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 Hörmander'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.
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.
Zane Li : Interpreting a classical argument for Vinogradovs Mean Value Theorem into decoupling language
- Applied Math and Analysis ( 155 Views )There are two proofs of Vinogradov's Mean Value Theorem (VMVT), the harmonic analysis decoupling proof by Bourgain, Demeter, and Guth from 2015 and the number theoretic efficient congruencing proof by Wooley from 2017. While there has been some work illustrating the relation between these two methods, VMVT has been around since 1935. It is then natural to ask: What does previous partial progress on VMVT look like in harmonic analysis language? How similar or different does it look from current decoupling proofs? We talk about a classical argument due to Karatsuba that shows VMVT "asymptotically" and interpret this in decoupling language. This is joint work with Brian Cook, Kevin Hughes, Olivier Robert, Akshat Mudgal, and Po-Lam Yung.
Qin Li : Stability of stationary inverse transport equation in diffusion scaling
- Applied Math and Analysis ( 149 Views )We consider the inverse problem of reconstructing the optical parameters for stationary radiative transfer equation (RTE) from velocity-averaged measurement. The RTE often contains multiple scales char- acterized by the magnitude of a dimensionless parameterthe Knudsen number (Kn). In the diffusive scaling (Kn ≪ 1), the stationary RTE is well approximated by an elliptic equation in the forward setting. However, the inverse problem for the elliptic equation is acknowledged to be severely ill-posed as compared to the well- posedness of inverse transport equation, which raises the question of how uniqueness being lost as Kn → 0. We tackle this problem by examining the stability of inverse problem with varying Kn. We show that, the discrepancy in two measurements is amplified in the reconstructed parameters at the order of Knp (p = 1 or 2), and as a result lead to ill-posedness in the zero limit of Kn. Our results apply to both continuous and discrete settings. Some numerical tests are performed in the end to validate these theoretical findings.
Lenya Ryzhik : $l_1$-minimization via a generalized Lagrange multiplier algorithm
- Applied Math and Analysis ( 137 Views )We consider the basis pursuit problem: find the solution of an underdetermined system $Ax=y$ that minimizes the $l_1$-norm. We formulate a min-max principle (that, as we learned, actually goes back to 1970's) based on a Largange multiplier, and propose an iterative shrinkage-thresholding type algorithm that seems to work quite well. We show that the numerical algorithm converges to the exact solution of the basis pursuit problem. We also discuss its application to array imaging in wave propagation. The analysis is based on ODE techniques, regularization and energy methods. This is a joint work with M. Moscoso, A. Novikov and G. Papanicolaou.
Benjamin Dodson : Concentration compactness for the L^2 critical nonlinear Schrodinger equation
- Applied Math and Analysis ( 136 Views )The nonlinear Schrodinger equation
i u_{t} + D u = m |u|^{(4/d)}u | (1) |
As time permits the talk will also discuss the energy - critical problem in R^{d} \ W,
i u_{t} + D u = |u|^{4/(d - 2)} u | , u|_{Bdry(W)} = 0, (2) |
Gotz Pfander : Sampling of Operators
- Applied Math and Analysis ( 135 Views )Sampling and reconstruction of functions is a central tool in science. A key result is given by the classical sampling theorem for bandlimited functions. We describe the recently developed sampling theory for operators. We call operators bandlimited if their Kohn-Nirenberg symbols are band limited. The addresses engineers and mathematicians and should be accessible for those who have some education in linear algebra and calculus. The talk reviews sampling of functions and introduces some terminology from the theory of pseudodifferential operators. We will also discuss sampling theorems for stochastic operators.
Vladimir Sverak : On long-time behavior of 2d flows
- Applied Math and Analysis ( 134 Views )Our knowledge of the long-time behavior of 2d inviscid flows is quite limited. There are some appealing conjectures based on ideas in Statistical Mechanics, but they appear to be beyond reach of the current methods. We will discuss some partial results concerning the dynamics, as well as some results for variational problems to which the Statistical Mechanics methods lead.
Sayan Mukherjee : Geometric Perspectives on Supervised Dimension Reduction
- Applied Math and Analysis ( 117 Views )The statistical problem of supervised dimension reduction (SDR) is given observations of high-dimensional data as explanatory variables and univariate response variable, find a submanifold or subspace of the explanatory variables that predict the response. It is generally assumed that the data is concentrated on a low dimensional manifold in the high-dimensional space of explanatory variables.
The gradient of the manifold will be shown to be a central quantity in the problem of SDR. We will present a regularization algorithm for inferring the gradient geiven data. We will prove the rate of convergence of the gradient estimate to the gradient on the manifold of the true function to be of the order of the dimension of the manifold and not the much larger
The second part of the talk will rephrase the problem of SDR in a classical probabilistic (Bayesian) setting of mixture models of multivariate normals. An interesting result of this procedure is that the subspaces relevant to prediction are drawn from a posterior distribution on Grassmannian manifolds. For both methods efficacy on simulated and real data will be shown. ambient space.
Bill Allard : Some new results on total variation regularization for image processing
- Applied Math and Analysis ( 112 Views )Total variation regularization has been used for image denoising and other purposes for about twenty years now. For the last few years I have been studying the geometric and regularity properties of minimizers for the associated variational problems with an eye toward a better understanding of them. In this talk I will describe some new results in this area including results on minimizers for anisotropic total variation. The motivation for this work is that some computational schemes naturally use a polygonal approximation to the standard Euclidean metric to define total variation.
Valery A. Kholodnyi : Foreign Exchange Option Symmetry and a Coordinate-Free Description of a Foreign Exchange Option Market
- Applied Math and Analysis ( 28 Views )In spite of the fact that symmetries play one of the major roles in physics, the ir usage in finance is relatively new and, to the best of our knowledge, can be traced to 1995 when Kholodnyi introduced the beliefs-preferences gauge symmetry. In this talk we present another symmetry, foreign exchange option symmetry, int roduced by Kholodnyi and Price in 1996. Foreign exchange option symmetry associa tes financially equivalent options on opposite sides of the foreign exchange mar ket. In a two-currency market, the foreign exchange option symmetry is formalized in terms of the one-dimensional Kelvin transform. In a multiple-currency market the foreign exchange option symmetry is formalized in terms of differential geometr y on graphs, that is, in terms of vector lattice bundles on graphs and connectio ns on these bundles. Foreign exchange option symmetry requires no assumptions on the nature of a prob ability distribution for exchange rates. In fact, it does not even require the a ssumptions of the existence of such a distribution. Furthermore, the symmetry is applicable not only to a foreign exchange market but to any financial market as well. The practical applications of the foreign exchange option symmetry range from th e detection of a new type of true arbitrage to the detection of inconsistent mod els of foreign exchange option markets and the development of algorithms and sof tware to value and analyze portfolios of foreign exchange options.