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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2016

Enhanced nearfield acoustic holography for larger distances of reconstructions using fixed parameter Tikhonov regularization

This paper evaluates the performance of various regularization parameter choice methods applied to different approaches of nearfield acoustic holography when a very nearfield measurement is not possible. For a fixed grid resolution, the larger the hologram distance, the larger the error in the naive nearfield acoustic holography reconstructions. These errors can be smoothed out by using an appropriate order of regularization. In conclusion, this study shows that by using a fixed/manual choice of regularization parameter, instead of automated parameter choice methods, reasonably accurate reconstructions can be obtained even when the hologram distance is 16 times larger than the grid resolution.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

TNet: A Model-Constrained Tikhonov Network Approach for Inverse Problems

Deep learning (DL), in particular deep neural networks, by default is purely data-driven and in general does not require physics. This is the strength of DL but also one of its key limitations when applied to science and engineering problems in which underlying physical properties—such as stability, conservation, and positivity—and accuracy are required. DL methods in their original forms are often not capable of respecting the underlying mathematical models or achieving desired accuracy even in big-data regimes. On the other hand, many data-driven science and engineering problems, such as inverse problems, typically have limited experimental or observational data, and DL would overfit the data in this case. Leveraging information encoded in the underlying mathematical models, we argue, not only compensates for missing information in low data regimes but also provides opportunities to equip DL methods with the underlying physics, hence promoting better generalization. This paper develops a model-constrained DL approach and its variant TNet—a Tikhonov neural network—which are capable of learning not only information hidden in the training data but also in the underlying mathematical models to solve inverse problems governed by partial differential equations in low data regimes. We provide the constructions and some theoretical results for the proposed approaches for both linear and nonlinear inverse problems. Since TNet is designed to learn inverse solutions with Tikhonov regularization, it is interpretable: in fact it recovers Tikhonov solutions for linear cases while potentially approximating Tikhonov solutions for nonlinear inverse problems. We also prove that data randomization can enhance not only the smoothness of the networks but also their generalizations. Comprehensive numerical results confirm the theoretical findings and show that with even as little as 1 training data sample for one-dimensional (1D) deconvolution, 5 for an inverse 2D heat conductivity problem, 100 for inverse initial conditions for a time-dependent 2D Burgers’s equation, and 50 for inverse initial conditions for 2D Navier–Stokes equations, TNet solutions can be as accurate as Tikhonov solutions while being several orders of magnitude faster. Furthermore, this is possible owing to the model-constrained term, replications, and randomization.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

The parameter choice rules for weighted Tikhonov regularization scheme

The well-known approach to solve the ill-posed problem is Tikhonov regularization scheme. But, the approximate solution of Tikhonov scheme may not contain all the details of the exact solution. To circumference this problem, weighted Tikhonov regularization has been introduced. In this article, we propose two a posteriori parameter choice rules to choose the regularization parameter for weighted Tikhonov regularization and establish the optimal rate of convergence O(δ{sup ((α+1)/(α+2))}) for the scheme based on these proposed rules. The numerical results are documented to demonstrate the significance of the theoretical results.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2012

Iterative Nonlinear Tikhonov Algorithm with Constraints for Electromagnetic Tomography

Low frequency electromagnetic tomography such as the capacitance tomography (ECT) has been proposed for monitoring and mass-gauging of gas-liquid two-phase system under microgravity condition in NASA's future long-term space missions. Due to the ill-posed inverse problem of ECT, images reconstructed using conventional linear algorithms often suffer from limitations such as low resolution and blurred edges. Hence, new efficient high resolution nonlinear imaging algorithms are needed for accurate two-phase imaging. The proposed Iterative Nonlinear Tikhonov Regularized Algorithm with Constraints (INTAC) is based on an efficient finite element method (FEM) forward model of quasi-static electromagnetic problem. It iteratively minimizes the discrepancy between FEM simulated and actual measured capacitances by adjusting the reconstructed image using the Tikhonov regularized method. More importantly, it enforces the known permittivity of two phases to the unknown pixels which exceed the reasonable range of permittivity in each iteration. This strategy does not only stabilize the converging process, but also produces sharper images. Simulations show that resolution improvement of over 2 times can be achieved by INTAC with respect to conventional approaches. Strategies to further improve spatial imaging resolution are suggested, as well as techniques to accelerate nonlinear forward model and thus increase the temporal resolution.

