation of the kernel followed by a numerical truncated singular value expansion, an error estimate is given. Numerical experiments illustrate that this approach is capable of efficiently compute good approximations of linear discrete ill-posed problems, even facing perturbed available data function, with no programming effort. Several test problems are used to evaluate the performance and reliability of the solvers. The final product of this paper is the numerical solution of a first-kind integral equation, which is constructed using only two inputs from the user: the kernel and the right-hand side. [ABSTRACT FROM AUTHOR]

The modeling of many problems of practical interest leads to nonlinear ill-posed equations (for example, the parameter identification problem (see the Numerical section)). In this article, we introduce a new source condition (SC) and a new parameter choice strategy (PCS) for the Tikhonov regularization (TR) method for nonlinear ill-posed problems. The new PCS is introduced using a new SC to compute the regularization parameter (RP) before computing the regularized solution. The theoretical results are verified using a numerical example.

This paper deals with speeding up the convergence of a class of two-step iterative methods for solving linear systems of equations. To implement the acceleration technique, the residual norm associated with computed approximations for each sub-iterate is minimized over a certain two-dimensional subspace. Convergence properties of the proposed method are studied in detail. The approach is further developed to solve (regularized) normal equations arising from the discretization of ill-posed problems. The results of numerical experiments are reported to illustrate the performance of exact and inexact variants of the method on several test problems from different application areas. [ABSTRACT FROM AUTHOR]

We consider infinite-dimensional generalized Hilbert matrices of the form H i , j = d i d j x i + x j , where di are nonnegative weights and xi are pairwise distinct positive numbers. We state sufficient and, for monotonically rearrangeable xi, also necessary conditions for di, xi such that the induced operator from l 2 → l 2 and related operators are well-defined, bounded, or compact. Furthermore, we give conditions, when this operator is injective and ill-posed. [ABSTRACT FROM AUTHOR]

Tikhonov regularization technique is widely recognized as one of the most prevalent and well-established approaches for solving linear discrete ill-posed problems. The present study introduces two novel randomized iterative algorithms for the computation of numerical solutions to large-scale Tikhonov regularization problems. The first one applies the randomized Kaczmarz algorithm to an augmented regularized normal system of equations, the second one is an accelerated version of the first one by means of greedy probability criterion. In theory, we establish some convergence results for these two algorithms. Numerical experiments demonstrate the convergence properties and illustrate the performances of these two algorithms. [ABSTRACT FROM AUTHOR]

The modeling of many problems of practical interest leads to nonlinear ill-posed equations (for example, the parameter identification problem (see the Numerical section)). In this article, we introduce a new source condition (SC) and a new parameter choice strategy (PCS) for the Tikhonov regularization (TR) method for nonlinear ill-posed problems. The new PCS is introduced using a new SC to compute the regularization parameter (RP) before computing the regularized solution. The theoretical results are verified using a numerical example. [ABSTRACT FROM AUTHOR]

This paper presents a universal numerical scheme tailored for tackling linear integral, integro-differential, and both initial and boundary value problems of ordinary differential equations. The numerical scheme is readily adapted for resolving ill-posed problems. Central to our approach is high-degree piecewise-polynomial approximation to the exact solution. We illustrate the accuracy and stability of our numerical solutions in the presence of noise through illustrative examples. Additionally, we demonstrate that proposed regularization being applied to high-degree interpolation, effectively eliminates Runge's phenomenon. [ABSTRACT FROM AUTHOR]

This study investigates a numerical method based on the Jacobi–Gauss quadrature for solving Fredholm integral equations of the first kind with a weakly singular kernel by combining the Tikhonov regularization and projection methods. This numerical method reduces the solution of the weakly singular integral equations of the first kind to the solution of a linear system of algebraic equations. The theoretical analysis of the proposed technique is provided. Finally, several tests are presented to show the validity and efficiency of this approach. [ABSTRACT FROM AUTHOR]

