Quasi-optimality Criterion Research Articles

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.

A New Hybrid Method for Solving Inverse Heat Conduction Problems

Solving the inverse problems, especially in the field of heat transfer, is one of the challenges of engineering due to its importance in industrial applications. It is well-known that inverse heat conduction problems (IHCPs) are severely ill-posed, which means that small disturbances in the input may cause extremely large errors in the solution. This paper introduces an accurate method for solving inverse problems by combining Tikhonov's regularization and the genetic algorithm. Finding the regularization parameter as the decisive parameter is modelled by this method, a few sample problems were solved to investigate the efficiency and accuracy of the proposed method. A linear sum of fundamental solutions with unknown constant coefficients assumed as an approximated solution to the sample IHCP problem and collocation method is used to minimize residues in the collocation points. In this contribution, we use Morozov's discrepancy principle and Quasi-Optimality criterion for defining the objective function, which must be minimized to yield the value of the optimum regularization parameter.

Current distribution reconstruction in switching arcs by means of regularization based on GSVD

The knowledge of the variation of arc current is helpful in improving the breaking performance of a vacuum circuit breaker. As a novel non-intrusive method, arc magnetic performance testing technology has attracted attention. In view of the widely used regularization method of the standard form in arc current reconstructions cannot fully overcome the ill-posed nature of the Biot-Savart equation, the regularization approach based on generalized singular value decomposition (GSVD) is proposed to improve the situation. Moreover, a discrete 3-D arc model with the help of the finite element technique incorporating the current divergence-free and boundary conditions is introduced to improve the authenticity of the model. The performance of the truncated generalized singular value decomposition (TGSVD) and general-form Tikhonov regularization with derivative operators L 1 and L 2 are studied in numerical tests. In addition, the effectiveness of generalized cross validation criterion, quasi optimality criterion and normalized cumulative periodic graph method in selecting regularization parameters are compared. The methodology can help improve arc control and guide the design of vacuum circuit breakers with improved performance parameters.

Q-Curve and Area Rules for Choosing Heuristic Parameter in Tikhonov Regularization

We consider choice of the regularization parameter in Tikhonov method if the noise level of the data is unknown. One of the best rules for the heuristic parameter choice is the quasi-optimality criterion where the parameter is chosen as the global minimizer of the quasi-optimality function. In some problems this rule fails. We prove that one of the local minimizers of the quasi-optimality function is always a good regularization parameter. For the choice of the proper local minimizer we propose to construct the Q-curve which is the analogue of the L-curve, but on the x-axis we use modified discrepancy instead of discrepancy and on the y-axis the quasi-optimality function instead of the norm of the approximate solution. In the area rule we choose for the regularization parameter such local minimizer of the quasi-optimality function for which the area of the polygon, connecting on Q-curve this minimum point with certain maximum points, is maximal. We also provide a posteriori error estimates of the approximate solution, which allows to check the reliability of the parameter chosen heuristically. Numerical experiments on an extensive set of test problems confirm that the proposed rules give much better results than previous heuristic rules. Results of proposed rules are comparable with results of the discrepancy principle and the monotone error rule, if the last two rules use the exact noise level.

Determination of the Fractional Order in Semilinear Subdiffusion Equations

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

A parameter choice rule for Tikhonov regularization based on predictive risk

In this work, we propose a new criterion for choosing the regularization parameter in Tikhonov regularization when the noise is white Gaussian. The criterion minimizes a lower bound of the predictive risk, when both data norm and noise variance are known, and the parameter choice involves minimizing a function whose solution depends only on the signal-to-noise ratio. Moreover, when neither noise variance nor data norm is given, we propose an iterative algorithm which alternates between a minimization step of finding the regularization parameter and an estimation step of estimating signal-to-noise ratio. Simulation studies on both small- and large-scale datasets suggest that the approach can provide very accurate and stable regularized inverse solutions and, for small sized samples, it outperforms discrepancy principle, balancing principle, unbiased predictive risk estimator, L-curve method, generalized cross validation, and quasi-optimality criterion, and achieves excellent stability hitherto unavailable.

The quasi-optimality criterion in the linear functional strategy

The linear functional strategy for the regularization of inverse problems is considered. For selecting the regularization parameter therein, we propose the heuristic quasi-optimality principle and some modifications including the smoothness of the linear functionals. We prove convergence rates for the linear functional strategy with these heuristic rules taking into account the smoothness of the solution and the functionals and imposing a structural condition on the noise. Furthermore, we study these noise conditions in both a deterministic and stochastic setup and verify that for mildly-ill-posed problems and Gaussian noise, these conditions are satisfied almost surely, where on the contrary, in the severely-ill-posed case and in a similar setup, the corresponding noise condition fails to hold. Moreover, we propose an aggregation method for adaptively optimizing the parameter choice rule by making use of improved rates for linear functionals. Numerical results indicate that this method yields better results than the standard heuristic rule.

