scholarly journals Maximal spaces for approximation rates in $$\ell ^1$$-regularization

2021 ◽  
Author(s):  
Philip Miller ◽  
Thorsten Hohage
Keyword(s):  
Regularization Parameter ◽  
Identification Problem ◽  
Rates Of Convergence ◽  
Parameter Choice ◽  
Penalty Term ◽  
Approximation Rates ◽  
Wavelet Regularization ◽  
Parameter Identification Problem ◽  
Theoretical Results ◽  
Forward Operator

AbstractWe study Tikhonov regularization for possibly nonlinear inverse problems with weighted $$\ell ^1$$ ℓ 1 -penalization. The forward operator, mapping from a sequence space to an arbitrary Banach space, typically an $$L^2$$ L 2 -space, is assumed to satisfy a two-sided Lipschitz condition with respect to a weighted $$\ell ^2$$ ℓ 2 -norm and the norm of the image space. We show that in this setting approximation rates of arbitrarily high Hölder-type order in the regularization parameter can be achieved, and we characterize maximal subspaces of sequences on which these rates are attained. On these subspaces the method also converges with optimal rates in terms of the noise level with the discrepancy principle as parameter choice rule. Our analysis includes the case that the penalty term is not finite at the exact solution (’oversmoothing’). As a standard example we discuss wavelet regularization in Besov spaces $$B^r_{1,1}$$ B 1 , 1 r . In this setting we demonstrate in numerical simulations for a parameter identification problem in a differential equation that our theoretical results correctly predict improved rates of convergence for piecewise smooth unknown coefficients.

A wavelet method for numerical fractional derivative with noisy data

2016 ◽  
Vol 14 (05) ◽  
pp. 1650038 ◽  
Author(s):  
Xiangtuan Xiong ◽  
Qiang Cheng ◽  
Yanfeng Kong ◽  
Jin Wen
Keyword(s):  
Fractional Derivative ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Large Change ◽  
Stability Estimates ◽  
Parameter Choice ◽  
Wavelet Regularization ◽  
Ill Posed ◽  
Meyer Wavelet ◽  
Parameter Choice Rules

Numerical fractional differentiation is a classical ill-posed problem in the sense that a small perturbation in the data can cause a large change in the fractional derivative. In this paper, we consider a wavelet regularization method for solving a reconstruction problem for numerical fractional derivative with noise. A Meyer wavelet projection regularization method is given, and the Hölder-type stability estimates under both apriori and aposteriori regularization parameter choice rules are obtained. Some numerical examples show that the method works well.

Simplified Iteratively Regularized Gauss–Newton Method in Banach Spaces Under a General Source Condition

2020 ◽  
Vol 20 (2) ◽  
pp. 321-341
Author(s):  
Pallavi Mahale ◽  
Sharad Kumar Dixit
Keyword(s):  
Newton Method ◽  
Regularization Parameter ◽  
A Priori ◽  
Identification Problem ◽  
Stopping Rule ◽  
Optimal Error Estimates ◽  
Source Condition ◽  
Parameter Identification Problem ◽  
Ill Posed ◽  
General Source

AbstractIn this paper, we consider a simplified iteratively regularized Gauss–Newton method in a Banach space setting under a general source condition. We will obtain order-optimal error estimates both for an a priori stopping rule and for a Morozov-type stopping rule together with a posteriori choice of the regularization parameter. An advantage of a general source condition is that it provides a unified setting for the error analysis which can be applied to the cases of both severely and mildly ill-posed problems. We will give a numerical example of a parameter identification problem to discuss the performance of the method.

Derivative Free Regularization Method for Nonlinear Ill-Posed Equations in Hilbert Scales

2019 ◽  
Vol 19 (4) ◽  
pp. 765-778
Author(s):  
Santhosh George ◽  
K. Kanagaraj
Keyword(s):  
Regularization Parameter ◽  
Adjoint Operator ◽  
Identification Problem ◽  
Optimal Order ◽  
Order Error ◽  
Adaptive Parameter ◽  
Derivative Free ◽  
Parameter Identification Problem ◽  
Ill Posed ◽  
Parameter Choice Strategy

AbstractIn this paper, we deal with nonlinear ill-posed operator equations involving a monotone operator in the setting of Hilbert scales. Our convergence analysis of the proposed derivative-free method is based on the simple property of the norm of a self-adjoint operator. Using a general Hölder-type source condition, we obtain an optimal order error estimate. Also we consider the adaptive parameter choice strategy proposed by Pereverzev and Schock (2005) for choosing the regularization parameter. Finally, we applied the proposed method to the parameter identification problem in an elliptic PDE in the setting of Hilbert scales and compare the results with the corresponding method in Hilbert space.

scholarly journals Regularization of backward time-fractional parabolic equations by Sobolev-type equations

