Tikhonov Regularization Parameter Research Articles

Magnetics only real-time equilibrium reconstruction on ASDEX Upgrade

Real-time reconstruction of the magnetic equilibrium provides fundamental control of plasma shape and position in a tokamak. Details of the implementation of the equilibrium reconstruction code developed for the ASDEX Upgrade tokamak (JANET++) are summarized. Cubic Hermite splines are introduced as current density basis functions for solving the Grad–Shafranov equation. The choice of the optimal Tikhonov regularization parameter is discussed. The code is validated by comparing the results of the equilibrium reconstruction with those of further equilibrium reconstructions available on ASDEX Upgrade (CLISTE and IDE). In a high time resolution study of a discharge with edge localized modes (ELM), the poloidal asymmetry of the fits and magnetic probe measurements suggest that the real-time equilibrium reconstruction captures the essential features of the current density redistribution in an ELMing edge plasma. An efficient algorithm to locate multiple X-points and identify the active one in advanced X-divertor and snowflake divertor configurations is presented.

Intelligent Particle Swarm Optimization Method for Parameter Selecting in Regularization Method for Integral Equation

We use the Tikhonov method as a regularization technique for solving the integral equation of the first kind with noisy and noise-free data. Following that, we go over how to choose the Tikhonov regularization parameter by implementing the Intelligent Piratical Swarm Optimization (IPOS) technique. The effectiveness of combining these two approaches IPOS and Tikhonov is demonstrated to be highly practicable.

Python programs to apply regularized derivatives in the magnetic tilt derivative and gradient intensity data processing: A graphical procedure to choose the regularization parameter

The Tikhonov regularization parameter is a key parameter controlling the smoothness degree and oscillations of a regularized unknown solution. Usual methods to determine a proper parameter (L-curve or the discrepancy principle, for example) are not readily applicable to the evaluation of regularized derivatives, since this formulation does not make explicit a set of model parameters that are necessary to implement these methods. We develop a procedure for the determination of the regularization parameter based on the graphical construction of a characteristic “staircase” function associated with the L2-norm of the regularized derivatives for a set of trial regularization parameters. This function is independent of model parameters and presents a smooth and monotonic variation. The regularization parameters at the upper step (low values) of the ''staircase'' function provide equivalent results to the non-regularized derivative, the parameters at the lower step (high values) leading to over-smoothed derivatives. For the evaluated data sets, the proper regularization parameter is located in the slope connecting these two flat end-members of the staircase curve, thus balancing noise amplification against the amplitude loss in the transformed fields. A set of Python programs are presented to evaluate the regularization procedure in a well-known synthetic model composed of multiple (bulk and elongated) magnetic sources. This numerical approach also is applied in gridded aeromagnetic data covering high-grade metamorphic terrains of the Anápolis-Itauçu Complex in the Brasília Fold Belt central portion of Tocantins Province, central Brazil, characterized by multiple magnetic lineaments with different directions and intersections which are associated with shear zones, geologic faults, and intrusive bodies. The results obtained from the regularization procedure show efficiency in improving the maps of filtered fields, better tracking the continuity of magnetic lineaments and general geological trends. The results from the application in the Brasília Fold Belt enhance the importance and broader coverage of the Pirineus Zone of High Strain.

Approximation of the Tikhonov regularization parameter through Aitken's extrapolation

In the present work, we study the determination of the regularization parameter and the computation of the regularized solution in Tikhonov regularization, by the Aitken's extrapolation method. In particular, this convergence acceleration method is adjusted for the approximation of quadratic forms that appear in regularization methods, such as the generalized cross-validation method, the quasi-optimality criterion, the Gfrerer/Raus method and the Morozov's discrepancy principle. We present several numerical examples to illustrate the effectiveness of the derived estimates for approximating the regularization parameter for several linear discrete ill-posed problems and we compare the described method with further existing methods, for the determination of the regularized solution.

On the solution stability of parabolic optimal control problems

The paper investigates stability properties of solutions of optimal control problems constrained by semilinear parabolic partial differential equations. Hölder or Lipschitz dependence of the optimal solution on perturbations are obtained for problems in which the equation and the objective functional are affine with respect to the control. The perturbations may appear in both the equation and in the objective functional and may nonlinearly depend on the state and control variables. The main results are based on an extension of recently introduced assumptions on the joint growth of the first and second variation of the objective functional. The stability of the optimal solution is obtained as a consequence of a more general result obtained in the paper–the metric subregularity of the mapping associated with the system of first order necessary optimality conditions. This property also enables error estimates for approximation methods. A Lipschitz estimate for the dependence of the optimal control on the Tikhonov regularization parameter is obtained as a by-product.

