Tikhonov Regularization Scheme Research Articles

Nonlinear Tikhonov regularization in Hilbert scales for inverse learning

In this paper, we study Tikhonov regularization scheme in Hilbert scales for a nonlinear statistical inverse problem with general noise. The regularizing norm in this scheme is stronger than the norm in the Hilbert space. We focus on developing a theoretical analysis for this scheme based on conditional stability estimates. We utilize the concept of the distance function to establish high probability estimates of the direct and reconstruction errors in the Reproducing Kernel Hilbert space setting. Furthermore, explicit rates of convergence in terms of sample size are established for the oversmoothing case and the regular case over the regularity class defined through an appropriate source condition. Our results improve upon and generalize previous results obtained in related settings.

Fuzzy Levenberg-Marquardt damping factor update strategy and Tikhonov regularization applied to a coupled conduction-radiation inverse heat transfer problem

The recently proposed Fuzzy Levenberg-Marquardt Damping Factor Updating Strategy (FLM) is used along with Tikhonov Regularization (TR) in order to solve a coupled conduction-radiation function estimation inverse problem. FLM consists of updating the Levenberg-Marquardt (LM) damping factor with the assist of Fuzzy Logic. To test the FLM algorithm combined with TR, a conduction-radiation problem is addressed, and it serves as test bed to the numerical experiments, where the objective is to obtain estimatives of parameters, such as the spatially variable scattering albedo, the thermal conductivity and the optical thickness, simultaneously, which is often performed separately. Two main test cases are presented, one considering only the radiation problem, used as benchmark, in order to investigate noise levels and hyperparameters. The second main test case deals with the coupled conduction-radiation heat transfer problem for different functional forms of the spatially variable scattering albedo. For all presented numerical experiments, good estimates were achieved for the sought functions and parameters. These results demonstrate that the methodology presented in this work, through the combination of the Fuzzy LM dampting factor updating strategy and the Tikhonov regularization scheme, may provide valuable tools for inverse heat transfer problems.

Adaptive Tikhonov regularization and dynamic control points for accurate shape parameter control of plasmas

Precise control of plasma shape parameters, such as elongation and triangularity is duly needed to achieve high-performance tokamak plasmas, for which we propose adaptive search schemes of (1) optimum regularization parameter for the Tikhonov regularization, and (2) control points to specify the key shape parameters such as elongation and triangularity. Many control points that exceed the number of actuator coils become an ill-conditioned problem, which is successfully resolved by Tikhonov regularization with adaptively optimized free parameters. Furthermore, we develop dynamically changed control points by using the Cauchy Condition Surface scheme, where elongation and triangularity become direct control values. By virtue of both the adaptive Tikhonov regularization scheme and the dynamic control point, we achieved accurate shape control with elongation up to 1.93 and triangularity up to 0.65 in JT-60SA, where the allowable speed for the change of the elongation is 0.1 s−1. We also verified the resilience of our developed our logics to the noises. The sequence of the result will contribute to enhance equilibrium controllability in upcoming JT-60SA experiments and provide the robust shape parameter control scheme for ITER and DEMO.

Simplified REGINN-IT method in Banach spaces for nonlinear ill-posed operator equations

Abstract In 2021, Z. Fu, Y. Chen and B. Han introduced an inexact Newton regularization (REGINN-IT) using an idea involving the non-stationary iterated Tikhonov regularization scheme for solving nonlinear ill-posed operator equations. In this paper, we suggest a simplified version of the REGINN-IT scheme by using the Bregman distance, duality mapping and a suitable parameter choice strategy to produce an approximate solution. The method is comprised of inner and outer iteration steps. The outer iterates are stopped by a Morozov-type stopping rule, while the inner iterate is executed by making use of the non-stationary iterated Tikhonov scheme. We have studied convergence of the proposed method under some standard assumptions and utilizing tools from convex analysis. The novelty of the method is that it requires computation of the Fréchet derivative only at an initial guess of an exact solution and hence can be identified as more efficient compared to the method given by Z. Fu, Y. Chen and B. Han. Further, in the last section of the paper, we discuss test examples to inspect the proficiency of the method.

