Tikhonov Regularization Approach Research Articles

Identification Source Term of the Distributed Order Time-Space Fractional Diffusion Equation

Abstract The inverse source problem for time-distributed order spatial fractional diffusion equations is examined in this study. The forward problem is solved by combining finite difference methods with matrix transformation techniques. The Tikhonov regularization approach is then applied to convert the inverse problem into a variational problem. The optimal perturbation approach is used to find the minimizer of the variational problem, leading to an approximation solution that depends solely on the time-dependent source term. The algorithm’s effectiveness and stability are demonstrated using numerical examples in one-dimensional domains.

Ill-Posedness and the Bias-Variance Tradeoff in Residual Stress Measurement Inverse Solutions

BackgroundRelaxation methods determine residual stresses by measuring the deformations produced by incremental removal of a subdomain of the specimen. Measured strains at any given increment, determined by the cumulative effect of the relieved stresses, appear as an integral equation, which must be inverted to obtain residual stresses. In practice, stress distributions are discretized by a finite-dimensional basis, to transform the integral equations into a linear system of equations, which is often ill-conditioned.ObjectiveThis article demonstrates that the problem is actually ill-posed and comes with an inherent bias-variance tradeoff.MethodsThe hole drilling method is used as an example application, and the practical effects of ill-posedness are illustrated.ResultsTraditional regularization of the solution by limiting the resolution of the discretization reduces solution variance (noise) at the expense of increased bias and often results in the ultimately harmful practice of taking fewer data points. A careful analysis including the alternate Tikhonov regularization approach shows that the highest number of measurements should always be taken to reduce the variance for a given regularization scheme. Unfortunately, the variability of a regularized solution cannot be used to build a valid confidence interval, since an unknown bias term is always present in the true overall error.ConclusionsThe mathematical theory of ill-posed problems provides tools to manage the bias-variance tradeoff on a reasonable statistical basis, especially when the statistical properties of measurement errors are known. In the long run, physical arguments that provide constraints on the true solution would be of utmost importance, as they could regularize the problem without introducing an otherwise unknown bias. Constraining the minimum length scale to some physically meaningful value is one promising possibility.

Tensorial conditional gradient method for solving multidimensional ill-posed problems

We consider solving a class of tensorial ill-conditioned problems. This problem is treated as a convex constrained minimization problem. This kind of ill-conditioned problems appears in several applications as color images and video restoration. A new tensor degradation model to recover color image and video from blur and noise is given. Tikhonov regularization approach is used to reduce the effect of noise in the computed solution. This approach leads to a convex tensor minimization problem, that can be solved basing on the conditional gradient method. We adapted the Generalized Cross Validation method to the tensorial model for appropriate choice of the regularization parameter. Numerical tests for image and video restoration are given to illustrate the effectiveness of the proposed approach compared with the classical method.

Recognition of partially occluded faces using regularized ICA

Face recognition approaches that use subspace projection are heavily related to basis images, especially in the case of partial occlusion. To improve the recognition performance, the occlusion should be excluded from the test image during the recognition process. In terms of similarity with image reconstruction, the proposed approach aims at representing the whole face image based on facial subregion. In this respect, face representation can be considered as an inverse problem. The Tikhonov regularization approach is combined with independent component analysis (ICA) in order to obtain the image parameter from the occluded image, where this parameter is compared with that trained by ICA. The combined algorithm is named as RegICA and is performed on face images in the AR Face Database. Cumulative match characteristics was taken as a measure for evaluating the performance of RegICA with occlusion. Compared with ICA on facial occlusion problem, it was found that the proposed approach outperforms ICA. In addition, it has the ability to recognize faces using any facial subregion even if it is small. Furthermore, it is shown that RegICA is not time-consuming, and it outperforms some of the recent approaches in terms of accuracy.

Machine Learning-Based Statistical Approach to Analyze Process Dependencies on Threshold Voltage in Recessed Gate AlGaN/GaN MIS-HEMTs

In this work, we demonstrate the use of a machine learning (ML)-based statistical approach to model and analyze the impact of the fabrication processes on the threshold voltage in recessed gate AlGaN/GaN metal-insulator-semiconductor high electron mobility transistors. First, we employ a ML-based Tikhonov regularization approach using the input of 19 different processing splits to generate a multivariable analytical equation that considers four critical processing parameters, such as the remaining AlGaN depth, ex-situ wet clean, in-situ plasma, and gate dielectric. Furthermore, the artificial neural network-based approach, which cannot be used to further analyze the impact of the process, is implemented for the comparison. Second, the results from this analytical equation show a nice agreement with the measured threshold voltage, indicating a successful correlation between the processing parameters and the threshold voltage. Finally, the impact of each process on the threshold voltage can be analyzed through the coefficient value related to each process, which can be a useful guidance for the device optimization toward the targeted performance.