Structural finite element model updating using incomplete ambient vibration modal data

This paper presents an effective approach for updating finite element dynamic model from incomplete modal data identified from ambient vibration measurements. The proposed method is based on the relationship between the perturbation of structural parameters such as stiffness and mass changes and the modal data measurements of the tested structure such as measured mode shape readings. Structural updating parameters including both stiffness and mass parameters are employed to represent the differences in structural parameters between the finite element model and the associated tested structure. These updating parameters are then evaluated by an iterative solution procedure, giving optimised solutions in the least squares sense without requiring an optimisation technique. In order to reduce the influence of modal measurement uncertainty, the truncated singular value decomposition regularization method incorporating the quasi-optimality criterion is employed to produce reliable solutions for the structural updating parameters. Finally, the numerical investigations of a space frame structure and the practical applications to the Canton Tower benchmark problem demonstrate that the proposed method can correctly update the given finite element model using the incomplete modal data identified from the recorded ambient vibration measurements.

Old and new parameter choice rules for discrete ill-posed problems

Linear discrete ill-posed problems are difficult to solve numerically because their solution is very sensitive to perturbations, which may stem from errors in the data and from round-off errors introduced during the solution process. The computation of a meaningful approximate solution requires that the given problem be replaced by a nearby problem that is less sensitive to disturbances. This replacement is known as regularization. A regularization parameter determines how much the regularized problem differs from the original one. The proper choice of this parameter is important for the quality of the computed solution. This paper studies the performance of known and new approaches to choosing a suitable value of the regularization parameter for the truncated singular value decomposition method and for the LSQR iterative Krylov subspace method in the situation when no accurate estimate of the norm of the error in the data is available. The regularization parameter choice rules considered include several L-curve methods, Regińska’s method and a modification thereof, extrapolation methods, the quasi-optimality criterion, rules designed for use with LSQR, as well as hybrid methods.

A new choice rule for regularization parameters in Tikhonov regularization

This article proposes and analyses a novel heuristic rule for choosing regularization parameters in Tikhonov regularization for inverse problems. The existence of solutions to the regularization parameter equation is shown, and a variational characterization of the inverse solution is provided. Some a posteriori error estimates of the approximate solutions to the inverse problem are also established. An iterative algorithm is suggested for the efficient numerical realization of the new choice rule, which is shown to have a practically desirable monotone convergence. Numerical experiments for both mildly and severely ill-posed benchmark inverse problems with various regularizing functionals of Tikhonov type, e.g. L 2–L 2 with constraints, L 2–ℓ1 and L 1–TV, are presented to demonstrate its effectiveness, and compared with the quasi-optimality criterion.

Parameter choice methods using minimization schemes

Regularization is typically based on the choice of some parametric family of nearby solutions, and the choice of this family is a task in itself. Then, a suitable parameter must be chosen in order to find an approximation of good quality. We focus on the second task. There exist deterministic and stochastic models for describing noise and solutions in inverse problems. We will establish a unified framework for treating different settings for the analysis of inverse problems, which allows us to prove the convergence and optimality of parameter choice schemes based on minimization in a generic way. We show that the well known quasi-optimality criterion falls in this class. Furthermore we present a new parameter choice method and prove its convergence by using this newly established tool.

Comparison of parameter choices in regularization algorithms in case of different information about noise level

We consider linear ill-posed problems in Hilbert space with noisy data. The noise level may be given exactly or approximately or there may be no information about the noise level. We regularize the problem using the Landweber method, the Tikhonov method or the extrapolated version of the Tikhonov method. For all three cases of noise information we propose rules for choice of the regularization parameter. Extensive numerical experiments show the advantage of the proposed rules over known rules, including the discrepancy principle, the quasioptimality criterion, the Hanke-Raus rule, the Brezinski-Rodriguez-Seatzu rule and others. Numerical comparison also shows at which information about the noise level our rules for approximately given noise level should be preferred to other rules.

Heuristic Parameter-Choice Rules for Convex Variational Regularization Based on Error Estimates

In this paper, we are interested in heuristic parameter choice rules for general convex variational regularization which are based on error estimates. Two such rules are derived and generalize those from quadratic regularization, namely the Hanke-Raus rule and quasi-optimality criterion. A posteriori error estimates are shown for the Hanke-Raus rule, and convergence for both rules is also discussed. Numerical results for both rules are presented to illustrate their applicability.

