Browsing by Author "Chern, I-Liang"

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A KINETIC ENERGY REDUCTION TECHNIQUE AND CHARACTERIZATIONS OF THE GROUND STATES OF SPIN-1 BOSE-EINSTEIN CONDENSATES
(2014-06-01)
Lin, Liren
;
Chern, I-Liang
;
應用數學系
;
數學建模與科學計算所(含中心)
;
Department of Applied Mathematics
We justify some characterizations of the ground states of spin-1 Bose-Einstein condensates exhibited from numerical simulations. For ferromagnetic systems, we show the validity of the single-mode approximation (SMA). For an antiferromagnetic system with nonzero magnetization, we prove the vanishing of the m(F) = 0 component. In the end of the paper some remaining degenerate situations are also discussed. The proofs of the main results are all based on a simple observation, that a redistribution of masses among different components will reduce the kinetic energy.
A Perfect Match Condition for Point-Set Matching Problems Using the Optimal Mass Transport Approach
(2013)
Chen, Pengwen
;
Lin, Ching-Long
;
Chern, I-Liang
;
應用數學系
;
數學建模與科學計算所(含中心)
We study the performance of optimal mass transport-based methods applied to point-set matching problems. The present study, which is based on the L2 mass transport cost, states that perfect matches always occur when the product of the point-set cardinality and the norm of the curl of the nonrigid deformation field does not exceed some constant. This analytic result is justified by a numerical study of matching two sets of pulmonary vascular tree branch points whose displacement is caused by the lung volume changes in the same human subject. The nearly perfect match performance verifies the effectiveness of this mass transport-based approach.
Accurate gradient approximation for complex interface problems in 3D by an improved coupling interface method
(2014-10-15)
Shu, Yu-Chen
;
Chern, I-Liang
;
Chang, Chien C.
;
應用數學系
;
Department of Applied Mathematics
Most elliptic interface solvers become complicated for complex interface problems at those "exceptional points" where there are not enough neighboring interior points for high order interpolation. Such complication increases especially in three dimensions. Usually, the solvers are thus reduced to low order accuracy. In this paper, we classify these exceptional points and propose two recipes to maintain order of accuracy there, aiming at improving the previous coupling interface method [26]. Yet the idea is also applicable to other interface solvers. The main idea is to have at least first order approximations for second order derivatives at those exceptional points. Recipe 1 is to use the finite difference approximation for the second order derivatives at a nearby interior grid point, whenever this is possible. Recipe 2 is to flip domain signatures and introduce a ghost state so that a second-order method can be applied. This ghost state is a smooth extension of the solution at the exceptional point from the other side of the interface. The original state is recovered by a post-processing using nearby states and jump conditions. The choice of recipes is determined by a classification scheme of the exceptional points. The method renders the solution and its gradient uniformly second-order accurate in the entire computed domain. Numerical examples are provided to illustrate the second order accuracy of the presently proposed method in approximating the gradients of the original states for some complex interfaces which we had tested previous in two and three dimensions, and a real molecule (1D63) which is double-helix shape and composed of hundreds of atoms. (C) 2014 Elsevier Inc. All rights reserved.
Accurate gradient approximation for complex interface problems in 3D by an improved coupling interface method
(2014-10-15)
Shu, Yu-Chen
;
Chern, I-Liang
;
Chang, Chien C.
;
應用數學系
;
Department of Applied Mathematics
Most elliptic interface solvers become complicated for complex interface problems at those "exceptional points" where there are not enough neighboring interior points for high order interpolation. Such complication increases especially in three dimensions. Usually, the solvers are thus reduced to low order accuracy. In this paper, we classify these exceptional points and propose two recipes to maintain order of accuracy there, aiming at improving the previous coupling interface method [26]. Yet the idea is also applicable to other interface solvers. The main idea is to have at least first order approximations for second order derivatives at those exceptional points. Recipe 1 is to use the finite difference approximation for the second order derivatives at a nearby interior grid point, whenever this is possible. Recipe 2 is to flip domain signatures and introduce a ghost state so that a second-order method can be applied. This ghost state is a smooth extension of the solution at the exceptional point from the other side of the interface. The original state is recovered by a post-processing using nearby states and jump conditions. The choice of recipes is determined by a classification scheme of the exceptional points. The method renders the solution and its gradient uniformly second-order accurate in the entire computed domain. Numerical examples are provided to illustrate the second order accuracy of the presently proposed method in approximating the gradients of the original states for some complex interfaces which we had tested previous in two and three dimensions, and a real molecule (1D63) which is double-helix shape and composed of hundreds of atoms. (C) 2014 Elsevier Inc. All rights reserved.
