Wednesdays at 4:00 PM in 384H.
Nonlocality is ubiquitous in nature. While partial differential equations (PDE) have been used as effective models of many physical processes, nonlocal balanced laws are also attracting more and more attentions as alternatives to model singularities and anomalous behavior. In this talk, we discuss the mathematical structure of some nonlocal models by exploiting a recently developed nonlocal vector calculus that offers an analogy of classical vector calculus for local PDEs. The latter, when physically valid, may be derived as local limits of their nonlocal counterparts. We also present asymptotically compatible discretizations of nonlocal models that can provide convergent approximations to the local limit. Such discretizations can be more robust for multiscale problems with varying length scales.
High frequency wave propagation has been a longstanding challenge in scientific computing. For the time-harmonic problems, integral formulations and/or efficient numerical discretization often lead to dense linear systems. Such linear systems are extremely difficult to solve for standard iterative methods since they are highly indefinite. In this talk, we consider several such examples. For each one, we construct a sparsifying preconditioner that reduces the dense linear system to a sparse one and solves the problem within a small number of iterations.
The need for artificial computational boundaries in the solution of exterior wave problems, called “absorbing boundaries” among other names, arises quite often in various fields of application. In solid-earth geophysics they are needed for practically every simulation. The problem of devising good absorbing boundaries may seem very easy: we simply want to do nothing at the boundary, and let waves leave the computational domain. However, this is actually a difficult problem that has occupied researchers for many years.
During the last two decades, two classes of methods have emerged as especially powerful: the Perfectly Matched Layer (PML) method and the method of using high-order Absorbing Boundary Conditions (ABCs), which are local and involve no high derivatives. The use of ABCs has been very popular since the early 1970's, but the term “high-order ABCs” relates to the ability to implement local ABCs of an arbitrarily high order.
Recently, a new technique, called the Double Absorbing Boundary (DAB) method, has been devised for solving wave problems in unbounded domains. It has common features to high-order ABC and to PML. However, it is different from both and enjoys relative advantages with respect to both. The DAB method is first introduced in general terms, and then it is applied to the scalar wave equation in a wave guide, and to elastodynamics problems in homogeneous and heterogeneous media. Stability issues, which are especially delicate in the case of elastodynamics, are discussed in some detail.
Ongoing work and remaining challenges will be described, and in particular some problems concerning anisotropic elasticity.
This is work with Pierre Collet and Maher Younan. I will start by explaining some topological dynamics (obtained by T1-flips (Pachner moves) for a simple energy function on the set of all triangulations with a fixed number of nodes on the 2-sphere. On the 2-sphere, "everything" is known about the combinatorial aspects since Tutte's work (1962). In 3 dimensions, the problem consists mostly of unsolved, but challenging problems. For example, it is not known whether the number of triangulations of the 3-sphere grows exponentially or faster in the number of tetrahedra. I will review the current status of both the combinatorics and (if time permits) the dynamics for 3d.
Since its introduction by P.L. Lions in his lectures and seminars at the College de France, see also the very helpful notes of Cardialaguet on Lions' lectures, the Master equation has attracted a lot of interest, and various points of view have been expressed, see Carmona-Delarue, Bensoussan-Frehse-Yam , Buckdahn-Li-Peng-Rainer . There are several ways to introduce this type of equation. It involves an argument which is a probability measure, and P.L. Lions has proposed the idea of working with the Hilbert space of random variables which are square integrable. So writing the equation is an issue. Another issue is its origin. We discuss in this paper these various aspects, and for the modeling rely heavily on a seminar at College de France delivered by P.L. Lions on November 14, 2014.
In the early days of scientific computing, Goldstine and Von Neumman suggested that it would be fruitful to study the "typical" performance of Gaussian elimination on random input. This approach lay dormant for decades until Alan Edelman’s 1989 thesis on the condition numbers of random matrices. Since then numerical linear algebraists have made basic contributions to random matrix theory and the study of condition numbers of random matrices has proven to be a rich subject. We approach the symmetric eigenvalue problem from a similar viewpoint. The underlying mathematical issue is to analyze the number of iterations required for an eigenvalue algorithm to converge. Our study focuses on the QR algorithm, the Toda algorithm and a version of the matrix-sign algorithm. All three algorithms have intimate ties with completely integrable Hamiltonian systems. Our results stress an empirical discovery of "universality in computation". We also show that this problems admits an elegant mathematical formulation and suggests interesting new questions in integrable systems, kinetic theory and random matrix theory. Very little background in these areas will be presumed, and the talk will be self-contained. This is joint work with Percy Deift and Tom Trogdon (Courant Institute), Christian Pfrang (JP Morgan) and Enrique Pujals (IMPA).
We discuss a family of stochastic differential equations with two real parameters, interpolating one parameter subgroups of a Lie group G and hypoelliptic diffusions on a compact sub Lie group H. These models are motivated by inhomogeneous scaling of Riemannian metrics and collapsing of Berger’s spheres. We take the parameter 1/ epsilon to infinity and study the singular perturbation problem associated with L_epsilon= (1/epsilon) (A_k)^2 +Y_0. This is viewed as perturbation to the conservation law. We explore the fact that the projection map, that takes a group element to the coset containing it, is a conservation law for the unperturbed system and discuss the effective limits on both G and on the orbit manifolds G/H, the latter are not necessarily Markov processes. We hope to include examples on the spheres, the Grassmanian manifolds and the non-compact hyperbolic space.
We will survey recent progress on the KPZ universality, and describe recent results on the domains of attraction for the large time limits and robustness of the weakly asymmetric limit.
We demonstrate that the integration of data-driven dynamical systems and machine learning strategies with adaptive control are capable of producing efficient and optimal self-tuning algorithms for many complex systems arising in the engineering, physical and biological sciences. The adaptive controller, based upon a multi-parameter extremum-seeking control algorithm, is capable of obtaining and maintaining optimal states while the machine learning and sparse sensing techniques characterize the system itself for rapid state identification and improved optimization. Additionally, we can use the data directly to construct, in an adaptive manner, governing equations, even nonlinear dynamics, that best model the system measured using sparsity-promoting techniques. Recent innovations also allow for handling multi-scale physics phenomenon in an adaptive and robust way. The overall architecture provides an equation-free model reduction technique where the dynamics and control protocols are discovered directly from data acquired from sensors. The theory developed is demonstrated on two example physical systems: (i) a mode-locked laser driven by nonlinear polarization rotation in conjunction with waveplates and a polarizer and (ii) a metamaterial antenna array. We suggest how such a mathematical architecture can be ideal for controlling emerging neuro-stimulation devices as well.
Statistical inference problems arising within signal processing, data mining, and machine learning naturally give rise to hard combinatorial optimization problems. These problems become intractable when the dimensionality of the data is large, as is often the case for modern datasets. A popular idea is to construct convex relaxations of these combinatorial problems, which can be solved efficiently for large scale datasets. Semidefinite programming (SDP) relaxations are among the most powerful methods in this family, and are surprisingly well-suited for a broad range of problems where data take the form of matrices or graphs. It has been observed several times that, when the `statistical noise' is small enough, SDP relaxations correctly detect the underlying combinatorial structures. I will present a few asymptotically exact predictions for the `detection thresholds' of SDP relaxations, with applications to synchronization and community detection. [Based on Joint work with Adel Javanmard, Federico Ricci-Tersenghi and Subhabrata Sen]
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