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Sino-French Conference on Computational and Applied Mathematics

1

Abstracts

Recent progress on very high order schemes for

compressible flows

Speaker: Remi Abgrall, University of Zurich

email: [email protected]

Abstract: We are interested in the development of very high order schemes

for compressible fluid dynamics governed by the Navier-Stokes equations.

These approximations use unstructured conformal meshes.

In the talk, we will describe a family of nonlinear schemes that are high

order and non-oscillatory. We will show extensions to the Navier Stokes

equations, and also how the unsteady case can be handled.

Several examples including two and three dimensional flows will be

discussed, showing the performance of the method.

On some recent advances for the indirect

controllability of coupled systems of PDE's

Speaker: Fatiha Alabau-Boussouira, University of Lorraine


Abstract: We consider controlled coupled systems of reversible hyperbolic

PDE’s. A challenging issue for such systems is to understand whether if

controllability can hold when the number of controls is strictly less than the

number of equations. If the answer is positive, one says that the equations

which are free of controls are indirectly controlled.

Such questions naturally arise in the built-in of in sensitizing controls for

scalar PDE’s, in simultaneous controllability or in engineering applications.

We present some recent advances for the indirect control of systems

coupled in cascade based on a multi-levels energy method. We also give

recent results showing the importance of the coupling operator properties

leading to positive as well as negative observability, controllability and unique

continuation results.


2

Geometrical constraints in the level-set method for

shape and topology optimization

Speaker: Gregoire Allaire, Ecole Polytechnique


Abstract: In the context of structural optimization via a level-set method we

propose a framework to handle geometric constraints related to a notion of

local thickness. The local thickness is calculated using the signed distance

function to the shape. We formulate global constraints using integral functions

and compute their shape derivatives. We discuss different strategies and

possible approximations to handle the geometric constraints. We implement

our approach in two and three space dimensions for a model of linearized

elasticity. As can be expected, the resulting optimized shapes are strongly

dependent on the initial guesses and on the specific treatment of the

constraints since, in particular, some topological changes may be prevented

by those constraints. This is a joint work with Francois Jouve and Georgios

Michailidis.

Inverse problems in wave propagation

Speaker: Gang Bao, Zhejiang University


Abstract: Our recent progress in mathematical analysis and computational

studies of the inverse boundary value problems in wave propagation will be

reported. Several classes of inverse problems will be studied, namely inverse

medium problems, inverse source problems, inverse obstacle problems,

inverse rough surface scattering problems, and inverse waveguide problems.

Issues on ill-posedness and its remedy for the inverse problems will be

addressed. Computational methods will be discussed for the inverse problems.

New stability results for the inverse problem will be presented. In particular, our

most recent stability result on inverse problems for the related time-domain

wave equation with possible caustics will be highlighted.


3

Multiscale methods and analysis for the nonlinear

Klein-Gordon equation in the nonrelativistic limit

regime

Speaker: Weizhu Bao, National University of Singapore


Abstract: In this talk, I will review our recent works on numerical methods and

analysis for solving the nonlinear Klein-Gordon (KG) equation in the

nonrelativistic limit regime, involving a small dimensionless parameter which is

inversely proportional to the speed of light. In this regime, the solution is highly

oscillating in time and the energy becomes unbounded, which bring significant

difficulty in analysis and heavy burden in numerical computation. We begin

with four frequently used finite difference time domain (FDTD) methods and

obtain their rigorous error estimates in the nonrelativistic limit regime by paying

particularly attention to how error bounds depend explicitly on mesh size and

time step as well as the small parameter.

Then we consider a numerical method by using spectral method for spatial

derivatives combined with an exponential wave integrator (EWI) in the

Gautschi-type for temporal derivatives to discretize the KG equation. Rigorious

error estimates show that the EWI spectral method show much better temporal

resolution than the FDTD methods for the KG equation in the nonrelativistic

limit regime. In order to design a multiscale method for the KG equation, we

establish error estimates of FDTD and EWI spectral methods for the nonlinear

Schrodinger equation perturbed with a wave operator. Finally, a multiscale

method is presented for discretizing the nonlinear KG equation in the

nonrelativistic limit regime based on large-small amplitude wave

decomposition. This multiscale method converges uniformly in

spatial/temporal discretization with respect to the small parameter for the

nonlinear KG equation in the nonrelativistic limite regime. Finally, applications

to several high oscillatory dispersive partial differential equations will be

discussed.


