Jiwen He

Finite element approximation of multi-scale elliptic problems using patches of elements

In this paper we present a method for the numerical solution of elliptic problems with multi-scale data using multiple levels of not necessarily nested grids. The method consists in calculating successive corrections to the solution in patches whose discretizations are not necessarily conforming. This paper provides proofs of the results published earlier (see C. R. Acad. Sci. Paris, Ser. I 337 (2003) 679–684), gives a generalization of the latter to more than two domains and contains extensive numerical illustrations. New results including the spectral analysis of the iteration operator and a numerical method to evaluate the constant of the strengthened Cauchy-Buniakowski-Schwarz inequality are presented.

Neumann control of unstable parabolic systems: numerical approach

The present article is concerned with the Neumann control of systems modeled by scalar or vector parabolic equations of reaction-advection-diffusion type with a particular emphasis on systems which are unstable if uncontrolled. To solve these problems, we use a combination of finite-difference methods for the time discretization, finite-element methods for the space discretization, and conjugate gradient algorithms for the iterative solution of the discrete control problems. We apply then the above methodology to the solution of test problems in two dimensions, including problems related to nonlinear models.

Diffeomorphic Matching and Dynamic Deformable Surfaces in 3d Medical Imaging

Abstract We consider optimal matching of submanifolds such as curves and surfaces by a variational approach based on Hilbert spaces of diffeomorphic transformations. In an abstract setting, the optimal matching is formulated as a minimization problem involving actions of diffeomorphisms on regular Borel measures considered as supporting measures of the reference and the target submanifolds. The objective functional consists of two parts measuring the elastic energy of the dynamically deformed surfaces and the quality of the matching. To make the problem computationally accessible, we use reproducing kernel Hilbert spaces with radial kernels and weighted sums of Dirac measures which gives rise to diffeomorphic point matching and amounts to the solution of a finite dimensional minimization problem. We present a matching algorithm based on the first order necessary optimality conditions which include an initial-value problem for a dynamical system in the trajectories describing the deformation of the surfaces and a final-time problem associated with the adjoint equations. The performance of the algorithm is illustrated by numerical results for examples from medical image analysis.

Steady Bingham fluid flow in cylindrical pipes: a time dependent approach to the iterative solution

The main goal of this article is to discuss a novel iterative method for the numerical simulation of a steady Bingham fluid flow in a cylindrical pipe. The method is of the primal-dual type and can be interpreted as an implicit scheme of backward Euler type, applied to a well chosen time dependent variant of the problem under consideration. A key ingredient of the algorithm is a kind of dynamical Tychonoff regularization of the fixed point relation verified by the dual solution. After proving the convergence of the method, we apply it to the solution of test problems and verify its anticipated good convergence properties. Copyright

On Shape Optimization and Related Issues

The main goal of this article is to review, briefly, some of the issues associated with shape optimization for systems modeled by partial differential equations. The practical calculation of the objective function gradient is one of these issues and it clearly includes the use of Automatic Differentiation (AD) techniques for derivative computations. Also, we shall take advantage of this article to describe some recent results concerning the controllability of the Kuramoto-Sivashinsky equation since these results seem to justify the well-known claim that under certain conditions “chaos may enhance controllability.”

A phase equilibrium model for atmospheric aerosols containing inorganic electrolytes and organic compounds (UHAERO), with application to dicarboxylic acids

Computation of phase and chemical equilibria of water-organic-inorganic mixtures is of significant interest in atmospheric aerosol modeling. A new version of the phase partitioning model, named UHAERO, is presented here, which allows one to compute the phase behavior for atmospheric aerosols containing inorganic electrolytes and organic compounds. The computational implementation of the model is based on standard minimization of the Gibbs free energy using a primal-dual method, coupled to a Newton iteration. Water uptake and deliquescence properties of mixtures of aqueous solutions of salts and dicarboxylic acids, including oxalic, malonic, succinic, glutaric, maleic, malic, or methyl succinic acids, are based on a hybrid thermodynamic approach for the modeling of activity coefficients (Clegg and Seinfeld, 2006a, 2006b). UHAERO currently considers ammonium salts and the neutralization of dicarboxylic acids and sulfuric acid. Phase diagrams for sulfate/ammonium/water/dicarboxylic acid systems are presented as a function of relative humidity at 298.15 K over the complete space of compositions.

