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An efficient parallel algorithm of variational nodal method for heterogeneous neutron transport problems
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作者 Han Yin Xiao-Jing Liu Teng-Fei Zhang 《Nuclear Science and Techniques》 SCIE EI CAS CSCD 2024年第4期29-45,共17页
The heterogeneous variational nodal method(HVNM)has emerged as a potential approach for solving high-fidelity neutron transport problems.However,achieving accurate results with HVNM in large-scale problems using high-... The heterogeneous variational nodal method(HVNM)has emerged as a potential approach for solving high-fidelity neutron transport problems.However,achieving accurate results with HVNM in large-scale problems using high-fidelity models has been challenging due to the prohibitive computational costs.This paper presents an efficient parallel algorithm tailored for HVNM based on the Message Passing Interface standard.The algorithm evenly distributes the response matrix sets among processors during the matrix formation process,thus enabling independent construction without communication.Once the formation tasks are completed,a collective operation merges and shares the matrix sets among the processors.For the solution process,the problem domain is decomposed into subdomains assigned to specific processors,and the red-black Gauss-Seidel iteration is employed within each subdomain to solve the response matrix equation.Point-to-point communication is conducted between adjacent subdomains to exchange data along the boundaries.The accuracy and efficiency of the parallel algorithm are verified using the KAIST and JRR-3 test cases.Numerical results obtained with multiple processors agree well with those obtained from Monte Carlo calculations.The parallelization of HVNM results in eigenvalue errors of 31 pcm/-90 pcm and fission rate RMS errors of 1.22%/0.66%,respectively,for the 3D KAIST problem and the 3D JRR-3 problem.In addition,the parallel algorithm significantly reduces computation time,with an efficiency of 68.51% using 36 processors in the KAIST problem and 77.14% using 144 processors in the JRR-3 problem. 展开更多
关键词 Neutron transport variational nodal method PARALLELIZATION KAIST JRR-3
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Variational iteration method for solving the mechanism of the Equatorial Eastern Pacific El Nino-Southern Oscillation 被引量:35
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作者 莫嘉祺 王辉 +1 位作者 林万涛 林一骅 《Chinese Physics B》 SCIE EI CAS CSCD 2006年第4期671-675,共5页
A class of coupled system for the E1 Nifio-Southern Oscillation (ENSO) mechanism is studied. Using the method of variational iteration for perturbation theory, the asymptotic expansions of the solution for ENSO mode... A class of coupled system for the E1 Nifio-Southern Oscillation (ENSO) mechanism is studied. Using the method of variational iteration for perturbation theory, the asymptotic expansions of the solution for ENSO model are obtained and the asymptotic behaviour of solution for corresponding problem is considered. 展开更多
关键词 nonlinear method of variational iteration perturbation theory El Nino- Southern Oscillation model
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Variational regularization method of solving the Cauchy problem for Laplace's equation: Innovation of the Grad–Shafranov(GS) reconstruction 被引量:4
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作者 颜冰 黄思训 《Chinese Physics B》 SCIE EI CAS CSCD 2014年第10期650-655,共6页
The simplified linear model of Grad-Shafranov (GS) reconstruction can be reformulated into an inverse boundary value problem of Laplace's equation. Therefore, in this paper we focus on the method of solving the inv... The simplified linear model of Grad-Shafranov (GS) reconstruction can be reformulated into an inverse boundary value problem of Laplace's equation. Therefore, in this paper we focus on the method of solving the inverse boundary value problem of Laplace's equation. In the first place, the variational regularization method is used to deal with the ill- posedness of the Cauchy problem for Laplace's equation. Then, the 'L-Curve' principle is suggested to be adopted in choosing the optimal regularization parameter. Finally, a numerical experiment is implemented with a section of Neumann and Dirichlet boundary conditions with observation errors. The results well converge to the exact solution of the problem, which proves the efficiency and robustness of the proposed method. When the order of observation error δ is 10-1, the order of the approximate result error can reach 10-3. 展开更多
