scholarly journals Restoration of Hardware Function of Spectral Colorimeters Using Regularization Methods

2021 ◽  
Vol 20 (6) ◽  
pp. 487-492
Author(s):  
M. A. Radjabova ◽  
B. I. Eshmatova ◽  
N. K. Babanazarova
Keyword(s):  
Regularization Parameter ◽  
Scattered Light ◽  
A Priori ◽  
Light Flux ◽  
Experimental Methods ◽  
Electrical Signal ◽  
Regularization Methods ◽  
Special Equipment ◽  
Spectral Device ◽  
Hardware Function

The problem of determining the spectral characteristic of a controlled sample under conditions of limited a priori information using regularization methods is considered in the paper. A change in the state of the surface of optical elements significantly increases the light scattering, so it is necessary regularly to take into account the amount of scattered light in the light flux reflected from the surface and the measured and comparative samples. The conversion of the light flux into the electrical signal of the photodetector can also occur non-linearly. This requires the development of such measurement method that considers both the scattered light and various non-linearities of the measuring circuit. It is known that the mathematical model of measurement is described by the Fredholm integral equation of the first kind, its solution under the accepted assumptions is recommended to be sought in the form of a matrix equation using a recurring procedure. With regard to the fact that the estimation of the initial data errors in the equation is associated with certain difficulties, in the case under consideration, it is advisable to determine the regularization parameter based on the method of quasi-optimality. A characteristic disadvantage of the known analytical and experimental methods for determining the hardware function of a spectral device is that they do not take into account its change during operation. Since the actual hardware function of the device usually differs from the Gaussian curve, the use of hardware functions in the form of analytical dependencies does not always give the desired result, and for experimental methods, special equipment with a quasi-monochromatic radiation source is required. An algorithm for restoring the hardware function of a spectral device based on regular methods for solving ill-posed problems is proposed. The estimation of the matrix operator of the hardware function is proposed to be obtained on the basis of explicit least squares estimation algorithms. The expediency of choosing a value of the regularization parameter that minimizes the accepted characteristic of the accuracy of the solution is indicated.

CROSS-VALIDATION BASED ADAPTATION FOR REGULARIZATION OPERATORS IN LEARNING THEORY

2010 ◽  
Vol 08 (02) ◽  
pp. 161-183 ◽  
Author(s):  
ANDREA CAPONNETTO ◽  
YUAN YAO
Keyword(s):  
Cross Validation ◽  
Regularization Parameter ◽  
Broad Class ◽  
A Priori ◽  
Regularization Methods ◽  
A Posteriori ◽  
Effective Dimension ◽  
Truncated Svd ◽  
Universal Consistency ◽  
Minimax Rate

We consider learning algorithms induced by regularization methods in the regression setting. We show that previously obtained error bounds for these algorithms, using a priori choices of the regularization parameter, can be attained using a suitable a posteriori choice based on cross-validation. In particular, these results prove adaptation of the rate of convergence of the estimators to the minimax rate induced by the "effective dimension" of the problem. We also show universal consistency for this broad class of methods which includes regularized least-squares, truncated SVD, Landweber iteration and ν-method.

scholarly journals Two fractional regularization methods for identifying the radiogenic source of the Helium production-diffusion equation

2021 ◽  
Vol 6 (10) ◽  
pp. 11425-11448
Author(s):  
Xuemin Xue ◽  
◽  
Xiangtuan Xiong ◽  
Yuanxiang Zhang ◽  
Keyword(s):  
Diffusion Equation ◽  
Source Term ◽  
Regularization Parameter ◽  
A Priori ◽  
Regularization Methods ◽  
A Posteriori ◽  
Parameter Choice ◽  
Ill Posed ◽  
Parameter Choice Rules ◽  
Regularized Solution

<abstract><p>The predication of the helium diffusion concentration as a function of a source term in diffusion equation is an ill-posed problem. This is called inverse radiogenic source problem. Although some classical regularization methods have been considered for this problem, we propose two new fractional regularization methods for the purpose of reducing the over-smoothing of the classical regularized solution. The corresponding error estimates are proved under the a-priori and the a-posteriori regularization parameter choice rules. Some numerical examples are shown to display the necessarity of the methods.</p></abstract>

