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47
Computer simulation of glioma growth and morphology. Neuroimage 37
 Suppl
, 2007
"... Despite major advances in the study of glioma, the quantitative links between intratumor molecular/cellular properties, clinically observable properties such as morphology, and critical tumor behaviors such as growth and invasiveness remain unclear, hampering more effective coupling of tumor physic ..."
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Cited by 29 (2 self)
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Despite major advances in the study of glioma, the quantitative links between intratumor molecular/cellular properties, clinically observable properties such as morphology, and critical tumor behaviors such as growth and invasiveness remain unclear, hampering more effective coupling of tumor physical characteristics with implications for prognosis and therapy. Although molecular biology, histopathology, and radiological imaging are employed in this endeavor, studies are severely challenged by the multitude of different physical scales involved in tumor growth, i.e., from molecular nanoscale to cell microscale and finally to tissue centimeter scale. Consequently, it is often difficult to determine the underlying dynamics across dimensions. New techniques are needed to tackle these issues. Here, we address this multiscalar problem by employing a novel predictive threedimensional mathematical and computational model based on firstprinciple equations (conservation laws of physics) that describe mathematically
Local discontinuous Galerkin methods for the Cahn–Hilliard type equations
 J. Comput. Phys
"... In this paper, we develop, analyze and test a local discontinuous Galerkin (LDG) method for solving the CamassaHolm equation which contains nonlinear high order derivatives. The LDG method has the flexibility for arbitrary h and p adaptivity. We prove the L2 stability for general solutions and give ..."
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Cited by 26 (6 self)
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In this paper, we develop, analyze and test a local discontinuous Galerkin (LDG) method for solving the CamassaHolm equation which contains nonlinear high order derivatives. The LDG method has the flexibility for arbitrary h and p adaptivity. We prove the L2 stability for general solutions and give a detailed error estimate for smooth solutions, and provide numerical simulation results for different types of solutions of the nonlinear CamassaHolm equation to illustrate the accuracy and capability of the LDG method.
DIFFUSE INTERFACE MODELS ON GRAPHS FOR CLASSIFICATION OF HIGH DIMENSIONAL DATA
, 2012
"... There are currently several communities working on algorithms for classification of high dimensional data. This work develops a class of variational algorithms that combine recent ideas from spectral methods on graphs with nonlinear edge/region detection methods traditionally used in in the PDEba ..."
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Cited by 22 (10 self)
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There are currently several communities working on algorithms for classification of high dimensional data. This work develops a class of variational algorithms that combine recent ideas from spectral methods on graphs with nonlinear edge/region detection methods traditionally used in in the PDEbased imaging community. The algorithms are based on the GinzburgLandau functional which has classical PDE connections to total variation minimization. Convexsplitting algorithms allow us to quickly find minimizers of the proposed model and take advantage of fast spectral solvers of linear graphtheoretic problems. We present diverse computational examples involving both basic clustering and semisupervised learning for different applications. Case studies include feature identification in images, segmentation in social networks, and segmentation of shapes in high dimensional datasets.
A Level Set Method for Interfacial Flows with Surfactant
"... A levelset method for the simulation of fluid interfaces with insoluble surfactant is presented in twodimensions. The method can be straightforwardly extended to threedimensions and to soluble surfactants. The method couples a semiimplicit discretization for solving the surfactant transport equa ..."
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Cited by 17 (1 self)
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A levelset method for the simulation of fluid interfaces with insoluble surfactant is presented in twodimensions. The method can be straightforwardly extended to threedimensions and to soluble surfactants. The method couples a semiimplicit discretization for solving the surfactant transport equation recently developed by Xu and Zhao [62] with the immersed interface method originally developed by LeVeque and Li and [31] for solving the fluid flow equations and the LaplaceYoung boundary conditions across the interfaces. Novel techniques are developed to accurately conserve component mass and surfactant mass during the evolution. Convergence of the method is demonstrated numerically. The method is applied to study the effects of surfactant on single drops, dropdrop interactions and interactions among multiple drops in Stokes flow under a steady applied shear. Due to Marangoni forces and to nonuniform Capillary forces, the presence of surfactant results in larger drop deformations and more complex dropdrop interactions compared to the analogous cases for clean drops. The effects of surfactant are found to be most significant in flows with multiple drops. To our knowledge, this is the first time that the levelset method has been used to simulate fluid interfaces with surfactant.