关键词 Grad-Shafranov reconstruction variational regularization method Cauchy problem
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A RELAXED INERTIAL FACTOR OF THE MODIFIED SUBGRADIENT EXTRAGRADIENT METHOD FOR SOLVING PSEUDO MONOTONE VARIATIONAL INEQUALITIES IN HILBERT SPACES 被引量:2
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作者 Duong Viet THONG Vu Tien DUNG 《Acta Mathematica Scientia》 SCIE CSCD 2023年第1期184-204,共21页
In this paper,we investigate pseudomonotone and Lipschitz continuous variational inequalities in real Hilbert spaces.For solving this problem,we propose a new method that combines the advantages of the subgradient ext... In this paper,we investigate pseudomonotone and Lipschitz continuous variational inequalities in real Hilbert spaces.For solving this problem,we propose a new method that combines the advantages of the subgradient extragradient method and the projection contraction method.Some very recent papers have considered different inertial algorithms which allowed the inertial factor is chosen in[0;1].The purpose of this work is to continue working in this direction,we propose another inertial subgradient extragradient method that the inertial factor can be chosen in a special case to be 1.Under suitable mild conditions,we establish the weak convergence of the proposed algorithm.Moreover,linear convergence is obtained under strong pseudomonotonicity and Lipschitz continuity assumptions.Finally,some numerical illustrations are given to confirm the theoretical analysis. 展开更多
关键词 subgradient extragradient method inertial method variational inequality problem pseudomonotone mapping strong convergence convergence rate
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Research on the discrete variational method for a Birkhoffian system
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作者 刘世兴 花巍 郭永新 《Chinese Physics B》 SCIE EI CAS CSCD 2014年第6期328-332,共5页
In this paper, we present a new integration algorithm based on the discrete Pfaff-Birkhoff principle for Birkhoffian systems. It is proved that the new algorithm can preserve the general symplectic geometric structure... In this paper, we present a new integration algorithm based on the discrete Pfaff-Birkhoff principle for Birkhoffian systems. It is proved that the new algorithm can preserve the general symplectic geometric structures of Birkhoffian systems. A numerical experiment for a damping oscillator system is conducted. The result shows that the new algorithm can better simulate the energy dissipation than the R-K method, which illustrates that we can numerically solve the dynamical equations by the discrete variational method in a Birkhoffian framework for the systems with a general symplectic structure. Furthermore, it is demonstrated that the results of the numerical experiments are determined not by the constructing methods of Birkhoffian functions but by whether the numerical method can preserve the inherent nature of the dynamical system. 展开更多
关键词 Birkhoff's equations discrete variational methods general symplectic structure discrete Birkhoff's equations
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Normalized Solutions of Nonlinear Choquard Equations with Nonconstant Potential
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作者 LI Nan XU Liping 《应用数学》 北大核心 2025年第1期14-29,共16页
In this paper,we mainly focus on a type of nonlinear Choquard equations with nonconstant potential.Under appropriate hypotheses on potential function and nonlinear terms,we prove that the above Choquard equation with ... In this paper,we mainly focus on a type of nonlinear Choquard equations with nonconstant potential.Under appropriate hypotheses on potential function and nonlinear terms,we prove that the above Choquard equation with prescribed 2-norm has some normalized solutions by introducing variational methods. 展开更多