In this paper, we apply the discrete Legendre Galerkin and multi-Galerkin methods to find the approximate solution of the Tikhonov regularized equation of the Fredholm integral equations of the first kind. We evaluate the error bounds for the approximate solutions with the exact solution in the infinity norm. We provide an a priori parameter choice strategy to find the convergence rates under the infinity norm. Since smoothness of the solution is not known in applied problems, we discuss an adaptive parameter choice rule to choose the regularization parameter, and then using this regularization parameter, we obtain the order of convergence in infinity norm. We give test examples to justify the theoretical estimates. [ABSTRACT FROM AUTHOR]

A weakly infeasible semidefinite program (SDP) has no feasible solution, but it has approximate solutions whose constraint violation is arbitrarily small. These SDPs are ill-posed and numerically often unsolvable. They are also closely related to "bad" linear projections that map the cone of positive semidefinite matrices to a nonclosed set. We describe a simple echelon form of weakly infeasible SDPs with the following properties: (i) it is obtained by elementary row operations and congruence transformations, (ii) it makes weak infeasibility evident, and (iii) it permits us to construct any weakly infeasible SDP or bad linear projection by an elementary combinatorial algorithm. Based on our echelon form, we generate a library of computationally very difficult SDPs. Finally, we show that some SDPs in the literature are in our echelon form, for example, the SDP from the sum-of-squares relaxation of minimizing Motzkin's famous polynomial. [ABSTRACT FROM AUTHOR]

According to common knowledge, the first kind of Volterra integral equation is an example of a problem that is not well-posed. This kind of equation is encountered in several problems of science, and it is useful in a variety of fields, including control theory, nuclear reactors, and ecological systems, where it can be used to evolutionary processes. In this article, we'll present an effective and accurate technique to converting VK1 into those of the second kind (VK2) and the kernel must not be zero for the conversion process to be effective, and we will find the approximation solutions of them by using the decomposition of Taylor series with Nyström method (Trapezoidal and Simpson's rules). In finality, we will present a variety of numerical examples to demonstrate that the conversion that has been proposed is both successful and stable. [ABSTRACT FROM AUTHOR]

In this manuscript we propose and analyze an implicit two-point type method (or inertial method) for obtaining stable approximate solutions to linear ill-posed operator equations. The method is based on the iterated Tikhonov (iT) scheme. We establish convergence for exact data, and stability and semi-convergence for noisy data. Regarding numerical experiments we consider: (i) a 2D inverse potential problem, (ii) an image deblurring problem; the computational efficiency of the method is compared with standard implementations of the iT method. [ABSTRACT FROM AUTHOR]

The aim of this study is to present a new regularized extreme learning machine (ELM) algorithm that can perform variable selection based on the simultaneous use of both ridge and Liu regressions in order to cope with some disadvantages of ELM and its variants such as instability and poor generalization performance and lack of sparsity. The proposed algorithm was compared with the classical ELM as well as the variants based on ridge, Liu, Lasso and Elastic Net approaches by cross-validation process and best tuning parameter over seven different real-world applications and their performances were presented comparatively. The proposed algorithm outperformed ridge, Lasso and Elastic Net algorithms in training performance prediction (average 40%) and stability (average 80%) and in test performance prediction (average 20%) and stability (60%) in the majority of the data. In addition, the proposed ELM was found to be more compact (better sparsity capability) with lower norm values. The results confirmed that the proposed ELM presents more stable and sparse solutions with better generalization performance than any other algorithm under favorable conditions. The findings based on experimental study via real-world applications indicate that the proposed ELM provides effective solutions to the mentioned drawbacks and yields more stable and sparse performance with better generalization capability than its competitors. Consequently, the proposed algorithm represents a powerful alternative both regression and classification tasks in machine learning field due to its theoretical flexibility. [ABSTRACT FROM AUTHOR]