4

On the master equation in mean field theory

Speaker: Alain Bensoussan, University of Texas & University Hong Kong


Abstract: One of the major founders of Mean Field Games, P.L. Lions has

introduced in his lectures at College de France the concept of Master Equation.

It is obtained through a formal analogy with the set of partial differential

equations derived for the Nash equilibrium of a differential game with a large

number of players. The objective of this lecture is to explain its derivation, not

by analogy, but through its interpretation. We do that for both Mean Field Type

Control and Mean Field Games. We obtain complete solutions in the linear

quadratic case. We analyze the connection with Nash equilibrium.

When it rains on a sand beach: the Richards equations

Speaker: Christine Bernardi, Pierre and Marie Curie University


Abstract: Richards equation models the water flow in a partially saturated

underground porous medium under the surface. When it rains on the surface,

boundary conditions of Signorini type must be considered on this part of the

boundary. We first study this problem which results into a variational equality or

inequality according to the type of boundary conditions. We propose and study

a discretization by an implicit Euler's scheme in time and finite elements in

space.

The initial- and boundary-value problem for the

transport equation with low regularity data

Speaker: Franck Boyer, Aix-Marseille University


Abstract: The first part of the talk will be dedicated to well-posedness results

for the initial- and boundary-value problem for the transport equation

associated with a velocity field possessing the same Sobolev regularity as the

one required in the usual DiPerna-Lions theory. I will in particular discuss trace

theorems adapted to this framework and some applications.

In a second part, I will briefly describe some convergence results, uniform

in time, for the upwind finite volume approximation of this problem on general

unstructured grids, without any additional regularity assumptions on the data


5

Analysis and control of 2D turbulence

Speaker: Charles-Henri Bruneau, University of Bordeaux


Abstract: We first show what are the physical structures responsibles of the

two turbulence cascades in two-dimensions. Then we explore the ability to

control the drag coefficient of a blunt body using polymers in solution. The

numerical solutions of Navier-Stokes equations and Oldroyd-B model are

compared to soap films experiments.

Exact divergence free H(div) basis and Feymann-kac

formula based boundary element methods

Speaker: Wei Cai, University of North Carolina at Charlotte & Shanghai

Jiaotong University


Abstract: In this talk, we will discuss two results on numerical methods for

solving PDEs, one on designing high order divergence free basis for MHD

equations and another on Feymann-Kac formula based boundary element

method for elliptic PDEs. In the first, we give the construction of a new high

order hierarchical basis functions in the H(div) space and show how to use

interior bubble functions in imposing exact divergence-free conditions for MHD

problems. While for the second, we will present a communication-free domain

decomposition boundary element method for Laplace equations by combining

Feyman-Kac formula and local time of Brownian motions and local integral

equations.


6

A Legendre-Galerkin spectral method for optimal

control problems governed by Stokes equations

Speaker: Yanping Chen, South China Normal University


Abstract: In this work, we study the Legendre–Galerkin spectral

approximation of distributed optimal control problems governed by Stokes

equations. We show that the discretized control problems satisfy the

well-known Babuska–Brezzi conditions by choosing an appropriate pair of

discretization spaces for the velocity and the pressure. Constructing suitable

base functions of the discretization spaces leads to sparse coefficient matrices.

We first derive a priori error estimates in both H^1 and L^2 norms for the

Legendre–Galerkin approximation of the unconstrained control problems.

Then both a priori and a posteriori error estimates are obtained for control

problems with the constraints of an integral type, thanks to the higher regularity

of the optimal control. Finally, some illustrative numerical examples are

presented to demonstrate the error estimates.

Reverse time migration for reconstructing extended

obstacles in planar acoustic waveguides

Speaker: Zhiming Chen, LSEC & Institute of Mathematics and System

Sciences, CAS


Abstract: We propose a new reverse time migration method for reconstructing

extended obstacles in the planar waveguide using acoustic waves at a fixed

frequency. We prove the resolution of the reconstruction method in terms of

the aperture and the thickness of the waveguide. The resolution analysis

implies that the imaginary part of the cross-correlation imaging function is

always positive and thus may have better stability properties. Numerical

experiments are included to illustrate the powerful imaging quality and to

confirm our resolution results. This is a joint work with Guanghui Huang.


