**Wednesdays** at **4:15 PM** in **380C**.

- April 1, (Special time and room TBA)
**Liliana Borcea,**University of Michigan

A model reduction approach to inversion for a parabolic partial differential equation- April 1
**Konstantin Khanin,**University of Toronto

KPZ Universality and Random Hamilton-Jacobi Equation- April 8,
**Ming Gu,**University of California, Berkeley

Efficient Partition Computations for Kadison-Singer Problems- April 15
**Hongkai Zhao,**University of California, Irvine

Approximate separability of Green’s function and intrinsic complexity of differential operators- April 22
**Vlad Vicol,**Princeton University

Almost global existence for the Prandtl boundary layer equations- April 29
**Anil Damle,**Stanford University

Compressed representation of Kohn-Sham orbitals via selected columns of the density matrix- May 6
**Steve Lalley,**University of Chicago

TBA- May 13
**Special room and time TBA** **Boris Rozovsky,**Brown University

On distribution free Malliavin calculus- May 13
**Eric Dunham,**Stanford University

TBA- May 20
**Chao Yang,**Lawrence Berkeley Laboratory

TBA- May 27
**Special room and time TBA** **Gautam Iyer,**

TBA- May 27
**Diego Cordoba,**Instituto de Ciencias Matemáticas

TBA

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.

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.

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.

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.

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.

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.

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)

For questions, contact

**Lenya Ryzhik**, ryzhik@math.stanford.edu