Accurate Spectral Collocation Computation of High Order Eigenvalues for Singular Schrödinger Equations

We are concerned with the study of some classical spectral collocation methods, mainly Chebyshev and sinc as well as with the new software system Chebfun in computing high order eigenpairs of singular and regular Schrödinger eigenproblems. We want to highlight both the qualities as well as the shortcomings of these methods and evaluate them in conjunction with the usual ones. In order to resolve a boundary singularity, we use Chebfun with domain truncation. Although it is applicable with spectral collocation, a special technique to introduce boundary conditions as well as a coordinate transform, which maps an unbounded domain to a finite one, are the special ingredients. A challenging set of “hard”benchmark problems, for which usual numerical methods (f. d., f. e. m., shooting, etc.) fail, were analyzed. In order to separate “good”and “bad”eigenvalues, we have estimated the drift of the set of eigenvalues of interest with respect to the order of approximation and/or scaling of domain parameter. It automatically provides us with a measure of the error within which the eigenvalues are computed and a hint on numerical stability. We pay a particular attention to problems with almost multiple eigenvalues as well as to problems with a mixed spectrum.

Quantitative analysis of mixed pigments for Chinese paintings using the improved method of ratio spectra derivative spectrophotometry based on mode

Background Mixed pigment analysis is an important and complex subject in preserving and restoring Chinese paintings since the colors observed by our naked eyes or instruments such as hyperspectral cameras are usually a mixture of several kinds of pigments. The purpose of this study was to explore a more effective method to confirm the type of every pure pigment and their proportion in pigment mixtures on the surface of paintings. Methods Two endmember extraction algorithms were adopted to identify the types of pigments and an improved method of ratio spectra derivative spectrophotometry was used to determine their proportion. Main works (1) Extracting the pure pigment components from mixed spectrum by adopting two blind source separation algorithms: Independent Component Analysis and Non-negative Matrix Factorization; (2) matching the separated pure spectrum with the pigment spectral library built in our laboratory to determine the pigment type; and (3) calculating the proportions of mixed pigments using the Ratio Spectra Derivative Spectrophotometry based on Mode, which is improved based on the original algorithm. Finally, a comparison was made with two abundance inversion algorithms: Least Squares Algorithm and Minimum Volume Simplex Analysis. The correlation coefficient and root mean square error were used to provide evidence for accuracy evaluation. Conclusions (1) Non-negative matrix factorization is more suitable for endmember extraction; and (2) Ratio spectra derivative spectrophotometry based on mode is more suitable for abundance inversion.

Dose Dependence of Micro-Voids Distributions in Low-Temperature Neutron Irradiated Eurofer97 Steel

The microstructural effects of mixed spectrum neutron irradiation at 250 °C and 300 °C, for 2.7 dpa, 8.4 dpa, and 16.3 dpa doses, have been investigated in standard Eurofer97 (0.12 C, 9 Cr, 0.48 Mn, 0.2 V, 1.08 W, 0.14 Ta wt%) by means of small-angle neutron scattering (SANS) compared with un-irradiated Eurofer97. The observed SANS effects are attributed to the development of micro-voids, also detected by electron microscopy. The micro-voids distributions have been obtained by an improved transformation method of the SANS cross-sections providing consistent results both before and after subtraction of the un-irradiated reference. Mono-disperse micro-voids distributions are found, with average radii increasing with the dose, namely 4.4 Å for irradiation to 2.7 dpa at 300 °C, 6.6 Å for 8.4 dpa at 300 °C, and 12.9 Å for 16.4 dpa at 250 °C; the corresponding volume fractions are 0.001, 0.006, and 0.004, respectively. The differences in such distributions might reflect different damage evolution mechanisms for the different irradiation conditions, as also suggested by the comparison with a Eurofer97 sample irradiated under fast spectrum. A good resistance of Eurofer97 to micro-structural radiation damage, at least under these irradiation conditions, is suggested by the analysis of these experimental results.