7

The mathematical model for the contamination

problems and related inverse problems

Speaker: Jin Cheng, Fudan University


Abstract: In this talk, we discuss the motivation for the study of the abnormal

diffusion models for the contamination problems. From the practical point of

view, several inverse problems proposed and studied Theoretic results, for

example, the uniqueness and stability are shown. The possibility of the

application of these studies is mentioned.

A new approach to elasticity problems and their finite

element discretizations

Speaker: Philippe G. Ciarlet, City University of Hong Kong


Abstract: We describe and analyze an approach to the pure Neumann problem

of three-dimensional linearized elasticity, whose novelty consists in

considering the strain tensor field as the sole unknown, instead of the

displacement vector field as is customary. This approach leads to a well-posed

minimization problem of a new type, constrained by a weak form of the

classical Saint Venant compatibility conditions, the justification of which

essentially rests on J.L. Lions lemma. Interestingly, this approach also

provides a new proof of Korn's inequality.

We also describe and analyze a natural finite element approximation of this

problem.


8

Mathematical models for tumor growth: construction,

validation and clinical applications

Speaker: Thierry Colin, Bordeaux Institute of Technology


Abstract: In the last few years there have been dramatic increases in the

range and quality of information available from non-invasive medical imaging

methods, so that several potentially valuable imaging measurements are now

available to quantitatively measure tumor growth, assess tumor status as well

as anatomical or functional details. Using different methods such as the CT

scan, magnetic resonance imaging (MRI), or positron emission tomography

(PET), it is now possible to evaluate and define tumor status at different levels:

physiological, molecular and cellular.

These multimodal data help the decision process of oncologists in the

definition of therapeutic protocols. At present, this decision process is mainly

based on previously acquired statistical evidence and on the practitioner

experience. The quality of the response to a treatment is decided according to

the OMS criteria by estimating the length of the two main axis of the tumor in

the largest cut. There are two blocking difficulties in this approach that we

want to attack: i) previous statistical information is not patient specific; ii) there

exist no quantitative mean of summarizing and using as predicting tools the

multimodal patient-specific data presently available thanks to CT scans, MRI,

PET scans and molecular biology data. The aim of this talk is to show how we

can provide a simulation framework based on quantitative patient-specific data

by using nonlinear model based on PDE. I will present how one can build such

model in order to describe the may features of tumor growth and how one can

expect to obtain a suitable parametrization of tumor growth by solving inverse

problem. I will present some mathematical results on the models.

The applications will concern lung and lever metastasis, meningiomas and

brain tumor.


9

On the control of the Korteweg-de Vries equation

Speaker: Jean-Michel Coron, Pierre and Marie Curie University


Abstract: The Korteweg-de Vries equation allows to describe approximately

long waves in water of relatively shallow depth. We study this equation on a

finite interval. The control is acting on the boundary of this interval. Depending

on these boundary conditions we get different control systems. We present

some results and methods to study the controllability and the stabilization of

these control systems.

Some Korn inequalities originating in a periodic

homogenization problem of linearized elasticity with

inclusions and contact conditions

Speaker: Alain Damlamian, University Paris-Est


Abstract: In the study of the homogenization of a contact problem in linearized

elasticity with inclusions, some new unilateral Korn inequalities are needed to

establish the coercivity of the functional to be minimized. The talk will mainly

explain how to prove various versions of the Korn inequality in this context.

The type of inequality is strongly dependent on the fact that the boundary of

each inclusion is invariant under some group of rotations or not.


10

An AMR method for systems arising in multifluid

dynamics

Speaker: Jean-Michel Ghidaglia, ENS Cachan


Abstract: Computing transient solutions with very strong spatial gradients is a

well known challenge in numerical analysis. Automatic Mesh Refinement

(AMR) is a way for addressing this question. It is indeed intuitively clear that, if

we can dynamically adapt the mesh in order that spatial regions where the

gradient are large are well resolved, then the computed solution will be more

accurate.

In this talk we describe the derivation of an AMR method for a non

conservative system of equations arising in CmFD (Computational multi Fluid

Dynamics). Then we shall present some numerical results on the classical

Ransom's faucet benchmark.