Comparisons of regularization methods and regularization parameter selection methods in sound source identification using inverse boundary element method

2019 ◽  
Vol 67 (3) ◽  
pp. 219-227
Author(s):  
Youhong Xiao ◽  
Qingqing Song ◽  
Shaowei Li ◽  
Guoxue Lv ◽  
Zhenlin Ji
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Transfer Matrix ◽  
Sound Source ◽  
Source Identification ◽  
Regularization Parameter ◽  
Vibration Velocity ◽  
Regularization Methods ◽  
Inverse Boundary ◽  
Element Method

In noise source identification based on the inverse boundary element method (IBEM), the boundary vibration velocity is predicted based on the field pressure through a transfer matrix of the vibration velocity and field pressure established on the Helmholtz integral equation. Because the matrix is often ill-posed, it needs to be regularized before reconstructing the vibration velocity. Two regularization methods and two methods of selecting the regularization parameter are investigated through the simulation analysis of a pulsating sphere. The result of transfer matrix regularization is further verified through the reconstruction of the vibration of an aluminum plate. Additionally, to reduce the large errors at some frequencies in the reconstruction result, increasing the number of measuring points is more effective than reducing the distance between the measurement plane and the sound source.

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 A Tikhonov-Type Regularization Method for Identifying the Unknown Source in the Modified Helmholtz Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Jinghuai Gao ◽  
Dehua Wang ◽  
Jigen Peng
Keyword(s):  
Helmholtz Equation ◽  
Regularization Method ◽  
Regularization Parameter ◽  
A Priori ◽  
Inverse Source Problem ◽  
A Posteriori ◽  
Modified Helmholtz Equation ◽  
Numerical Tests ◽  
Theoretical Frame ◽  
Set Up

An inverse source problem in the modified Helmholtz equation is considered. We give a Tikhonov-type regularization method and set up a theoretical frame to analyze the convergence of such method. A priori and a posteriori choice rules to find the regularization parameter are given. Numerical tests are presented to illustrate the effectiveness and stability of our proposed method.

scholarly journals Technical Note: Regularization performances with the error consistency method in the case of retrieved atmospheric profiles

2006 ◽  
Vol 6 (6) ◽  
pp. 13307-13321
Author(s):  
S. Ceccherini ◽  
C. Belotti ◽  
B. Carli ◽  
P. Raspollini ◽  
M. Ridolfi
Keyword(s):  
Regularization Parameter ◽  
Technical Note ◽  
Field Of View ◽  
Regularization Methods ◽  
Vertical Profiles ◽  
Vertical Resolution ◽  
Vertical Field ◽  
The Stability ◽  
First Time ◽  
Consistency Method

Abstract. The retrieval of concentration vertical profiles of atmospheric constituents from spectroscopic measurements is often an ill-conditioned problem and regularization methods are frequently used to improve its stability. Recently a new method, that provides a good compromise between precision and vertical resolution, was proposed to determine analytically the value of the regularization parameter. This method is applied for the first time to real measurements with its implementation in the operational retrieval code of the satellite limb-emission measurements of the MIPAS instrument and its performances are quantitatively analyzed. The adopted regularization improves the stability of the retrieval providing smooth profiles without major degradation of the vertical resolution. In the analyzed measurements the retrieval procedure provides a vertical resolution that, in the troposphere and low stratosphere, is smaller than the vertical field of view of the instrument.

scholarly journals Laser Doppler Flowmetry versatile acquisition system with low quantization error

2018 ◽  
Vol 4 (1) ◽  
pp. 25
Author(s):  
Francesco Marrazzi ◽  
Frederic Truffer ◽  
Martial Geiser
Keyword(s):  
Laser Doppler Flowmetry ◽  
Analog Circuit ◽  
Scattered Light ◽  
Blood Perfusion ◽  
Electrical Signal ◽  
Laser Doppler ◽  
Physical Models ◽  
Acquisition System ◽  
Invasive Technique ◽  
Doppler Flowmetry