关键词 Nonlinear Choquard equation Potential function variational method Normalized solution
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Potential-dependent insights into the origin of high ammonia yield rate on copper surface via nitrate reduction:A computational and experimental study
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作者 Yangge Guo Nannan Sun +5 位作者 Liuxuan Luo Xiaojing Cheng Xueying Chen Xiaohui Yan Shuiyun Shen Junliang Zhang 《Journal of Energy Chemistry》 SCIE EI CAS CSCD 2024年第9期272-281,共10页
Focusing on revealing the origin of high ammonia yield rate on Cu via nitrate reduction(NO3RR),we herein applied constant potential method via grand-canonical density functional theory(GC-DFT)with implicit continuum s... Focusing on revealing the origin of high ammonia yield rate on Cu via nitrate reduction(NO3RR),we herein applied constant potential method via grand-canonical density functional theory(GC-DFT)with implicit continuum solvation model to predict the reaction energetics of NO3RR on pure copper surface in alkaline media.The potential-dependent mechanism on the most prevailing Cu(111)and the minor(100)and(110)facets were established,in consideration of NO_(2)_(−),NO,NH_(3),NH_(2)OH,N_(2),and N_(2)O as the main products.The computational results show that the major Cu(111)is the ideal surface to produce ammonia with the highest onset potential at 0.06 V(until−0.37 V)and the highest optimal potential at−0.31 V for ammonia production without kinetic obstacles in activation energies at critical steps.For other minor facets,the secondary Cu(100)shows activity to ammonia from−0.03 to−0.54 V with the ideal potential at−0.50 V,which requires larger overpotential to overcome kinetic activation energy barriers.The least Cu(110)possesses the longest potential range for ammonia yield from−0.27 to−1.12 V due to the higher adsorption coverage of nitrate,but also with higher tendency to generate di-nitrogen species.Experimental evaluations on commercial Cu/C electrocatalyst validated the accuracy of our proposed mechanism.The most influential(111)surface with highest percentage in electrocatalyst determined the trend of ammonia production.In specific,the onset potential of ammonia production at 0.1 V and emergence of yield rate peak at−0.3 V in experiments precisely located in the predicted potentials on Cu(111).Four critical factors for the high ammonia yield and selectivity on Cu surface via NO3RR are summarized,including high NO3RR activity towards ammonia on the dominant Cu(111)facet,more possibilities to produce ammonia along different pathways on each facet,excellent ability for HER inhibition and suitable surface size to suppress di-nitrogen species formation at high nitrate coverage.Overall,our work provides comprehensive potential-dependent insights into the reaction details of NO3RR to ammonia,which can serve as references for the future development of NO3RR electrocatalysts,achieving higher activity and selectivity by maximizing these characteristics of copper-based materials. 展开更多
关键词 Nitrate reduction to ammonia Copper surface Density functional theory Constant electrode potential method Experimental validation
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MULTIPLICITY OF NORMALIZED SOLUTIONS FOR THE FRACTIONAL SCHR?DINGER-POISSON SYSTEM WITH DOUBLY CRITICAL GROWTH
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作者 孟禹希 贺小明 《Acta Mathematica Scientia》 SCIE CSCD 2024年第3期997-1019,共23页
In this paper,we are concerned with solutions to the fractional Schrodinger-Poisson system■ with prescribed mass ∫_(R^(3))|u|^(2)dx=a^(2),where a> 0 is a prescribed number,μ> 0 is a paremeter,s ∈(0,1),2 <... In this paper,we are concerned with solutions to the fractional Schrodinger-Poisson system■ with prescribed mass ∫_(R^(3))|u|^(2)dx=a^(2),where a> 0 is a prescribed number,μ> 0 is a paremeter,s ∈(0,1),2 <q <2_(s)^(*),and 2_(s)^(*)=6/(3-2s) is the fractional critical Sobolev exponent.In the L2-subcritical case,we show the existence of multiple normalized solutions by using the genus theory and the truncation technique;in the L^(2)-supercritical case,we obtain a couple of normalized solutions by developing a fiber map.Under both cases,to recover the loss of compactness of the energy functional caused by the doubly critical growth,we need to adopt the concentration-compactness principle.Our results complement and improve upon some existing studies on the fractional Schrodinger-Poisson system with a nonlocal critical term. 展开更多