Eddy currents nondestructive evaluation

Speaker: Olivier Goubet, University of Picardie Jules Verne


Abstract: Eddy current nondestructive evaluation is commonly used to track a

defect inside a conductor. Eddy currents are a quasi-static approximation of

Maxwell equations. We address here a mathematical method to solve these

equations as a transmission problem at the interface of the conductor and the

coil that provides the electromagnetic source.


11

A second-order maximum principle preserving

continuous finite element technique for nonlinear

scalar conservation equations

Speaker: Jean Luc Guermond, Texas A&M University


Abstract: In the first part of the talk I will introduces a first-order viscosity

method for the explicit approximation of scalar conservation equations with

Lipschitz fluxes using continuous finite elements on arbitrary grids in any

space dimension. Provided the lumped mass matrix is positive definite, the

method is shown to satisfy the local maximum principle under a usual CFL

condition. The method is independent of the cell type; for instance, the mesh

can be a combination of tetrahedra, hexahedra, and prisms in three space

dimensions. An a priori convergence estimate is given provided the initial data

is BV.

In the second part of the talk I will extend the accuracy of the method to

second-order (at least). The technique is based on mass-lumping correction, a

high-order entropy viscosity method, and the Boris-Book-Zalesak flux

correction technique. The algorithm works for arbitrary meshes in any space

dimension and for all Lipschitz fluxes. The formal second-order accuracy of the

method and its convergence properties are tested on a series of linear and

nonlinear benchmark problems.

Some results on spectral element method

Speaker: Benyu Guo, Shanghai Normal University


Abstract: In this talk, we present the basic results on the generalized Jacobi

quasi-orthogonal approximation, which plays an important role in spectral

element methods. As examples of its applications, we consider the spectral

element methods for high order problems in one dimension, fourth order

problems in two dimensions and some problems defined on non-rectangular

domains. The numerical results indicate the high accuracy of suggested

algorithms.


12

Analysis and numerical methods in RTE based

medical imaging

Speaker: Weimin Han, University of Iowa & Xi'an Jiaotong University


Abstract: The radiative transfer equation (RTE) arises in a wide range of

applications. Recently, there has been much interest in the RTE due to its

application in biomedical imaging. It is challenging to solve RTE numerically

because of its integro-differential form, high dimension, and numerical

singularity in highly forward-peaked media. In this talk, adaptive solution

algorithms will be discussed for solving the RTE based on rigorously derived a

posteriori error estimates, and a family of differential approximations of the

RTE will be explored and analyzed. In addition, parameter identification

problems and source identification problems will be briefly discussed.

Stochastic symplectic methods for stochastic

Hamiltonian systems

Speaker: Jialin Hong, LSEC, Academy of Mathematics and Systems Science,

CAS


Abstract: The phase flow of stochastic Hamiltonian systems (SHSs)

preserves stochastic symplectic structure. A numerical method applied SHSs

is called stochastic symplectic if it preserves the structure. In this talk we

present some results on stochastic sympelctic methods for SHSs, including

stochastic generating functions for stochastic symplectic methods, and

stochastic Hamilton-Jacobi theory. We investigate the canonical form and the

stochastic symplectic structure of stochastic Schroedinger equations (SSEs),

and show that the symplectic Runge-Kutta semidiscretization of SSEs in time

preserves charge conservation law. We give a fundamental convergence

theorem, in mean-square sense, on the semidiscretization of SSEs in time,

and an application to a symplectic semidiscretization.


13

On the dynamical Rayleigh-Taylor instability in

compressible viscous flows without heat conductivity

Speaker: Song Jiang, Institute of Applied Physics and Computational

Mathematics, Beijing


Abstract: We investigate the instability of a smooth Rayleigh-Taylor

steady-state solution to Compressible viscous flows without heat conductivity

in the presence of a uniform gravitational field in a bounded domain with

smooth boundary. We show that the steady-state is linearly unstable by

constructing a suitable energy functional and exploiting arguments of the

modified variational method. Then, based on the constructed linearly unstable

solutions and a local well-posedness result of classical solutions to the original

nonlinear problem, we further reconstruct the initial data of linearly unstable

solutions to be the one of the original nonlinear problem and establish an

appropriate energy estimate of Gronwall-type. With the help of the established

energy estimate, we show that the steady-state is nonlinearly unstable in the

sense of Hadamard by a careful bootstrap argument. As a byproduct of our

analysis, we verify the destabilizing effect of compressibility in the linearized

problem for compressible viscous flows without heat conductivity.