The Laser Doppler Flowmetry (LDF) is a non-invasive technique used to evaluate blood perfusion of various human tissues like the skin or the fundus of the eye. It is based on the scattering of light on moving red blood cells in tissue. Frequency shifted scattered light is detected and provide an electrical signal. Physical models for LDF use the DC and AC components of this signal. If AC is small relative to the DC, digitalization becomes an issue, and if more than two LDF signal acquisitions and analysis have to be done simultaneously, the device becomes expensive and bulky. We propose here a versatile and inexpensive acquisition system, which overcomes quantization errors issue by first separating DC from AC, then amplifying AC and finally recombining both signals before digitalization. We designed an analog circuit combined with a 12 bit analog-to-digital converter, a microcontroller unit and a Raspberry Pi2 (Rpi2) for the signal processing. Results are accessed remotely from the Rpi2 through HTTP protocol. Multiple systems can easily be used simultaneously for multichannel acquisitions.

scholarly journals The Impact of the Discrepancy Principle on the Tikhonov-Regularized Solutions with Oversmoothing Penalties

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 331
Author(s):  
Bernd Hofmann ◽  
Christopher Hofmann
Keyword(s):  
Case Studies ◽  
Convergence Rates ◽  
Regularization Parameter ◽  
A Priori ◽  
Discrepancy Principle ◽  
Parameter Choice ◽  
Operator Equations ◽  
Analytical Results ◽  
Ill Posed ◽  
The Impact

This paper deals with the Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. One focus is on the application of the discrepancy principle for choosing the regularization parameter and its consequences. Numerical case studies are performed in order to complement analytical results concerning the oversmoothing situation. For example, case studies are presented for exact solutions of Hölder type smoothness with a low Hölder exponent. Moreover, the regularization parameter choice using the discrepancy principle, for which rate results are proven in the oversmoothing case in in reference (Hofmann, B.; Mathé, P. Inverse Probl. 2018, 34, 015007) is compared to Hölder type a priori choices. On the other hand, well-known analytical results on the existence and convergence of regularized solutions are summarized and partially augmented. In particular, a sketch for a novel proof to derive Hölder convergence rates in the case of oversmoothing penalties is given, extending ideas from in reference (Hofmann, B.; Plato, R. ETNA. 2020, 93).

Analysis of Order of the Sequential Regularization Solutions of Inverse Heat Conduction Problems

1989 ◽  
Vol 111 (2) ◽  
pp. 218-224 ◽  
Author(s):  
E. P. Scott ◽  
J. V. Beck
Keyword(s):  
Heat Conduction ◽  
Regularization Method ◽  
Heat Conduction Problem ◽  
Regularization Parameter ◽  
Inverse Heat Conduction Problem ◽  
Regularization Methods ◽  
Inverse Heat Conduction ◽  
Random Errors ◽  
Internal Temperature ◽  
Inverse Heat Conduction Problems

Various methods have been proposed to solve the inverse heat conduction problem of determining a boundary condition at the surface of a body from discrete internal temperature measurements. These include function specification and regularization methods. This paper investigates the various components of the regularization method using the sequential regularization method proposed by Beck and Murio (1986). Specifically, the effects of the regularization order and the influence of the regularization parameter are analyzed. It is shown that as the order of regularization increases, the bias errors decrease and the variance increases. Comparatively, the zeroth regularization has higher bias errors and the second-order regularization is more sensitive to random errors. As the regularization parameter decreases, the sensitivity of the estimator to random errors is shown to increase; on the other hand, the bias errors are shown to decrease.

A derivative-free iterative method for nonlinear ill-posed equations with monotone operators

2017 ◽  
Vol 25 (5) ◽  
pp. 543-551 ◽  
Author(s):  
Santhosh George ◽  
M. Thamban Nair
Keyword(s):  
Iterative Method ◽  
Convergence Analysis ◽  
Monotone Operator ◽  
Regularization Parameter ◽  
A Priori ◽  
Monotone Operators ◽  
Adaptive Procedure ◽  
Operator Equations ◽  
Derivative Free ◽  
Ill Posed

AbstractRecently, Semenova [12] considered a derivative free iterative method for nonlinear ill-posed operator equations with a monotone operator. In this paper, a modified form of Semenova’s method is considered providing simple convergence analysis under more realistic nonlinearity assumptions. The paper also provides a stopping rule for the iteration based on an a priori choice of the regularization parameter and also under the adaptive procedure considered by Pereverzev and Schock [11].

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