We investigate \texorpdfstring{$L_2$}{}-approximation problems in the worst case setting in the weighted Hilbert spaces \texorpdfstring{$H(K_{R_{d,\a,\g}})$}{} with weights \texorpdfstring{$R_{d,\a,{\bm \ga}}$}{} under parameters \texorpdfstring{$1\ge \ga_1\ge \ga_2\ge \cdots\ge 0$ and $1<\az_1\le \az_2\le \cdots$}{}. Several interesting weighted Hilbert spaces \texorpdfstring{$H(K_{R_{d,\a,\g}})$}{} appear in this paper. We consider the worst case error of algorithms that use finitely many arbitrary continuous linear functionals. Multivariate approximation; information complexity; tractability; weighted Hilbert spacesWe discuss tractability of \texorpdfstring{$L_2$}{}-approximation problems for the involved Hilbert spaces, which describes how the information complexity depends on \texorpdfstring{$d$}{} and \texorpdfstring{$\va^{-1}$}{}. As a consequence we study the strongly polynomial tractability (SPT), polynomial tractability (PT), weak tractability (WT), and \texorpdfstring{$(t_1,t_2)$}{}-weak tractability (\texorpdfstring{$(t_1,t_2)$}{}-WT) for all \texorpdfstring{$t_1>1$}{} and \texorpdfstring{$t_2>0$}{} in terms of the introduced weights under the absolute error criterion or the normalized error criterion.

In this paper, a range-gated lidar system utilizing an LN crystal as the electro-optical switch and a SCMOS (Scientific Complementary Metal Oxide Semiconductor) imaging device is designed. To achieve range-gated, we utilize two polarizers and a LN (LiNbO3) crystal to form an electro-optical switch. The optical switch is realized by applying a pulse voltage at both ends of the crystal due to the crystal's conoscopic interference effect and electro-optical effect. The advantage of this system is that low-bandwidth detectors such as CMOS and CCD (Charge-coupled Device) can be used to replace conventional high-bandwidth detectors such as ICCD (Intensified Charge Coupled Device), and time it has better imaging performance under specific conditions at the same. However, after using an electro-optical crystal as an optical switch, a new inhomogeneity error will be introduced due to the conscopic interference effect of the electro-optical crystal, resulting in range error of the lidar system. To reduce the influence of inhomogeneity error on the system, this paper analyzes the sources of inhomogeneity error caused by the electro-optical crystal and gives the crystal inhomo-geneity mathematical expression. A compensation method is proposed based on the above inho-mogeneity mathematical expression. An experimental lidar system is constructed in this paper to verify the validity of the compensation method. The experimental results of the range-gated lidar system show that in a specific field of view (2.6mrad), the lidar system has a good imaging per-formance, its ranging standard deviation is 3.86cm and further decreased to 2.86cm after com-pensation, which verifies the accuracy of the compensation method.

We designed a bidirectional reflection distribution function (BRDF) measurement system to research the characteristics of space target materials. This system measures the BRDF data of an aluminum plate and gold foil. A hybrid biogeography-based optimization-firefly (BBO-Firefly) algorithm was proposed to optimize the five-parameter BRDF model. To check the performance of the hybrid BBO-Firefly algorithm, we set up a contrast experiment to optimize the BRDF parameters using the BBO algorithm, Firefly algorithm, and hybrid BBO-Firefly algorithm. The experimental results prove that the hybrid BBO-Firefly algorithm surpasses the BBO algorithm and Firefly algorithm in convergence speed and precision in the same situation and has a better optimization effect. Finally, we substitute the optimization results of the hybrid BPO-Firefly algorithm into the BRDF model to analyze the reflection characteristics of an aluminum plate and gold foil.

We study L_2-approximation problems in the weighted Hilbert spaces in the worst case setting. Three interesting weighted Hilbert spaces appear in this paper, whose weights are equipped with two positive parameters $\ga_j$ and $\az_j$ for $j=1,\,2,\dots,\,d$. We consider the worst case error of algorithms that use finitely many arbitrary continuous linear functionals. We discuss the exponential convergence-(t,s)-weak tractability (EC-(t,s)-WT) of these L_2-approximation problems under the absolute or normalized error criterion. In particular, we obtain the sufficient and necessary conditions for EC-(1,1)-WT and EC-(t,1)-WT with t