Objective: This work presents an implementation of a stable algorithm that recovers sources located at the boundary separating two homogeneous media in field-programmable gate arrays. Two loop unrolling architectures were developed and analyzed for this purpose. This inverse source problem is ill-posed due to numerical instability, i.e., small errors in the measurement can produce significant changes in the source location. Methodology: To handle the numerical instability when recovering these sources, the Tikhonov regularization method in combination with the Fourier series truncation method are applied in the stable algorithm. This stable algorithm is implemented in two different architectures developed in this work: The first architecture (Mode 1) allows for different operating speeds, which is an advantage depending on whether we work with fast or slow signals. The second one (Mode 2) reduces resource consumption by exploiting the characteristics of the source identification algorithm, which is an advantage for multichannel problems such as inverse electrocardiography or electroencephalography. Results: The architectures were tested on four devices of the 7 Series of Xilinx: Spartan-7 xc7s100fgga484, Virtex-7 xc7v585tffg1157, Kintex-7 xc7k70tfbg484, and Artix-7 xc7a35tcpg236. The two hardware implementations of the stable algorithm were validated using synthetic examples implemented in MATLAB, which shows the advantages of each architecture. Contributions: We developed two efficient architectures based on a loop unrolling design for source identification problems. These are effective strategies to divide and assign tasks to the configurable hardware, and they appear as an appropriate technique for implementing the algorithm. The first one is simple and allows for different operating speeds. The second one uses a control system based on multiplexors that reduce resource consumption and complexity of the design and can be used for multichannel problems. From the numerical test, we found the regularization parameters. The synthetic examples developed here can be considered for similar problems and can be extended to concentric spheres. [ABSTRACT FROM AUTHOR]

Let X be a Banach space with norm ∥ ⋅ ∥ . Let A : D ⁢ (A) ⊂ X → X be an (possibly unbounded) operator that generates a uniformly bounded holomorphic semigroup. Suppose that ε > 0 and T > 0 are two given constants. The backward parabolic equation of finding a function u : [ 0 , T ] → X satisfying u t + A ⁢ u = 0 , 0 < t < T , ∥ u ⁢ (T) - φ ∥ ⩽ ε , for φ in X, is regularized by the generalized Sobolev equation u α ⁢ t + A α ⁢ u α = 0 , 0 < t < T , u α ⁢ (T) = φ , where 0 < α < 1 and A α = A ⁢ (I + α ⁢ A b ) - 1 with b ⩾ 1 . Error estimates of the method with respect to the noise level are proved. [ABSTRACT FROM AUTHOR]

It is well known that the backward heat conduction problem of recovering the temperature u ⁢ (⋅ , t) at a time t ≥ 0 from the knowledge of the temperature at a later time, namely g := u ⁢ (⋅ , τ) for τ > t , is ill-posed, in the sense that small error in g can lead to large deviation in u ⁢ (⋅ , t) . However, in the case of a time fractional backward heat conduction problem (TFBHCP), the above problem is well-posed for t > 0 and ill-posed for t = 0 . We use this observation to obtain stable approximate solutions as solutions for t ∈ (0 , τ ] with t as regularization parameter for approximating the solution at t = 0 , and derive error estimates under suitable source conditions. We shall also provide some numerical examples to illustrate the approximation properties of the regularized solutions. [ABSTRACT FROM AUTHOR]

Background. The article is devoted to approximate methods for solving the phase problem for one-dimensional and two-dimensional signals. The cases of continuous and discrete signals are considered. The solution of the phase problem consists of two stages. At the first stage, the original signal is restored. At the second stage, the Fourier transform of the reconstructed signal is calculated and the phase of the signal spectrum is approximately calculated. Materials and methods. The construction and justification of the computing scheme is based on a continuous method for solving nonlinear operator equations, based on the theory of stability of solutions to ordinary differential equation systems. The method is stable under perturbations of the parameters of the mathematical model and, when solving nonlinear operator equations, does not require the reversibility of the Gato (or Freshe) derivatives of nonlinear operators. Results and conclusions. To restore the original signal, spline-collocation schemes with splines of zero and first orders are proposed. Computing schemes are implemented by a continuous method for solving nonlinear operator equations.