Efficient numerical methods for computing ground states of spin-1 Bose-Einstein condensates based on their characterizations
(2013-11-15)
Bao, Weizhu
;
Chern, I-Liang
;
Zhang, Yanzhi
;
應用數學系
;
數學建模與科學計算所(含中心)
In this paper, we propose efficient numerical methods for computing ground states of spin-1 Bose-Einstein condensates (BECs) with/without the Ioffe-Pritchard magnetic field B (X). When B (X) not equal 0, a numerical method is introduced to compute the ground states and it is also applied to study properties of ground states. Numerical results suggest that the densities of m(F) = +/- 1 components in ground states are identical for any nonzero B(x). In particular, if B(x) B not equal 0 is a constant, the ground states satisfy the single-mode approximation. When B(x) 0, efficient and simpler numerical methods are presented to solve the ground states of spin-1 BECs based on their ferromagnetic/antiferromagnetic characterizations. Numerical simulations show that our methods are more efficient than those in the literature. In addition, some conjectures are made from our numerical observations. (C) 2013 Elsevier Inc. All rights reserved.
Numerical Method of Fabric Dynamics Using Front Tracking and Spring Model
(2013-11-01)
Li, Yan
;
Chern, I-Liang
;
Kim, Joung-Dong
;
Li, Xiaolin
;
數學建模與科學計算所(含中心)
We use front tracking data structures and functions to model the dynamic evolution of fabric surface. We represent the fabric surface by a triangulated mesh with preset equilibrium side length. The stretching and wrinkling of the surface are modeled by the mass-spring system. The external driving force is added to the fabric motion through the "Impulse method" which computes the velocity of the point mass by superposition of momentum. The mass-spring system is a nonlinear ODE system. Added by the numerical and computational analysis, we show that the spring system has an upper bound of the eigen frequency. We analyzed the system by considering two spring models and we proved in one case that all eigenvalues are imaginary and there exists an upper bound for the eigen-frequency. This upper bound plays an important role in determining the numerical stability and accuracy of the ODE system. Based on this analysis, we analyzed the numerical accuracy and stability of the nonlinear spring mass system for fabric surface and its tangential and normal motion. We used the fourth order Runge-Kutta method to solve the ODE system and showed that the time step is linearly dependent on the mesh size for the system.
SIMULATING BINARY FLUID-SURFACTANT DYNAMICS BY A PHASE FIELD MODEL
(2012-06-01)
Teng, Chun-Hao
;
Chern, I-Liang
;
Lai, Ming-Chih
;
數學建模與科學計算所(含中心)
;
Graduate Program of Mathematical Modeling and Scientific Computing, Department of Applied Mathematics
In this paper, the dynamics of a binary fluid-surfactant system described by a phenomenological phase field model is investigated through analytical and numerical computations. We first consider the case of one-dimensional planar interface and prove the existence of the equilibrium solution. Then we derive the analytical equilibrium solution for the order parameter and the surfactant concentration in a particular case. The results show that the present phase field formulation qualitatively mimics the surfactant adsorption on the binary fluid interfaces. We further study the time-dependent solutions of the system by numerical computations based on the pseudospectral Fourier computational framework. The present numerical results are in a good agreement with the previous theoretical study in the way that the surfactant favors the creation of interfaces and also stabilizes the formation of phase regions.
String-averaging expectation-maximization for maximum likelihood estimation in emission tomography
(2014-05-01)
Helou, Elias Salomao
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Censor, Yair
;
Chen, Tai-Been
;
Chern, I-Liang
;
De Pierro, Alvaro Rodolfo
We study the maximum likelihood model in emission tomography and propose a new family of algorithms for its solution, called string-averaging expectation maximization (SAEM). In the string-averaging algorithmic regime, the index set of all underlying equations is split into subsets, called \'strings\', and the algorithm separately proceeds along each string, possibly in parallel. Then, the end-points of all strings are averaged to form the next iterate. SAEM algorithms with several strings present better practical merits than the classical row-action maximum-likelihood algorithm. We present numerical experiments showing the effectiveness of the algorithmic scheme, using data of image reconstruction problems. Performance is evaluated from the computational cost and reconstruction quality viewpoints. A complete convergence theory is also provided.