Xu, Feng↗
DOE OSTI2018

Applying Legendre wavelet method with Tikhonov regularization for one-dimensional time-fractional diffusion equations

In this paper, a new method based on two-dimensional Legendre wavelet basis is proposed to solve time-fractional diffusion equations. This technique is used to convert the problem into a system of linear algebraic equations via expanding the required approximation based on the Legendre wavelet basis. The effectiveness of the proposed method is examined by comparing the numerical results with other methods. When the desired system is large, it is ill-conditioned; in this case, Tikhonov regularization method, with discrepancy principle for finding the regularization parameter, is applied to stabilize the solution.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

Phillips-Tikhonov regularization with a priori information for neutron emission tomographic reconstruction on Joint European Torus

A method of tomographic reconstruction of the neutron emissivity in the poloidal cross section of the Joint European Torus (JET, Culham, UK) tokamak was developed. Due to very limited data set (two projection angles, 19 lines of sight only) provided by the neutron emission profile monitor (KN3 neutron camera), the reconstruction is an ill-posed inverse problem. The aim of this work consists in making a contribution to the development of reliable plasma tomography reconstruction methods that could be routinely used at JET tokamak. The proposed method is based on Phillips-Tikhonov regularization and incorporates a priori knowledge of the shape of normalized neutron emissivity profile. For the purpose of the optimal selection of the regularization parameters, the shape of normalized neutron emissivity profile is approximated by the shape of normalized electron density profile measured by LIDAR or high resolution Thomson scattering JET diagnostics. In contrast with some previously developed methods of ill-posed plasma tomography reconstruction problem, the developed algorithms do not include any post-processing of the obtained solution and the physical constrains on the solution are imposed during the regularization process. The accuracy of the method is at first evaluated by several tests with synthetic data based on various plasma neutron emissivity models (phantoms). Then, the method is applied to the neutron emissivity reconstruction for JET D plasma discharge #85100. It is demonstrated that this method shows good performance and reliability and it can be routinely used for plasma neutron emissivity reconstruction on JET.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2022

Deblurring galaxy images with Tikhonov regularization on magnitude domain

Abstract We propose a regularization-based deblurring method that works efficiently for galaxy images. The spatial resolution of a ground-based telescope is generally limited by seeing conditions and is much worse than space-based telescopes. This circumstance has generated considerable research interest in the restoration of spatial resolution. Since image deblurring is a typical inverse problem and often ill-posed, solutions tend to be unstable. To obtain a stable solution, much research has adopted regularization-based methods for image deblurring, but the regularization term is not necessarily appropriate for galaxy images. Although galaxies have an exponential or Sérsic profile, the conventional regularization assumes the image profiles to behave linearly in space. The significant deviation between the assumption and real situations leads to blurring of the images and smoothing out the detailed structures. Clearly, regularization on logarithmic domain, i.e., magnitude domain, should provide a more appropriate assumption, which we explore in this study. We formulate a problem of deblurring galaxy images by an objective function with a Tikhonov regularization term on a magnitude domain. We introduce an iterative algorithm minimizing the objective function with a primal–dual splitting method. We investigate the feasibility of the proposed method using simulation and observation images. In the simulation, we blur galaxy images with a realistic point spread function and add both Gaussian and Poisson noise. For the evaluation with the observed images, we use galaxy images taken by the Subaru HSC-SSP. Both of these evaluations show that our method successfully recovers the spatial resolution of the deblurred images and significantly outperforms the conventional methods. The code is publicly available from the GitHub 〈https://github.com/kzmurata-astro/PSFdeconv_amag〉.