Influence of the Tikhonov Regularization Parameter on the Accuracy of the Inverse Problem in Electrocardiography.

The electrocardiogram (ECG) is the standard method in clinical practice to non-invasively analyze the electrical activity of the heart, from electrodes placed on the body's surface. The ECG can provide a cardiologist with relevant information to assess the condition of the heart and the possible presence of cardiac pathology. Nonetheless, the global view of the heart's electrical activity given by the ECG cannot provide fully detailed and localized information about abnormal electrical propagation patterns and corresponding substrates on the surface of the heart. Electrocardiographic imaging, also known as the inverse problem in electrocardiography, tries to overcome these limitations by non-invasively reconstructing the heart surface potentials, starting from the corresponding body surface potentials, and the geometry of the torso and the heart. This problem is ill-posed, and regularization techniques are needed to achieve a stable and accurate solution. The standard approach is to use zero-order Tikhonov regularization and the L-curve approach to choose the optimal value for the regularization parameter. However, different methods have been proposed for computing the optimal value of the regularization parameter. Moreover, regardless of the estimation method used, this may still lead to over-regularization or under-regularization. In order to gain a better understanding of the effects of the choice of regularization parameter value, in this study, we first focused on the regularization parameter itself, and investigated its influence on the accuracy of the reconstruction of heart surface potentials, by assessing the reconstruction accuracy with high-precision simultaneous heart and torso recordings from four dogs. For this, we analyzed a sufficiently large range of parameter values. Secondly, we evaluated the performance of five different methods for the estimation of the regularization parameter, also in view of the results of the first analysis. Thirdly, we investigated the effect of using a fixed value of the regularization parameter across all reconstructed beats. Accuracy was measured in terms of the quality of reconstruction of the heart surface potentials and estimation of the activation and recovery times, when compared with ground truth recordings from the experimental dog data. Results show that values of the regularization parameter in the range (0.01-0.03) provide the best accuracy, and that the three best-performing estimation methods (L-Curve, Zero-Crossing, and CRESO) give values in this range. Moreover, a fixed value of the regularization parameter could achieve very similar performance to the beat-specific parameter values calculated by the different estimation methods. These findings are relevant as they suggest that regularization parameter estimation methods may provide the accurate reconstruction of heart surface potentials only for specific ranges of regularization parameter values, and that using a fixed value of the regularization parameter may represent a valid alternative, especially when computational efficiency or consistency across time is required.

Prediction identification method of time-varying internal heat source based on inverse heat transfer problem

Purpose This paper aims to discuss the inverse problems that arise in various practical heat transfer processes. The purpose of this paper is to provide an identification method for predicting the internal boundary conditions for thermal analysis of mechanical structure. A few examples of heat transfer systems are given to illustrate the applicability of the method and the challenges that must be addressed in solving the inverse problem. Design/methodology/approach In this paper, the thermal network method and the finite difference method are used to model the two-dimensional heat conduction inverse problem of the tube structure, and the heat balance equation is arranged into an explicit form for heat load prediction. To solve the matrix ill-conditioned problem in the process of solving the inverse problem, a Tikhonov regularization parameter selection method based on the inverse computation-contrast-adjustment-approach was proposed. Findings The applicability of the proposed method is illustrated by numerical examples for different dynamically varying heat source functions. It is proved that the method can predict dynamic heat source with different complexity. Practical implications The modeling calculation method described in this paper can be used to predict the boundary conditions for the inner wall of the heat transfer tube, where the temperature sensor cannot be placed. Originality/value This paper presents a general method for the direct prediction of heat sources or boundary conditions in mechanical structure. It can directly obtain the time-varying heat flux load and thtemperature field of the machine structure.

A generalized adaptive Levenberg–Marquardt method for solving nonlinear ill-posed problems

We investigate the method of regularization for the stable approximate solution to nonlinear ill-posed problems whose forward operators may not be Gâteaux differentiable. The method is designed by combining the classical Levenberg–Marquardt method with the two-point gradient iteration, and the adaptive stepsize which is related to the Tikhonov regularization parameter and the structure of the forward operator. In order to further enhance the acceleration effect, we employ a modified discrete backtracking search algorithm to determine the combination parameters involved. With the help of the concept of asymptotic stability and a generalized tangential cone condition, the convergence analysis of the proposed method is studied. Moreover, several numerical experiments are performed to illustrate the effectiveness and acceleration effect.