Stable and efficient finitedifference nonlinearmultigrid schemes for the phasefield crystal equation
 J. Comput. Phys
"... In this paper we present two unconditionally energy stable finite difference schemes for the Modified Phase Field Crystal (MPFC) equation, a sixthorder nonlinear damped wave equation, of which the purely parabolic Phase Field Crystal (PFC) model can be viewed as a special case. The first is a conve ..."
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Cited by 15 (5 self)
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In this paper we present two unconditionally energy stable finite difference schemes for the Modified Phase Field Crystal (MPFC) equation, a sixthorder nonlinear damped wave equation, of which the purely parabolic Phase Field Crystal (PFC) model can be viewed as a special case. The first is a convex splitting scheme based on an appropriate decomposition of the discrete energy and is first order accurate in time and second order accurate in space. The second is a new, fully secondorder scheme that also respects the convex splitting of the energy. Both schemes are nonlinear but may be formulated from the gradients of strictly convex, coercive functionals. Thus, both are uniquely solvable regardless of the time and space step sizes. The schemes are solved by efficient nonlinear multigrid methods. Numerical results are presented demonstrating the accuracy, energy stability, efficiency, and practical utility of the schemes. In ∗Currently at Haerbin Inst. Techn., China 1 ar
Conservative Multigrid Methods for ternary CahnHilliard Fluids, in preparation
"... Abstract. We develop a conservative, second order accurate fully implicit discretization of ternary (threephase) CahnHilliard (CH) systems that has an associated discrete energy functional. This is an extension of our work for twophase systems [13]. We analyze and prove convergence of the scheme. ..."
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Cited by 14 (4 self)
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Abstract. We develop a conservative, second order accurate fully implicit discretization of ternary (threephase) CahnHilliard (CH) systems that has an associated discrete energy functional. This is an extension of our work for twophase systems [13]. We analyze and prove convergence of the scheme. To efficiently solve the discrete system at the implicit timelevel, we use a nonlinear multigrid method. The resulting scheme is efficient, robust and there is at most a 1st order time step constraint for stability. We demonstrate convergence of our scheme numerically and we present several simulations of phase transitions in ternary systems. Key words. ternary CahnHilliard system, nonlinear multigrid method 1.
Numerical schemes for a three component CahnHilliard model
"... Abstract. In this article, we investigate numerical schemes for solving a three component CahnHilliard model. The space discretization is performed by using a Galerkin formulation and the finite element method. Concerning the time discretization, the main difficulty is to write a scheme ensuring, a ..."
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Cited by 9 (3 self)
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Abstract. In this article, we investigate numerical schemes for solving a three component CahnHilliard model. The space discretization is performed by using a Galerkin formulation and the finite element method. Concerning the time discretization, the main difficulty is to write a scheme ensuring, at the discrete level, the decrease of the free energy and thus the stability of the method. We study three different schemes and prove existence and convergence theorems. Theoretical results are illustrated by various numerical examples showing that the new semiimplicit discretization that we propose seems to be a good compromise between robustness and accuracy.
An Efficient, Energy Stable Scheme for the CahnHilliardBrinkman System
"... We present an unconditionally energy stable and uniquely solvable finite difference scheme for the CahnHilliardBrinkman (CHB) system, which is comprised of a CahnHilliardtype diffusion equation and a generalized Brinkman equation modeling fluid flow. The CHB system is a generalization of the Cah ..."
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Cited by 6 (0 self)
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We present an unconditionally energy stable and uniquely solvable finite difference scheme for the CahnHilliardBrinkman (CHB) system, which is comprised of a CahnHilliardtype diffusion equation and a generalized Brinkman equation modeling fluid flow. The CHB system is a generalization of the CahnHilliardStokes model and describes two phase very viscous flows in porous media. The scheme is based on a convex splitting of the discrete CH energy and is semiimplicit. The equations at the implicit time level are nonlinear, but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable, regardless of time stepsize. Owing to energy stability, we show that the scheme is stable in the time and space discrete ℓ ∞ ( 0, T; H1) 2 h and ℓ ( 0, T; H2) h norms. We also present an efficient, practical nonlinear multigrid method – comprised of a standard FAS method for the CahnHilliard part, and a method based on the Vanka smoothing strategy for the Brinkman part – for solving these equations. In particular, we provide evidence that the solver has nearly optimal complexity in typical situations. The solver is applied to simulate spinodal decomposition of a viscous fluid in a porous medium, as well as to the more general problems of buoyancy and boundarydriven flows.