Murata, Kazumi↗
DOE OSTI2018

Constructing the Pareto front for multi-objective Markov chains handling a strong Pareto policy approach

In this paper, we present a multi-objective solution of the Pareto front for a certain class of discrete-time ergodic controllable Markov chains. For solving the problem, we transform the original multi-objective problem into an equivalent nonlinear programming problem implementing the Lagrange principle. We then propose to adopt the Tikhonov’s regularization method to ensure the convergence of the cost functions to a unique point into the Pareto front. We show that Pareto policies are characterized as optimal policies. One of the fundamental problems related to the construction of the Pareto front is the existence and characterization of both, Pareto and strong Pareto policies. By introducing the Tikhonov’s regularizator, we ensure the existence of strong Pareto policies. This paper proposes a real solution to this problem. We formulate the original problem considering several constraints: (a) we employ the c-variable method for the introduction of linear constraints over the nonlinear problem and (b) restrict the cost functions, allowing points in the Pareto front to have a small distance from one another. The constraints imposed by the c-variable method make the problem computationally tractable and the restriction imposed by the small distance change ensures the continuation of the Pareto front. The resulting equation in this nonlinear system is an optimization problem for which the necessary and efficient condition of a minimum is solved using the projected gradient method. Moreover, we also prove the convergence conditions and compute the estimate rate of convergence of variables corresponding to the Lagrange principle and the Tikhonov’s regularization, respectively. We provide all the details needed to implement the proposed method in an efficient way. The usefulness of the method is successfully demonstrated by a numerical example.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

DIAS: A Data-Informed Active Subspace Regularization Framework for Inverse Problems

This paper presents a regularization framework that aims to improve the fidelity of Tikhonov inverse solutions. At the heart of the framework is the data-informed regularization idea that only data-uninformed parameters need to be regularized, while the data-informed parameters, on which data and forward model are integrated, should remain untouched. We propose to employ the active subspace method to determine the data-informativeness of a parameter. The resulting framework is thus called a data-informed (DI) active subspace (DIAS) regularization. Four proposed DIAS variants are rigorously analyzed, shown to be robust with the regularization parameter and capable of avoiding polluting solution features informed by the data. They are thus well suited for problems with small or reasonably small noise corruptions in the data. Furthermore, the DIAS approaches can effectively reuse any Tikhonov regularization codes/libraries. Though they are readily applicable for nonlinear inverse problems, we focus on linear problems in this paper in order to gain insights into the framework. Various numerical results for linear inverse problems are presented to verify theoretical findings and to demonstrate advantages of the DIAS framework over the Tikhonov, truncated SVD, and the TSVD-based DI approaches.

Nguyen, Hai↗
DOE OSTI2021

Revisiting matrix-based inversion of scanning mobility particle sizer (SMPS) and humidified tandem differential mobility analyzer (HTDMA) data

