## Jacob Bedrossian : Positive Lyapunov exponents for 2d Galerkin-Navier-Stokes with stochastic forcing

- Applied Math and Analysis ( 436 Views )In this talk we discuss our recently introduced methods for obtaining strictly positive lower bounds on the top Lyapunov exponent of high-dimensional, stochastic differential equations such as the weakly-damped Lorenz-96 (L96) model or Galerkin truncations of the 2d Navier-Stokes equations (joint with Alex Blumenthal and Sam Punshon-Smith). This hallmark of chaos has long been observed in these models, however, no mathematical proof had previously been made for any type of deterministic or stochastic forcing. The method we proposed combines (A) a new identity connecting the Lyapunov exponents to a Fisher information of the stationary measure of the Markov process tracking tangent directions (the so-called "projective process"); and (B) an L1-based hypoelliptic regularity estimate to show that this (degenerate) Fisher information is an upper bound on some fractional regularity. For L96 and GNSE, we then further reduce the lower bound of the top Lyapunov exponent to proving that the projective process satisfies HÃ¶rmander's condition. I will also discuss the recent work of Sam Punshon-Smith and I on verifying this condition for the 2d Galerkin-Navier-Stokes equations in a rectangular, periodic box of any aspect ratio using some special structure of matrix Lie algebras and ideas from computational algebraic geometry.

## Joe Kileel : Inverse Problems, Imaging, and Tensor Decomposition

- Applied Math and Analysis ( 374 Views )Perspectives from computational algebra and numerical optimization are brought to bear on a scientific application and a data science application. In the first part of the talk, I will discuss cryo-electron microscopy (cryo-EM), an imaging technique to determine the 3-D shape of macromolecules from many noisy 2-D projections, recognized by the 2017 Chemistry Nobel Prize. Mathematically, cryo-EM presents a particularly rich inverse problem, with unknown orientations, extreme noise, big data and conformational heterogeneity. In particular, this motivates a general framework for statistical estimation under compact group actions, connecting information theory and group invariant theory. In the second part of the talk, I will discuss tensor rank decomposition, a higher-order variant of PCA broadly applicable in data science. A fast algorithm is introduced and analyzed, combining ideas of Sylvester and the power method.

## Ben Krause : Dimension independent bounds for the spherical maximal function on products of finite groups

- Applied Math and Analysis ( 272 Views )The classical Hardy-Littlewood maximal operators (averaging over families of Euclidean balls and cubes) are known to satisfy L^p bounds that are independent of dimension. This talk will extend these results to spherical maximal functions acting on Cartesian products of cyclic groups equipped with the Hamming metric.

## Xiaochuan Tian : Analysis and computation of nonlocal models

- Applied Math and Analysis ( 249 Views )Nonlocal models are experiencing a firm upswing recently as more realistic alternatives to the conventional local models for studying various phenomena from physics and biology to materials and social sciences. In this talk, I will describe our recent effort in taming the computational challenges for nonlocal models. I will first highlight a family of numerical schemes -- the asymptotically compatible schemes -- for nonlocal models that are robust with the modeling parameter approaching an asymptotic limit. Second, fast algorithms will be presented to reduce the high computational cost from the numerical implementation of the nonlocal operators. Although new nonlocal models have been gaining popularity in various applications, they often appear as phenomenological models, such as the peridynamics model in fracture mechanics. Here we will try to provide better perspectives of the origin of nonlocality from multiscale modeling and homogenization, which in turn may help the development of more effective numerical methods for homogenization.

## Min Kang : Tropically Linear Interface Growth Models

- Applied Math and Analysis ( 249 Views )We first discuss a general method to derive macroscopic laws from certain microscopic interactions that can be applied to a large class of particle systems. In particular we consider a broad class of systems that are linear in a special algebra, so-called tropical algebra. Some natural connections among the scaling limits of these random systems, the solutions to specific partial differential equations (Hamilton-Jacobi type) and combinatorial optimization problems have been noticed. If time allows, we further discuss a useful application of the variational formula (microscopic version of Hopf-Lax formula) to a well-known interacting particle system, totally asymmetric simple exclusion process.

## Jun Kitagawa : A convergent Newton algorithm for semi-discrete optimal transport

- Applied Math and Analysis ( 246 Views )The optimal transport (Monge-Kantorovich) problem is a variational problem involving transportation of mass subject to minimizing some kind of energy, and it arises in connection with many parts of math, both pure and applied. In this talk, I will discuss a numerical algorithm to approximate solutions in the semi-discrete case. We propose a damped Newton algorithm which exploits the structure of the associated dual problem, and using geometric implications of the regularity theory of Monge-Amp{\`e}re equations, we are able to rigorously prove global linear convergence and local superlinear convergence of the algorithm. This talk is based on joint work with Quentin M{\â??e}rigot and Boris Thibert.

