Amic Frouvelle : Macroscopic limits of a system of self-propelled particles with phase transition
- Applied Math and Analysis ( 99 Views )The Vicsek model, describing alignment and self-organisation in large systems of self-propelled particles, such as fish schools or flocks of birds, has attracted a lot of attention with respect to its simplicity and its ability to reproduce complex phenomena. We consider here a time-continuous version of this model, in the spirit of the one proposed by P. Degond and S. Motsch, but where the rate of alignment is proportional to the mean speed of the neighboring particles. In the hydrodynamic limit, this model undergoes a phase transition phenomenon between a disordered and an ordered phase, when the local density crosses a threshold value. We present the two different macroscopic limits we can obtain under and over this threshold, namely a nonlinear diffusion equation for the density, and a first-order non-conservative hydrodynamic system of evolution equations for the local density and orientation. (joint work with Pierre Degond and Jian-Guo Liu).
Mark Hoefer : Eulerian Dispersive Shock Waves and Instabilities
- Applied Math and Analysis ( 116 Views )Recent experimental and theoretical research in Bose-Einstein condensation and nonlinear optics have demonstrated novel supersonic, fluid-like phenomena. Shock waves in these and other systems are modeled by a dispersive regularization of Euler's equations, implemented by use of the Whitham averaging technique. Normal and oblique dispersive shock waves (DSWs) connecting two constant states are constructed. Numerical computations of supersonic, dispersive flow over a corner in the special case of systems modeled by the Nonlinear Schrodinger equation (NLS) exhibit stable pattern formation (oblique DSWs) or instability (turbulent-like behavior) depending on the flow parameters. A combination of analytical and computational approaches are used to demonstrate that this change in behavior can be identified with the transition from convective to absolute instability of dark solitons. The linearized NLS behavior about the dark soliton DSW trailing edge is studied in detail to identify the separatrix between convective and absolute instabilities.
Saverio Eric Spagnolie : Hydrodynamics of Self-Propulsion Near a Boundary
- Applied Math and Analysis ( 99 Views )The swimming kinematics and trajectories of many microorganisms are altered by the presence of nearby boundaries, be they solid or deformable, and often in perplexing fashion. When an organism's swimming dynamics vary near such boundaries a question arises naturally: is the change in behavior fluid mechanical, biological, or perhaps due to other physical laws? We isolate the first possibility by exploring a far-field description of swimming organisms, providing a general framework for studying the fluid-mediated modifications to swimming trajectories. Using the simplified model we consider trapped/escape trajectories and equilibria for model organisms of varying shape and propulsive activity. This framework may help to explain surprising behaviors observed in the swimming of many microorganisms and synthetic microswimmers.
Ellen R. Peterson : A Droplet Spreading on a Thin Liquid Film: a Gradient Flow Formulation
- Applied Math and Analysis ( 100 Views )We consider a droplet spreading on a thin liquid film where both fluids are Newtonian, incompressible, and immiscible. Rather than following the typical asymptotic derivation for a thin film, we formulate the model through a gradient flow approach. The sign of the spreading parameter indicates the spreading behavior (complete or partial spreading) and is a relation between the three interfacial tensions: fluid/air, fluid/drop, and drop/air. We are particularly interested in the case where the spreading parameter is negative. In this case, the drop is expected to spread to a static lens and we find the corresponding equilibrium solution. Finally, we make a comparison between the theoretical model and experimental results.
Chi-Wang Shu : Discontinuous Galerkin Finite Element Method for Multiscale Problems
- Applied Math and Analysis ( 98 Views )In this talk, we first give a brief introduction to the discontinuous Galerkin method, which is a finite element method using completely discontinuous basis functions, for solving hyperbolic conservation laws and parabolic and elliptic equations. We will then survey the progress in developing discontinuous Galerkin methods for multiscale problems, in three different approaches, namely using the heterogeneous multiscale method (HMM) framework, using domain decompositions, and using multiscale basis in the discontinuous Galerkin method. Numerical results will be shown to demonstrate the effectiveness of the multiscale discontinuous Galerkin methods.