The paper describes a grid method for solving an ill-posed problem for the Fredholm equation of the first kind using the A. N. Tikhonov regularizer. The convergence theorem for this method was formulated and proved. A procedure for thickening grids with a simultaneous increase in digit capacity of calculations is proposed.

Doppler positioning, as an early form of positioning, has regained significant research interest in the context of Low Earth Orbit (LEO) satellites.Given the LEO mega-constellation scenario, the objective function of Doppler positioning manifests significant nonlinearity, leading to ill-conditioning challenges for prevalent algorithms like iterative least squares (LS) estimation, especially in cases where inappropriate initial values are selected. In this study, we investigate the causes of ill-posed problems from two perspectives. Firstly, we analyze the linearization errors of the Doppler observation equations in relation to satellite orbital altitude and initial value errors, revealing instances where traditional algorithms may fail to converge. Secondly, from an optimization theory perspective, we demonstrate the occurrence of convergence to locally non-unique solutions for Doppler positioning. Subsequently, to address these ill-conditioning issues, we introduce Tikhonov regularization terms in the objective function to constrain algorithm divergence, with a fitted model for the regularization coefficient. Finally, we conduct comprehensive simulation experiments in both dynamic and static scenarios to validate the performance of the proposed algorithm. On the one hand, when the initial values are set to 0, our algorithm achieves high-precision positioning, whereas the iterative LS fails to converge. On the other hand, in certain simulation scenarios, the iterative LS converges to locally non-unique solutions, resulting in positioning errors exceeding 50 km in the north and east directions, several hundred kilometers in the vertical direction, and velocity errors surpassing 120 m/s. In contrast, our algorithm demonstrates typical errors of a position error of 6.8462 m, velocity error of 0.0137 m/s, and clock drift error of 8.3746 × 10−6 s/s. This work provides an effective solution to the sensitivity issue of initial points in Doppler positioning and can serve as a reference for the algorithm design of Doppler positioning receivers with LEO mega-constellations.

The present paper is devoted to identifying an inaccessible boundary condition for a fractional elliptic problem of Tricomi-Gellerstedt-Keldysh-type. Using the expansion Fourier method, the considered problem can be reformulated as an operator equation of the first kind. To construct a stabilized approximate solution we employ a variant of the iterative method. We also present error estimates between the exact solution and the regularized solution by the a priori and the a posteriori parameter choice rules. Finally, some numerical verifications on the efficiency and accuracy of the proposed algorithm is presented. [ABSTRACT FROM AUTHOR]

We consider the ill-posed problem of finding the position of the discontinuity lines of a function of two variables. It is assumed that the function is smooth outside the lines of discontinuity but has a discontinuity of the first kind on the line. At each node of a uniform grid with step 휏 , the mean values of the perturbed function on a square with side 휏 are known. The perturbed function approximates the exact function in the space subscript 퐿 2 superscript ℝ 2 . The perturbation level 훿 is assumed to be known. Previously, the authors investigated (accuracy estimates were obtained) global discrete regularizing algorithms for approximating the set of lines of discontinuity of a noisy function provided that the line of discontinuity of the exact function satisfies the local Lipschitz condition. In this paper, we introduce a one-sided Lipschitz condition and formulate a new, wider correctness class. New methods for localizing discontinuity lines are constructed that work on an extended class of functions. A convergence theorem is proved, and estimates of the approximation error and other important characteristics of the algorithms are obtained. It is shown that the new methods determine the position of the discontinuity lines with guarantee in situations where the standard methods do not work. [ABSTRACT FROM AUTHOR]