Observational constraints on methane emissions from Polish coal mines using a ground-based remote sensing network

Abstract. Given its abundant coal mining activities, the Upper Silesian Coal Basin (USCB) in southern Poland is one of the largest sources of anthropogenic methane (CH4) emissions in Europe. Here, we report on CH4 emission estimates for coal mine ventilation facilities in the USCB. Our estimates are driven by pairwise upwind–downwind observations of the column-average dry-air mole fractions of CH4 (XCH4) by a network of four portable, ground-based, sun-viewing Fourier transform spectrometers of the type EM27/SUN operated during the CoMet campaign in May–June 2018. The EM27/SUN instruments were deployed in the four cardinal directions around the USCB approximately 50 km from the center of the basin. We report on six case studies for which we inferred emissions by evaluating the mismatch between the observed downwind enhancements and simulations based on trajectory calculations releasing particles out of the ventilation shafts using the Lagrangian particle dispersion model FLEXPART. The latter was driven by wind fields calculated by WRF (Weather Research and Forecasting model) under assimilation of vertical wind profile measurements of three co-deployed wind lidars. For emission estimation, we use a Phillips–Tikhonov regularization scheme with the L-curve criterion. Diagnosed by the emissions averaging kernels, we find that, depending on the catchment area of the downwind measurements, our ad hoc network can resolve individual facilities or groups of ventilation facilities but that inspecting the emissions averaging kernels is essential to detect correlated estimates. Generally, our instantaneous emission estimates range between 80 and 133 kt CH4 a−1 for the southeastern part of the USCB and between 414 and 790 kt CH4 a−1 for various larger parts of the basin, suggesting higher emissions than expected from the annual emissions reported by the E-PRTR (European Pollutant Release and Transfer Register). Uncertainties range between 23 % and 36 %, dominated by the error contribution from uncertain wind fields.

Nitrous Oxide Profiling from Infrared Radiances (NOPIR): Algorithm Description, Application to 10 Years of IASI Observations and Quality Assessment

Nitrous oxide (N2O) is the third most abundant anthropogenous greenhouse gas (after carbon dioxide and methane), with a long atmospheric lifetime and a continuously increasing concentration due to human activities, making it an important gas to monitor. In this work, we present a new method to retrieve N2O concentration profiles (with up to two degrees of freedom) from each cloud-free satellite observation by the Infrared Atmospheric Sounding Interferometer (IASI), using spectral micro-windows in the N2O ν3 band, the Radiative Transfer for TOVS (RTTOV) tools and the Tikhonov regularization scheme. A time series of ten years (2011–2020) of IASI N2O profiles and integrated partial columns has been produced and validated with collocated ground-based Network for the Detection of Atmospheric Composition Change (NDACC) and Total Carbon Column Observing Network (TCCON) data. The importance of consistency in the ancillary data used for the retrieval for generating consistent time series has been demonstrated. The Nitrous Oxide Profiling from Infrared Radiances (NOPIR) N2O partial columns are of very good quality, with a positive bias of 1.8 to 4% with respect to the ground-based data, which is less than the sum of uncertainties of the compared values. At high latitudes, the comparisons are a bit worse, due to either a known bias in the ground-based data, or to a higher uncertainty in both ground-based and satellite retrievals.

Active manipulation of Helmholtz scalar fields in an ocean of two homogeneous layers of constant depth

In this work, we prove the possibility of actively controlling the acoustic field in an ocean consisting of two homogeneous layers of constant depth using a surface source embedded in one of the layers. For a class of prescribed fields on some bounded control regions in either layer, we show the existence of a boundary input on the source, either acoustic pressure or normal velocity so that the propagated field approximates the prescribed fields. Also, we provide a constructive scheme to find the required boundary input using a Morozov discrepancy principle-based Tikhonov regularization scheme. Several numerical simulations are also provided to support our analyses.

Lsforce: A Python-Based Single-Force Seismic Inversion Framework for Massive Landslides

Abstract We present an open-source Python package, lsforce, for performing single-force source inversions of long-period (tens to hundreds of seconds) seismic signals. Although the software is designed primarily for landslides, it can be used for any single-force seismic source. The package allows users to produce estimates of the three-component time series of forces exerted on the Earth by a landslide with postprocessing options to estimate the trajectory of its center of mass. Green’s functions for a user-selected 1D Earth model are obtained automatically from the Incorporated Research Institutions for Seismology Synthetics Engine webservice or can be computed for custom 1D Earth models using Computer Programs in Seismology. lsforce implements the two most commonly used source parameterizations: a fully flexible, high-resolution approach and a more stable but lower-resolution method of overlapping triangle sources. Regularization options include a blended zeroth-, first-, and second-order semiautomated Tikhonov regularization scheme, as well as additional optional constraints on start times, end times, and on the sum of forces. Uncertainty due to data selection can be assessed using either a leave-one-out approach or a modified jackknife technique that randomly excludes subsets of the data for multiple re-inversions. Numerous built-in plotting methods allow for easy quality control and assessment of results. In this article, we briefly outline the theory and methodology, describe our implementation, and demonstrate the usage of lsforce using the well-studied 28 June 2016 Lamplugh rock avalanche in Alaska. Despite the rapidly increasing prevalence of landslide single-force inversions in the landslide and seismology literature over the past decade, to our knowledge this is the first open-source code for performing such inversions.

