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In this work, we consider the problem of identifying the time independent source for full parabolic equations in Rn from noisy data. This is an ill-posed problem in the sense of Hadamard. To compensate the factor that causes the instability, a family of parametric regularization operators is introduced, where the rule to select the value of the regularization parameter is included. This rule, known as regularization parameter choice rule, depends on the data noise level and the degree of smoothness that it is assumed for the source. The proof for the stability and convergence of the regularization criteria is presented and a Hölder type bound is obtained for the estimation error. Numerical examples are included to illustrate the effectiveness of this regularization approach.

This communication describes version 4.0 of Regularization Tools, a Matlab package for analysis and solution of discrete ill-posed problems. The new version allows for under-determined problems, and it is expanded with several new iterative methods, as well as new test problems and new parameter-choice methods.

The focus of this paper is the nonparametric estimation of an instrumental regression function defined by conditional moment restrictions that stem from a structural econometric model E[Y — (Z) | W] = 0, and involve endogenous variables Y and Z and instruments W. The function is the solution of an ill-posed inverse problem and we propose an estimation procedure based on Tikhonov regularization. The paper analyzes identification and overidentification of this model, and presents asymptotic properties of the estimated nonparametric instrumental regression function.

Outlier removal is a fundamental data processing task to ensure the quality of scanned point cloud data (PCD), which is becoming increasing important in industrial applications and reverse engineering. Acquired scanned PCD is usually noisy, sparse and temporarily incoherent. Thus the processing of scanned data is typically an ill-posed problem. In the paper, we present a simple and effective method based on two geometrical characteristics constraints to trim the noisy points. One of the geometrical characteristics is the local density information and another is the deviation from the local fitting plane. The local density based method provides a preprocessing step, which could remove those sparse outlier and isolated outlier. The non-isolated outlier removal in this paper depends on a local projection method, which placing those points onto objects. There is no doubt that the deviation of any point from the local fitting plane should be a criterion to reduce the noisy points. The experimental results demonstrate the ability to remove the noisy point from various man-made objects consisting of complex outlier.

This paper is devoted to identifying an unknown source for a time-fractional diffusion equation in a general bounded domain. First, we prove the problem is non-well posed and the stability of the source function. Second, by using the Modified Fractional Landweber method, we present regularization solutions and show the convergence rate between regularization solutions and sought solution are given under a priori and a posteriori choice rules of the regularization parameter, respectively. Finally, we present an illustrative numerical example to test the results of our theory.

This paper identifies a source term depending on spatial variable in a heat equation from just part of the boundary conditions. The measurement data are specified at an internal moment of time. The ill-posedness of the problem is higher than most of the previous source identification problems. This is because the problem becomes a noncharacteristic Cauchy problem for the heat equation if the source term is given, which is known as severely ill-posed. The method of fundamental solutions (MFS) in conjunction with the classical Tikhonov regularization method is proposed to reconstruct a stable approximation. The fundamental solutions for the heat equation are spherically symmetric in spatial variable and satisfy the equation automatically, and thus only the boundary conditions need to be satisfied. This characteristic allows the discretization to be performed only on boundary-like geometry and improve the computational efficiency. In this paper, several numerical examples are listed to show the feasibility and effectiveness of the suggested method.

This paper describes solution methods for linear discrete ill-posed problems defined by third order tensors and the t-product formalism introduced in (Linear Algebra Appl 435:641–658, 2011). A t-product Arnoldi (t-Arnoldi) process is defined and applied to reduce a large-scale Tikhonov regularization problem for third order tensors to a problem of small size. The data may be represented by a laterally oriented matrix or a third order tensor, and the regularization operator is a third order tensor. The discrepancy principle is used to determine the regularization parameter and the number of steps of the t-Arnoldi process. Numerical examples compare results for several solution methods, and illustrate the potential superiority of solution methods that tensorize over solution methods that matricize linear discrete ill-posed problems for third order tensors.

Linear discrete ill-posed problems are difficult to solve numerically because their solution is very sensitive to perturbations, which may stem from errors in the data and from round-off errors introduced during the solution process. The computation of a meaningful approximate solution requires that the given problem be replaced by a nearby problem that is less sensitive to disturbances. This replacement is known as regularization. A regularization parameter determines how much the regularized problem differs from the original one. The proper choice of this parameter is important for the quality of the computed solution. This paper studies the performance of known and new approaches to choosing a suitable value of the regularization parameter for the truncated singular value decomposition method and for the LSQR iterative Krylov subspace method in the situation when no accurate estimate of the norm of the error in the data is available. The regularization parameter choice rules considered include several L-curve methods, Regińska’s method and a modification thereof, extrapolation methods, the quasi-optimality criterion, rules designed for use with LSQR, as well as hybrid methods.

Acoustic Doppler current profilers (ADCPs) are a global standard in observing flow fields in rivers, estuaries and the coastal ocean. To date, it remains a labor intensive challenge to isolate mean flow fields governed by river discharge, tides and atmospheric forcing on the one hand, from small‐scale turbulence, positioning imprecision, Doppler noise and erroneous backscatter, on the other hand. Here, we introduce a generic, new method of combining raw shipborne ADCP transect data with continuity and smoothness constraints to obtain better estimates of turbulence‐averaged three‐dimensional flow velocities in any type of open water body. The physical constraints are enforced with variable relative importance via generalized Tikhonov regularization. We demonstrate that in complex estuarine flow, this procedure allows for more reliable estimates of tidal amplitudes, phases and their gradients than what is possible with a purely data‐based approach, by testing the method's generalization capabilities and robustness to turbulence and measurement noise on a data set retrieved at a tidal channel junction. The increased adherence to mass conservation and robustness to noise of various kinds allows for more reliable and verifiable estimates of Reynolds‐averaged flow components, and subsequently, of terms in the Navier‐Stokes equations. Plain Language Summary Vessel‐mounted Acoustic Doppler Current Profilers (ADCPs) are often used to measure open‐channel flow and derived quantities such as river discharge, bed shear stress and tidal flow amplitudes. Depending on the measurement setup and the detail of processes and patterns to be resolved, this estimation procedure can be heavily influenced by measurement noise, positioning inaccuracy and turbulent fluctuations, reducing the accuracy of flow estimates. In the present work, we use physical laws to improve the reliability and robustness of the inversion algorithm that converts raw ADCP data to flow vectors. As a proof of concept, the method is applied to a real data set that was collected for cross‐river transects in a tidal channel. Key Points A generic, new framework for analyzing vessel‐mounted ADCP data is introduced and validated Flow velocity estimation is improved using physics‐informed generalized Tikhonov regularization As a proof of concept, the method is applied to cross‐sectional data collected in complex estuarine flow

Nonlinear inverse problems appear in many applications, and typically they lead to mathematical models that are ill-posed, i.e., they are unstable under data perturbations. Those problems require a regularization, i.e., a special numerical treatment. This book presents regularization schemes which are based on iteration methods, e.g., nonlinear Landweber iteration, level set methods, multilevel methods and Newton type methods.