Rayan Saab : Quantization of compressed sensing measurements and frame expansions
- Applied Math and Analysis ( 101 Views )Compressed sensing, as a signal acquisition technique, has been shown to be highly effective for dimensionality reduction. On the other hand, reconstruction from compressed sensing measurements is highly non-linear and requires digital computers. Thus, quantizing (i.e., digitizing) compressed sensing measurements is an important, albeit under-addressed topic. In this talk, we show that by using $\Sigma\Delta$ quantization instead of the most commonly assumed approach (uniform quantization), a significant reduction in the reconstruction error is possible. In particular, we prove error decay rates of $\lambda^{-c r}$ where $\lambda$ is the ratio of the number of measurements to the sparsity of the underlying signal, and $r$ is the order of the $\Sigma\Delta$ scheme. In addition to the compressed sensing scenario we also consider the quantization of frame expansions, where one collects more measurements than the ambient dimension. We show state of the art results for certain frames (including random frames) and $\Sigma\Delta$ schemes. In particular, we prove error rates of $e^{-c\sqrt{\lambda}}$, where $\lambda$ is the oversampling ratio.
Jill Pipher : Geometric discrepancy theory: directional discrepancy in 2-D
- Applied Math and Analysis ( 90 Views )Discrepancy theory originated with some apparently simple questions about
sequences of numbers. The discrepancy of an infinite sequence is a
quantitative measure of how far it is from being uniformly distributed.
Precisely, an infinite sequence { a1,a2, ...} is said to be uniformly
distributed in [0, 1] if
lim_{n\to\infty} (1/n|{a1, a2,...an} intersect [s,t]|) = t-s.
If a sequence {ak} is uniformly distributed, then it is also the case
that for all (Riemann) integrable functions f on [0, 1],
lim_{n\to\infty} (1/n\sum_{k=1}^n f(ak))=\int_0^1 f(x)dx.
Thus, uniformly distributed sequences provide good numerical schemes
for approximating integrals. For example, if alpha is any irrational
number in [0, 1], then the fractional part {alphak}:=ak is uniformly
distributed. Classical Fourier analysis enters here, in the form of
Weyl's criterion.
The discrepancy of a sequence with respect to its first n entries is
D({ak},n) := sup_{s
See PDF.
Massimo Fornasier : The projection method for dynamical systems and kinetic equations modelling interacting agents in high-dimension
- Applied Math and Analysis ( 100 Views )In this talk we explore how concepts of high-dimensional data compression via random projections onto lower-dimensional spaces can be applied for tractable simulation of certain dynamical systems modeling complex interactions. In such systems, one has to deal with a large number of agents (typically millions) in spaces of parameters describing each agent of high dimension (thousands or more). Even with todays powerful computers, numerical simulations of such systems are prohibitively expensive. We propose an approach for the simulation of dynamical systems governed by functions of adjacency matrices in high dimension, by random projections via Johnson-Lindenstrauss embeddings, and recovery by compressed sensing techniques. We show how these concepts can be generalized to work for associated kinetic equations, by addressing the phenomenon of the delayed curse of dimension, known in information-based complexity for optimal measure quantization in high dimension. This is a joint work with Jan Haskovec and Jan Vybiral.
See PDF
Nathan Totz : A Rigorous Justification of the Modulation Approximation to the 2D Full Water Wave Problem
- Applied Math and Analysis ( 133 Views )In this joint work with Sijue Wu (U. Mich.), we consider solutions to the 2D
inviscid infinite depth water wave problem neglecting surface tension which
are to leading order wave packets of the form $\alpha + \epsilon B(\epsilon
\alpha, \epsilon t, \epsilon^2 t)e^{i(k\alpha + \omega t)}$ for $k > 0$.