2020 ◽  
Vol 28 (5) ◽  
pp. 659-676
Author(s):  
Dinh Nho Hào ◽  
Nguyen Van Duc ◽  
Nguyen Van Thang ◽  
Nguyen Trung Thành
Keyword(s):  
Error Estimates ◽  
Parabolic Equations ◽  
Regularization Parameter ◽  
A Priori ◽  
Sobolev Type ◽  
A Posteriori ◽  
Parameter Choice ◽  
Numerical Tests ◽  
Ill Posed ◽  
Parameter Choice Rules

AbstractThe problem of determining the initial condition from noisy final observations in time-fractional parabolic equations is considered. This problem is well known to be ill-posed, and it is regularized by backward Sobolev-type equations. Error estimates of Hölder type are obtained with a priori and a posteriori regularization parameter choice rules. The proposed regularization method results in a stable noniterative numerical scheme. The theoretical error estimates are confirmed by numerical tests for one- and two-dimensional equations.

scholarly journals Multilayer method for solving a problem of metals rupture under creep conditions

2019 ◽  
Vol 23 (Suppl. 2) ◽  
pp. 575-582 ◽  
Author(s):  
Evgenii Kuznetsov ◽  
Sergey Leonov ◽  
Dmitry Tarkhov ◽  
Alexander Vasilyev
Keyword(s):  
Neural Network ◽  
Experimental Data ◽  
Parameter Identification ◽  
Uniaxial Tension ◽  
Network Modeling ◽  
Identification Problem ◽  
Neural Network Modeling ◽  
Creep Theory ◽  
Parameter Identification Problem ◽  
Kinetic Creep

The paper deals with a parameter identification problem for creep and fracture model. The system of ordinary differential equations of kinetic creep theory is applied for describing this model. As for solving the parameter identification problem, we proposed to use the technique of neural network modeling, as well as the multilayer approach. The procedures of neural network modeling and multilayer approximation constructing application is demonstrated by the example of finding parameters for uniaxial tension model for isotropic steel 45 specimens at creep conditions. The solution corresponding to the obtained parameters agrees well with theoretical strain-damage characteristics, experimental data, and results of other authors.

Electrostatic lattice energy in ionic crystals: optimization of the convergence of Ewald series

1978 ◽  
Vol 34 (6) ◽  
pp. 974-979 ◽  
Author(s):  
M. Catti
Keyword(s):  
Lattice Energy ◽  
Rates Of Convergence ◽  
Unit Cell ◽  
Group Symmetry ◽  
Cube Root ◽  
Cell Dimensions ◽  
Madelung Energy ◽  
Direct Lattice ◽  
Unit Cell Dimensions ◽  
Theoretical Results

Ewald's method is reconsidered to express the dependence of Madelung energy on the ionic charges explicitly, also taking into account the space-group symmetry of the structure. Upper bounds for the residues of the two partial series have been calculated by integral approximation; that relative to the direct-lattice series is shown to depend on the cube root of the unit-cell volume. The optimum value of the parameter A, which equalizes the rates of convergence of the two sums and minimizes the total number of terms, has been determined numerically for a given termination error and for a range of unit-cell dimensions. Theoretical results are tested by calculations on some specific crystal structures.

scholarly journals A Novel Improved Maximum Entropy Regularization Technique and Application to Identification of Dynamic Loads on the Coal Rock

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Chunsheng Liu ◽  
Chunping Ren
Keyword(s):  
Maximum Entropy ◽  
Dynamic Load ◽  
Regularization Parameter ◽  
Estimation Method ◽  
Identification Problem ◽  
Iteration Algorithm ◽  
Load Identification ◽  
Signal Processing Algorithm ◽  
Rock Structure ◽  
Coal Rock

A new signal processing algorithm was proposed to identify the dynamic load acting on the coal-rock structure. First, the identification model for dynamic load is established through the relationship between the uncertain load vector, and the assembly matrix of the responses was measured by the machinery dynamic system. Then, the entropy item of maximum entropy regularization (MER) is redesigned using the robust estimation method, and the elongated penalty function according to the ill-posedness characteristics of load identification, which was named as a novel improved maximum entropy regularization (IMER) technique, was proposed to process the dynamic load signals. Finally, the load identification problem is transformed into an unconstrained optimization problem and an improved Newton iteration algorithm was proposed to solve the objective function. The result of IMER technique is compared with MER technique, and it is found that IMER technique is available for analyzing the dynamic load signals due to higher signal-noise ratio, lower restoration time, and fewer iterative steps. Experiments were performed to investigate the effect on the performance of dynamic load signals identification by different regularization parameters and calculation parameters, pi, respectively. Experimental results show that the identified dynamic load signals are closed to the actual load signals using IMER technique combined with the proposed PSO-L regularization parameter selection method. Selecting optimal calculated parameters pi is helpful to overcome the ill-condition of dynamic load signals identification and to obtain the stable and approximate solutions of inverse problems in practical engineering. Meanwhile, the proposed IMER technique can also play a guiding role for the coal-rock interface identification.

Sign in / Sign up

Export Citation Format

Share Document