## Wuchen Li : Mean-Field Games for Scalable Computation and Diverse Applications

- Applied Math and Analysis ( 235 Views )Mean field games (MFGs) study strategic decision-making in large populations where individual players interact via specific mean-field quantities. They have recently gained enormous popularity as powerful research tools with vast applications. For example, the Nash equilibrium of MFGs forms a pair of PDEs, which connects and extends variational optimal transport problems. This talk will present recent progress in this direction, focusing on computational MFG and engineering applications in robotics path planning, pandemics control, and Bayesian/AI sampling algorithms. This is based on joint work with the MURI team led by Stanley Osher (UCLA).

## Linfeng Zhang : Neural network models and concurrent learning schemes for multi-scale molecular modelling

- Applied Math and Analysis ( 233 Views )We will discuss two issues in the context of applying deep learning methods to multi-scale molecular modelling: 1) how to construct symmetry-preserving neural network models for scalar and tensorial quantities; 2) how to efficiently explore the relevant configuration space and generate a minimal set of training data. We show that by properly addressing these two issues, one can systematically develop deep learning-based models for electronic properties and interatomic and coarse-grained potentials, which greatly boost the ability of ab-initio molecular dynamics; one can also develop enhanced sampling techniques that are capable of using tens or even hundreds of collective variables to drive phase transition and accelerate structure search

## Yian Ma : Bridging MCMC and Optimization

- Applied Math and Analysis ( 223 Views )In this talk, I will discuss three ingredients of optimization theory in the context of MCMC: Non-convexity, Acceleration, and Stochasticity.

I will focus on a class of non-convex objective functions arising from mixture models. For that class of objective functions, I will demonstrate that the computational complexity of a simple MCMC algorithm scales linearly with the model dimension, while optimization problems are NP-hard.

I will then study MCMC algorithms as optimization over the KL-divergence in the space of measures. By incorporating a momentum variable, I will discuss an algorithm which performs "accelerated gradient descent" over the KL-divergence. Using optimization-like ideas, a suitable Lyapunov function is constructed to prove that an accelerated convergence rate is obtained.

Finally, I will present a general recipe for constructing stochastic gradient MCMC algorithms that translates the task of finding a valid sampler into one of choosing two matrices. I will then describe how stochastic gradient MCMC algorithms can be applied to applications involving temporally dependent data, where the challenge arises from the need to break the dependencies when considering minibatches of observations.

## Mark Stern : Monotonicity and Betti Number Bounds

- Applied Math and Analysis ( 200 Views )In this talk I will discuss the application of techniques initially developed to study singularities of Yang Mill's fields and harmonic maps to obtain Betti number bounds, especially for negatively curved manifolds.

## Suncica Canic : Mathematical modeling for cardiovascular stenting

- Applied Math and Analysis ( 193 Views )The speaker will talk about several projects that are taking place in an interdisciplinary endeavor between the researchers in the Mathematics Department at the University of Houston, the Texas Heart Institute, Baylor College of Medicine, the Mathematics Department at the University of Zagreb, and the Mathematics Department of the University of Lyon 1. The projects are related to non-surgical treatment of aortic abdominal aneurysm and coronary artery disease using endovascular prostheses called stents and stent-grafts. Through a collaboration between mathematicians, cardiovascular specialists and engineers we have developed a novel mathematical model to study blood flow in compliant (viscoelastic) arteries treated with stents and stent-grafts. The mathematical tools used in the derivation of the effective, reduced equations utilize asymptotic analysis and homogenization methods for porous media flows. The existence of a unique solution to the resulting fluid-structure interaction model is obtained by using novel techniques to study systems of mixed, hyperbolic-parabolic type. A numerical method, based on the finite element approach, was developed, and numerical solutions were compared with the experimental measurements. Experimental measurements based on ultrasound and Doppler methods were performed at the Cardiovascular Research Laboratory located at the Texas Heart Institute. Excellent agreement between the experiment and the numerical solution was obtained. This year marks a giant step forward in the development of medical devices and in the development of the partnership between mathematics and medicine: the FDA (the United States Food and Drug Administration) is getting ready to, for the first time, require mathematical modeling and numerical simulations to be used in the development of peripheral vascular devices. The speaker acknowledges research support from the NSF, NIH, and Texas Higher Education Board, and donations from Medtronic Inc. and Kent Elastomer Inc.