Enhance the delivery of light energy ultra-deep into turbid medium by controlling multiple scattering photons to travel in open channels

Multiple light scattering is considered as the major limitation for deep imaging and focusing in turbid media. In this paper, we present an innovative method to overcome this limitation and enhance the delivery of light energy ultra-deep into turbid media with significant improvement in focusing. Our method is based on a wide-field reflection matrix optical coherence tomography (RM-OCT). The time-reversal decomposition of the RM is calibrated with the Tikhonov regularization parameter in order to get more accurate reversal results deep inside the scattering sample. We propose a concept named model energy matrix, which provides a direct mapping of light energy distribution inside the scattering sample. To the best of our knowledge, it is the first time that a method to measure and quantify the distribution of beam intensity inside a scattering sample is demonstrated. By employing the inversion of RM to find the matched wavefront and shaping with a phase-only spatial light modulator, we succeeded in both focusing a beam deep (~9.6 times of scattering mean free path, SMFP) inside the sample and increasing the delivery of light energy by an order of magnitude at an ultra-deep (~14.4 SMFP) position. This technique provides a powerful tool to understand the propagation of photon in a scattering medium and opens a new way to focus light inside biological tissues.

Research on Tikhonov regularization parameter selection in dynamic light scattering measurement of flowing particles

Research on Tikhonov regularization parameter selection in dynamic light scattering measurement of flowing particles

On an unsupervised method for parameter selection for the elastic net

<abstract><p>Despite recent advances in regularization theory, the issue of parameter selection still remains a challenge for most applications. In a recent work the framework of statistical learning was used to approximate the optimal Tikhonov regularization parameter from noisy data. In this work, we improve their results and extend the analysis to the elastic net regularization. Furthermore, we design a data-driven, automated algorithm for the computation of an approximate regularization parameter. Our analysis combines statistical learning theory with insights from regularization theory. We compare our approach with state-of-the-art parameter selection criteria and show that it has superior accuracy.</p></abstract>

Computation analysis of regularization methods and parameter selection for acoustics radiation modes source reconstruction of vibrating plates

Source localization and power estimation is a topic of great interest in acoustics and vibration. Acoustic source radiation modes reconstruction is a method of particular interest. Solutions leads to determinate sound/vibration power and surface velocity distribution from sparse acoustics samples. It has been shown that the quality of the results depends on Tikhonov regularization parameter. This inverse method is based on the radiation resistance matrix and we show that a single instruction multiple threads computing approach for graphics processing unit device exhibit better speed performance than common approaches to achieve the solution. We compare four regularization and three estimating methods for regularization parameters. We use a similarity measure to the simulated cases in three frequencies. Tikhonov regularization exhibits best reconstruction results. However, truncated singular vector decomposition also shows good performance with the advantage of not using a regularization parameter. Graphics processing unit implementation reduce reconstruction's computing time at least in a half.

A machine learning approach to optimal Tikhonov regularization I: Affine manifolds

Despite a variety of available techniques, such as discrepancy principle, generalized cross validation, and balancing principle, the issue of the proper regularization parameter choice for inverse problems still remains one of the relevant challenges in the field. The main difficulty lies in constructing an efficient rule, allowing to compute the parameter from given noisy data without relying either on any a priori knowledge of the solution, noise level or on the manual input. In this paper, we propose a novel method based on a statistical learning theory framework to approximate the high-dimensional function, which maps noisy data to the optimal Tikhonov regularization parameter. After an offline phase where we observe samples of the noisy data-to-optimal parameter mapping, an estimate of the optimal regularization parameter is computed directly from noisy data. Our assumptions are that ground truth solutions of the inverse problem are statistically distributed in a concentrated manner on (lower-dimensional) linear subspaces and the noise is sub-gaussian. We show that for our method to be efficient, the number of previously observed samples of the noisy data-to-optimal parameter mapping needs to scale at most linearly with the dimension of the solution subspace. We provide explicit error bounds on the approximation accuracy from noisy data of unobserved optimal regularization parameters and ground truth solutions. Even though the results are more of theoretical nature, we present a recipe for the practical implementation of the approach. We conclude with presenting numerical experiments verifying our theoretical results and illustrating the superiority of our method with respect to several state-of-the-art approaches in terms of accuracy or computational time for solving inverse problems of various types.