Multiscale calculations formally suggest that such solutions have
modulations $B$ that evolve on slow time scales according to a focusing
cubic NLS equation. Justifying this rigorously is a real problem, since
standard existence results do not yield solutions which exist for long
enough to see the NLS dynamics. Nonetheless, given initial data within
$O(\epsilon^{3/2})$ of such wave packets in $L^2$ Sobolev space, we show
that there exists a unique solution to the water wave problem which remains
within $O(\epsilon^{3/2})$ to the approximate solution for times of order
$O(\epsilon^{-2})$. This is done by using a version of the evolution
equations for the water wave problem developed by Sijue Wu with no quadratic nonlinearity.
See arXiv:1101.0545
Shi Jin : An Eulerian surface hopping method for the Schr\{o}dinger equation with conical crossings
- Applied Math and Analysis ( 104 Views )In a nucleonic propagation through conical crossings of electronic energy levels, the codimension two conical crossings are the simplest energy level crossings, which affect the Born-Oppenheimer approximation in the zeroth order term. The purpose of this paper is to develop the surface hopping method for the Schr\"{o}dinger equation with conical crossings in the Eulerian formulation. The approach is based on the semiclassical approximation governed by the Liouville equations, which are valid away from the conical crossing manifold. At the crossing manifold, electrons hop to another energy level with the probability determined by the Landau-Zener formula. This hopping mechanics is formulated as an interface condition, which is then built into the numerical flux for solving the underlying Liouville equation for each energy level. While a Lagrangian particle method requires the increase in time of the particle numbers, or a large number of statistical samples in a Monte Carlo setting, the advantage of an Eulerian method is that it relies on fixed number of partial differential equations with a uniform in time computational accuracy. We prove the positivity and $l^{1}$-stability and illustrate by several numerical examples the validity and accuracy of the proposed method.
Eitan Tadmor : A new model for self-organized dynamics
- Applied Math and Analysis ( 116 Views )We introduce a new particle-based model for self-organized dynamics which, we argue, addresses several drawbacks of the celebrated Cucker-Smale (C-S) model. The new model does not involve any explicit dependence on the number of agents: only their self-driven geometry in phase space enters the model. It lacks, however, the symmetry property, which is the key for the various recent studies of C-S flocking behavior. To this end, we introduce here a new unifying framework to analyze the phenomenon of flocking for a general class of dynamical systems in the presence of non-symmetric influence matrices. In particular, we prove the emerging behavior of flocking in the proposed model, when the pairwise long-range interactions between its agents decays sufficiently slow.
The methodology presented in this paper is based on the notion of active sets, which carries over from the particle to the kinetic and hydrodynamic descriptions. In particular, we discuss the hydrodynamic description of our new model for self-organized dynamics, and we prove its unconditional flocking for sufficiently slowly decaying influence functions.
Lek-Heng Lim : Multilinear Algebra and Its Applications
- Applied Math and Analysis ( 109 Views )In mathematics, the study of multilinear algebra is largely limited to properties of a whole space of tensors --- tensor products of k vector spaces, modules, vector bundles, Hilbert spaces, operator algebras, etc. There is also a tendency to take an abstract coordinate-free approach. In most applications, instead of a whole space of tensors, we are often given just a single tensor from that space; and it usually takes the form of a hypermatrix, i.e.\ a k-dimensional array of numerical values that represents the tensor with respect to some coordinates/bases determined by the units and nature of measurements. How could one analyze this one single tensor then? If the order of the tensor k = 2, then the hypermatrix is just a matrix and we have access to a rich collection of tools: rank, determinant, norms, singular values, eigenvalues, condition number, etc. This talk is about the case when k > 2. We will see that one may often define higher-order analogues of common matrix notions rather naturally: tensor ranks, hyperdeterminants, tensor norms (Hilbert-Schmidt, spectral, Schatten, Ky Fan, etc), tensor eigenvalues and singular values, etc. We will discuss the utility as well as difficulties of various tensorial analogues of matrix problems. In particular we shall look at how tensors arise in a variety of applications including: computational complexity, control engineering, mathematical biology, neuroimaging, quantum computing, signal processing, spectroscopy, and statistics.