Abstract. Tikhonov regularization is a tool for reducing noise amplification during data inversion. This work introduces RegularizationTools.jl, a general-purpose software package for applying Tikhonov regularization to data. The package implements well-established numerical algorithms and is suitable for systems of up to ∼ 1000 equations. Included is an abstraction to systematically categorize specific inversion configurations and their associated hyperparameters. A generic interface translates arbitrary linear forward models defined by a computer function into the corresponding design matrix. This obviates the need to explicitly write out and discretize the Fredholm integral equation, thus facilitating fast prototyping of new regularization schemes associated with measurement techniques. Example applications include the inversion involving data from scanning mobility particle sizers (SMPSs) and humidified tandem differential mobility analyzers (HTDMAs). Inversion of SMPS size distributions reported in this work builds upon the freely available software DifferentialMobilityAnalyzers.jl. The speed of inversion is improved by a factor of ∼ 200, now requiring between 2 and 5 ms per SMPS scan when using 120 size bins. Previously reported occasional failure to converge to a valid solution is reduced by switching from the L-curve method to generalized cross-validation as the metric to search for the optimal regularization parameter. Higher-order inversions resulting in smooth, denoised reconstructions of size distributions are now included in DifferentialMobilityAnalyzers.jl. This work also demonstrates that an SMPS-style matrix-based inversion can be applied to find the growth factor frequency distribution from raw HTDMA data while also accounting for multiply charged particles. The outcome of the aerosol-related inversion methods is showcased by inverting multi-week SMPS and HTDMA datasets from ground-based observations, including SMPS data obtained at Bodega Marine Laboratory during the CalWater 2/ACAPEX campaign and co-located SMPS and HTDMA data collected at the US Department of Energy observatory located at the Southern Great Plains site in Oklahoma, USA. Results show that the proposed approaches are suitable for unsupervised, nonparametric inversion of large-scale datasets as well as inversion in real time during data acquisition on low-cost reduced-instruction-set architectures used in single-board computers. The included software implementation of Tikhonov regularization is freely available, general, and domain-independent and thus can be applied to many other inverse problems arising in atmospheric measurement techniques and beyond.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

MODFLOW6 models used to evaluate potential stresses and hydrologic conditions driving water-level fluctuations in well ER-5-3-2, Frenchman Flat, Southern Nevada

The hydrograph for well ER-5-3-2 in Frenchman Flat, southern Nevada, has previously unexplained water-level fluctuations. Four, three-dimensional, groundwater models (MODFLOW 6) were developed to evaluate potential stresses and hydrologic conditions affecting the well ER-5-3-2 hydrograph. Four model scenarios were developed that simulated: (1) wellbore leakage without recharge, (2) wellbore leakage with recharge, (3) shallow (low transmissivity) and deep (high transmissivity) carbonate rocks, and (4) lateral heterogeneity of carbonate rocks. Input and output files for the four model scenarios are in the model and output directories, respectively. Hydraulic conductivity, specific storage, and wellbore-leakage rates (when simulated) were estimated with parameter estimation (PEST) by minimizing a weighted composite, sum-of-squares objective function. The objective function was informed by measurement and Tikhonov regularization observations. Measurement observations included drawdowns from the constant-rate aquifer test and water-level altitudes measured in well ER-5-3-2 from 2001-2021. Tikhonov regularization informed hydraulic conductivity and specific storage parameters that were insensitive to measurement observations, where homogeneity was the preferred relation. Batch files, executables, and MODFLOW 6, PEST, and post-processing utilities are in the ancillary directory. Supplementary data also are included in the ancillary directory, including site information, high-frequency water-level and aquifer-test data, transmissivity estimates, water-chemistry data, and water-temperature analyses. This USGS data release contains data, analyses, and model files for the simulations and analysis results described in U.S. Geological Survey Scientific Investigations Report (https://doi.org/10.3133/sir20225132).

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2013

Optical tomography reconstruction algorithm with the finite element method: An optimal approach with regularization tools

Highlights: •New strategies to improve the accuracy of the reconstruction through mesh and finite element parameterization. •Use of gradient filtering through an alternative inner product within the adjoint method. •An integral form of the cost function is used to make the reconstruction compatible with all finite element formulations, continuous and discontinuous. •Gradient-based algorithm with the adjoint method is used for the reconstruction. -- Abstract: Optical tomography is mathematically treated as a non-linear inverse problem where the optical properties of the probed medium are recovered through the minimization of the errors between the experimental measurements and their predictions with a numerical model at the locations of the detectors. According to the ill-posed behavior of the inverse problem, some regularization tools must be performed and the Tikhonov penalization type is the most commonly used in optical tomography applications. This paper introduces an optimized approach for optical tomography reconstruction with the finite element method. An integral form of the cost function is used to take into account the surfaces of the detectors and make the reconstruction compatible with all finite element formulations, continuous and discontinuous. Through a gradient-based algorithm where the adjoint method is used to compute the gradient of the cost function, an alternative inner product is employed for preconditioning the reconstruction algorithm. Moreover, appropriate re-parameterization of the optical properties is performed. These regularization strategies are compared with the classical Tikhonov penalization one. It is shown that both the re-parameterization and the use of the Sobolev cost function gradient are efficient for solving such an ill-posed inverse problem.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