PMHSS iteration method and preconditioners for Stokes control PDE-constrained optimization problems

The preconditioned modified Hermitian/skew-Hermitian splitting (PMHSS) iteration method and the corresponding preconditioning technique can achieve satisfactory results for solving optimal control problems governed by Poisson’s equation. We explore the feasibility of such a method and preconditioner for solving optimization problems constrained by the more complicated Stokes system. Theoretical results demonstrate that the PMHSS iteration method is convergent because the spectral radius of the iterative matrix is less than $\frac {\sqrt {2}}{2}$ . Additionally, the PMHSS preconditioner still clusters eigenvalues on a unitary segment. It guarantees that the convergence of the PMHSS iteration method and preconditioning is independent of not only discretizing mesh size, but also of the Tikhonov regularization parameter. A more effective preconditioner is proposed based on the PMHSS preconditioner. The proposed preconditioner avoids the inner iterations when solving saddle point systems appearing in the generalized residual equations. Furthermore, it is still convergent and maintains its independence of parameter and mesh size.

Quasi-best approximation in optimization with PDE constraints

We consider finite element solutions to quadratic optimization problems, where the state depends on the control via a well-posed linear partial differential equation. Exploiting the structure of a suitably reduced optimality system, we prove that the combined error in the state and adjoint state of the variational discretization is bounded by the best approximation error in the underlying discrete spaces. The constant in this bound depends on the inverse square root of the Tikhonov regularization parameter. Furthermore, if the operators of control action and observation are compact, this quasi-best approximation constant becomes independent of the Tikhonov parameter as the mesh size tends to 0 and we give quantitative relationships between mesh size and Tikhonov parameter ensuring this independence. We also derive generalizations of these results when the control variable is discretized or when it is taken from a convex set.

Development of beam arrangement design for tunable diode laser absorption tomography reconstruction based on Tikhonov regularization parameter matrix

Beam arrangement with limited projections is a critical part of research on tunable diode laser absorption tomography reconstruction for combustion diagnosis. Based on the efforts to regularize this rank-deficient and ill-posed problem with Tikhonov regularization, a novel approach to using the regularization parameter matrix is developed for designing optical component layout and predicting the reconstruction accuracy. Objective function of beam arrangement is established by the rigorous mathematical derivation, and genetic algorithm is adopted to realize the optimization of function to overcome the difficulty associated with the multimodal nature of the problem. Nonuniform distribution properties of matrix elements in physical space relate to location and alignment of the laser/detector pairs, and form a basis for adjusting the weight between measurement and regularization to improve the reconstruction performance. A mathematical model of double Gauss distributions is established in a 10 × 10 element discrete tomography domain, and typically 20 measurement beams scanning the H<sub>2</sub>O transition at 7185.6 cm<sup>–1</sup> are available to probe the domain of interest. The systematic comparison between optimized beam array here and four existing beam arrangements in the literature is analyzed to validate the method. The reconstruction with Tikhonov regularization parameter matrix shows obvious advantages of reducing errors especially under the condition of fewer projections. The validation of reconstruction performance of the optimized beam array is also examined by simulating the laser absorption measurement which is carried out on phantoms generated using a simulation of external flow field of an air-gasoline pulsed detonation engine. The result shows that the optimized beam array consistently outperforms other arrangements reported in complicated fluid field. A demonstration reconstruction experiment is performed on the distribution from small gas burners. Both locations and amplitudes are in good agreement with those in the actual case. This proposed design method will be valuable in broadening the scope of applications of tunable diode laser absorption tomography reconstruction for engine diagnosis and combustion efficiency improvement.

Quasi-best approximation in optimization with PDE constraints

We consider finite element solutions to quadratic optimization problems, where the state depends on the control via a well-posed linear partial differential equation. Exploiting the structure of a suitably reduced optimality system, we prove that the combined error in the state and adjoint state of the variational discretization is bounded by the best approximation error in the underlying discrete spaces. The constant in this bound depends on the inverse square root of the Tikhonov regularization parameter. Furthermore, if the operators of control action and observation are compact, this quasi-best approximation constant becomes independent of the Tikhonov parameter as the mesh size tends to 0 and we give quantitative relationships between mesh size and Tikhonov parameter ensuring this independence. We also derive generalizations of these results when the control variable is discretized or when it is taken from a convex set.