Gitta Kutyniok : Frames and Sparsity
- Applied Math and Analysis ( 121 Views )Frames are nowadays a standard methodology in applied mathematics, computer science, and engineering when redundant, yet stable expansions are required. Sparsity is a new paradigm in signal processing, which allows for significantly reduced measurements yet still highly accurate reconstruction. In this talk, we will focus on the main two links between these exciting, rapidly growing areas. Firstly, the redundancy of a frame promotes sparse expansions of signals, thereby strongly supporting sparse recovery methods such as Compressed Sensing. After providing an overview of sparsity methodologies, we will discuss new results on sparse recovery for structured signals, in particular, which are a composition of `distinct' components. Secondly, in very high dimensions, frame decompositions might be intractable in applications with limited computing budget. This problem can be addressed by requiring sparsity of the frame itself, and we will show how to derive optimally sparse frames. Finally, we will discuss how some of the presented results generalize to the novel notion of fusion frames, which was introduced a few years ago for modeling distributed processing applications.
Mark Levi : Arnold diffusion in a chain of coupled pendula
- Applied Math and Analysis ( 127 Views )A chain of pendula connected by nearest neighbor coupling is a near-- integrable system if the coupling is weak. As a consequence of KAM, for most initial data the energy of each pendulum stays near its initial value for all time. We show that these KAM motions coexist with ``diffusing" motions for which the energy can leak from any pendulum to any other pendulum, and it can do so with a prescribed itinerary. This is joint work with Vadim Kaloshin.
Gadi Fibich : Aggregate Diffusion Dynamics in Agent-Based Models with a Spatial Structure
- Applied Math and Analysis ( 115 Views )The diffusion or adoption of new products (such as fax machines, skype, facebook, Ipad, etc.) is one of the key problems in Marketing research. In recent years, this problem was often studied numerically, using agent-based models (ABMs). In this talk I will focus on analysis of the aggregate diffusion dynamics in ABMs with a spatial structure. In one-dimensional ABMs, the aggregate diffusion dynamics can be explicitly calculated, without using the mean-field approximation. In multidimensional ABMs, we introduce a clusters-dynamics approach, and use it to derive an analytic approximation of the aggregate diffusion dynamics. The clusters-dynamics approximation shows that the aggregate diffusion dynamics does not depend on the average distance between individuals, but rather on the expansion rate of clusters of adopters. Therefore, the grid dimension has a large effect on the aggregate adoption dynamics, but a small-world structure and heterogeneity among individuals have only a minor effect. Our results suggest that the one-dimensional model and the fully-connected Bass model provide a lower bound and an upper bound, respectively, for the aggregate diffusion dynamics in agent-based models with "any" spatial structure. This is joint work with Ro'i Gibori and Eitan Muller
Dongho Chae : On the presure conditions for the regularity and the triviality in the 3D Euler equations
- Applied Math and Analysis ( 116 Views )In this talk we present some observations regarding the pressure conditions leading to the vanishing of velocity in the Euler and the Navier-Stokes equations. In the case of axisymmetric 3D Euler equations with special initial data we find that the unformicity condition for the derivatives of the pressure is not consistent with the global regularity.