Nuclear three-body problem in the complex energy plane: Complex-scaling Slater method

The physics of open quantum systems is an interdisciplinary area of research. The nuclear “openness” manifests itself through the presence of the many-body continuum representing various decay, scattering, and reaction channels. As the radioactive nuclear beam experimentation extends the known nuclear landscape toward the particle drip lines, the coupling to the continuum space becomes exceedingly more important. Of particular interest are weakly bound and unbound nuclear states appearing around particle thresholds. Theories of such nuclei must take into account their open quantum nature. To describe open quantum systems, we introduce a complex-scaling (CS) approach in the Slater basis. We benchmark it with the complex-energy Gamow shell model (GSM) by studying energies and wave functions of the bound and unbound states of the two-neutron halo nucleus 6 He viewed as an $α + n + n$ cluster system. Both CS and GSM approaches are applied to a translationally invariant Hamiltonian with the two-body interaction approximated by the finite-range central Minnesota force. In the CS approach, we use the Slater basis, which exhibits the correct asymptotic behavior at large distances. To extract particle densities from the back-rotated CS solutions, we apply the Tikhonov regularization procedure, which minimizes the ultraviolet numerical noise. In this paper, we show that the CS-Slater method is both accurate and efficient. Its equivalence to the GSM approach has been demonstrated numerically for both energies and wave functions of 6 He. One important technical aspect of our calculation was to fully retrieve the correct asymptotic behavior of a resonance state from the complex-scaled (square-integrable) wave function. While standard applications of the inverse complex transformation to the complex-rotated solution provide unstable results, the stabilization method fully reproduces the GSM benchmark. We also propose a method to determine the smoothing parameter of the Tikhonov regularization. The combined suite of CS-Slater and GSM techniques has many attractive features when applied to nuclear problems involving weakly bound and unbound states. While both methods can describe energies, total widths, and wave functions of nuclear states, the CS-Slater method—if it can be applied—can provide additional information about partial energy widths associated with individual thresholds.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2013

The inverse problem of bimorph mirror tuning on a beamline

One of the challenges of tuning bimorph mirrors with many electrodes is that the calculated focusing voltages can be different by more than the safety limit (such as 500 V for the mirrors used at 17-ID at the Advanced Photon Source) between adjacent electrodes. A study of this problem at 17-ID revealed that the inverse problem of the tuning in situ, using X-rays, became ill-conditioned when the number of electrodes was large and the calculated focusing voltages were contaminated with measurement errors. Increasing the number of beamlets during the tuning could reduce the matrix condition number in the problem, but obtaining voltages with variation below the safety limit was still not always guaranteed and multiple iterations of tuning were often required. Applying Tikhonov regularization and using the L-curve criterion for the determination of the regularization parameter made it straightforward to obtain focusing voltages with well behaved variations. Some characteristics of the tuning results obtained using Tikhonov regularization are given in this paper.

43 PARTICLE ACCELERATORS↗
DOE OSTI2025

Improved Bayesian regularization of inverse problems in vibrations and acoustics using noise-only measurements

Here, this paper studies Tikhonov regularization (ridge regression) parameter selection for problems in vibrations and acoustics. The selection method is based on a popular Bayesian method, but it incorporates measurements of sensor noise. The regularization parameter is closely related to the ratio of system input energy to noise energy, so noise measurements inform the inference procedure and improve parameter identification. In cases where standard Bayesian regularization identifies zero as the optimal regularization parameter, noise measurements guarantee a unique nonzero optimum. Sufficient theoretical criteria are developed for this guarantee. The method is verified in even-determined and under-determined configurations in an acoustic source localization simulation and a vibration load identification experiment. It is shown to yield significant improvements over existing empirical Bayesian regularization. Improvements are larger in the even-determined case and smaller in the under-determined case, wherein the inverse solution is less sensitive to the regularization parameter.