Non-Negative Iterative Convex Refinement Approach for Accurate and Robust Reconstruction in Cerenkov Luminescence Tomography.

Cerenkov luminescence tomography (CLT) is a promising imaging tool for obtaining three-dimensional (3D) non-invasive visualization of the in vivo distribution of radiopharmaceuticals. However, the reconstruction performance remains unsatisfactory for biomedical applications because the inverse problem of CLT is severely ill-conditioned and intractable. In this study, therefore, a novel non-negative iterative convex refinement (NNICR) approach was utilized to improve the CLT reconstruction accuracy, robustness as well as the shape recovery capability. The spike and slab prior information was employed to capture the sparsity of Cerenkov source, which could be formalized as a non-convex optimization problem. The NNICR approach solved this non-convex problem by refining the solutions of the convex sub-problems. To evaluate the performance of the NNICR approach, numerical simulations and in vivo tumor-bearing mice models experiments were conducted. Conjugated gradient based Tikhonov regularization approach (CG-Tikhonov), fast iterative shrinkage-thresholding algorithm based Lasso approach (Fista-Lasso) and Elastic-Net regularization approach were used for the comparison of the reconstruction performance. The results of these experiments demonstrated that the NNICR approach obtained superior reconstruction performance in terms of location accuracy, shape recovery capability, robustness and in vivo practicability. It was believed that this study would facilitate the preclinical and clinical applications of CLT in the future.

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.

Compact terahertz spectrometer based on sequential modulation of disordered rough surfaces.

We present herein a compact terahertz (THz) spectrometer in which transmission intensity distribution associated with dispersive interference effects in disordered random surfaces are used for reconstructing the frequency contents of an incoming THz beam. The device sweeps the frequency-dependent parameter of a roughened transmission plate through lateral displacement or electro-optic modulation. 2D transmission intensities are sequentially captured by a single detector for a range of modulation depths. With a calibration data set as the reference, one can reconstruct the spectra of the probe THz beam by solving a system of simultaneous linear equations. A smoothing Tikhonov regularization approach has been implemented to improve the accuracy of the spectral reconstruction. The reported compact, broadband, high-resolution THz spectrometer is well suited for portable THz spectroscopy applications.

Quasi-boundary value methods for regularizing the backward parabolic equation under the optimal control framework**Jun Liu’s research was supported by a Seed Grants for the Transitional and Exploratory Projects (STEP) Award (FY2019) from the SIUE Graduate School. Mingqing Xiao’s research was supported in part by the NSF-DMS140928, 1854638 of the United States.

In this paper, we propose to solve the classical backward parabolic equation under the optimal control framework associated with the Tikhonov regularization formulation, where the initial condition is set to be the control variable while the misfit of the final condition is formulated as the Tikhonov objective functional to be minimized. The corresponding first-order necessary optimality system is discretized in a one-shot manner by a second-order finite difference scheme in space and time. The proposed optimal control setting, as discussed in the paper, provides more general framework in terms of solving this type of ill-posed inverse problem. In particular, several existing nonlocal quasi-boundary value methods appear to be the special cases of our proposed optimal control framework with appropriately chosen generalized regularization setting respectively. The relationship between our Tikhonov regularization approach and the quasi-boundary value methods is discussed in detail. The optimal control approach based on Tikhonov regularization is shown to deliver the known optimal order convergence rate. Numerical examples are provided to demonstrate both effective accuracy of approximation as well as excellent stability under the optimal control realization, comparing with three recent quasi-boundary value methods in literature.