## Greg Forest : An overview of the Virtual Lung Project at UNC, and whats math got to do with it?

- Applied Math and Analysis ( 192 Views )An effort at UNC is involved in understanding key mechanisms in the lung related to defense against pathogens. In diseases ranging from Cystic Fibrosis to asthma, these mechanisms are highly compromised, requiring therapeutic strategies that one would like to be able to quantify or even predict in some way. The Virtual Lung Project has focused on one principal component of lung defense: "the mucus escalator" as it is called in physiology texts. My goal in this lecture, with apologies to Tina Turner, is to give a longwinded answer to "what's math got to do with it?", and at the same time to convey how this collaboration is influencing the applied mathematics experience at UNC.

## Casey Rodriguez : The Radiative Uniqueness Conjecture for Bubbling Wave Maps

- Applied Math and Analysis ( 191 Views )One of the most fundamental questions in partial differential equations is that of regularity and the possible breakdown of solutions. We will discuss this question for solutions to a canonical example of a geometric wave equation; energy critical wave maps. Break-through works of Krieger-Schlag-Tataru, Rodnianski-Sterbenz and Rapha Ì?el-Rodnianski produced examples of wave maps that develop singularities in finite time. These solutions break down by concentrating energy at a point in space (via bubbling a harmonic map) but have a regular limit, away from the singular point, as time approaches the final time of existence. The regular limit is referred to as the radiation. This mechanism of breakdown occurs in many other PDE including energy critical wave equations, Schro Ì?dinger maps and Yang-Mills equations. A basic question is the following: â?¢ Can we give a precise description of all bubbling singularities for wave maps with the goal of finding the natural unique continuation of such solutions past the singularity? In this talk, we will discuss recent work (joint with J. Jendrej and A. Lawrie) which is the first to directly and explicitly connect the radiative component to the bubbling dynamics by constructing and classifying bubbling solutions with a simple form of prescribed radiation. Our results serve as an important first step in formulating and proving the following Radiative Uniqueness Conjecture for a large class of wave maps: every bubbling solution is uniquely characterized by itâ??s radiation, and thus, every bubbling solution can be uniquely continued past blow-up time while conserving energy.

## Greg Baker : Accelerating Liquid Layers

- Applied Math and Analysis ( 188 Views )A pressure difference across a liquid layer will accelerate it. For incompressible and inviscid motion, it is possible to describe the motion of the surfaces through boundary integral techniques. In particular, dipole distributions can be used together with an external flow that specifies the acceleration. The classical Rayleigh-Taylor instability and the creation of bubbles at an orifice are two important applications. A new method for the numerical approximation of the boundary integrals removes the difficulties associate with surfaces in close proximity.

## Wenjun Ying : Recent developments of the kernel-free boundary integral method

- Applied Math and Analysis ( 188 Views )The kernel-free boundary integral method is a Cartesian grid based method for solving elliptic partial differential equations (PDEs). It solves elliptic PDEs in the framework of boundary integral equations (BIEs). The method evaluates boundary and volume integrals by solving equivalent simple interface problems on Cartesian grids. It takes advantages of the well-conditioning properties of the BIE formulation, the convenience of grid generation with Cartesian grids and the availability of fast and efficient elliptic solvers for the simple interface problems. In this talk, I will present recent developments of the method for the reaction-diffusion equations in computational cardiology, the nonlinear Poisson-Boltzmann equation in biophysics, the Stokes equation in fluid dynamics as well as some free boundary and moving interface problems.

## Ioannis Kevrekidis : No Equations, No Variables, No Parameters, No Space, No Time -- Data, and the Crystal Ball Modeling of Complex/Multiscale Systems

- Applied Math and Analysis ( 184 Views )Obtaining predictive dynamical equations from data lies at the heart of science and engineering modeling, and is the linchpin of our technology. In mathematical modeling one typically progresses from observations of the world (and some serious thinking!) first to selection of variables, then to equations for a model, and finally to the analysis of the model to make predictions. Good mathematical models give good predictions (and inaccurate ones do not) --- but the computational tools for analyzing them are the same: algorithms that are typically operating on closed form equations.

