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A NEW APPROACH AND ERROR ANALYSIS FOR RECONSTRUCTING THE SCATTERED WAVE BY THE POINT SOURCE METHOD

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Journal of Computational Mathematics 2007年第2期25卷 113-130页

作者： Jijun Liu Gen Nakamura Roland Potthast Department of Mathematics Southeast University Nanjing 210096 China Department of Mathematics Graduate School of Science Hokkaido University Sapporo 060-0810 Japan Institute for Numerical and Applied Mathematics University of Gottingen Lotzestr 16-1837085 Gottingen Germany

Consider an inverse scattering problem by an obstacle D belong to R^2 with impedance boundary. We investigate the reconstruction of the scattered field u^s from its far field pattern u^∞ using the point source method. First, by applying the boundary integral equation method, we provide a new approach to the point-source method of Potthast by classical potential theory. This extends the range of the point source method from plane waves to scattering of arbitrary waves. Second, by analyzing the behavior of the Hankel function, we obtain an improved strategy for the choice of the regularizing parameter from which an improved convergence rate （compared to the result of [15]） is achieved for the reconstruc- tion of the scattered wave. Third, numerical implementations are given to test the validity and stability of the inversion method for the impedance obstacle.

关键词： Inverse scattering Regularization Error estimate numerics

Breather Dynamics of the Sine-Gordon Equation

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Communications in Theoretical Physics 2013年第6期59卷 664-670页

作者： Stephen Johnson Anjan Biswas Stephen Johnson and Anjan Biswas Department of Mathematical Sciences Delaware State University Department of Mathematics Faculty of Science King Abdulaziz University

This paper studies the adiabatic dynamics of the breather soliton of the sine-Gordon equation. The integrals of motion are found and then used in soliton perturbation theory to derive the differential equation governing the soliton velocity. Time-dependent functions arise and their properties are studied. These functions are found to be bounded and periodic and affect the soliton velocity. The soliton velocity is numerically plotted against time for different combinations of initial velocities and perturbation terms.

关键词： solitons phonons conservation laws perturbations numerics

RECONSTRUCTION OF SCATTERED FIELD FROM FAR-FIELD BY REGULARIZATION

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Journal of Computational Mathematics 2004年第3期22卷 389-402页

作者： Ji-junLiu JinCheng G.Nakamura DepartmentofMathematics SoutheastUniversityNanjing210096China DepartmentofMathematics FudanUniversityShanghai200433China DepartmentofMathematics HokkaidoUniversitySapporo060-0810Japan

In this paper, we consider an inverse scattering problem for an obstacle D R^2 with Robin boundary condition. By applying the point source, we give a regularizing method to recover the scattered field from the far-fie... 详细信息

关键词： Inverse scattering Point source Regularization numerics

ON DECOMPOSITION METHOD FOR ACOUSTIC WAVE SCATTERING BY MULTIPLE OBSTACLES

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Acta Mathematica Scientia 2013年第1期33卷 1-22页

作者：王海兵刘继军 Department of Mathematics Southeast University School of Mathematics and Computational Science Hunan University of Science and Technology

Consider acoustic wave scattering by multiple obstacles with different sound properties on the boundary, which can be modeled by a mixed boundary value problem for the Helmholtz equation in frequency domain. Compared with the standard scattering problem for one obstacle, the difficulty of such a new problem is the interaction of scattered wave by different obstacles. A decomposition method for solving this multiple scattering problem is developed. Using the boundary integral equation method, we decompose the total scattered field into a sum of contributions by separated obstacles. Each contribution corresponds to scattering problem of single obstacle. However, all the single scattering problems are coupled via the boundary conditions, representing the physical interaction of scattered wave by different obstacles. We prove the feasibility of such a decomposition. To compute these contributions efficiently, an iteration algorithm of Jacobi type is proposed, decoupling the interaction of scattered wave from the numerical points of view. Under the well-separation assumptions on multiple obstacles, we prove the convergence of iteration sequence generated by the Jacobi algorithm, and give the error estimate between exact scattered wave and the iteration solution in terms of the obstacle size and the minimal distance of multiple obstacles. Such a quantitative description reveals the essences of wave scattering by multiple obstacles. Numerical examples showing the accuracy and convergence of our method are presented.