Indirect health monitoring of bridges using Tikhonov regularization scheme and signal averaging technique

This paper presents a novel damage identification technique for bridges based on the dynamic response of a moving vehicle. A quarter car vehicle model instrumented with two accelerometers and an inertial profilometer is used for this purpose. The method involves the coupling of Tikhonov regularization scheme with signal averaging technique to handle the problem of measurement noise and road roughness. In the first stage, a damage-dependent road roughness profile is estimated from measured acceleration response by minimizing a Tikhonov regularized least squares cost function. The second stage involves the minimization of the profile roughness residual function that depends on the location and magnitude of damage. This objective function compares the roughness profile computed from the first stage and that measured by the inertial profilometer. It is proved that efficient damage detection ensues from considering multiple runs of the vehicle and choosing the appropriate regularization parameter. The present study considers various aspects, such as the uniqueness of results, robustness of results to measurement noise, effect of modelling error, effect of vehicle speed, and uncertainty in estimation. Numerical results show that the approach is capable of detecting the magnitude as well as the location of damage. In addition, the problem of road roughness and measurement noise is handled competently.

On Regularization of an Optimal Control Problem for Ill-Posed Nonlinear Elliptic Equations

We discuss the existence issue to an optimal control problem for one class of nonlinear elliptic equations with an exponential type of nonlinearity. We deal with the control object when we cannot expect to have a solution of the corresponding boundary value problem in the standard functional space for all admissible controls. To overcome this difficulty, we make use of a variant of the classical Tikhonov regularization scheme. In particular, we eliminate the PDE constraints between control and state and allow such pairs run freely by introducing an additional variable which plays the role of “compensator” that appears in the original state equation. We show that this fictitious variable can be determined in a unique way. In order to provide an approximation of the original optimal control problem, we define a special family of regularized optimization problems. We show that each of these problems is consistent, well-posed, and their solutions allow to attain an optimal solution of the original problem as the parameter of regularization tends to zero. As a consequence, we prove the existence of optimal solutions to the original problem and propose a way for their approximation.

Determination of the Fractional Order in Semilinear Subdiffusion Equations

Abstract We analyze the inverse boundary value-problem to determine the fractional order ν of nonautonomous semilinear subdiffusion equations with memory terms from observations of their solutions during small time. We obtain an explicit formula reconstructing the order. Based on the Tikhonov regularization scheme and the quasi-optimality criterion, we construct the computational algorithm to find the order ν from noisy discrete measurements. We present several numerical tests illustrating the algorithm in action.

Sparse additive machine with ramp loss

Sparse additive machines (SAMs) have attracted increasing attention in high dimensional classification due to their representation flexibility and interpretability. However, most of existing methods are formulated under Tikhonov regularization scheme with the hinge loss, which are susceptible to outliers. To circumvent this problem, we propose a sparse additive machine with ramp loss (called ramp-SAM) to tackle classification and variable selection simultaneously. Misclassification error bound is established for ramp-SAM with the help of detailed error decomposition and constructive hypothesis error analysis. To solve the nonsmooth and nonconvex ramp-SAM, a proximal block coordinate descent method is presented with convergence guarantees. The empirical effectiveness of our model is confirmed on simulated and benchmark datasets.

Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems

In this paper, we consider the nonlinear ill-posed inverse problem with noisy data in the statistical learning setting. The Tikhonov regularization scheme in Hilbert scales is considered to reconstruct the estimator from the random noisy data. In this statistical learning setting, we derive the rates of convergence for the regularized solution under certain assumptions on the nonlinear forward operator and the prior assumptions. We discuss estimates of the reconstruction error using the approach of reproducing kernel Hilbert spaces.

Application of force reconstruction based on an improved Tikhonov regularization scheme

Application of force reconstruction based on an improved Tikhonov regularization scheme

Convergence analysis of Tikhonov regularization for non-linear statistical inverse problems

We study a non-linear statistical inverse problem, where we observe the noisy image of a quantity through a non-linear operator at some random design points. We consider the widely used Tikhonov regularization (or method of regularization) approach to estimate the quantity for the non-linear ill-posed inverse problem. The estimator is defined as the minimizer of a Tikhonov functional, which is the sum of a data misfit term and a quadratic penalty term. We develop a theoretical analysis for the minimizer of the Tikhonov regularization scheme using the concept of reproducing kernel Hilbert spaces. We discuss optimal rates of convergence for the proposed scheme, uniformly over classes of admissible solutions, defined through appropriate source conditions.