Uncertainty quantification for transport equation with

uncertain coefficients in the diffusive regimes

Speaker: Shi Jin, Shanghai Jiaotong University & University of Wisconsin


Abstract: In this talk we will study generalized polynomial chaos (gPC)

approach to transport equation with uncertain cross-sections and show that

they can be made asymptotic-preserving, in the sense that in the diffusion limit

the gPC scheme for the transport equation approaches to the gPC scheme for

the diffusion equation with random diffusion coefficient. This allows the

implemention of the gPC method without numerically resolving (by space, time,

and gPC modes) the small mean free path for transport equation in the

diffusive regime.


14

The energy landscape of cellular systems from the

large deviation point of view

Speaker: Tiejun Li, Peking University


Abstract: The dynamics of cellular systems is of non-gradient type in general.

To understand its robustness, adaptivity and related properties with respect to

noise perturbation is a fundamental question. In this talk, we will show how to

construct the energy landscape of cellular systems by incorporating the large

deviation theory and characterize the robustness of the dynamical process.

The considered system will include the genetic switching and a simplified

budding yeast cell cycle process.

A numerical method for the quasi-incompressible

NSCH model for variable density two-phase flows with

a discrete energy law

Speaker: Ping Lin, University of Dundee


Abstract: We will present some recent work on phase-field models for

two-phase fluids with variable densities. The Quasi-Incompressible

Navier-Stokes-Cahn-Hilliard model with the gravitational force being

incorporated in the thermodynamically consistent framework will be

investigated. Under a minor reformulation of the system we show that there is

a continuous energy law underlying the system, assuming that all variables

have reasonable regularities. For the reformulated system we then design a

continuous finite element method and a special temporal scheme such that the

energy law is accurately preserved at the discrete level. Such a discrete

energy law for a variable density two-phase flow model has never been

established before with continuous finite element. Some numerical results will

be presented to demonstrate the capabilities of our numerical schemes. We

will also show an example that an energy law preserving method will perform

better for multiphase flow problems. Finally, we will extend the model to

account for the thermocapillary effects. It allows for the different properties

(densities, viscosities and heat conductivities) of each component while

maintaining thermodynamic consistency. To our knowledge such a model is

new. Numerical validation is provided too.


15

Modeling and simulation of Gaseous microflows

Speaker: Li-Shi Luo, Old Dominion University


Abstract: We study gaseous flows in micro-scales by using molecular

dynamics (MD), kinetic equation, and hydrodynamic equations. First, we solve

the linearized Boltzmann equation in a wide range of Knudsen number by

using an efficient high-order collocation method the singular integral equation.

Based the solution of the integral equation, we construct various approximated

solutions which can be modeled by macroscopic equations. We also simulate

molecular flows by using molecular dynamics. We propose macroscopic model

to simulate molecular and kinetic microflows. We use Couette flow in

two-dimensions as the specific example to illustrate our ideas.

A priori and a posteriori analysis for electronic

structure calculation

Speaker: Yvon Maday, Pierre and Marie Curie University


Abstract: In this talk I shall provide a priori and a posteriori estimates for the

approximation of nonlinear eigenvalue problems that are encountered in

electronic structure calculations.I shall start by reviewing the definition of the

models (Hartree Fock and Kohn-Sham) then present the complete results

(both on eigenvectors and eigenvalues) on the a priori error analysis for such

problems.

The analysis will include plane wave calculations and finite element

approximation together with DG approximations. We shall also provide a

complete analysis of a posteriori type where error due to space discretization,

incomplete iterative solution methods and models are considered and

balanced in order to get an optimal convergence.

This presentation assembles joined contributions with Eric Cances,

Rachida Chakir, Genevieve Dusson, Benjamin Stamm and Martin Vohralik.


16

Normal forms for semi-linear equations with

non-dense domain

Speaker: Pierre Magal, University of Bordeaux


Abstract: This presentation is devoted to bifurcation problems for some

classes of PDE arising in the context of population dynamics. The main

difficulty in such a context is to understand the dynamical properties of a PDE

with non-linear and non-local boundary conditions. A typical class of examples

is the so called age structured models. Age structured models have been well

understood in terms of existence, uniqueness, and stability of equilibria since

the 80's. Nevertheless, up to recently, the bifurcation properties of the

semiflow generated by such a system has been only poorly understood.

