Order Tikhonov Regularization Research Articles

Decade‐Long Ozone Profile Record From Suomi NPP OMPS Limb Profiler: Assessment of Version 2.6 Data

AbstractWe evaluate a decadal ozone profile record derived from the Suomi National Polar‐orbiting Partnership (SNPP) Ozone Mapping and Profiler Suite (OMPS) Limb Profiler (LP) satellite instrument. In 2023, the OMPS LP data were re‐processed with the new version 2.6 retrieval algorithm that combines measurements from the ultraviolet (UV) and visible (VIS) parts of the spectra. It employs the second order Tikhonov regularization to retrieve a single vertical ozone profile between 12.5 km (or cloud tops) and 57.5 km with a vertical resolution of about 1.9–2.5 km between 20 and 55 km. The algorithm uses radiances measured at six UV ozone‐sensitive wavelengths (295, 302, 306, 312, 317, and 322 nm) paired with 353 nm, and one VIS wavelength at 606 nm combined with 510 and 675 nm to form a triplet. Each wavelength pair or triplet is used over a limited range of tangent altitudes where the sensitivity to ozone change is strongest. A new aerosol correction scheme was implemented based on a gamma‐function particle size distribution. In addition, numerous calibration changes that affected ozone retrievals were applied to measured LP radiances, including updates in altitude registration, radiometric calibration, stray light, and spectral registration. The key version 2.6 enhancement is the improved stability of the LP ozone record confirmed by the reduction in relative drifts between LP ozone and correlative measurements, linked previously to a drift in the version 2.5 LP altitude registration.

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.

Tikhonov Regularization and Perturbation-Level Tuning for the CNM in Pharmacokinetics

In pharmacokinetics, the clinical information collected from the patient is often much less than the complexity of the patient’s internal operations, hence the undetermined inverse problem has emerged as a challenge to solve it and find multiple possible point sets for considering the many possible implications of drug kinetics in the patient’s body. This paper suggests two enhanced schemes for the early cluster Newton method (CNM) to concomitantly explore a great solutions number for the inverse parameter determination in pharmacokinetics. The first scheme is the application of Tikhonov regularization in order to deal with the overdetermined system for hyperplane fitting in the CNM, and the second is an effective iterative strategy by tuning perturbation-level for the CNM. As a result of Tikhonov’s filtering operation, the singular values of lower order than the regularization parameter, which are to blame for the instability of the matrix equation, are efficiently eliminated. With perturbation-level tuning, following every iteration, as the point cluster (PoC) gets near the solution manifold (SoM), it is essential to lessen the level of perturbation in the patient’s clinical measurement data and this is suited for a numerical stabilization. Numerical simulation scenarios of two schemes have revealed that these suggested schemes can lower the iterations number and computed time, and PoC move more steadily towards the solutions manifold.

Estimating cardiac active tension from wall motion-An inverse problem of cardiac biomechanics.

The contraction of the human heart is a complex process as a consequence of the interaction of internal and external forces. In current clinical routine, the resulting deformation can be imaged during an entire heart beat. However, the active tension development cannot be measured in vivo but may provide valuable diagnostic information. In this work, we present a novel numerical method for solving an inverse problem of cardiac biomechanics-estimating the dynamic active tension field, provided the motion of the myocardial wall is known. This ill-posed non-linear problem is solved using second order Tikhonov regularization in space and time. We conducted a sensitivity analysis by varying the fiber orientation in the range of measurement accuracy. To achieve RMSE <20% of the maximal tension, the fiber orientation needs to be provided with an accuracy of 10°. Also, variation was added to the deformation data in the range of segmentation accuracy. Here, imposing temporal regularization led to an eightfold decrease in the error down to 12%. Furthermore, non-contracting regions representing myocardial infarct scars were introduced in the left ventricle and could be identified accurately in the inverse solution (sensitivity >0.95). The results obtained with non-matching input data are promising and indicate directions for further improvement of the method. In future, this method will be extended to estimate the active tension field based on motion data from clinical images, which could provide important insights in terms of a new diagnostic tool for the identification and treatment of diseased heart tissue.

A New Hybrid Inversion Method for 2D Nuclear Magnetic Resonance Combining TSVD and Tikhonov Regularization.