While the skeleton of the process remains the same, today we witness the development of mathematical techniques that operate directly on observations --- data, and appear to circumvent the serious thinking that goes into selecting variables and parameters and deriving accurate equations. The process then may appear to the user a little like making predictions by "looking into a crystal ball". Yet the "serious thinking" is still there and uses the same --- and some new --- mathematics: it goes into building algorithms that "jump directly" from data to the analysis of the model (which is now not available in closed form) so as to make predictions. Our work here presents a couple of efforts that illustrate this "new" path from data to predictions. It really is the same old path, but it is traveled by new means.

## Franca Hoffmann : Gradient Flows: From PDE to Data Analysis.

- Applied Math and Analysis ( 184 Views )Certain diffusive PDEs can be viewed as infinite-dimensional gradient flows. This fact has led to the development of new tools in various areas of mathematics ranging from PDE theory to data science. In this talk, we focus on two different directions: model-driven approaches and data-driven approaches. In the first part of the talk we use gradient flows for analyzing non-linear and non-local aggregation-diffusion equations when the corresponding energy functionals are not necessarily convex. Moreover, the gradient flow structure enables us to make connections to well-known functional inequalities, revealing possible links between the optimizers of these inequalities and the equilibria of certain aggregation-diffusion PDEs. We present recent results on properties of these equilibria and long-time asymptotics of solutions in the setting where attractive and repulsive forces are in competition. In the second part, we use and develop gradient flow theory to design novel tools for data analysis. We draw a connection between gradient flows and Ensemble Kalman methods for parameter estimation. We introduce the Ensemble Kalman Sampler - a derivative-free methodology for model calibration and uncertainty quantification in expensive black-box models. The interacting particle dynamics underlying our algorithm can be approximated by a novel gradient flow structure in a modified Wasserstein metric which reflects particle correlations. The geometry of this modified Wasserstein metric is of independent theoretical interest.

## Hongkai Zhao : Approximate Separability of Greens Function for Helmholtz Equation in the High Frequency Limit

- Applied Math and Analysis ( 183 Views )Approximate separable representations of GreenÂ?s functions for differential operators is a basic and important question in the analysis of differential equations, the development of efficient numerical algorithms and imaging. Being able to approximate a GreenÂ?s function as a sum with few separable terms is equivalent to low rank properties of corresponding numerical solution operators. This will allow for matrix compression and fast solution techniques. Green's functions for coercive elliptic differential operators have been shown to be highly separable and the resulting low rank property for discretized system was explored to develop efficient numerical algorithms. However, the case of Helmholtz equation in the high frequency limit is more challenging both mathematically and numerically. We introduce new tools based on the study of relation between two GreenÂ?s functions with different source points and a tight dimension estimate for the best linear subspace approximating a set of almost orthogonal vectors to prove new lower bounds for the number of terms in the representation for the Green's function for Helmholtz operator in the high frequency limit. Upper bounds are also derived. We give explicit sharp estimates for cases that are common in practice and present numerical examples. This is a joint work with Bjorn Engquist.

## Xiaoqian Xu : Suppression of chemotactic explosion by mixing

- Applied Math and Analysis ( 178 Views )Chemotaxis plays a crucial role in a variety of processes in biology and ecology. One of the most studied PDE models of chemotaxis is given by Keller-Segel equation, which describes a population density of bacteria or mold which attract chemically to substance they secrete. However, solution of Keller-Segel equation can exhibit dramatic collapsing behavior. In other words, there exist initial data leading to finite time blow up. In this talk, we will discuss the possible effects resulting from interaction of chemotactic and fluid transport processes, namely we will consider the Keller-Segel equation with additional advection term modeling ambient fluid flow. We will prove that the presence of fluid can prevent the singularity formation. We will discuss two classes of flows that have the explosion arresting property. Both classes are known as very efficient mixers.

## P-E Jabin : Quantitative estimates of propagation of chaos for stochastic systems

- Applied Math and Analysis ( 175 Views )We derive quantitative estimates proving the propagation of chaos for large stochastic systems of interacting particles. We obtain explicit bounds on the relative entropy between the joint law of the particles and the tensorized law at the limit. Technically, the heart of the argument are new laws of large numbers at the exponential scale, proved through an explicit combinatorics approach. Our result only requires weak regularity on the interaction kernel in negative Sobolev spaces, thus including the Biot-Savart law and the point vortices dynamics for the 2d incompressible Navier-Stokes. For dissipative gradient flows, we may allow any singularity lower than the Poisson kernel. This talk corresponds to a joint work with Z. Wang and an upcoming work with D. Bresch and Z. Wang.