In this presentation, we will start with some results about existence and

smoothness of the center manifold. We will mention some Hopf’s bifurcation

results as well as some applications. We will also present some recent results

about normal form theory in such a context. As a consequence the normal

form theory, we will be in position to investigate the Bogdanov-Takens

bifurcations for a class of age-structured models.

Quasicontinuum methods for a crack model

Speaker: Pingbing Ming, LSEC, CAS


Abstract: In this talk, we study three quasicontinuum approximations of a

lattice model for crack propagation. The influence of the approximation on the

bifurcation patterns is investigated. The estimate of the modeling error is

applicable to near and beyond bifurcation points, which enables us to evaluate

the approximation over a finite range of loading and multiple mechanical

equilibria. Our results indicate that the so-called ghost forces has a nonlocal

effect on the system un study, which differs very much from that on the defect

free system. This is a joint work with Xiantao Li.


17

Global existence and parabolic limit of first-order

quasilinear hyperbolic systems

Speaker: Yue-Jun Peng, Blaise Pascal University


Abstract: We consider the Cauchy problem for first-order quasilinear partially

dissipative hyperbolic systems with a small parameter. Typically, the small

parameter is the relaxation time in physical models. Under stability conditions,

we show the global existence of smooth solutions in a uniform neighbourhood

of a constant equilibrium state with respect to the parameter. In a slow time,

this allows to obtain the global-in-time convergence of the systems to parabolic

equations as the parameter goes to zero. For large initial data, the

convergence is justified in a uniform local-in-time interval.

Some new results of global existence in

reaction-diffusion-advection systems

Speaker: Michel Pierre, ENS Rennes & IRMAR


Abstract: We will present several new results of global existence for

reaction-diffusion-advection systems for which preservation of positivity and

control of mass hold.It is well-known that blow up may occur infinite time even

if the total mass is uniformly bounded in time. Positive known results are

essentially of two kinds: either we assume some so-called triangular structure

of the reactive part and we

aim at classical solutions, or we look for weak global solutions. We will present

some new contributions in the following directions: anisotropic time dependent

diffusion together with not so regular advection terms; nonlinear diffusions;

Wentzell boundary conditions; electro-diffusion-reaction systems. They have

been obtained in collaboration with D. Bothe, A. Fischer, G. Rolland, G.&J.

Goldstein, M. Meyries.


18

Dynamic programming for mean-field type control: two

applications

Speaker: Olivier Pironneau, Pierre and Marie Curie University


Abstract: Mean-field games with an infinite number of players yield in the limit

to a new type of stochastic control problem where the coefficients are function

of the probability measure of the mean stochastic process. Assuming that a

PDF exists, the problem is standard for calculus of variations but non-standard

for dynamic programming. Using derivatives with respect to measures as in

Lions1 we shall derive the HJB equations. We shall present two numerical

applications of the method, one for a portfolio optimization and another for the

systemic risk problem studied by J. Garnier and G. Papanicolaou2. Using

freefem++, we have computed the most probable transition, among the rare

events, that passes from a state of "stability" (left hump in figure 1) to a state of

"crisis" (right hump) for a situation with many agents who optimize their utility

which is itself function of the average of the position of all agents. This is a joint

work with Mathieu Lauriere.

Figure 1: Left: Time snapshots of PDF (left) at 0,T/3,2T/3 and T.

Right:Optimization history of J : Red log( J ). Green same minus penalty part.

Blue 2 log |||| Jgradu .


19

Controllability of fluid flows

Speaker: Jean-Pierre Puel, University of Versailles


Abstract: This presentation will give an overview on the controllability for fluid

flows. There will be no technical proof. First of all we will describe in an

abstract situation the various concepts of controllability for evolution equations.

We will then present some problems and results concerning the controllability

of systems modeling fluid flows. First of all we will consider the Euler equation

describing the motion of an incompressible inviscid fluid. Then we will give

some results concerning the Navier-Stokes equations, modeling an

incompressible viscous fluid, and some related systems, in particular the case

of what is called Lagrangian control and which might lead to a lot of

developments. Finally, we will present a first result of controllability for the case

of a compressible fluid (in dimension 1) and some important open problems.