This paper is concerned with the reconstruction of relaxation time distributions in Nuclear Magnetic Resonance (NMR) relaxometry. This is a large-scale and ill-posed inverse problem with many potential applications in biology, medicine, chemistry, and other disciplines. However, the large amount of data and the consequently long inversion times, together with the high sensitivity of the solution to the value of the regularization parameter, still represent a major issue in the applicability of the NMR relaxometry. We present a method for two-dimensional data inversion (2DNMR) which combines Truncated Singular Value Decomposition and Tikhonov regularization in order to accelerate the inversion time and to reduce the sensitivity to the value of the regularization parameter. The Discrete Picard condition is used to jointly select the SVD truncation and Tikhonov regularization parameters. We evaluate the performance of the proposed method on both simulated and real NMR measurements.

A numerical differentiation method based on legendre expansion with super order Tikhonov regularization

The aim of this paper is to develop a method based on Legendre expansion to compute numerical derivatives of a function from its perturbed data. The Tikhonov regularization combined with a new penalty term is used to deal with the ill posed-ness of the problem. It has been shown that the solution process is uniform for various smoothness of functions. Moreover, the convergence rates can be obtained self-adaptively when we choose the regularization parameter by a discrepancy principle. Numerical tests show that the method gives good results.

A Bayesian 3-D linear gravity inversion for complex density distributions: application to the Puysegur subduction system

SUMMARY We have developed a linear 3-D gravity inversion method capable of modelling complex geological regions such as subduction margins. Our procedure inverts satellite gravity to determine the best-fitting differential densities of spatially discretized subsurface prisms in a least-squares sense. We use a Bayesian approach to incorporate both data error and prior constraints based on seismic reflection and refraction data. Based on these data, Gaussian priors are applied to the appropriate model parameters as absolute equality constraints. To stabilize the inversion and provide relative equality constraints on the parameters, we utilize a combination of first and second order Tikhonov regularization, which enforces smoothness in the horizontal direction between seismically constrained regions, while allowing for sharper contacts in the vertical. We apply this method to the nascent Puysegur Trench, south of New Zealand, where oceanic lithosphere of the Australian Plate has underthrust Puysegur Ridge and Solander Basin on the Pacific Plate since the Miocene. These models provide insight into the density contrasts, Moho depth, and crustal thickness in the region. The final model has a mean standard deviation on the model parameters of about 17 kg m–3, and a mean absolute error on the predicted gravity of about 3.9 mGal, demonstrating the success of this method for even complex density distributions like those present at subduction zones. The posterior density distribution versus seismic velocity is diagnostic of compositional and structural changes and shows a thin sliver of oceanic crust emplaced between the nascent thrust and the strike slip Puysegur Fault. However, the northern end of the Puysegur Ridge, at the Snares Zone, is predominantly buoyant continental crust, despite its subsidence with respect to the rest of the ridge. These features highlight the mechanical changes unfolding during subduction initiation.

Inversion methods to determine two-dimensional aerosol mass-mobility distributions: A critical comparison of established methods

This paper provides a critical review of methods used to invert tandem measurements to determine the two-dimensional distribution of particle mass and mobility. We consider the performance of weighted least-squares analysis, Twomey-type approaches, a maximum entropy method, Tikhonov regularization (over a range of regularization parameters), and statistical inversion. A detailed analysis is performed on a bimodal phantom to demonstrate the typical characteristics of reconstructions resulting from the different inversion techniques, before the Euclidean error between the phantom and reconstructions are evaluated for a wider range of phantoms. It is found that 1st order Tikhonov regularization generally outperforms the other established inversion methods, even for narrow phantoms, where the finite representation of the mass-mobility distribution becomes a larger contributor to reconstruction accuracy. Twomey-type approaches, while not as robust, are shown to be an acceptable alternative.

A new fast algorithm to achieve the dose uniformity around high dose rate brachytherapy stepping source using Tikhonov regularization.

The dose optimization algorithm based on anatomical points is developed to produce rapidly uniform doses over target distances generated on the target volume edges in high-dose-rate (HDR) brachytherapy stepping source application for a treatment length of 6cm. Monte Carlo modeling of the 60Co HDR brachytherapy source and the surrounding medium were performed using PHITS code. The source dwell times were optimized using Tikhonov regularization in order to obtain uniform dose distribution at the anatomical points located at predefined target distances. The computed dose rates at distances from 0.25 up to 20cm away from the source were first verified with the literature data sets. Then, the simulation results of the optimization process were compared to the calculations of commercial treatment planning system (TPS) SagiPlan. As a result, the dose uniformity was observed in the isodose curves at the target distances of 10 and 15mm of the treatment length and the prescribed dose achieved the anatomical points uniformly. The algorithm developed in the present study can be applied for achieving the dose uniformity around the brachytherapy stepping source as a quicker tool for different treatment lengths and different target distances while maintaining the high quality of the treatment plans, saving time by avoiding the manual isodose shaping and then better suitable treatment for patients.