Three-dimensional transient inverse heat conduction problems in the enhanced pool boiling heat transfer

In this paper, the solution methods of three-dimensional transient inverse heat conduction problems (3D IHCP) arising from pool boiling applications that need to reconstruct Neumann boundary conditions are summarized. Due to their mathematical ill-posedness, it is still non-trivial to solve these 3D IHCP at a reasonable cost. Firstly, we introduce the optimization-based regularization methods, such as Tikhonov regularization and iterative regularization. In general, it requires to solve a large number of optimization problems constrained by 3D partial differential equations (PDE). In this case, an improved Tikhonov regularization method is developed to quickly obtain optimal Tikhonov regularization parameter and thereby can accelerate the entire inverse calculation process. Furthermore, an iterative regularization method that is based on the conjugate gradient method is proposed to handle the high-resolution measurement data and many pointwise measurement data. In pool boiling experiments, they can be obtained by an infrared camera and mirco-thermocouples, respectively. Secondly, a completely different approach, a direct two-step strategy is presented to solve the same 3D transient ICHP on thin-plates that commonly occur in engineering applications. The main idea of the approach is to transform the inverse problem into a forward problem that is easier to solve by replacing the unknown Neumann boundary condition with an approximated Dirichlet boundary condition. The method has great advantage in efficiency compared with the optimization-based method that often requires to solve dozens or hundreds of direct, adjoint and sensitivity PDE problems, and therefore has the potential to be developed as efficient online heat-flux soft sensors for many practical applications. Thirdly, to further speed up the numerical solution procedure we develop a time-space dependent adaptive mesh refinement strategy due to the fact that the heat-flux and temperature dynamics related to bubble structures rapidly change in space and time on the boiling surface. A few classical engineering benchmark problems are considered to validate the aforementioned solution approaches, and the proposed approaches are also successfully applied to reconstruct the transient heat-flux distributions on the boiling surface from the noisy experimental data obtained in a series of real pool boiling experiments along the entire boiling curve. On the one hand, the corresponding research works reviewed can be used to explore the heat-transfer mechanism and establish a more accurate heat transfer model enabling the fast, robust and accurate reconstruction of local heat fluxes, which can be considered as realistic boundary conditions on the boiling surface to further study the two-phase flow and heat transfer process of boiling fluid above the heater. On the other hand, it can be also applied to explore the local details of the enhanced pool boiling heat transfer for the design of new heaters such as micro/nano structured surfaces, provide predictions related to the boiling process and conduct optimal experimental design in a systematic way. In the future work, a large number of 3D IHCP for different test regularization parameters will be calculated through a high-throughput parallel computational strategy. Besides, for thick heaters the direct two-step strategy can be used to solve the 3D ICHP accurately with an initial heat-flux guess obtained by the proposed optimization-based inversion approach. This way, the computational efficiency can be further improved. Furthermore, other new inversion methods, for example, which are based on artificial neural network method are being developed to solve 3D transient IHCP for more complex structured heater surfaces such as the micro/nano structures.

Unbiased predictive risk estimation of the Tikhonov regularization parameter: convergence with increasing rank approximations of the singular value decomposition

The truncated singular value decomposition may be used to find the solution of linear discrete ill-posed problems in conjunction with Tikhonov regularization and requires the estimation of a regularization parameter that balances between the sizes of the fit to data function and the regularization term. The unbiased predictive risk estimator is one suggested method for finding the regularization parameter when the noise in the measurements is normally distributed with known variance. In this paper we provide an algorithm using the unbiased predictive risk estimator that automatically finds both the regularization parameter and the number of terms to use from the singular value decomposition. Underlying the algorithm is a new result that proves that the regularization parameter converges with the number of terms from the singular value decomposition. For the analysis it is sufficient to assume that the discrete Picard condition is satisfied for exact data and that noise completely contaminates the measured data coefficients for a sufficiently large number of terms, dependent on both the noise level and the degree of ill-posedness of the system. A lower bound for the regularization parameter is provided leading to a computationally efficient algorithm. Supporting results are compared with those obtained using the method of generalized cross validation. Simulations for two-dimensional examples verify the theoretical analysis and the effectiveness of the algorithm for increasing noise levels, and demonstrate that the relative reconstruction errors obtained using the truncated singular value decomposition are less than those obtained using the singular value decomposition. This is a pre-print of an article published in BIT Numerical Mathematics. The final authenticated version is available online at: https://doi.org/10.1007%2Fs10543-019-00762-7.

Mapping nearly continuous function by Newton-like operators in Baire's space

The paper presents a mapping 𝐹𝑛: 𝑋 → 𝑌, where 𝐹acts on a topological space 𝑋 into a regular topological space 𝑌 that is nearly continuous. We use Newton-like operators in Baire’s space in the presence of singularities of Jacobian matrices with contributory factor of the Tikhonov regularization parameter introduced on the discontinuous system in order to enforce the homeomorphic image to be continuous under the induced closed graph theorem. Keywords: nonlinear system; Baire’s space; Newton-like methods; strange attractor.