It is known that the standard Tikhonov regularization methods oversmoothen the solution x ^ of the ill-posed equation T (x) = y , so the computed approximate solution lacks many inherent details that are expected in the desired solution. To rectify this problem, Fractional Tikhonov Regularization (FTR) method have been introduced. Kanagaraj et al. (J Appl Math Comput 63(1):87–105, 2020), studied FTR method for solving ill-posed problems. Techniques are developed to study the Finite Dimensional FTR (FDFTR) method. We also study Raus and Gfrerer type discrepancy principle for FDFTR method and compare the numerical results with other discrepancy principles of the same type. [ABSTRACT FROM AUTHOR]

The present paper is devoted to identifying an inaccessible boundary condition for a fractional elliptic problem of Tricomi-Gellerstedt-Keldysh-type. Using the expansion Fourier method, the considered problem can be reformulated as an operator equation of the first kind. To construct a stabilized approximate solution we employ a variant of the iterative method. We also present error estimates between the exact solution and the regularized solution by the a priori and the a posteriori parameter choice rules. Finally, some numerical verifications on the efficiency and accuracy of the proposed algorithm is presented.

Numerous methods have been studied and discussed for solving ill-posed Volterra integral equations and ill-posed Fredholm integral equations, but rarely for both simultaneously. In this study, we focus on numerically solving the ill-posed Volterra-Fredholm integral equation of the first kind by replacing it with its perturbed counterpart. We employ Chebyshev polynomials of the first kind to solve the perturbed equation. Our findings suggest that this technical approach is superior to the regularization method of Tikhonov. It is simpler, less cumbersome, and this simplicity is demonstrated through various examples.

A data-driven type Kaczmarz method with uniformly convex constraints is derived and analyzed for solving ill-posed problems. By using a black box strategy, the training data are utilized to learn an operator, which is used to construct a data-driven term in iterative scheme. The introduced convex penalty is allowed to include L1 and total variation functionals, which contributes to recovering special features of solutions such as sparsity and piecewise constancy. In addition, the acceleration version by two-point strategy of the proposed method is also considered. The convergence and regularity of the method is established under suitable assumptions. The proposed method is validated through numerical experiments on both linear problems and nonlinear problems. The numerical results show that the introduced data-driven term helps to reduce the numerical consumptions and improve the reconstruction accuracy, especially for the case that the desired solution is well approximated by the training data. Meanwhile, the convex penalty of the proposed algorithm efficiently captures the special features of the solutions and removes the artifacts brought by the data-driven term when data information is not close to the true solution. [ABSTRACT FROM AUTHOR]

We analyse the Krylov solvability of inverse linear problems on Hilbert space H where the underlying operator is compact and normal. Krylov solvability is an important feature of inverse linear problems that has profound implications in theoretical and applied numerical analysis as it is critical to understand the utility of Krylov based methods for solving inverse problems. Our results explicitly describe for the first time the Krylov subspace for such operators given any datum vector g ∈ H , as well as prove that all inverse linear problems are Krylov solvable provided that g is in the range of such an operator. We therefore expand our knowledge of the class of Krylov solvable operators to include the normal compact operators. We close the study by proving an isomorphism between the closed Krylov subspace for a general bounded normal operator and an L 2 -measure space based on the scalar spectral measure. [ABSTRACT FROM AUTHOR]

In this study, we present an advanced numerical model for solving the 2D first kind Fredholm integral equations, which are well known to be ill-posed problems. This numerical approach is built on the quadrature formula with the Lavrentiev regularization method. Under some essential assumptions, a comprehensive theoretical explanation of the presented numerical approach is provided. Finally, various numerical examples support the theoretical findings and demonstrate the accuracy of our method. [ABSTRACT FROM AUTHOR]