Dave Schaeffer : Finite-length effects in Taylor-Couette flow
- Applied Math and Analysis ( 131 Views )Taylor-Couette flow provides one of the pre-eminent examples of bifurcation in fluid dynamics. This phrase refers to the flow between concentric rotating cylinders. If the rotation speed is sufficiently rapid, the primary rotary flow around the axis becomes unstable, leading to a steady secondary flow in approximately periodic cells. Assuming infinite cylinders and exact periodicity in his theory, Taylor obtained remarkable agreement with experiment for the onset of instability, agreement that remains unsurpassed in fluid mechanics to this day. This talk is concerned with incorporating the effect of finite-length cylinders into the theory, an issue whose importance was emphasized by Benjamin. Numerous experiments and simulations of the Navier Stokes equations all support to the following, seemingly paradoxical, observations: No matter how long the apparatus, finite-length effects greatly perturb many of the bifurcating flows but, provided the cylinders are long, hardly perturb others. We understand this paradox as a result of symmetry breaking. The relevant symmetry, which is only approximate, is a symmetry between two normal-mode flows with large, and nearly equal, numbers of cells.
Ravi Srinivasan : Kinetic theory for shock clustering and Burgers turbulence
- Applied Math and Analysis ( 94 Views )A remarkable model of stochastic coalescence arises from considering shock statistics in scalar conservation laws with random initial data. While originally rooted in the study of Burgers turbulence, the model has deep connections to statistics, kinetic theory, random matrices, and completely integrable systems. The evolution takes the form of a Lax pair which, in addition to yielding interesting conserved quantities, admits some rather intriguing exact solutions. We will describe several distinct derivations for the evolution equation and, time-permitting, discuss properties of the corresponding kinetic system. This talk consists of joint work with Govind Menon (Brown).
Manoj Gopalkrishnan : On catalysis in biochemical networks
- Applied Math and Analysis ( 121 Views )It is a common expectation in chemistry that a chemical transformation which takes place in the presence of a catalyst must also take place in its absence, though perhaps at a much slower rate. A reaction network will be called ``saturated'' if it satisfies such an expectation. I propose a mathematical definition for saturated networks and show that the associated dynamical systems have no boundary equilibria in positive stoichiometric classes, and are therefore permanent. This result is independent of the specific rates, and generalizes previous results for complete networks by Gnacadja, atomic event-systems by Adleman et al. and constructive networks by Shinar et al. I require no assumption of complex balance or deficiency restrictions. The question of permanence for weakly-reversible reaction networks remains a long-standing open problem.
Mark Iwen : Signal Recovery via Discrete Measurement Matrices
- Applied Math and Analysis ( 132 Views )We will discuss a class of binary measurement matrices having a simple discrete incoherence property. These matrices can be shown to have both useful analytic (i.e., restricted isometry and l1-approximation properties) and combinatorial (i.e., group testing and expander graph related) structure which allows them to be utilized for sparse signal approximation in the spirit of compressive sensing. In addition, their structure allows the actual signal recovery process to be carried out by highly efficient algorithms once measurements have been taken. One application of these matrices and their related recovery algorithms is their application to the development of sublinear-time Fourier methods capable of accurately approximating periodic functions using far fewer samples and run time than required by standard Fourier transform techniques.
Ilya Timofeyev : Sub-sampling in Parametric Estimation of Effective Stochastic Models from Discrete Data
- Applied Math and Analysis ( 136 Views )It is often desirable to derive an effective stochastic model for the physical process from observational and/or numerical data. Various techniques exist for performing estimation of drift and diffusion in stochastic differential equations from discrete datasets. In this talk we discuss the question of sub-sampling of the data when it is desirable to approximate statistical features of a smooth trajectory by a stochastic differential equation. In this case estimation of stochastic differential equations would yield incorrect results if the dataset is too dense in time. Therefore, the dataset has to sub-sampled (i.e. rarefied) to ensure estimators' consistency. Favorable sub-sampling regime is identified from the asymptotic consistency of the estimators. Nevertheless, we show that estimators are biased for any finite sub-sampling time-step and construct new bias-corrected estimators.
