# Stanford Applied Math Seminar

Wednesdays at 4:15 PM in 380C.

## Spring 2015 Schedule

April 1, (Special time and room TBA)
Liliana Borcea, University of Michigan
A model reduction approach to inversion for a parabolic partial differential equation

I will describe a novel numerical inversion method for a parabolic partial differential equation arising in applications of control source electromagnetic exploration. The unknown is the electrical resistivity in the earth and the data are time resolved measurements of the magnetic field. The method described uses model reduction ideas and has been implemented in one and two dimensions.

April 1
Konstantin Khanin, University of Toronto
KPZ Universality and Random Hamilton-Jacobi Equation

Universal properties related to the KPZ equation is an extremely active research field right now. It is connected to many different areas in mathematics and mathematical physics: probability theory and statistical mechanics, PDEs, random matrices, integrable systems, and many others. While it was a big recent progress, mainly in deriving exact formulas for correlation functions, the problem of universality is still largely open. We shall discuss some recent results in this direction, as well as possible approaches to establishing of the full universality.

April 8,
Ming Gu, University of California, Berkeley
Efficient Partition Computations for Kadison-Singer Problems

In their seminal work in 2013, Marcus, Spielman and Srivastava showed the existence of the generalized weaver partition (GWP). Their work immediately implies that the Kadison-Singer conjecture is true, but leaves the question of computing the GWP unanswered. In this talk, we develop efficient algorithms for computing the GWP and for solving a number of other related Kadison-Singer problems such as the Bourgain-Tzafriri problem and the Feichtinger problem. Our numerical experimental results suggest that the partitions computed by our algorithms are close to optimal.

April 15
Hongkai Zhao, University of California, Irvine
Approximate separability of Green’s function and intrinsic complexity of differential operators

Approximate separable representation of the Green’s functions for differential operators is a basic and important question in analysis of differential equations and development of efficient numerical algorithms. It can reveal intrinsic complexity, e.g., Kolmogorov n-width or degrees of freedom of the corresponding differential equation. Computationally, being able to approximate a Green’s function as a sum with few separable terms is equivalent to the existence of low rank approximation of the discretized system which can be explored for matrix compression and fast solution techniques such as in fast multiple method and direct matrix inverse solver. In this talk, we will mainly focus on Helmholtz equation in the high frequency limit for which we developed a new approach to study the approximate separability of Green’s function based on an geometric characterization of the relation between two Green's functions and a tight dimension estimate for the best linear subspace approximating a set of almost orthogonal vectors. We derive both lower bounds and upper bounds and show their sharpness and implications for computation setups that are commonly used in practice. We will also make comparisons with other types of differential operators such as coercive elliptic differential operator with rough coefficients in divergence form and hyperbolic differential operator. This is a joint work with Bjorn Engquist.

April 22
Almost global existence for the Prandtl boundary layer equations

We consider the Prandtl boundary layer equations on the half plane, with initial datum that lies in a weighted $H^1$ space with respect to the normal variable, and is real-analytic with respect to the tangential variable. The boundary trace of the horizontal Euler flow is taken to be a constant. We prove that if the Prandtl datum lies within $\epsilon$ of a stable profile, then the unique solution of the Cauchy problem can be extended at least up to time $T_\epsilon \geq \exp( \epsilon^{-1} / \log(\epsilon^{-1}))$. This is joint work with Mihaela Ignatova.

April 29
Anil Damle, Stanford University
Compressed representation of Kohn-Sham orbitals via selected columns of the density matrix

Kohn−Sham density functional theory is a widely used electronic structure theory for molecules and systems in condensed phase. Given a set of Kohn-Sham orbitals from an insulating system, it is often desirable to build a set of localized basis functions for the associated subspace. We present a simple, robust, efficient, and parallelizable method to construct a set of (optionally orthogonal) localized basis functions. The basis is constructed directly from a set of selected columns of the density matrix (SCDM) without the use of an optimization procedure. We also demonstrate a procedure for combining the orthogonalized SCDM with Hockney’s algorithm to efficiently perform Hartree−Fock exchange energy calculations with near-linear scaling. Finally, we briefly discuss an extension of the SCDM procedure to solids with k-point sampling.

May 6
Steve Lalley, University of Chicago
Hybrid Percolation, Spatial Epidemics, and Dawson-Watanabe Processes

Hybrid percolation is a random graph on the vertex set V= Z^d x [N] in which vertices (x,k) and (y,l) are connected with probability p_N if x and y are neighboring vertices of the lattice Z^d. The percolation graph is the skeleton of a spatial epidemic that evolves as follows: (a) infected individuals, represented by vertices of the graph, remain infected for one unit of time, after which they recover and acquire permanent immunity from further infection; and (b) infected individuals infect susceptible individuals (vertices at neighboring sites of Z^d) with probability p_N. We shall describe results concerning the scaling limits of both the percolation graph and the associated epidemic process as N becomes large in the critical regime p_N = 1/(2dN) .

May 13 Special room and time: 384I at 12:30pm
Boris Rozovsky, Brown University
On distribution free Malliavin calculus

Traditional stochastic analysis, including Ito and Malliavin versions of stochastic calculus, depend on the type of random parameters, e.g. random forcing, random initial data, etc. For example, stochastic analysis for Gaussian and Poisson calculus appear to be quite di§erent. In this talk I will discuss a unifed approach to stochastic calculus independent of the types of randomness involved. The is starting point of the current presentation is a sequence of uncorrelated random variables. The distribution functions of these variables are assumed to be given but no assumptions on the types or the structure of these distributions are made. The above setting constitutes the so called "distribution free" paradigm. Under these assumptions, a version of Skorokhod-Malliavin calculus and its applications to stochastic PDEs will be discussed. Joint work with R. Mikulevicius (USC)

May 13
Eric Dunham, Stanford University
Efficient simulation of wave propagation through networks of fluid-filled cracks and conduits

In both volcanoes and oil/gas industry hydraulic fracturing operations, fluids are present within networks of pipe-like conduits and cracks in an elastic solid. Waves propagating through this system provide one means of inferring the geometry and fluid properties. An efficient approach to the forward modeling problem is developed using high-order finite differences that provide an energy estimate for the semi-discrete problem. Computationally expensive simulations involving simultaneous solution of the elastic wave equation and an approximate version of the Navier-Stokes equation are first solved for the crack response, in isolation of other components of the system. The response of the overall, coupled system of cracks and conduits can then be constructed in the frequency domain using numerically or analytically derived transfer functions.

May 20
Chao Yang, Lawrence Berkeley Laboratory
Absorption Spectrum Estimation via Linear Response Time-dependent Density Functional Theory

In the time-dependent density functional theory framework, the absorption spectrum of a molecular system can be estimated from the trace of the dynamic polarizability associated with the linear response of the charge density to an external potential perturbation of the ground state Hamiltonian. Although an accurate description of the absorption spectrum requires the diagonalization of the so-called Casida Hamiltonian, there are more efficient ways to obtain a good approximation to the overall profile of the absorption spectrum without computing eigenvalues and eigenvectors. We will describe these methods that only require multipling the Casida Hamiltonian with a number of vectors. When highly accurate oscillator strength is required for a few selected excitation energy levels, we can use a special iterative method to obtain the eigenvalues and eigenvectors associated with these energy levels efficiently.

May 27 Special room and time: 384I at 12:30pm
Gautam Iyer,
TBA

May 27
Diego Cordoba, Instituto de Ciencias Matemáticas
TBA

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