关键词 fractional Schrodinger-Poisson system normalized solutions variational methods L^(2)-subcritical L^(2)-supercritical
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The Existence of Solutions for Kirchhoff-Type Equations with General Singular Terms
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作者 WANG Ji-nan SUN Da-wei 《Chinese Quarterly Journal of Mathematics》 2024年第3期315-323,共9页
We study the existence of solutions for Kirchhoff-type equations.Firstly,we use the Sobolev inequality and the weakly lower semi-continuity of the norm to prove that the corresponding function can reach the global min... We study the existence of solutions for Kirchhoff-type equations.Firstly,we use the Sobolev inequality and the weakly lower semi-continuity of the norm to prove that the corresponding function can reach the global minimum.Then,we use the variational method and some analytical techniques to obtain the existence of the positive solution of the equation whenλis small enough. 展开更多
关键词 Kirchhoff-type equation the existence Positive solution variational method
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A NEW ITERATIVE METHOD FOR FINDING COMMON SOLUTIONS OF GENERALIZED EQUILIBRIUM PROBLEM,FIXED POINT PROBLEM OF INFINITE k-STRICT PSEUDO-CONTRACTIVE MAPPINGS,AND QUASI-VARIATIONAL INCLUSION PROBLEM 被引量:5
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作者 刘敏 张石生 《Acta Mathematica Scientia》 SCIE CSCD 2012年第2期499-519,共21页
In this article, we introduce a hybrid iterative scheme for finding a common element of the set of solutions for a generalized equilibrium problems, the set of common fixed point for a family of infinite k-strict pseu... In this article, we introduce a hybrid iterative scheme for finding a common element of the set of solutions for a generalized equilibrium problems, the set of common fixed point for a family of infinite k-strict pseudo-contractive mappings, and the set of solutions of the variational inclusion problem with multi-valued maximal monotone mappings and inverse-strongly monotone mappings in Hilbert space. Under suitable conditions, some strong convergence theorems are proved. Our results extends the recent results in G.L.Acedo and H.K.Xu [2], Zhang, Lee and Chan [8], Wakahashi and Toyoda [9], Takahashi and Takahashi [I0] and S. S. Chang, H. W. Joseph Lee and C. K. Chan [II], S.Takahashi and W.Takahashi [12]. Moreover, the method of proof adopted in this article is different from those of [4] and [12]. 展开更多
关键词 k-strict pseudo-contractive mappings generalized equilibrium problem vis-cosity approximation method variational inclusion problem multi-valuedmaximal monotone mappings s-inverse-strongly monotone mapping
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Application of the Tikhonov regularization method to wind retrieval from scatterometer data I.Sensitivity analysis and simulation experiments 被引量:1
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作者 钟剑 黄思训 +1 位作者 杜华栋 张亮 《Chinese Physics B》 SCIE EI CAS CSCD 2011年第3期274-283,共10页
Scatterometer is an instrument which provides all-day and large-scale wind field information, and its application especially to wind retrieval always attracts meteorologists. Certain reasons cause large direction erro... Scatterometer is an instrument which provides all-day and large-scale wind field information, and its application especially to wind retrieval always attracts meteorologists. Certain reasons cause large direction error, so it is important to find where the error mainly comes. Does it mainly result from the background field, the normalized radar cross-section (NRCS) or the method of wind retrieval? It is valuable to research. First, depending on SDP2.0, the simulated 'true' NRCS is calculated from the simulated 'true' wind through the geophysical