## Peter Mucha : Hierarchical Structure in Networks: From Football to Congres

- Applied Math and Analysis ( 174 Views )The study of various questions about networks have increased dramatically in recent years across a number of areas of application, including communications, sociology, and phylogenetic biology. Important questions about communities and groupings in networks have led to a number of competing techniques for identifying communities, structures and hierarchies. We discuss results about the networks of (1) NCAA Division I-A college football matchups and (2) committee assignments in the U.S. House of Representatives. In college football, the underlying structure of the network strongly influences the computer rankings that contribute to the Bowl Championship Series standings. In Congress, the changes of the hierarchical structure from one Congress to the next can be used to investigate major political events, such as the "Republican Revolution" of 1994 and the introduction of the Select Committee on Homeland Security following September 11th. While many structural elements in each case are seemingly robust, we include attention to variations across identification algorithms as we investigate the roles of such structures.

## Peter Smereka : The Gaussian Wave Packet Transform: Efficient Computation of the Semi-Classical Limit of the Schroedinger Equation

- Applied Math and Analysis ( 170 Views )An efficient method for simulating the propagation of a localized solution of the Schroedinger equation near the semiclassical limit is presented. The method is based on a time dependent transformation closely related to Gaussian wave packets and yields a Schroedinger type equation that is very ammenable to numerical solution in the semi-classical limit. The wavefunction can be reconstructed from the transformed wavefunction whereas expectation values can easily be evaluated directly from the transformed wavefunction. The number of grid points needed per degree of freedom is small enough that computations in dimensions of up to 4 or 5 are feasible without the use of any basis thinning procedures. This is joint work with Giovanni Russo.

## Mark Levi : Arnold diffusion in physical examples

- Applied Math and Analysis ( 170 Views )Arnold diffusion is the phenomenon of loss of stability of a completely integrable Hamiltonian system: an arbitrarily small perturbation can cause action to change along some orbit by a finite amount. Arnold produced the first example of diffusion and gave an outline of the proof. After a brief overview of related results I will describe the simplest example of Arnold diffusion which we found recently with Vadim Kaloshin. We consider geodesics on the 3-torus, or equivalently rays in a periodic optical medium in $ {\mathbb R} ^3 $ (or equivalently a point mass in a periodic potential in $ {\mathbb R} ^3 $.) Arnold diffusion has a transparent intuitive explanation and a simple proof. Resonances and the so-called ``whiskered tori" acquire a clear geometrical interpretation as well. I will conclude with a sketch of a different but related manifestation of Arnold diffusion as acceleration of a particle by a pulsating potential. This is joint work with Vadim Kaloshin.

## Ying Cui : Modern ``Non-Optimization for Data Science

- Applied Math and Analysis ( 169 Views )We have witnessed a lot of exciting development of data science in recent years. From the perspective of optimization, many modern data-science problems involve some basic ``nonâ??â??-properties that lack systematic treatment by the current approaches for the sake of the computation convenience. These non-properties include the coupling of the non-convexity, non-differentiability and non-determinism. In this talk, we present rigorous computational methods for solving two typical non-problems: the piecewise linear regression and the feed-forward deep neural network. The algorithmic framework is an integration of the first order non-convex majorization-minimization method and the second order non-smooth Newton methods. Numerical experiments demonstrate the effectiveness of our proposed approach. Contrary to existing methods for solving non-problems which provide at best very weak guarantees on the computed solutions obtained in practical implementation, our rigorous mathematical treatment aims to understand properties of these computed solutions with reference to both the empirical and the population risk minimizations. This is based on joint work with Jong-Shi Pang, Bodhisattva Sen and Ziyu He.

## Costas Pozrikidis : Biofluid-dynamics of blood cells

- Applied Math and Analysis ( 168 Views )Blood is a concentrated suspension of red cells, white cells, and platelets, each having a unique constitution and serving a different function. Red cells are highly deformable liquid capsules enclosed by a thin incompressible membrane whose resting shape is a biconcave disk. White cells are viscoelastic spherical particles enclosed by a cortical shell. In the unactivated state, platelets are oblate spheroids with an average aspect ratio approximately equal to 0.25. Like red cells, platelets lack a nucleus; like white cells, platelets exhibit a low degree of flow-induced deformation. In this talk, the biomechanics and biofluid-dynamics of these three types of cells will be discussed, recent progress in modeling and simulation methods will be reviewed, and open problems will be outlined.

