The inverse problem of bimorph mirror tuning on a beamline

2011 ◽  
Vol 18 (6) ◽  
pp. 930-937 ◽  
Author(s):  
Rong Huang
Keyword(s):  
Inverse Problem ◽  
Tikhonov Regularization ◽  
Measurement Errors ◽  
Regularization Parameter ◽  
X Rays ◽  
The Matrix ◽  
Matrix Condition Number ◽  
Matrix Condition ◽  
Safety Limit

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 tuningin 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.

scholarly journals INVERSE PROBLEM OF A ONE-DIMENSIONAL MODEL IN MULTILAYER HEAT CONDUCTION

2020 ◽  
Vol 19 (1) ◽  
pp. 42
Author(s):  
G. C. Oliveira ◽  
S. S. Ribeiroa ◽  
G. Guimarães
Keyword(s):  
Inverse Problem ◽  
Measurement Errors ◽  
Input Data ◽  
Temperature Measurements ◽  
Dimensional Model ◽  
One Dimensional ◽  
Ill Posed ◽  
Measurements Errors ◽  
Chip Chip

The inverse problem in conducting heat is related to the determination of the boundary condition, rate of heat generation, or thermophysical properties, using temperature measurements at one or more positions of the solid. The inverse problem in conducting heat is mathematically one of the ill-posed problems, because its solution extremely sensitive to measurement errors. For a well-placed problem the following conditions must be satisfied: the solution must exist, it must be unique and must be stable on small changes of the input data. The objective of the work is to estimate the heat flux generated at the tool-chip-chip interface in a manufacturing process. The term "estimation" is used because in the temperature measurements, errors are always present and these affect the accuracy of the calculation of the heat flow.

Determination of the Distribution of the Relaxation Times from Dielectric Spectra

2004 ◽  
Vol 9 (1) ◽  
pp. 75-88 ◽  
Author(s):  
J. Macutkevic ◽  
J. Banys ◽  
A. Matulis
Keyword(s):  
Integral Equations ◽  
Tikhonov Regularization ◽  
Regularization Parameter ◽  
Relaxation Times ◽  
Dielectric Susceptibility ◽  
Dynamical Processes ◽  
Regularization Technique ◽  
Dielectric Spectra

The dielectric susceptibility measurements are usually interpreted in terms of the relaxation times of various dynamical processes. Using the simple examples of the simulated spectra it is shown how the distribution of these relaxation times can be obtained by means of the integral equations solved with the Tikhonov regularization technique, and the criteria for the choice of the regularization parameter is discussed.

Tuning the axial singularity method for accurate calculation of potential flow around axisymmetric bodies

1994 ◽  
Vol 98 (976) ◽  
pp. 215-226
Author(s):  
M. F. Zedan
Keyword(s):  
Condition Number ◽  
Intensity Variation ◽  
Axial Line ◽  
Control Points ◽  
Singularity Method ◽  
The Matrix ◽  
Matrix Condition Number ◽  
Order Of Magnitude ◽  
Matrix Condition ◽  
Solution Parameters

Abstract The performance of axial line singularity methods has been investigated systematically for various solution parameters using carefully chosen test cases. The results indicate that increasing the number of elements and using stretched node distribution improves the solution accuracy until the matrix becomes near-singular. The matrix condition number increases with these parameters as well as with the order of intensity variation and profile thickness. For moderate fineness ratios, the linear methods outperform zero-order methods. The linear doublet method performs best with control points at the x-locations of nodes while the source methods perform best with control points mid-way between nodes. The doublet method has a condition number an order of magnitude lower than the source method and generally provides more accurate results and handles a wider range of bodies. With appropriate solution parameters, the method provides excellent accuracy for bodies without slope discontinuity. The smoothing technique proposed recently by Hemsch has been shown to reduce the condition number of the matrix; however it should be used with caution. It is recommended to use it only when the solution is highly oscillatory with a near-singular matrix. A criterion for the optimum value of the smoothing parameter is proposed.

The Choice of Lattice Planes in X-Ray Strain Measurements of Single Crystals

1985 ◽  
Vol 29 ◽  
pp. 113-118 ◽  
Author(s):  
Balder Ortner
Keyword(s):  
Measurement Errors ◽  
Linear Equations ◽  
Strain Tensor ◽  
Equation System ◽  
Necessary Condition ◽  
X Rays ◽  
Distance Measurements ◽  
The Matrix ◽  
Strain Components ◽  
Lattice Planes

It is well known that all of the six independent components of the strain tensor can be calculated if the linear strains in six appropriate directions are known (e.g.). That calculation is to solve a system of linear equations, whose coefficients are defined by the orientations of the measured planes. The strains are determined by lattice plane distance measurements using X-rays.The linear equation system can only be solved if the matrix of coefficients has rank. Whether this condition is met or not can be decided without calculating a determinant just from geometric relationships among the planes to be measured. A demand beyond that necessary condition is to make the matrix of coefficients so that the accuracy of the calculated strain tensor is best. From error calculation we know that there exist distinct ratios between the inevitable measurement errors and the errors of the calculated strain components. These ratios depend strongly on the geometric relationship among the lattice planes. It is the purpose of this paper to show how lattice planes should be chosen in order to get these ratios as small as possible i.e. to get a maximum of accuracy at a given number of measurements, or a minimum of experimental effort if a distinct limit of error is to be reached.

A COMBINED REGULARIZATION ALGORITHM FOR ELECTRICAL IMPEDANCE TOMOGRAPHY SYSTEM USING RECTANGULAR ELECTRODES ARRAY

2012 ◽  
Vol 24 (04) ◽  
pp. 313-322 ◽  
Author(s):  
Wei He ◽  
Bing Li ◽  
Zheng Xu ◽  
Haijun Luo ◽  
Peng Ran
Keyword(s):  
Inverse Problem ◽  
Electrical Impedance Tomography ◽  
Tikhonov Regularization ◽  
Electrical Impedance ◽  
Regularization Parameter ◽  
Impedance Tomography ◽  
Regularization Algorithm ◽  
Tomography System ◽  
One Step ◽  
Mean Square Distance

A novel Electrical Impedance Tomography system with rectangular electrodes array and back electrode is proposed. This system could reconstruct a deeper target and is easy to operate. By studying different reconstructed algorithms: Tikhonov regularization and the Newton's One-step Error Reconstructor (NOSER), a combined regularization algorithm is proposed. The L-curve and posteriori method are used to choose Tikhonov and NOSER regularization parameter. Two evaluation parameters of reconstructed algorithm: normalization mean square distance criterion (NMSD), normalized mean absolute distance criterion (NMAD) are used to evaluate the result's precision of inverse problem quantificationally. The comparison among Tikhonov regularization, NOSER and the combined regularization shows that the ill-condition and the error of inverse problem are reduced. This new algorithm can decrease condition number by 70%, NMSD by 51%, and NMAD by 41% at least. Simulate results show that the combined regularization algorithm could reconstructed the target image in the depth from 10–40 mm. The experimental results show that a 15 mm × 9 mm × 9 mm cuboids whose depth is 35 mm could be distinguished. The performance of this system and the combined regularization algorithm demonstrate significantly better spatial resolution and minor reconstructed error.

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