Evaluation of Fifteen Algorithms for the Resolution of the Electrocardiography Imaging Inverse Problem Using ex-vivo and in-silico Data.

The electrocardiographic imaging inverse problem is ill-posed. Regularization has to be applied to stabilize the problem and solve for a realistic solution. Here, we assess different regularization methods for solving the inverse problem. In this study, we assess (i) zero order Tikhonov regularization (ZOT) in conjunction with the Method of Fundamental Solutions (MFS), (ii) ZOT regularization using the Finite Element Method (FEM), and (iii) the L1-Norm regularization of the current density on the heart surface combined with FEM. Moreover, we apply different approaches for computing the optimal regularization parameter, all based on the Generalized Singular Value Decomposition (GSVD). These methods include Generalized Cross Validation (GCV), Robust Generalized Cross Validation (RGCV), ADPC, U-Curve and Composite REsidual and Smoothing Operator (CRESO) methods. Both simulated and experimental data are used for this evaluation. Results show that the RGCV approach provides the best results to determine the optimal regularization parameter using both the FEM-ZOT and the FEM-L1-Norm. However for the MFS-ZOT, the GCV outperformed all the other regularization parameter choice methods in terms of relative error and correlation coefficient. Regarding the epicardial potential reconstruction, FEM-L1-Norm clearly outperforms the other methods using the simulated data but, using the experimental data, FEM based methods perform as well as MFS. Finally, the use of FEM-L1-Norm combined with RGCV provides robust results in the pacing site localization.

An inverse eigenvalue problem. Application: Graded-index optical fibers

The main objective of this paper is to solve an inverse eigenvalue problem that consists (for a given frequency) in seeking for a suitable graded-index optical fiber able to carry electromagnetic fields having known propagation constants. This is an ill-posed problem. We present an original iterative method, based on Tikhonov regularization in order to demonstrate the existence of a unique solution (the true refractive index or at least a better approximation of it). Otherwise, we compute the optimal regularization parameter thanks to L-curve method. Numerical results from various test cases show that the regularized (unique reconstructed) solution, which corresponds simultaneously to the corner of the L-curve and minimum of relative error (and even together with the minimum of absolute error at this corner), is in good agreement with the true solution.

The inversion of 2D NMR relaxometry data using L1 regularization

NMR relaxometry has been used as a powerful tool to study molecular dynamics. Many algorithms have been developed for the inversion of 2D NMR relaxometry data. Unlike traditional algorithms implementing L2 regularization, high order Tikhonov regularization or iterative regularization, L1 penalty term is involved to constrain the sparsity of resultant spectra in this paper. Then fast iterative shrinkage-thresholding algorithm (FISTA) is proposed to solve the L1 regularization problem. The effectiveness, noise vulnerability and practical utility of the proposed algorithm are analyzed by simulations and experiments. The results demonstrate that the proposed algorithm has a more excellent capability to reveal narrow peaks than traditional inversion algorithms. The L1 regularization implemented by our algorithm can be a useful complementary to the existing algorithms.

Comparison of L-curve and LOOCV depth profiles from TAARXPS data

Time and angle resolved X-ray photoelectron spectroscopy (TAARXPS) data, obtained from polystyrene samples exposed to an oxygen/helium plasma, have been interpreted using 1st order Tikhonov regularization to smooth the extracted depth profiles. Two methods for the choice of the regularization parameter, namely the L-curve method and leave-one-out cross-validation (LOOCV), are compared and contrasted.

Comparison of parameter choice methods for the analytical inversion of ARXPS data

Using 0th order Tikhonov regularization, we recovered concentration-depth profiles from simulated noisy ARXPS data using nine candidate methods for the selection of the regularization parameter α. Calculations were performed on 1000 noisy data sets at 10 levels of noise ranging from 0.1% to 9%. By comparison with the input profile, the parameter choice methods were rated in terms of the accuracy and reproducibility of the extracted profiles. We discuss the results obtained in the context of the inputs and constraints required by each method.