Exact synchronization for a coupled system of wave

equations with Dirichlet boundary controls

Speaker： Bopeng Rao, University of Strasbourg


Abstract: In this talk, the exact synchronization for a coupled system of wave

equations with Dirichlet boundary controls and some related concepts are

introduced. By means of the exact null controllability of a reduced coupled

system, under certain conditions of compatibility, the exact synchronization,

the exact synchronization by groups, and the exact null controllability and

synchronization by groups are all realized by suitable boundary controls. This

is a joint work with Tatsien Li.


20

Image restoration: wavelet frame approach, total

variation and beyond

Speaker: Zuowei Shen, National University of Singapore


Abstract: This talk is about the wavelet frame-based image and video

restorations. We start with some of main ideas of wavelet frame based image

restorations. Some of applications of wavelet frame based models image

analysis and restorations will be shown. Examples of such applications include

image and video inpainting, denoising, decomposition, image deblurring and

blind deblurring, segmentation, CT image reconstruction, 3D reconstruction

in electronmicroscopy, and etc. In all of these applications, spline wavelet

frames derived from the unitary extension principle are used. This allows us

to establish connections between wavelet frame base method and various

PDE based methods, that include the total variation model, nonlinear

diffusion PDE based methods, and model of Mumford-Shah. In fact, we will

show that when spline wavelet frames are used, right chosen models of a

wavelet frame method can be viewed as a discrete approximation at a given

resolution to the corresponding PDE based models. A convergence analysis in

terms of objective functionals and their approximate minimizers as image

resolution increases will be discussed.

Some aspects of finite element approximation for

Reissner-Mindlin plates

Speaker: Zhong-Ci Shi, Institute of Computational Mathematics, CAS


Abstract: The Reissner-Mindlin plate model is one of the most commonly

used models of a moderate-thick to thin elastic plate. However, a direct finite

element approximation usually yields very poor results, which is referred to

LOCKING phenomenon.

In the past two decades, many efforts have been devoted to the design of

locking free finite elements to resolve this model, most of these work focus on

triangular or rectangular elements, the latter may be extended to parallelogra

ms, but very few on quadrilaterals.

In this talk we will give an overview of the recent development of low order

quadrilateral elements and present some new results.


21

Viscosity solutions of path-dependent PDEs

Speaker: Nizar Touzi, Ecole polytechnique


Abstract: Partial differential equation with non-anticipative dependence on the

path arise in many applications as stochastic control and stochastic differential

games in non-Markov systems, backward stochastic differential equations,

large deviations for non-Markov diffusions, path-dependent branching

diffusions and the related simulation. We introduce an adaptation of the

Crandall-Ishii notion of viscosity solutions which bypasses the difficulty due to

the non local compacteness of the underlying space of paths. Our definition

replaces the pointwise tangency by the tangency in mean, thus admitting a

larger class of test functions. This notion of viscosity solution satisfies a

stability result, similar to the standard notion in finite-dimensional spaces.

Wellposedness results are established in the general fully nonlinear case. In

the semilinear case, the comparison result is obtained by simpler arguments,

as compared to the standard theory, by taking advantage of the larger class of

test functions.

Mathematics and swimming of aquatic organisms

Speaker: Marius Tucsnak, University of Lorraine


Abstract: We consider several mathematical models describing the motions of solids Immersed in an incompressible fluid. We begin by emphasising that, depending on the flow regime, the governing equations may exhibit a wide range of properties, leading to a rich mathematical structure. We next discuss wellposedness issues, where the major difficulty to be solved consists in tackling the presence of free boundaries. Finally we study the displacement of the solids (under the action of an exterior force or in a self-propelled manner) from a control theoretical perspective.


22

Analysis and computation of a polar active liquid

crystal models

Speaker: Qi Wang, University of South Carolina & Beijing Computational

Science Research Center


Abstract: Active liquid crystals encompasses a class of live and man-made

materials that consist of self-propelled molecules, nano- or micro-particles.

They can form orientation orders at sufficiently high concentration and produce

spontaneous flows, long-range spatial and temporal patterns. We will study the

polar active liquid crystal system using a continuum model and explore

mechanisms for the generation of spatial as well as temporal patterns. Stability

and special solutions in channel flows, capillary flows, and 2-D confined flows

will be analyzed. Transient flow simulation will be conducted using a finite

difference method.