mode] function NSCAT2. The simulated background field is configured by adding a noise to the simulated 'true' wind with the non-divergence constraint. Also, the simulated 'measured' NRCS is formed by adding a noise to the simulated 'true' NRCS. Then, the sensitivity experiments are taken, and the new method of regularization is used to improve the ambiguity removal with simulation experiments. The results show that the accuracy of wind retrieval is more sensitive to the noise in the background than in the measured NRCS; compared with the two-dimensional variational (2DVAR) ambiguity removal method, the accuracy of wind retrieval can be improved with the new method of Tikhonov regularization through choosing an appropriate regularization parameter, especially for the case of large error in the background. The work will provide important information and a new method for the wind retrieval with real data. 展开更多
关键词 SCATTEROMETER variational optimization analysis wind retrieval regularization method
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STRONG CONVERGENCE OF AN INERTIAL EXTRAGRADIENT METHOD WITH AN ADAPTIVE NONDECREASING STEP SIZE FOR SOLVING VARIATIONAL INEQUALITIES 被引量:1
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作者 Nguyen Xuan LINH Duong Viet THONG +2 位作者 Prasit CHOLAMJIAK Pham Anh TUAN Luong Van LONG 《Acta Mathematica Scientia》 SCIE CSCD 2022年第2期795-812,共18页
In this work,we investigate a classical pseudomonotone and Lipschitz continuous variational inequality in the setting of Hilbert space,and present a projection-type approximation method for solving this problem.Our me... In this work,we investigate a classical pseudomonotone and Lipschitz continuous variational inequality in the setting of Hilbert space,and present a projection-type approximation method for solving this problem.Our method requires only to compute one projection onto the feasible set per iteration and without any linesearch procedure or additional projections as well as does not need to the prior knowledge of the Lipschitz constant and the sequentially weakly continuity of the variational inequality mapping.A strong convergence is established for the proposed method to a solution of a variational inequality problem under certain mild assumptions.Finally,we give some numerical experiments illustrating the performance of the proposed method for variational inequality problems. 展开更多
关键词 Inertial method Tseng’s extragradient viscosity method variational inequality problem pseudomonotone mapping strong convergence
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Variational analysis of the disc-loaded waveguide slow-wave structures
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作者 李建清 莫元龙 《Chinese Physics B》 SCIE EI CAS CSCD 2005年第11期2300-2304,共5页
The variational method is applied to calculate the dispersion characteristics of disc-loaded waveguide slow-wave structures. The parameters describing the waveguide discontinuities in disc-loaded waveguide are calcula... The variational method is applied to calculate the dispersion characteristics of disc-loaded waveguide slow-wave structures. The parameters describing the waveguide discontinuities in disc-loaded waveguide are calculated by the variational method. Then the dispersion characteristics of slow-wave structures are obtained using lossless microwave quadrupole theory. Good agreement was observed between results of the Variational method and those of field matching method and high frequency structure simulator. In the case of broad band, results of the variational method are better than those of field matching method. 展开更多
关键词 variational method disc-loaded waveguide slow-wave structures dispersion characteristics
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Transmissivity of electromagnetic wave propagating in magnetized plasma sheath using variational method
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作者 Jianyi CHEN Chengxun YUAN +1 位作者 Xiudong SUN Lei HUO 《Plasma Science and Technology》 SCIE EI CAS CSCD 2019年第12期1-5,共5页