Yoshiaki Teramoto : Benard-Marangoni problem of heat convection with free surface
- Applied Math and Analysis ( 95 Views )When a fluid layer is heated from below with temperature larger than a certain critical value, the convective motion appears in the fluid. The convection caused by the thermocapillary effect is called Benard-Marangoni heat convection. The thermocapillary effect is the dependence of the surface tension on the temperature. Near a hot spot on a free surface of fluid a thermocapillary tangential stress generates a fluid motion. In this talk the mathematical model system for this convection is explained. The Oberbeck-Boussinesq approximation is used for the system and the upper boundary is a free surface with surface tension which depends on the temperature. After formulating the linearized problem around the conductive state, we derive the resolvent estimates which guarantee the sectorial property. Stationary and Hopf bifurcations (periodic solutions) are proved to exist depending on the parameters (Raylegh and Marangoni numbers).
Yossi Farjoun : Solving Conservation Law and Balance Equations by Particle Management
- Applied Math and Analysis ( 101 Views )Conservation equations are at the heart of many interesting and important problems. Examples come from physics, chemistry, biology, traffic and many more. Analytically, hyperbolic equations have a beautiful structure due to the existence of characteristics. These provide the possibility of transforming a conservation PDE into a system of ODE and thus greatly reducing the computational effort required to solve such problems. However, even in one dimension, one encounters problems after a short time.
The most obvious difficulty that needs to be dealt with has to do with the creation of shocks, or in other words, the crossing of characteristics. With a particle based method one would like to avoid a situation when one particle overtakes a neighboring one. However, since shocks are inherent to many hyperbolic equations and relevant to the problems that one would like to solve, it would be good not to ``smooth away'' the shock but rather find a good representation of it and a good solution for the offending particles.
In this talk I will present a new particle based method for solving (one dimensional, scalar) conservation law equations. The guiding principle of the method is the conservative property of the underlying equation. The basic method is conservative, entropy decreasing, variation diminishing and exact away from shocks. A recent extension allows solving equations with a source term, and also provides ``exact'' solutions to the PDE. The method compares favorably to other benchmark solvers, for example CLAWPACK, and requires less computation power to reach the same resolution. A few examples will be shown to illustrate the method, with its various extensions. Due to the current limitation to 1D scalar, the main application we are looking at is traffic flow on a large network. Though we still hope to manage to extend the method to either systems or higher dimensions (each of these extensions has its own set of difficulties), I would be happy to discuss further possible applications or suggestions for extensions.
Wenjun Ying : A Fast Accurate Boundary Integral Method for the Laplace Equation
- Applied Math and Analysis ( 163 Views )Boundary value and interface problems for the Laplace equation are often solved by boundary integral methods due to the reduction of dimensionality and its flexibility in domain geometry. However, there are two well-known computational issues with the boundary integral method: (a) evaluation of boundary integrals at points close to domain boundaries usually has low order accuracy; (b) the method typically yields dense coefficient matrices in the resulting discrete systems, which makes the matrix vector multiplication very expensive when the size of the system is very large. In this talk, I will describe a fast accurate boundary integral method for the Laplace boundary value and interface problems. The algorithm uses the high order accurate method proposed by (Beale and Lai 2001) for evaluation of the boundary integrals and applies the fast multipole method for the dense matrix vector multiplication. Numerical results demonstrating the efficiency and accuracy of the method will be presented.
Laura Miller : Scaling effects in heart development: Changes in bulk flow patterns and the resulting forces
- Applied Math and Analysis ( 92 Views )When the heart tube first forms, the Reynolds number describing intracardial flow is only about 0.02. During development, the Reynolds number increases to roughly 1000. The heart continues to beat and drive the fluid during its entire development, despite significant changes in fluid dynamics. Early in development, the atrium and ventricle bulge out from the heart tube, and valves begin to form through the expansion of the endocardial cushions. As a result of changes in geometry, conduction velocities, and material properties of the heart wall, the fluid dynamics and resulting spatial patterns of shear stress and transmural pressure change dramatically. Recent work suggests that these transitions are significant because fluid forces acting on the cardiac walls, as well as the activity of myocardial cells which drive the flow, are necessary for correct chamber and valve morphogenesis.