Non Parametric Calibration of the Local Volatility Surface for European Options Using a Second Order Tikhonov Regularization

We calibrate the local volatility surface for European options across all strikes and maturities of the same underlying. There is no interpolation or extrapolation of either the option prices or the volatility surface. We do not make any assumption regarding the shape of the volatility surface except to assume that it is smooth. Due to the smoothness assumption, we apply a second order Tikhonov regularization. We choose the Tikhonov regularization parameter as one of the singular values of the Jacobian matrix of the Dupire model. Finally we perform extensive numerical tests to assess and verify the aforementioned techniques for both volatility models with known analytical solutions of European option prices and real market option data.

Estimation of boundary conditions in the presence of unknown moving boundary caused by ablation

Ablative materials can sustain very high temperatures in which surface thermochemical processes are significant enough to cause surface recession. Existence of moving boundary over a wide range of temperatures, temperature-dependent thermophysical properties of ablators, and no prior knowledge about the location of the moving surface augment the difficulty for predicting the exposed heat flux at the receding surface of ablators. In this paper, the conjugate gradient method is proposed to estimate the unknown surface recession and time-varying net surface heat flux for these kinds of problems. The first order Tikhonov regularization is employed to stabilize the inverse solution. Considering the complicated phenomena that are taking place, it is shown via simulated experiment that unknown quantities can be obtained with reasonable accuracy using this method despite existing noises in the measurement data.

A Latent Factor Model of Multivariate Conditional Heteroscedasticity

This papers examines the joint dynamics of a system of asset returns. I describe and implement a multivariate factor stochastic volatility (MVFSV) model. I follow closely the work of Doz and Renault (2006), with two important changes. First, I design a sequential testing procedure to determine the dimensions of the appropriate factor structure needed to accommodate the conditional heteroscedasticity among a system of returns. Second, I employ a form of Tikhonov Regularization in order to overcome a near singularity among the moment conditions used for estimation. Simulation studies suggest that the MVFSV model is able to recover accurately the latent factors that drive the conditional volatility of returns. Moreover, the model estimates can be used to construct conditionally homoscedastic portfolios as linear combinations of the conditionally heteroscedastic assets. An empirical application to portfolios representing the twelve sectors of the U.S. economy

RESINVM3D: A 3D resistivity inversion package

We have developed an open source 3D, MATLAB based, resistivity inversion package. The forward solution to the governing partial differential equation is efficiently computed using a second-order finite volume discretization coupled with a preconditioned, biconjugate, stabilized gradient algorithm. Using the analytical solution to a potential field in a homogeneous half space, we evaluate the accuracy of our numerical forward solution and, subsequently, develop a source correction factor that reduces forward modeling errors associated with boundary effects and source electrode singularities. For the inversion algorithm we have implemented an inexact Gauss-Newton solver, with the model update being calculated using a preconditioned conjugate gradient algorithm. The inversion uses a combination of zero and first order Tikhonov regularization. Two synthetic examples demonstrate the usefulness of this code. The first example considers a surface resistivity survey with 3813 measurements. The discretized model space contains 19,040 cells. For this example, the inversion package converges in approximately [Formula: see text] on a [Formula: see text] Pentium 4, with [Formula: see text] of RAM. The second example considers the case of borehole based data acquisition. For this example there were 4704 measurements and 13,200 model cells. The inversion for this example requires [Formula: see text] of computational time.

Identification of a Temperature Dependent Heat Conductivity from Single Boundary Measurements

Considering the identification of a temperature dependent conductivity in a quasilinear elliptic heat equation from single boundary measurements, we proof uniqueness in dimensions $n \ge 2$. Taking noisy data into account, we apply Tikhonov regularization in order to overcome the instabilities. By using a problem-adapted adjoint, we give convergence rates under substantially weaker and more realistic conditions than required by the general theory. Our theory is supported by numerical tests.

Improved parameter choice methods and temporal filtering for the generalized eigensystem method applied to the inverse problem of electrocardiography

We have previously proposed the generalized eigensystem (GES) method as a modal expansion method for estimating electrical potentials on the heart outer surface from measurements of electrical potentials on the body surface. In this paper, we present an alternative formulation of GES more like that of classical Tikhonov regularization where a single continuous parameter needs to be chosen. We then compare this formulation of GES with zero order Tikhonov regularization on data collected from a swine experiment with the swine heart paced from six different sites. Of the two algorithms used to estimate the regularization parameters, the composite residual and smoothing operator (CRESO) generally outperforms the generalized cross validation (GCV) for both GES and Tikhonov. Although the inverse problems are solved at each time instant independently, we also incorporate temporal information by moving average filtering the estimates, and this is more effective for the GES methods than for Tikhonov. In general, the epicardial estimates using the GES method are more accurate than the estimates generated using zero order Tikhonov regularization.