Online regularized pairwise learning with least squares loss

In this paper, we study online algorithm for pairwise problems generated from the Tikhonov regularization scheme associated with the least squares loss function and a reproducing kernel Hilbert space (RKHS). This work establishes the convergence for the last iterate of the online pairwise algorithm with the polynomially decaying step sizes and varying regularization parameters. We show that the obtained error rate in [Formula: see text]-norm can be nearly optimal in the minimax sense under some mild conditions. Our analysis is achieved by a sharp estimate for the norms of the learning sequence and the characterization of RKHS using its associated integral operators and probability inequalities for random variables with values in a Hilbert space.

Reconstruction of vibration-deformation-induced pointing error via optimized acceleration measurement points for large reflector antennas

As large reflector antennas have a low natural frequency and primarily operate in open air without a radome, factors such as the stochastic wind disturbance and the moving platform obviously increase the vibration-deformation pointing error. Moreover, in the actual working operation of large reflector antennas, the traditional measurement method can only sense the pointing error at the hinge via encoders and cannot sense the vibration-deformation-induced pointing error of the main reflector. This paper presents a reconstruction method of the vibration-deformation-induced pointing error for the main reflector via optimized acceleration measurement points. First, an entire working condition model of the main reflector for vibration-deformation-induced pointing error observation is established to provide the general relations among the acceleration, location, number of measurement points, and vibration-deformation-induced pointing error. In this model, the vibration-deformation-induced pointing error is reconstructed based on the best-fitting paraboloid theory and the frequency-domain integration approach. Furthermore, a Tikhonov regularization scheme is adopted to mitigate the ill-posedness due to random noise in the measured accelerations. Second, to obtain a well-reconstructed vibration-deformation-induced pointing error with certain accuracy, this study proposes a novel four-symmetric-point dynamic optimization algorithm for optimization of the sensor layout is proposed. Finally, a wind disturbance simulation of a 7.3 m antenna is undertaken. The results show that under the accuracy constraint of 5%, a well-reconstructed vibration-deformation-induced pointing error is obtained from the 20 optimized measurement points. Compared with the effective independence method, four-symmetric-point dynamic optimization algorithm has higher precision in observing the vibration-deformation-induced pointing error and is more convenient for arranging the sensors.

A class of parameter choice rules for stationary iterated weighted Tikhonov regularization scheme

Regularization procedure involves the regularization parameter that plays a crucial role in the convergence analysis of the regularization scheme. Recently, Reddy (2017) has proposed two a posteriori parameter choice rules to choose the regularization parameter in the weighted Tikhonov regularization scheme. The primary purpose of this article is to introduce a class of parameter choice rules to choose the regularization parameter in the stationary iterated weighted Tikhonov (SIWT) regularization scheme and derive the optimal rate of convergence O(δj(α+1)1+j(α+1)) for a stationary iterated method based on these proposed rules. The numerical experiments support our theoretical results.

Accelerated image reconstruction using extrapolated Tikhonov filtering for photoacoustic tomography.

Development of simple and computationally efficient extrapolated Tikhonov filtering reconstruction methods for photoacoustic tomography. The model-based reconstruction algorithms in photoacoustic tomography typically utilize Tikhonov regularization scheme for the reconstruction of initial pressure distribution from the measured boundary acoustic data. The automated choice of regularization parameter in these cases is computationally expensive. Moreover, the Tikhonov scheme promotes the smooth features in the reconstructed image due to the smooth regularizer, thus leading to loss of sharp features. The proposed extrapolation method estimates the solution at zero regularization assuming the solution being a function of regularization parameter and thus posing it as a zero value problem. Thus, the numerically computed zero regularization solution is expected to have better features compared to standard Tikhonov solution, with an added advantage of removing the necessity of automated choice of regularization. The reconstructed results using this method were shown in three variants (Lanczos, traditional, and exponential) of Tikhonov filtering and were compared with the standard error estimate technique. Four numerical (including realistic breast phantom) and two experimental phantom data were utilized to show the effectiveness of the proposed method. It was shown that the proposed method performance was superior than the standard error estimate technique, being at least four times faster in terms of computation, and provides an improvement as high as 2.6 times in terms of standard figures of merit. The developed extrapolated Tikhonov filtering methods overcome the difficulty of obtaining a suitable regularization parameter and shown to be reconstructing high-quality photoacoustic images with additional advantage of being computationally efficient, making it more appealing in real-time applications.

Ill-posedness for fuzzy Fredholm integral equations of the first kind and regularization methods

In this study, the concept of ill-posedness for fuzzy Fredholm integral equations of the first kind was first proposed. An efficient regularization method based on classic Tikhonov regularization scheme to determine the approximate solution of the fuzzy Fredholm integral equations of the first kind is presented. To support our investigation, some examples are illustrated.