关键词： multiple obstacles scattering decomposition method Jacobi iteration con-vergence error estimate numerics

On Fluorophore Imaging by Diffusion Equation Model:Decompositions and Optimizations

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Acta Mathematicae Applicatae Sinica 2020年第1期36卷 203-222页

作者： Li-yan WANG Ji-jun LIU School of Mathematics S.T.Yau Center of Southeast UniversitySoutheast UniversityNanjing 210096China

Fluorescence imaging is a target-specific molecular imaging technology using absorption coefficient of *** this imaging model governed by an inverse problem for the coupled diffusion system,which describes the interaction of the excitation field from several boundary sources and the corresponding emission field,we reformulate it as an optimization *** solving this non-quadratic optimizing problem,we propose a decomposition scheme,which extracts the horizontal information of the target from the boundary measurement data *** realizability of this hybrid imaging scheme is rigorously proved mathematically for cubic and ellipsoid targets,by constructing an indicator function for the horizontal location of the target *** based on this horizonal location as a good initial guess for the iteration process,the cost functional is optimized efficiently using the trust domain *** implementations are provided to show the validity of the proposed scheme.

关键词： fluorescence imaging diffusion equation inverse problem decomposition optimization numerics

Convergence of the deep BSDE method for coupled FBSDEs

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Probability, Uncertainty and Quantitative Risk 2020年第1期5卷 102-134页

作者： Jiequn Han Jihao Long Department of Mathematics Princeton UniversityPrinceton 08544NJUSA School of Mathematical Sciences Peking UniversityBeijing 100871People’s Republic of China

The recently proposed numerical algorithm,deep BSDE method,has shown remarkable performance in solving high-dimensional forward-backward stochastic differential equations(FBSDEs)and parabolic partial differential equations(PDEs).This article lays a theoretical foundation for the deep BSDE method in the general case of coupled *** particular,a posteriori error estimation of the solution is provided and it is proved that the error converges to zero given the universal approximation capability of neural *** results are presented to demonstrate the accuracy of the analyzed algorithm in solving high-dimensional coupled FBSDEs.

关键词： Forward-backward SDE Weakly coupled condition Deep learning High dimension numerics

Study on Reconstruction of Heat Flux in Moving Boundary

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Study on Reconstruction of Heat Flux in Moving Boundary

An Analytical Solution to Neumann-Type Mixed Boundary Poiseuille Microfluidic Flow in Rectangular Channel Cross-Sections (Slip/No-Slip) including a Numerical Technique to Derive It

2012 National Conference on Information Technology and Computer Science

作者： Zhijian Wu Institute of Meteorology and Oceangraphy PLA University of Science Technology Nanjing,China Jing Feng Institute of Meteorology and Oceangraphy PLA University of Science Technology Nanjing,China Xueyan Zhang Institute of Meteorology and Oceangraphy PLA University of Science Technology Nanjing,China Aixia Zhou Institute of Meteorology and Oceangraphy PLA University of Science Technology Nanjing,China

The initial boundary value problem for parabolic equation with Neumann boundary condition is a kind of classical problem in partial differential *** this paper,we consider a heat inverse problem with moving *** introducing a density function for potential method,thisproblem is essentially converted into solving an integral equation of the first kind with respect to the density function,which can be solved by the Tikhonov regularizing *** function method is used to determine the regularizing *** results are presented to support our inversion schemes.

关键词： heat equation potential scheme heat flux regularization numerics

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Journal of Biomedical Science and Engineering 2017年第5期10卷 205-218页

作者： Christiane Richter Frederik Kotz N. Keller Tobias M. Nargang Kai Sachsenheimer Dorothea Helmer Bastian E. Rapp Karlsruhe Institute of Technology (KIT) Istitute of Microstructure Technology (IMT) Eggenstein-Leopoldshafen Germany

In most microfluidic applications, pressure-driven Poiseuille flow in a contained cross-section with no-slip boundary conditions is the underlying fluid-mechanical model. Solutions for this problem exist for many known cross-sections. We have recently demonstrated a simple method to solve the relevant Poisson equation using a finite difference scheme in a spreadsheet analysis tool such as Microsoft Excel. The numerical solutions obtained from such a spreadsheet are close-to-exact to the analytical solutions with errors on the order of only a few percent. However, there are numerous applications in microfluidics for which the no-slip boundary condition is not valid. Examples include drag-reducing air-retaining surfaces as well as open-channel flow. For these scenarios few to no analytical models exist. In this paper, we derive an analytical model for mixed boundary conditions (slip/no-slip) in two dimensions in a rectangular channel cross-section. We also demonstrate that the equivalent numerical solution can be derived conveniently by adaption of the spreadsheet. In general, mixed boundary-type flow scenarios are especially difficult to solve analytically whereas numerical solutions can be derived using Microsoft Excel within seconds.

关键词： Microfluidics numerics Navier-Stokes Equation Slip Condition No-Slip Condition