Objective: This work presents an implementation of a stable algorithm that recovers sources located at the boundary separating two homogeneous media in field-programmable gate arrays. Two loop unrolling architectures were developed and analyzed for this purpose. This inverse source problem is ill-posed due to numerical instability, i.e., small errors in the measurement can produce significant changes in the source location. Methodology: To handle the numerical instability when recovering these sources, the Tikhonov regularization method in combination with the Fourier series truncation method are applied in the stable algorithm. This stable algorithm is implemented in two different architectures developed in this work: The first architecture (Mode 1) allows for different operating speeds, which is an advantage depending on whether we work with fast or slow signals. The second one (Mode 2) reduces resource consumption by exploiting the characteristics of the source identification algorithm, which is an advantage for multichannel problems such as inverse electrocardiography or electroencephalography. Results: The architectures were tested on four devices of the 7 Series of Xilinx: Spartan-7 xc7s100fgga484, Virtex-7 xc7v585tffg1157, Kintex-7 xc7k70tfbg484, and Artix-7 xc7a35tcpg236. The two hardware implementations of the stable algorithm were validated using synthetic examples implemented in MATLAB, which shows the advantages of each architecture. Contributions: We developed two efficient architectures based on a loop unrolling design for source identification problems. These are effective strategies to divide and assign tasks to the configurable hardware, and they appear as an appropriate technique for implementing the algorithm. The first one is simple and allows for different operating speeds. The second one uses a control system based on multiplexors that reduce resource consumption and complexity of the design and can be used for multichannel problems. From the numerical test, we found the regularization parameters. The synthetic examples developed here can be considered for similar problems and can be extended to concentric spheres.

It is well known that ω -circulant matrices with ω ≠ 0 can be simultaneously diagonalized by a transform matrix, which can be factored as the product of a diagonal matrix, depending on ω , and of the unitary matrix F n associated to the Fast Fourier Transform. Hence, all the sets of ω -circulants form algebras whose computational power, in terms of complexity, is the same as the classical circulants with ω = 1 . However, stability is a delicate issue, since the condition number of the transform is equal to that of the diagonal part, tending to max { | ω | , | ω | − 1 } . For ω = 0 , the set of related matrices is still an algebra, which is the algebra of lower triangular matrices, but they do not admit a common transform since most of them (all except the multiples of the identity) are non-diagonalizable. In the present work, we review two modern applications, ranging from parallel computing in preconditioning of PDE approximations to algorithms for subdivision schemes, and we emphasize the role of such algebra. For the two problems, few numerical tests are conducted and critically discussed and the related conclusions are drawn. [ABSTRACT FROM AUTHOR]

We present an extension of the linear sampling method for solving the sound-soft inverse acoustic scattering problem with randomly distributed point sources. The theoretical justification of our sampling method is based on the Helmholtz-Kirchhoff identity, the cross-correlation between measurements, and the volume and imaginary near-field operators, which we introduce and analyze. Implementations in MATLAB using boundary elements, the SVD, Tikhonov regularization, and Morozov's discrepancy principle are also discussed. We demonstrate the robustness and accuracy of our algorithms with several numerical experiments in two dimensions. [ABSTRACT FROM AUTHOR]

This article considers a Cauchy problem of Helmholtz equations whose solution is well known to be exponentially unstable with respect to the inputs. In the framework of variational quasi-reversibility method, a Fourier truncation is applied to appropriately perturb the underlying problem, which allows us to obtain a stable approximate solution. The corresponding approximate problem is of a hyperbolic equation, which is also a crucial aspect of this approach. Error estimates between the approximate and true solutions are derived with respect to the noise level. From this analysis, the Lipschitz stability with respect to the noise level follows. Some numerical examples are provided to see how our numerical algorithm works well. [ABSTRACT FROM AUTHOR]

This manuscript considers a second-order semilinear control system in Hilbert space. By using Tikhonov regularization, we first construct the regularized control for the corresponding linear system and prove its approximate controllability. We then establish approximate controllability results using Gronwall's Lemma for the semilinear system. In addition, the error estimates and convergence rates involved in the regularization procedure are also obtained. Further, we give an example to demonstrate the results. [ABSTRACT FROM AUTHOR]