Multilevel Block-Incremental unknowns for general

anisotropic elliptic equations: diagonal

preconditioning and condition analysis

Speaker: Yu-Jiang Wu, Lanzhou University


Abstract: We give, at first, estimates of condition numbers of the system

produced by block incremental unknowns (BIU) with preconditioning for

anisotropic operators. Then a diagonal preconditioner is introduced to the

BIU-preconditioned system and new reduction of condition number is obtained.

The results will also be produced when we apply to more general anisotropic

elliptic equations. Finally, some numerical experiments and results are

presented to confirm our theoretical analysis. This is a joint work with Ai-Li

Yang and Lun-Ji Song.


23

Parallel adaptive finite element computations of

Poisson-Nernst-Planck equations and applications to

ion channel systems

Speaker: Lin-bo Zhang, LSEC, Chinese Academy of Sciences


Abstract: In this talk I will present our collaborative efforts on developing

parallel adaptive finite element codes for solving three-dimensional coupled

electro-diffusion equations, including the Poisson-Nernst-Planck equations

(PNP) and size-modified Poisson-Nernst-Planck equations (SMPNP), using

our parallel adaptive finite element toolbox Parallel Hierarchical Grid (PHG).

By integrating these codes with the pre- and post-processing tools that our

collaborators have been working on for many years, we have developed a

software package, ichannel, for numerical simulation of ion transport through

three-dimensional ion channel systems that consist of protein and membrane,

and succeeded in carrying out finite element simulations of some channels

such as the gA channel, the VDAC channel and the PA channel. The

numerical results obtained with ichannel agree well with existing Brownian

dynamics (BD) simulation results and experimental results.

Defects of liquid crystals

Speaker: Pingwen Zhang, Peking University


Abstract: Defects in liquid crystals (LCs) are of great practical and theoretical

importance. Recently there is a growing interest in LCs materials under

topological constrain and/or external force, but the defects pattern and

dynamics are still poorly understood. We investigate three-dimensional

spherical droplet within the Landau-de Gennes model under different boundary

conditions. When the Q-tensor is uniaxial, the model degenerates to vector

model (Oseen-Frank), but Q-tensor model is superior to vector model as the

former allows biaxial in the order parameter. Using numerical simulation, a rich

variety of defects pattern are found, and the results suggest that, line

disclinations always involve biaxial, or equivalently, uniaxial only admits point

defects. Then we believe that Q-tensor model is essential to include the

disclinations line which is a common phenomenon in LCs. The mathematical

implication of this observation will be discussed in this talk.


24

Some inverse problems for stochastic PDEs

Speaker: Xu Zhang, Sichuan University


Abstract: In this talk, I will present some recent results on the inverse

stochastic hyperbolic/parabolic problems. I will explain that both the

formulation of stochastic inverse problems and the tools to solve them may

differ considerably from their deterministic counterpart.

Superconvergence in polynomial spectral collocation:

an unclaimed territory

Speaker: Zhimin Zhang, Beijing Computational Science Research Center&

Wayne State


Abstract: Superconvergence phenomenon is well understood for the h

-version finite element method and researchers in this field have accumulated

a vast literature during the past 40 years. However, the relevant study for the p

-version finite element method and spectral methods is lacking. We believe

that the scientific community would also benefit from the study of

superconvergence phenomenon of those methods. Recently, some efforts

have been made to expand the territory of the superconvergence. In this talk,

we summarize some recent development on superconvergence study for the

spectral collocation methods.


25

Radial symmetry of entire solutions of a bi-harmonic

equation with exponential nonlinearity

Speaker: Feng Zhou, East China Normal University


Abstract: We will talk about necessary and sufficient conditions for an entire

solution $u$ of a biharmonic equation with exponential nonlinearity $eû$ to

be a radially symmetric solution. The standard tool to obtain the radial

symmetry for a system of equations is the Moving-Plane-Method (MPM). In

order to apply the MPM, we need to know the asymptotic expansions of

$u$ and $-\Delta u$ at $\infty$. The difficulties come from the fact that $eû$ is

supercritical for $N\geq5$, $eû \not \in L^{\frac{N}{4}} (\mathbb{R}^N)$. We

need to get the right decay rate of $u$ and $-\Delta u$ at $\infty$ in order to

start the moving plane procedure. This is a joint work with Z.M.Guo and X.

Huang.

program and abstract book - xiamen universitymath.xmu.edu.cn/conference/sfcam/abstracts.pdf ·...

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