A variational method is introduced to analyze the transmissivity of an electromagnetic wave propagating in the magnetized plasma sheath. The plasma density is modeled by two parabolic inhomogeneous regions separated b... A variational method is introduced to analyze the transmissivity of an electromagnetic wave propagating in the magnetized plasma sheath. The plasma density is modeled by two parabolic inhomogeneous regions separated by one homogeneous region. The Lagrangian density of the system is constructed based on the fluid energy density and the electromagnetic energy density.The total variation of the Lagrangian density is derived. The fluid and electromagnetic fields are numerically solved by expansion in piecewise polynomial function space. We investigate the effect of an external magnetic field on the transmissivity of the electromagnetic wave. It is found that the transmissivity is increased when an external magnetic field is applied. The dependence of transmissivity on the collision frequency between the electrons and the neutral particles has also been studied. We also show that the external magnetic field causes a shift in the critical frequency of the plasma sheath. 展开更多
关键词 electromagnetic WAVE PROPAGATION variational method WAVE theory WAVE PROPAGATION plasma sheath
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A General Projection Method for a System of Relaxed Coercive Variational Inequalities in Hilbert Spaces
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作者 杨峻 吴忠林 《Chinese Quarterly Journal of Mathematics》 CSCD 2009年第3期426-431,共6页
In this paper,we consider a new algorithm for a generalized system for relaxed coercive nonlinear inequalities involving three different operators in Hilbert spaces by the convergence of projection methods.Our results... In this paper,we consider a new algorithm for a generalized system for relaxed coercive nonlinear inequalities involving three different operators in Hilbert spaces by the convergence of projection methods.Our results include the previous results as special cases extend and improve the main results obtained by many others. 展开更多
关键词 relaxed coercive nonlinear variational inequality projection method relaxed coercive mapping
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带临界指数的Kirchhoff型线性耦合方程组正解的多重性
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作者 段雪亮 吴晓凡 +1 位作者 魏公明 杨海涛 《数学物理学报(A辑)》 CSCD 北大核心 2024年第3期699-716,共18页
该文研究了如下带Sobolev临界指数的Kirchhoff型线性耦合方程组{−(1+b_(1)∥u∥^(2))Δu+λ_(1)u=u5+βv,x∈Ω,−(1+b_(2)∥v∥^(2))Δv+λ_(2)v=v^(5)+βu,x∈Ω,u=v=0在∂Ω上,其中Ω⊂R^(3)是一个开球,∥⋅∥表示H_(0)^(1)(Ω)的范数,β... 该文研究了如下带Sobolev临界指数的Kirchhoff型线性耦合方程组{−(1+b_(1)∥u∥^(2))Δu+λ_(1)u=u5+βv,x∈Ω,−(1+b_(2)∥v∥^(2))Δv+λ_(2)v=v^(5)+βu,x∈Ω,u=v=0在∂Ω上,其中Ω⊂R^(3)是一个开球,∥⋅∥表示H_(0)^(1)(Ω)的范数,β∈R是一个耦合参数.常数b_(i)≥0和λ_(i)∈(−λ_(1)(Ω),−1/4λ_(1)(Ω)),i=1,2,这里λ_(1)(Ω)是(−Δ,H_(0)^(1)(Ω))的第一特征值.在含有Kirchhoff项的情形下,利用变分法证明了方程组有一个正基态解和一个高能量的正解,并研究了当β→0时这两个解的渐近行为. 展开更多
关键词 KIRCHHofF 型方程 线性耦合方程组 SOBOLEV 临界指数 变分法
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MULTIPLICITY AND CONCENTRATION BEHAVIOUR OF POSITIVE SOLUTIONS FOR SCHRDINGER-KIRCHHOFF TYPE EQUATIONS INVOLVING THE p-LAPLACIAN IN R^N 被引量:3
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作者 贾慧芳 李工宝 《Acta Mathematica Scientia》 SCIE CSCD 2018年第2期391-418,共28页
In this article, we study the multiplicity and concentration behavior of positive solutions for the p-Laplacian equation of SchrSdinger-Kirchhoff type -εpM(εp-N∫RN|△u|p)△pu+v(x|u|p-2u=f(u)in RN, where ... In this article, we study the multiplicity and concentration behavior of positive solutions for the p-Laplacian equation of SchrSdinger-Kirchhoff type -εpM(εp-N∫RN|△u|p)△pu+v(x|u|p-2u=f(u)in RN, where △p is the p-Laplacian operator, 1 〈 p 〈 N, M : R+ → R+ and V : RN →R+ are continuous functions, ε is a positive parameter, and f is a continuous function with subcritical growth. We assume that V satisfies the local condition introduced by M. del Pino and P. Felmer. By the variational methods, penalization techniques, and Lyusternik- Schnirelmann theory, we prove the existence, multiplicity, and concentration of solutions for the above equation. 展开更多
关键词 SchrSdinger-Kirchhoff type equation variational methods multiple positive solutions concentrating solution penalization method