In this presentation, computational fluid dynamics was used to explore how spatial distributions of the normal forces and shear stresses acting on the heart wall change as the endocardial cushions grow, as the Reynolds number increases, and as the cardiac wall increases in stiffness. The immersed boundary method was used to simulate the fluid-structure interaction between the cardiac wall and the blood in a simplified model of a two-dimensional heart. Numerical results are validated against simplified physical models. We find that the presence of chamber vortices is highly dependent upon cardiac cushion height and Reynolds number. Increasing cushion height also drastically increases the shear stress acting on the cushions and the normal forces acting on the chamber walls.
Leonid Berlyand : Flux norm approach to finite-dimensional homogenization approximation with non-separated scales and high contrast
- Applied Math and Analysis ( 149 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.
Junping Wang : Mathematics and Computation of Sediment Transport for Open Channels
- Applied Math and Analysis ( 106 Views )The purpose of this presentation is to communicate some mathematical and computational issues in sediment transport for open channels. The main topics are: (1) mathematical simulation for surface and subsurface fluid flow, (2) mathematical modeling of sediment transport in open channels as a 2D problem, and (3) numerical methods for fluid flow and sediment transport.
Badal Joshi : A coupled Poisson process model for sleep-wake cycling
- Applied Math and Analysis ( 114 Views )Sleep-wake cycling is an example of switching between discrete states in mammalian brain. Based on the experimental data on the activity of populations of neurons, we develop a mathematical model. The model incorporates several different time scales: firing of action potentials (milliseconds), sleep and wake bout times (seconds), developmental time (days). Bifurcation diagrams in a deterministic dynamical system gives the occupancy time distributions in the corresponding stochastic system. The model correctly predicts that forebrain regions help to stabilize wake state and thus modifies the wake bout distribution.
Cécile Piret : Overcoming the Gibbs Phenomenon Using a Modified Radial Basis Functions Method
- Applied Math and Analysis ( 119 Views )The Radial Basis Functions (RBF) method is not immune from the disastrous effects of the Gibbs phenomenon. When interpolating or solving PDEs whose solutions are piecewise smooth functions, the RBF method loses its notorious spectral accuracy. In this talk, a new method will be presented, based on the RBF method, which incorporates singularities using Heaviside functions and which keeps track of their location using the level set method. The resulting sharp interface method will be shown to recover the lost spectral accuracy and thus overcome the Gibbs phenomenon altogether.
Andrew Christlieb : A high order adaptive mesh refinement algorithm for hyperbolic conservation laws based on weighted essentially non-oscillatory methods
- Applied Math and Analysis ( 137 Views )In this work, we combine the adaptive mesh refinement (AMR) framework with high order finite difference weighted essentially non-oscillatory (WENO) method in space and TVD Runge-Kutta (RK) method in time (WENO-RK) for hyperbolic conservation laws. Our goal is to realize mesh adaptivity in the AMR framework, while maintaining very high (higher than second) order accuracy of the WENO-RK method in the finite difference setting. To maintain high order accuracy, we use high order prolongation in both space (WENO interpolation) and time (Hermite interpolation) from the coarse to find grid, and at ghost points. The resulting scheme is high order accuracy, robust and efficient, due to the mesh adaptivity and has high order accuracy in both space and time. We have experimented the third and fifth order AMR-finite difference WENO-RK schemes. The accuracy of the scheme is demonstrated by applying the method to several smooth test problems, and the quality and efficiency are demonstrated by applying the method to the shallow water and Euler equations with different challenging initial conditions. From our numerical experiment, we conclude a significant improvement of the fifth order AMR - WENO scheme over the third order one, not only in accuracy for smooth problems, but also in its ability in resolving complicated solution structures, which we think is due to the very low numerical diffusion of high order schemes. This work is in collaboration with Dr. Chaopeng Shen and Professor Jing-Mei Qiu.