Finite element methods with patches and applications

Theoretical and numerical aspects of multi-scale problems are investigated. On one hand, mathematical analysis is done on a new method for numerically solving problems with multi-scale behavior using multiple levels of not necessarily nested grids. A particularly flexible multiplicative Schwarz method is presented, requiring no conformity between the meshes at the different scales. The relaxed iterative method consists in calculating successive corrections to the solution in regions where the variations of a problem are too strong to be captured by a coarse initial mesh. In these sub-domains patches of finite elements are applied. A priori and a posteriori error estimates are given and an exact spectral analysis of the iteration operator describing the algorithm is presented. Computational issues are addressed and numerical methods to obtain optimal convergence are given. Crucial implementation matters are discussed with special regard to usage of memory and CPU-time. On the other hand, the efficiency of the introduced correction method is demonstrated on Laplace model problems, either with changing Dirichlet-Neumann boundary conditions or in a polygonal domain with entrant corner. The regularity of the solutions is studied as well as the improvement of the convergence order in the mesh size using various sizes of patches. The correction algorithm is also applied to improve the accuracy in the simulation of the stress field in glacier modeling. A simple model to obtain the effective stress field in the ice mass of a glacier is presented and concluding results are obtained using patches in the regions where changes in the basal boundary conditions are involved.

Patient-Specific Quantitation of Mitral Valve Strain by Computer Analysis of Three-Dimensional Echocardiography A Pilot Study

Background—A paucity of data exists on mitral valve (MV) deformation during the cardiac cycle in man. Real-time 3-dimensional (3D) echocardiography now allows dynamic volumetric imaging of the MV, thus enabling computerized modeling of MV function directly in health and disease. Methods and Results—MV imaging using 3D transesophageal echocardiography was performed in 10 normal subjects and 10 patients with moderate-to-severe or severe organic mitral regurgitation. Using proprietary 3D software, patient-specific models of the mitral annulus and leaflets were computed at mid- and end-systole. Strain analysis of leaflet deformation was derived from these models. In normals, mean strain intensity averaged 0.11±0.02 and was higher in the posterior leaflet than in the anterior leaflet (0.13±0.03 versus 0.10±0.02; P<0.05). Mean strain intensity was higher in patients with mitral regurgitation (0.15±0.03) than in normals (0.11±0.02; P=0.05). Higher mean strain intensity was noted for the posterior leaflet in both normal and organic valves. Regional valve analysis revealed that both anterior and posterior leaflets have the highest strain concentration in the commissural zone, and the boundary zone near the annulus and at the coaptation line, with reduced strain concentration in the central leaflet zone. Conclusions—In normals, MV strain is higher in the posterior leaflet, with the highest strain at the commissures, annulus, and coaptation zones. Patients with organic mitral regurgitation have higher strain than normals. Three-dimensional echocardiography allows noninvasive and patient-specific quantitation of strain intensities because of MV deformations and has the potential to improve noninvasive characterization and follow-up of MV disease.

Canonical Form and Mathematical Interpretation of Electrolyte Solution Systems

We describe a canonical form of electrolyte solution systems for the mathematical interpretation of solidliquid equilibrium. The canonical form is obtained from the analysis of the algebraic structure of electrolyte solution systems and the Karush-Kuhn-Tucker (KKT) conditions for the minimization of the total Gibbs free energy. As a result, the mathematical role of solid species in the solid-liquid equilibrium problem is explained as a Lagrange multiplier of a sort of the linearly constrained optimization problem. This finding will add to the development of an efficient numerical algorithm for the simulation of electrolyte solution systems.

Dirichlet feedback control for the stabilization of the wave equation: a numerical approach

Abstract In (J. Differential Equations 66 (1987) 340) a uniform stabilization method of the wave equation by boundary control a la Dirichlet has been discussed. In this article, we investigate the numerical implementation of the above stabilization process by a numerical scheme which mimics the energy decay properties of its continuous counterpart. The practical implementation of that scheme leads to a biharmonic problem of a new type which is solved by a method directly inspired by some related work of Glowinski and Pironneau on the solution of the Dirichlet problem for the biharmonic operator (SIAM Rev. 21(2) (1979) 167). Numerical experiments show that the decay properties of the energy are well-preserved by our numerical methodology.