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CSM Affiliates Modeling & Control Computer Research Affiliates ICES Reports Facilities
Research
Centers & Groups
Consortiums
CSM Affiliates Modeling & Control Computer Research Affiliates ICES Reports FacilitiesGroup Leader: Irene M. Gamba
The ICES group for Applied Mathematics is an interdisciplinary research effort from departments including Mathematics, Physics and Computer Sciences. It focuses its research on the modeling, analysis and numerical simulations of non-linear phenomena. Some of the current research projects include a wide range of theoretical and computational aspects of mathematical models for non-linear dynamics in quantum, kinetic and fluid transport. In particular we study some aspects of collisional and collisionless plasmas; integral diffusion (Levy) processes; multi-scale modeling in high frequency wave propagation, visibility optimization and image processing; non-linear elasticity in periodically or randomly heterogeneous media; multi-scale non-linear flows with applications to dispersive wave propagation, hydrodynamics; transport in porous media, macroscopic/mesoscopic and phenomenological models for phase transitions in spatial-temporal periodic and randomly heterogeneous media; geomechanics, aerodynamics, biological and molecular dynamics; and charged particle transport emphasizing the linking of quantum, statistical and fluid mechanical states; premixed and non-premixed turbulent combustion models in spatial-temporal periodic and randomly heterogeneous media, as well as computational number theory and combinatorics.
The research has direct applications to the formulation, interpretation, and assessment of model non-linear phenomena on multiple spatial and temporal scales in very diverse geometrical configurations, and to their accurate and efficient approximation using high-performance computing.
Individual contributions include the work of some members from the Department of Mathematics and physics as well as ICES. They have been involved in several scientific projects in areas of interest to Computational and Applied Mathematics (CAM), which are described below.
Professor Todd Arbogast’s research concerns the numerical approximation of partial differential equations as applied to subsurface fluid flow and transport. (1) In many physical systems, fine scale micro-scale effects are a major contributor to the overall macro-scale behavior of the system. Current projects include the modeling and simulation of highly heterogeneous porous media, where fine details of the medium strongly affect fluid flow in the system over tens to hundreds of meters. Numerical subgrid upscaling techniques have been developed that result in a course grid approximation to the solution that can be post-processed to provide a highly accurate fine grid representation of the solution. The technique involves a locally mass conservative, two-scale decomposition of both the solution and the differential system itself. The technique is being adapted by graduate student James Rath to precondition the full fine grid system of equations to improve the efficiency of fine-scale modeling when it is desired. The approach involves finding an optimal finite element basis. With Mary Wheeler, Ivan Yotov, and Gergina Pencheva, a similar multiscale method was devised using high order mortar finite elements to combine the blocks of a course grid accurately and efficiently. (2) A vuggy porous medium is one with many small cavities, and so is extremely heterogeneous. The vugs facilitate fluid flow and make simulation over large distances very difficult. Steve Bryant of the petroleum and geosystems engineering department, Jim Jennings of the Bureau of Economic Geology, and graduate student Mario San Martin Gomez are also working on this project. Appropriate computational algorithms are being developed to approximate the governing differential equations, and simulations will be conducted to determine the micro-scale properties of flow through the medium. These results will be used to validate our mathematical homogenization work on the system, which should result in accurate modeling of the system on the macro-scale. Experimental and computational investigation is proceeding on determining transport properties of the system. (3) Characteristic methods allow transport to be simulated over long time steps, produce little numerical dispersion, and can conserves mass locally. Since the shape of a trace-back region is approximate, its volume may be incorrect, giving inaccurate concentration densities. With Chieh-Sen Huang, a simple modification was developed that conserves both mass and volume of the transported fluid regions. With student Wenhao Wang, the technique is being extended to compressible and two-phase problems.
Professor William Beckner’s research is directed toward analysis and understanding for PDE’s and differential operators in model cases that exhibit characteristic geometric and symmetry structure for problems with mixed homogeneity which arise naturally in fluid dynamics and some cases of non-linear Schrodinger phenomena. His research focuses on the nature of geometric information drawn from sharp Sobolev embedding estimates, both exact and numerical, which provides insight into PDEs for physical processes, and to understand how such estimates may be controlled.
Professor Luis A. Caffarelli’s contributions in Computational and Applied Mathematics include non-linear analysis on optimal allocation problems and applications to statistical mechanics, phase transition and free boundary problems, and homogenization in periodic and random media. Specifically:
Free boundary problems. In combustion, free boundary problems arise, for instance, in premixed flames, where depending on the type of mixture, combustion may start, or not, and be complete or partial. In finance, pricing models constrain the optimization process to particular classes of pricing strategies.
Random and periodic homogenization: A good example is the shape of a capillary drop inside a three-dimensional periodic or random media. If the scale of the medium were comparable to the volume of the drop, the surface of the drop would be “wiggly”. As the scale of the medium becomes much smaller, the drop would take a smooth but non-spherical shape, dictated by the “effective area” of the medium. Similar mathematical issues surface for stationary flames in a periodic media (i.e. we inject reactant at the origin and it produces a stationary flame in the media). Some work is in collaboration with I. Babuska.