42 ENGINEERING↗
DOE OSTI2015

Regularization of soft-X-ray imaging in the DIII-D tokamak

We developed an image inversion scheme for the soft X-ray imaging system (SXRIS) diagnostic at the DIII-D tokamak in order to obtain the local soft X-ray emission at a poloidal cross-section from the spatially line-integrated image taken by the SXRIS camera. The scheme uses the Tikhonov regularization method since the inversion problem is generally ill-posed. The regularization technique uses the generalized singular value decomposition to determine a solution that depends on a free regularization parameter. The latter has to be chosen carefully, and the so called {\it L-curve} method to find the optimum regularization parameter is outlined. A representative test image is used to study the properties of the inversion scheme with respect to inversion accuracy, amount/strength of regularization, image noise and image resolution. Moreover, the optimum inversion parameters are identified, while the L-curve method successfully computes the optimum regularization parameter. Noise is found to be the most limiting issue, but sufficient regularization is still possible at noise to signal ratios up to 10%-15%. Finally, the inversion scheme is applied to measured SXRIS data and the line-integrated SXRIS image is successfully inverted.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2017

An experimental comparison of various methods of nearfield acoustic holography

An experimental comparison of four different methods of nearfield acoustic holography (NAH) is presented in this study for planar acoustic sources. The four NAH methods considered in this study are based on: (1) spatial Fourier transform, (2) equivalent sources model, (3) boundary element methods and (4) statistically optimized NAH. Two dimensional measurements were obtained at different distances in front of a tonal sound source and the NAH methods were used to reconstruct the sound field at the source surface. Reconstructed particle velocity and acoustic pressure fields presented in this study showed that the equivalent sources model based algorithm along with Tikhonov regularization provided the best localization of the sources. Reconstruction errors were found to be smaller for the equivalent sources model based algorithm and the statistically optimized NAH algorithm. Effect of hologram distance on the performance of various algorithms is discussed in detail. The study also compares the computational time required by each algorithm to complete the comparison. Four different regularization parameter choice methods were compared. The L-curve method provided more accurate reconstructions than the generalized cross validation and the Morozov discrepancy principle. Finally, the performance of fixed parameter regularization was comparable to that of the L-curve method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

A generalized framework for reduced-order modeling of a wind turbine wake

A reduced-order model for a wind turbine wake is sought from large eddy simulation data. Fluctuating velocity fields are combined in the correlation tensor to form the kernel of the proper orthogonal decomposition (POD). Proper orthogonal decomposition modes resulting from the decomposition represent the spatially coherent turbulence structures in the wind turbine wake; eigenvalues delineate the relative amount of turbulent kinetic energy associated with each mode. Back-projecting the POD modes onto the velocity snapshots produces dynamic coefficients that express the amplitude of each mode in time. A reduced-order model of the wind turbine wake (wakeROM) is defined through a series of polynomial parameters that quantify mode interaction and the evolution of each POD mode coefficients. The resulting system of ordinary differential equations models the wind turbine wake composed only of the large-scale turbulent dynamics identified by the POD. Tikhonov regularization is used to recalibrate the dynamical system by adding additional constraints to the minimization seeking polynomial parameters, reducing error in the modeled mode coefficients. The wakeROM is periodically reinitialized with new initial conditions found by relating the incoming turbulent velocity to the POD mode coefficients through a series of open-loop transfer functions. The wakeROM reproduces mode coefficients to within 25.2%, quantified through the normalized root-mean-square error. Furthermore, a high-level view of the modeling approach is provided as a platform to discuss promising research directions, alternate processes that could benefit stability and efficiency, and desired extensions of the wakeROM.

17 WIND ENERGY↗