In this text, we presented the Nachaoui's iterative alternating method for solving the Cauchy problem governed by the convection-diffusion equation. The method is an iterative algorithm that alternates between solving two subproblems of the same type with boundary conditions of the Dirichlet and Neuman type on the inaccessible part of the boundary. The algorithm continues iterating until a convergence criterion is met. We discussed the convergence and computational efficiency of the method. The numerical results show that the method is computationally efficient and that the relaxation parameter can greatly reduce the number of iterations. [ABSTRACT FROM AUTHOR]

In this article we propose and analyze a Levenberg–Marquardt–Kaczmarz-type (LMK) method for obtaining stable approximate solutions to systems of ill-posed equations modeled by non-linear operators acting between Hilbert spaces. We extend to the LMK iteration the strategy proposed in [A. Leitão, F. Margotti and B. F. Svaiter, Range-relaxed criteria for choosing the Lagrange multipliers in the Levenberg–Marquardt method, IMA J. Numer. Anal. 41 2021, 4, 2962–2989] for choosing the Lagrange multipliers in the Levenberg–Marquardt (LM) method. Our main goal is to devise a simple (and easy to implement) strategy for computing the multiplier in each iterative step, such that the resulting LMK iteration is both stable and numerically efficient. Convergence analysis for the proposed LMK type method is provided, including convergence for exact data, stability and semi-convergence. Numerical experiments using real data are presented for a 2D parameter identification problem, namely the Electrical Impedance Tomography (EIT) problem. The mathematical model known as complete electrode model (EIT-CEM) is considered. The obtained numerical results validate the efficiency of the proposed LMK-type method. [ABSTRACT FROM AUTHOR]

We use piecewise polynomial basis functions to obtain the stable approximation solution of the Tikhonov regularized equation of the Fredholm integral equation of the first kind by utilizing multi-projection (multi-Galerkin and multi-collocation) methods. We evaluate the error bounds for the approximate solution with the exact solution in infinity norm. We provide an a priori parameter choice strategy under infinity norm. In addition to determining the regularization parameter, we discuss Arcangeli's discrepancy principle and calculate the convergence rates in infinity norm. We give test examples to validate the theoretical results. [ABSTRACT FROM AUTHOR]

To solve the ill-posed integral equations, we use the regularized collocation method. This numerical method is a combination of the Legendre polynomials with non-stationary iterated Tikhonov regularization with fixed parameter. A theoretical justification of the proposed method under the required assumptions is detailed. Finally, numerical experiments demonstrate the efficiency of this method. [ABSTRACT FROM AUTHOR]

Recently, Reddy (2018) has studied a posteriori parameter choice rules for the weighted Tikhonov regularization scheme. In this article, we consider the finite dimensional weighted Tikhonov scheme and discuss the convergence analysis of the scheme. We introduce a class of parameter choice strategies to choose the regularization parameter in the finite dimensional weighted Tikhonov scheme; and derive the optimal rate of convergence O δ α + 1 α + 2 for the scheme based on the proposed strategies. Results of numerical experiments are documented to illustrate the performance of the scheme with an efficient finite dimensional approximation of an operator. [ABSTRACT FROM AUTHOR]

We consider the unique continuation (data assimilation) problem for the Helmholtz equation and study its numerical approximation based on physics-informed neural networks (PINNs). Exploiting the conditional stability of the problem, we first give a bound on the generalization error of PINNs. We then present numerical experiments in 2d for different frequencies and for geometric configurations with different stability bounds for the continuation problem. The results show that vanilla PINNs provide good approximations even for noisy data in configurations with robust stability (both low and moderate frequencies), but may struggle otherwise. This indicates that more sophisticated techniques are needed to obtain PINNs that are frequency-robust for inverse problems subject to the Helmholtz equation.