Fully non-linear equations that relate in a non-linear fashion the way a surface curves in different directions (the “principal” curvatures). In visualization, given a regular body with a small random noise superimposed, one wants to smooth the noise without rounding the edges. Thus, smoothing operators, which smooth “intermediate oscillations” of the curvature but not singular ones, seem appropriate. Similarly, simple examples show the limiting configuration “drop” in part b) would obey very discontinuous non-linear curvature equations. Some of this work is in collaboration with C. Bajaj.
Periodic solutions of the Monge Ampere equation arise in several contexts. The periodic quasi-geostrophic equations, for instance, describe how a periodic array of vortices evolve in the formation of weather patterns. Particularly interesting is the study of periodic “stationary” arrays of vortices and their stability.
Most recently, Professor Caffarelli has been focusing on the following topics:
Professor Irene M. Gamba’s research has focused on non-linear analysis and numerical methods for kinetic collisional equations (the Boltzmann Transport Equation) and the applications to non-equilibrium statistical mechanics such as energy dissipative flows modeling (granular flows, mixtures, charged transport). More specifically,
Professor Oscar Gonzalez research interests cover computational and applied mathematical problems related to the large-scale deformations of thin rods and ribbons, and more general three-dimensional bodies. Most of his current research efforts are directed toward understanding the mechanical properties of DNA at various length scales. He has contributed articles in various journals across mathematics, engineering and chemistry. Most notably: The Proceedings of the National Academy of Sciences, USA, Archive for Rational Mechanics and Analysis, Journal of Fluid Mechanics, Calculus of Variations and Partial Differential Equations, Theoretical Chemistry Accounts, and Computer Methods in Applied Mechanics and Engineering. He is also the author of a textbook together with A. Stuart entitled "A First Course in Continuum Mechanics" (Cambridge University Press, to appear Fall 2007) and regularly teaches courses in mathematical modeling and numerical analysis at both the graduate and undergraduate levels. Professor Gonzalez is currently working with an ICES PhD student (Jun Li) on using hydrodynamic models to estimate sequence-dependent curvature parameters for DNA. The sequence-dependent curvature and flexibility of DNA is critical for many biochemically important processes. However, few experimental methods are available for directly probing these properties at the basepair level. In this project we are developing an indirect method for estimating sequence-dependent curvature parameters for DNA from hydrodynamic data on short, stiff fragments. The method consists of minimizing a least-squares functional which quantifies the difference between theoretical and experimental sedimentation speeds over a given collection of fragments, and its numerical implementation requires the repeated solution of an exterior Stokes-type problem around various slender, three-dimensional domains. Our current efforts are centered on three areas: the modeling of a polymer in a Stokes-type fluid with thermal fluctuations, the design and analysis of fast, provably convergent boundary element methods, and the efficient implementation of various Quasi-Newton methods for large-scale optimization problems.
Professor Rafael de la Llave’s contributions in Computational and Appied Mathematics have been focused into devising methods to compute invariant manifolds, mostly in collaboration with A. Haro. Their program included theorems of validation, algorithms, computational analysis cpu time, storage, etc., as well as a numerical implementation and development of conjectures based on the numerical simulations. This program was the subject of an invited plenary talk in the biannual congress in Applied Dynamical Systems in Snowbird 2005. Resulting publications of R. de la Llave, A. Haro include:
Further publications in related areas, proposing another numerical exploration for the boundary of hyperbolicity and a method of proof of existence of invariant manifolds associated to neutral eigenvalues and numerical implementation, are:
The research of Professor Misha Vishik concentrated on the mathematical problems of fluid motion. For the incompressible Euler equations he obtained a uniqueness theorem that goes beyond the classical requirement of essentially bounded vorticity. For example, a solution with vorticity bounded in BMO is unique. For the existence problem he obtained the first (local) existence results in classes with unbounded vorticity in dimensions higher than 2. The idea of admissible symmetries plays an important role in the construction. In dimension 2 some of these results are global in time, and they produce unique solutions with the separation rate of liquid particles that is faster than that of the weak solutions of Yudovich. In fact, the flow map has regularity below any Holder class. In a joint paper of Friedlander, Vishik, and Yudovich, the stability problem for a plane ideal fluid flow with small scales in one direction was investigated. An averaged equation was obtained that describes stability spectrum in the limit when the small scale tends to zero.
In a joint paper of Latushkin and Vishik, it was proved that the growth bound for the linearized Euler equation is equal to the spectral bound in dimension 2. This means that the two natural definitions of stability , namely the one based on the spectrum of the generator and the one based on the exponential growth rate of an infinitesimal perturbation, give the same answer in dimension 2. This remains an important open question in dimension 3.
Professor Panagiotis Souganidis’ contributions include the analysis of homogenization in random media, turbulent combustion, and further development of a theory of solutions for fully nonlinear stochastic partial differential equations and their applications to phase transitions and statistical mechanics.