On Minimization Strategies for Choice of the Regularization Parameter in Ill-Posed Problems

We consider solving of linear ill-posed problems using the Tikhonov method (in self-adjoint case the Lavrentiev method), its iterated variant, Landweber method and conjugate gradient type methods. Several rules for a posteriori choice of the regularization parameter are proposed. In case of known noise level of data we propose to compute in Tikhonov method certain 2 parameters and take for regularization parameter minimal of them. In case of unknown noise level we consider family of rules where a certain function is minimized. The quasioptimality criterion and Hanke-Raus rule are included, error estimates are given. Extensive numerical experiments show an advantage of proposed rules over known rules.

Frequency-Dependent Diffusion Constant of Quantum Fluids from Path Integral Monte Carlo and Tikhonov's Regularizing Functional.

We present a novel implementation of the analytic continuation of the velocity autocorrelation function method that has been developed to study the transport properties of quantum liquids at finite temperatures. To invert the ill-posed linear Fredholm integral equation of the first kind, we combine Tikhonov's first-order regularizing functional with several methods used for automatic selection of the regularization parameter. Taking into account our results, we recommend two methods for automatic selection of the regularization parameter, namely: L-curve and quasi-optimality criterion. We found that the frequency-dependent diffusion power spectrum of normal liquid (4)He at T = 4 K and ρ = 0.01873 Å(-3) (ρ = 31.1 mmol cm(-3)) is characterized by a single asymmetric peak. The predicted self-diffusion coefficient of (4)He at this state point of 0.57-0.58 Å(2)/ps is in excellent agreement with previous works. We demonstrate that, within proposed mathematical treatment of the quantum transport at finite temperatures, the entire real-time frequency-dependent diffusion power spectrum of liquid normal (4)He, can be successfully reconstructed from the limited number of Trotter slices and without the knowledge of covariance matrix. Moreover, the small values of regularization parameters (i.e., order of 10(-7)) indicate that the information about quantum dynamics of normal liquid (4)He can be easily withdrawn from the high quality imaginary-time correlation function collected in the standard path integral Monte Carlo simulation.

Regularization independent of the noise level: an analysis of quasi-optimality

The quasi-optimality criterion chooses the regularization parameter in inverse problems without taking into account the noise level. This rule works remarkably well in practice, although Bakushinskii has shown that there are always counterexamples with very poor performance. We propose an average case analysis of quasi-optimality for spectral cut-off estimators (also known as truncated singular value decomposition, TSVD) and we prove that the quasi-optimality criterion determines estimators which are rate-optimal on average. Its practical performance is illustrated with a calibration problem from mathematical finance.

The quasi-optimality criterion for classical inverse problems

The quasi-optimality criterion chooses the regularization parameter in inverse problems without requiring knowledge about the noise level. It is well known that this cannot yield convergence for ill-posed problems in the worst case. In this paper, we establish conditions providing lower bounds on the approximation error and the propagated noise error, such that these terms can be estimated from above and below by a geometric series. Using these we can show convergence and optimal-order error bounds for Tikhonov regularization with the quasi-optimality criterion both in the case of deterministic problems as well as for stochastic noise.

On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization

In this paper we derive convergence and convergence rates results of the quasioptimality criterion for (iterated) Tikhonov regularization. We prove convergence and suboptimal rates under a qualitative condition on the decay of the noise with respect to the spectral family of $T$$T$*. Moreover, optimal rates are obtained if the exact solution satisfies a decay condition with respect to the spectral family of $T$*$T$.

Some considerations concerning regularization and parameter choice algorithms

Using a Bayesian-type approach to inverse problems many phenomena occurring in practice can be explained in a consistent way. In particular we can prove a discrete version of the quasi-optimality criterion for choosing the regularization parameter by simply imposing minor additional a priori assumptions about the solution and the measurement noise.

Reconstruction of the surface normal acoustic intensity from noisy pressure data by regularized Helmholtz equation least squares method

Reconstructing the normal acoustic intensity on a source surface from the field pressure measurements is critical in identifying the noise sources and their transmission paths. In this paper, we consider the reconstruction of time-averaged normal acoustic intensities from noisy pressure data using the Helmholtz equation least squares (HELS) method with regularization by truncated singular value decomposition or the method of Tikhonov. The regularization parameter is calculated by generalized cross validation, L-curve, and quasioptimality criterion and the best solution is chosen. Numerical results show that the normal velocity reconstruction is more sensitive to noise in the input data than the pressure reconstruction, and depends critically on the choice of the regularization parameter. It is found that the HELS-based Dirichlet-to-Neumann (DtN) map has a similar accuracy as compared to the Green’s formula-based DtN map with Tikhonov regularization for 1% noise or larger in the input data, provided that HELS converges sufficiently fast. Moreover, determining the surface normal velocity from the reconstructed surface pressure is more stable than reconstructing it directly from the field pressure data. It is concluded that a regularized HELS, when it converges fast enough, can be an easy-to-use and cost-effective tool for intensity reconstruction [Work supported by NSF.]