Reflection-Waveform Inversion Regularized with Structure-Oriented Smoothing Shaping

Restricted by the limited length of the receiver arrays, estimating the seismic properties of deep targets requires inverting reflection data. Reflection-waveform inversion (RWI) utilizes reflection data to recover the background velocity. RWI splits the Earth model into a long-wavelength background velocity and short-wavelength reflectors. Because of the limited apertures of reflection data illuminating targets, and the trade-off between the depth of reflectors and the average velocity above the reflectors, RWI is inherently ill-posed and contains a large null space. One manifestation of the ill-posedness is the existence of characteristic artificial blob-shaped or column-like anomalies in the velocity models estimated with RWI. That erroneous background velocity, which still fits the travel time of reflection arrivals in the recorded data, can lead to inaccurate velocity models and distorted migration images in the subsequent processes. In order to mitigate those artifacts and improve the conditioning of RWI, we incorporate a piece of prior information into RWI, i.e. slow property variation along geological structures. This prior information is expressed mathematically as structure-oriented smoothing. We achieved this smoothing by solving an anisotropic diffusion equation with a finite-difference method. Thus, we formulated a method for RWI regularized with structure-oriented smoothing shaping by alternating the following three steps: building temporary reflectors, updating background velocity, and solving the diffusion equation. We successfully tested this scheme on the Marmousi model and a modified overthrust model. The results show that the regularized RWI yields a stable and accurate background velocity, and it is more effective than conventional Tikhonov regularization approaches. The estimated background model leads to a high-quality final velocity model when used as an initial model in conventional full-waveform inversion.

Improved Spacecraft Magnetic Attitude Maneuvering

A magnetometer is essential in spacecraft magnetic attitude control due to the need for magnetic field information to compute the control command. The measurements of the magnetometer, however, are usually affected by other electric currents in the spacecraft, especially those of the magnetic coils when they are turned on during actuation. As a result, magnetic rods and magnetometers are usually turned on at alternate times, resulting in a reduced duty cycle of the magnetic rods, and hence longer maneuver times. This paper presents a magnetic attitude control system with extended duty cycle and low magnetometer measurements frequency. Instead of measurements, a computed magnetic field strength vector is used for updating the control command during the duty cycle. Using the measured spacecraft rotational motion, and knowing the control torque command, it is possible to compute the magnetic field strength vector. These computations are corrected using magnetometer measurements at a lower rate. The Tikhonov regularization approach is implemented to solve the singular magnetic torque system. This algorithm is demonstrated via Monte Carlo numerical simulations to have faster attitude maneuvers and lower power consumption by the magnetic rods. Real data obtained from the Cascade Smallsat and Ionospheric Polar Explorer spacecraft are used for validation of the proposed approach.

Modeling the one-dimensional inverse heat transfer problem using a Haar wavelet collocation approach

In this paper, a numerical method to solving the one-dimensional inverse heat transfer problem in Cartesian and Cylindrical coordinates, which is combination of the Haar wavelet collocation and Tikhonov regularization approach, has been used. The audit consisted in comparing the heat transfer in reality with numerical result as a noisy data, in order to identify errors of 1%–5% has been presented. In this study, the Haar functions, in addition to estimating the unknown functions, are also used to reduce output noises. Based on the obtained results, two main advantages of the repeated method have been proven, first, the precision of this method in estimating the unknown boundary condition and the second, processing speed due to the lack of need for wavelet functions to be collocated at low intervals. This suggested that this method, also has a high speed. According to the obtained results, it can be asserted that the present method by applying a small error in the input data, is preserved its stability.

A Joint Tikhonov Regularization and Augmented Lagrange Approach for Ill-Posed State Constrained Control Problems with Sparse Controls

We provide a modified augmented Lagrange method coupled with a Tikhonov regularization for solving ill-posed state constrained elliptic optimal control problems with sparse controls. We consider a linear quadratic optimal control problem without any additional L2 regularization terms. The sparsity is guaranteed by an additional L1 term. Here, the modification of the classical augmented Lagrange method guarantees us uniform boundedness of the multiplier that corresponds to the state constraints. We present a coupling between the regularization parameter introduced by the Tikhonov regularization and the penalty parameter from the augmented Lagrange method, which allows us to prove strong convergence of the controls and their corresponding states. Moreover, convergence results proving the weak convergence of the adjoint state and weak*-convergence of the multiplier are provided. Finally, we demonstrate our method in several numerical examples.

MoisturEC: A New R Program for Moisture Content Estimation from Electrical Conductivity Data.