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The splitting of low-lying states for hydroxyl molecule under spin orbit coupling 被引量:2
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作者 李桂霞 高涛 +3 位作者 陈东 李跃勋 张云光 朱正和 《Chinese Physics B》 SCIE EI CAS CSCD 2006年第5期998-1003,共6页
The splitting of potential energy curves for the states X^2Ⅱ3/2, ^2Ⅱ1/2 and A^2∑+ of hydroxyl OH under spin-orbit coupling (SOC) has been calculated by using the SO multi-configuration quasi-degenerate perturbat... The splitting of potential energy curves for the states X^2Ⅱ3/2, ^2Ⅱ1/2 and A^2∑+ of hydroxyl OH under spin-orbit coupling (SOC) has been calculated by using the SO multi-configuration quasi-degenerate perturbation theory (SO- MCQDPT). Their Murrell Sorbic (M-S) potential functions have been derived, then, the spectroscopic constants for X^2Ⅱ3/2, ^2Ⅱ1/2 and A^2∑+ have been derived from the M-S function. The calculated dissociation energies for the three states are Do[OH(X^2Ⅱ3/2)]=34966,632cm^-1, Do[OH(^2Ⅱ1/2)]=34922.802cm^-1, and Do[OH(A^2∑ )]=17469.794cm^-1, respectively. The vertical excitation energy u[^2Ⅱ1/2(v = 0) → X^2Ⅱ3/2(v = 0)] = 139.6cm^-1. All the spectroscopic data for the X^2Ⅱ3/2 and ^2Ⅱ1/2 are given for the first time except the dissociation energy of X^2Ⅱ3/2. 展开更多
关键词 SOC SO-MCQDPT method spectroscopic constants OH radical
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Kirchhoff型方程正规化解的多重性及渐近行为
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作者 靳振峰 孙红蕊 张为民 《数学物理学报(A辑)》 CSCD 北大核心 2024年第4期871-884,共14页
该文研究了如下Kirchhoff型方程{−(a+b∫R^(3)|∇u|^(2)dx)Δu=λu+|u|^(p−2)u,x∈R^(3),∥u∥22=ρ,其中a,b,ρ>0,λ∈R是与质量约束∥u∥22=ρ有关的Lagrange乘子.当p∈(2,103)或者p∈(143,6)时,利用亏格理论证明了上述方程L2-正规... 该文研究了如下Kirchhoff型方程{−(a+b∫R^(3)|∇u|^(2)dx)Δu=λu+|u|^(p−2)u,x∈R^(3),∥u∥22=ρ,其中a,b,ρ>0,λ∈R是与质量约束∥u∥22=ρ有关的Lagrange乘子.当p∈(2,103)或者p∈(143,6)时,利用亏格理论证明了上述方程L2-正规化解的多重性.此外,该文证明了上述解关于参数b→0^(+)时的渐近行为. 展开更多
关键词 KIRCHHofF方程 变分法 正规化解 渐近行为
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Simultaneous estimation of aerosol optical constants and size distribution from angular light-scattering measurement signals 被引量:3
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作者 Zhen-Zong He Dong Liang +1 位作者 Jun-Kui Mao Xing-Si Han 《Chinese Physics B》 SCIE EI CAS CSCD 2018年第5期564-572,共9页
The angular light-scattering measurement(ALSM) method combined with an improved artificial bee colony algorithm is introduced to determine aerosol optical constants and aerosol size distribution(ASD) simultaneousl... The angular light-scattering measurement(ALSM) method combined with an improved artificial bee colony algorithm is introduced to determine aerosol optical constants and aerosol size distribution(ASD) simultaneously. Meanwhile, an optimized selection principle of the ALSM signals based on the sensitivity analysis and principle component analysis(PCA)is proposed to improve the accuracy of the retrieval results. The sensitivity analysis of the ALSM signals to the optical constants or characteristic parameters in the ASD is studied first to find the optimized selection region of measurement angles. Then, the PCA is adopted to select the optimized measurement angles within the optimized selection region obtained by sensitivity analysis. The investigation reveals that, compared with random selection measurement angles, the optimized selection measurement angles can provide more useful measurement information to ensure the retrieval accuracy. Finally,the aerosol optical constants and the ASDs are reconstructed simultaneously. The results show that the retrieval accuracy of refractive indices is better than that of absorption indices, while the characteristic parameters in ASDs have similar retrieval accuracy. Moreover, the retrieval accuracy in studying L-N distribution is a little better than that in studying Gamma distribution for the difference of corresponding correlation coefficient matrixes of the ALSM signals. All the results confirm that the proposed technique is an effective and reliable technique in estimating the aerosol optical constants and ASD simultaneously. 展开更多
关键词 angular light-scattering measurement method aerosol optical constants aerosol size distribution inverse radiation problem
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