## Aaron Hoffman : Existence and Orbital Stability for Counterpropagating Waves in the FPU model

- Applied Math and Analysis ( 165 Views )The Fermi-Pasta-Ulam (FPU) model of coupled anharmonic oscillators has long been of interest in nonlinear science. It is only recently (Friesecke and Wattis 1994, Frieseck and Pego 1999-2003, and Mizumachi (submitted)) that the existence and stability of solitary waves in FPU has been completely understood. In light of the fact that the Korteweg-deVries (KdV) equation may recovered as a long wave limit of FPU and that the theory of soliton interaction is both beautiful and completely understood in KdV, it is of interest to describe the interaction of two colliding solitary waves in the FPU model. We show that the FPU model contains an open set of solutions which remain close to the linear sum of two long wave low amplitude solitions as time goes to either positive or negative infinity.

## Lucy Zhang : Modeling and Simulations of Fluid and Deformable-Structure Interactions in Bio-Mechanical Systems

- Applied Math and Analysis ( 164 Views )Fluid-structure interactions exist in many aspects of our daily lives. Some biomedical engineering examples are blood flowing through a blood vessel and blood pumping in the heart. Fluid interacting with moving or deformable structures poses more numerical challenges for its complexity in dealing with transient and simultaneous interactions between the fluid and solid domains. To obtain stable, effective, and accurate solutions is not trivial. Traditional methods that are available in commercial software often generate numerical instabilities.

In this talk, a novel numerical solution technique, Immersed Finite Element Method (IFEM), is introduced for solving complex fluid-structure interaction problems in various engineering fields. The fluid and solid domains are fully coupled, thus yield accurate and stable solutions. The variables in the two domains are interpolated via a delta function that enables the use of non-uniform grids in the fluid domain, which allows the use of arbitrary geometry shapes and boundary conditions. This method extends the capabilities and flexibilities in solving various biomedical, traditional mechanical, and aerospace engineering problems with detailed and realistic mechanics analysis. Verification problems will be shown to validate the accuracy and effectiveness of this numerical approach. Several biomechanical problems will be presented: 1) blood flow in the left atrium and left atrial appendage which is the main source of blood in patients with atrial fibrillation. The function of the appendage is determined through fluid-structure interaction analysis, 2) examine blood cell and cell interactions under different flow shear rates. The formation of the cell aggregates can be predicted when given a physiologic shear rate.

## Leonid Berlyand : Flux norm approach to finite-dimensional homogenization approximation with non-separated scales and high contrast

- Applied Math and Analysis ( 164 Views )PDF Abstract

Classical homogenization theory deals with mathematical models of strongly
inhomogeneous media described by PDEs with rapidly oscillating coefficients
of the form A(x/\epsilon), \epsilon → 0. The goal is to approximate this problem by a
homogenized (simpler) PDE with slowly varying coefficients that do not depend
on the small parameter \epsilon. The original problem has two scales: fine
O(\epsilon) and coarse O(1), whereas the homogenized problem has only a coarse
scale.
The homogenization of PDEs with periodic or ergodic coefficients and
well-separated scales is now well understood. In a joint work with H. Owhadi
(Caltech) we consider the most general case of arbitrary L∞ coefficients,
which may contain infinitely many scales that are not necessarily well-separated.
Specifically, we study scalar and vectorial divergence-form elliptic PDEs with
such coefficients. We establish two finite-dimensional approximations to the
solutions of these problems, which we refer to as finite-dimensional homogenization
approximations. We introduce a flux norm and establish the error
estimate in this norm with an explicit and optimal error constant independent
of the contrast and regularity of the coefficients. A proper generalization of
the notion of cell problems is the key technical issue in our consideration.
The results described above are obtained as an application of the transfer
property as well as a new class of elliptic inequalities which we conjecture.
These inequalities play the same role in our approach as the div-curl lemma
in classical homogenization. These inequalities are closely related to the issue
of H^2 regularity of solutions of elliptic non-divergent PDEs with non smooth
coefficients.

## Vita Rutka : EJIIM for Stationary Stokes Flow (Boundary Value Problems)

- Applied Math and Analysis ( 164 Views )The Explicit Jump Immersed Interface Method (EJIIM) is a finite difference method for elliptic partial differential equations that, like all Immersed Interface Methods, works on a regular grid in spite of non-grid aligned discontinuities in equation parameters and solution. The specific idea is to introduce jumps in function and its derivatives explicitely as additional variables. We present a finite difference based EJIIM for the stationary Stokes flow in saddle point formulation. Challenges related to staggered grid, fast Stokes solver and non-simply connected domains will be discussed.