Noninvasive geophysical estimation of soil moisture has potential to improve understanding of flow in the unsaturated zone for problems involving agricultural management, aquifer recharge, and optimization of landfill design and operations. In principle, several geophysical techniques (e.g., electrical resistivity, electromagnetic induction, and nuclear magnetic resonance) offer insight into soil moisture, but data-analysis tools are needed to "translate" geophysical results into estimates of soil moisture, consistent with (1) the uncertainty of this translation and (2) direct measurements of moisture. Although geostatistical frameworks exist for this purpose, straightforward and user-friendly tools are required to fully capitalize on the potential of geophysical information for soil-moisture estimation. Here, we present MoisturEC, a simple R program with a graphical user interface to convert measurements or images of electrical conductivity (EC) to soil moisture. Input includes EC values, point moisture estimates, and definition of either Archie parameters (based on experimental or literature values) or empirical data of moisture vs. EC. The program produces two- and three-dimensional images of moisture based on available EC and direct measurements of moisture, interpolating between measurement locations using a Tikhonov regularization approach.

Computation of Control for Linear Approximately Controllable System Using Tikhonov Regularization

ABSTRACTComputation of control for a controlled partial differential equation is a difficult task, especially when the control problem is ill posed. In this paper, we propose a method of computing the regularized control of a diffusion control system using Tikhonov regularization approach when the system is approximately controllable. The method proposed here for choosing regularization parameter guarantees the convergence of the proposed control.

Sensor fusion for structural tilt estimation using an acceleration-based tilt sensor and a gyroscope

A tilt sensor can provide useful information regarding the health of structural systems. Most existing tilt sensors are gravity/acceleration based and can provide accurate measurements of static responses. However, for dynamic tilt, acceleration can dramatically affect the measured responses due to crosstalk. Thus, dynamic tilt measurement is still a challenging problem. One option is to integrate the output of a gyroscope sensor, which measures the angular velocity, to obtain the tilt; however, problems arise because the low-frequency sensitivity of the gyroscope is poor. This paper proposes a new approach to dynamic tilt measurements, fusing together information from a MEMS-based gyroscope and an acceleration-based tilt sensor. The gyroscope provides good estimates of the tilt at higher frequencies, whereas the acceleration measurements are used to estimate the tilt at lower frequencies. The Tikhonov regularization approach is employed to fuse these measurements together and overcome the ill-posed nature of the problem. The solution is carried out in the frequency domain and then implemented in the time domain using FIR filters to ensure stability. The proposed method is validated numerically and experimentally to show that it performs well in estimating both the pseudo-static and dynamic tilt measurements.

High-resolution regional gravity field recovery from Poisson wavelets using heterogeneous observational techniques

We adopt Poisson wavelets for regional gravity field recovery using data acquired from various observational techniques; the method combines data of different spatial resolutions and coverage, and various spectral contents and noise levels. For managing the ill-conditioned system, the performances of the zero- and first-order Tikhonov regularization approaches are investigated. Moreover, a direct approach is proposed to properly combine Global Positioning System (GPS)/leveling data with the gravimetric quasi-geoid/geoid, where GPS/leveling data are treated as an additional observation group to form a new functional model. In this manner, the quasi-geoid/geoid that fits the local leveling system can be computed in one step, and no post-processing (e.g., corrector surface or least squares collocation) procedures are needed. As a case study, we model a new reference surface over Hong Kong. The results show solutions with first-order regularization are better than those obtained from zero-order regularization, which indicates the former may be more preferable for regional gravity field modeling. The numerical results also demonstrate the gravimetric quasi-geoid/geoid and GPS/leveling data can be combined properly using this direct approach, where no systematic errors exist between these two data sets. A comparison with 61 independent GPS/leveling points shows the accuracy of the new geoid, HKGEOID-2016, is around 1.1 cm. Further evaluation demonstrates the new geoid has improved significantly compared to the original model, HKGEOID-2000, and the standard deviation for the differences between the observed and computed geoidal heights at all GPS/leveling points is reduced from 2.4 to 0.6 cm. Finally, we conclude HKGEOID-2016 can be substituted for HKGEOID-2000 for engineering purposes and geophysical investigations in Hong Kong.Graphical abstract.

An improved current potential method for fast computation of stellarator coil shapes

Several fast methods for computing stellarator coil shapes are compared, including the classical NESCOIL procedure (Merkel 1987 Nucl. Fusion 27 867), its generalization using truncated singular value decomposition, and a Tikhonov regularization approach we call REGCOIL in which the squared current density is included in the objective function. Considering W7-X and NCSX geometries, and for any desired level of regularization, we find the REGCOIL approach simultaneously achieves lower surface-averaged and maximum values of both current density (on the coil winding surface) and normal magnetic field (on the desired plasma surface). This approach therefore can simultaneously improve the free-boundary reconstruction of the target plasma shape while substantially increasing the minimum distances between coils, preventing collisions between coils while improving access for ports and maintenance. The REGCOIL method also allows finer control over the level of regularization, it preserves convexity to ensure the local optimum found is the global optimum, and it eliminates two pathologies of NESCOIL: the resulting coil shapes become independent of the arbitrary choice of angles used to parameterize the coil surface, and the resulting coil shapes converge rather than diverge as Fourier resolution is increased. We therefore contend that REGCOIL should be used instead of NESCOIL for applications in which a fast and robust method for coil calculation is needed, such as when targeting coil complexity in fixed-boundary plasma optimization, or for scoping new stellarator geometries.

Knowing your slope on the track: Getting the most out of GPS and power data

Introduction When cycling on hilly tracks, gravitational forces have a major influence on the cycling performance. For uphill tracks overcoming altitude differences can consume over 90% of the rider’s total power output [1]. Therefore, a good estimate of the slope is essential for simulating, predicting and optimizing the rider’s power output on a hilly course. In this work different methods to estimate slope profiles from recorded altitude, speed and power data were applied. The resulting slope profiles are compared with reference values gathered by differential GPS to quantify the error on the corresponding modelled power output.  Methods Three rides were performed on a 5.3 km long course in good weather conditions. Altitude, speed and power data were recorded with a Garmin Edge 1000 and an SRM power meter. To account for drifts due to the barometric sensor, the altitude was corrected with radar based elevation map data provided by http://www.gpsies.com (SRTM1 dataset). Additionally, highly precise differential GPS data recorded by a Leica GPS900 was available on most parts of the course (accuracy < 5 cm). The first three methods rely only on the altitude measurements: smoothing with a Gaussian filter and applying the differential quotient (GAU), smoothing with a Savitzky-Golay filter (SGO) and using the Tikhonov regularization method [2] (TIK). The forth method incorporated only smoothed speed and power measurements, by calculating the slope with the P-v model [3] (MOD). Additionally, the Tikhonov regularization approach was extended to include altitude as well as speed and power measurements (TIM). For each method and ride, the best parameter set was determined individually.  Results and Discussion Since we want to quantify the impact of errors in the slope estimation on the rider’s power output, instead of presenting the error of the slope directly, we prefer the corresponding error in power needed to overcome altitude levels: The change of potential energy P pot can be described as P pot =mgsv, where m is the mass of rider and bike, g is the gravitational acceleration, s the slope of the course and v the speed [3]. The speed was comparable on all three rides with mean values of 18.6 ± 0.4 km/h. Figure 1 gives an overview of the root mean square errors (RMSE) and the mean errors (ME) of the different slope estimation methods compared to the reference slope, which was determined by slightly smoothing the differential GPS altitude and calculating the difference quotient. For the altitude based methods GAU, SGO and TIK the altitude correction eliminates the ME of around 5 W while the RMSE is not changed. This confirms the usefulness of the altitude correction to globally removing the drift while locally preserving the quality of the barometric data. The model based method MOD reduces the RMSE from over 40 W (GAU and SGO) to less than 30 W. It should be noted that this method is highly sensitive to external factors like wind or braking. TIM avoids those problems by additionally considering altitude measurements. This leads to a comparable RMSE as MOD but less variance between the three recordings. It can be observed, that the ME is not zero if we take model estimations into account. This indicates, that small errors in the model introduce a drift over time. Therefore, for future applications, a correction algorithm should be developed.  Conclusions By representing the error in terms of the power output, a direct measure of the error that is introduced to the P-v model for simulation and prediction purposes is provided. With the altitude correction, the mean error diminishes, which avoids that the impact of modelling errors adds up over time. Moreover, the local modelling error can be reduced, by considering speed and power data in addition to the recorded error

Regional regularization method for ECT based on spectral transformation of Laplacian.

Image reconstruction in electrical capacitance tomography is an ill-posed inverse problem, and regularization techniques are usually used to solve the problem for suppressing noise. An anisotropic regional regularization algorithm for electrical capacitance tomography is constructed using a novel approach called spectral transformation. Its function is derived and applied to the weighted gradient magnitude of the sensitivity of Laplacian as a regularization term. With the optimum regional regularizer, the a priori knowledge on the local nonlinearity degree of the forward map is incorporated into the proposed online reconstruction algorithm. Simulation experimentations were performed to verify the capability of the new regularization algorithm to reconstruct a superior quality image over two conventional Tikhonov regularization approaches. The advantage of the new algorithm for improving performance and reducing shape distortion is demonstrated with the experimental data.