Speaker: Tim Davis

Department of Computer Science and Engineering, TAMU


Sparse Matrix Algorithms : Combinatorics + Numerical Methods + Applications

Time / Date:   4:00 - 5:00 PM Monday, November 16

Location: Blocker 117


Sparse matrix algorithms lie in the intersection of graph theory and numerical linear algebra, and are a key component of high-performance combinatorial scientific computing. This talk highlights four of my contributions in this domain, ranging from theory and algorithms to reliable mathematical software and its impact on applications:
    (1) Sparse Cholesky update/downdate
    (2) Approximate minimum degree
    (3) Unsymmetric multifrontal method for sparse LU factorization
    (4) Multifrontal sparse QR factorization, including my current work in GPU-based heterogeneous high-performance parallel computing.

Speaker: Natalia Kopteva

University of Limerick, Ireland


Maximum norm a posteriori error estimates for parabolic partial differential equations

Time / Date:   12:00 - 1:00 PM Thursday, October 22

Location: Blocker 220


Solutions of partial differential equations frequently exhibit corner singularities and/or sharp boundary and interior layers. To obtain reliable numerical approximations of such solutions in an efficient way, one may want to use meshes that are adapted to solution singularities. Such meshes can be constructed using a priori information on the solutions, however it is rarely available in real-life applications. Therefore the best hope seems to be offered by the automated mesh construction by adaptive techniques. This approach requires no initial asymptotic understanding of the nature of the solutions and the solution singularity locations. Reliable adaptive algorithms are based on a posteriori error estimates, i.e. estimates of the error in terms of values obtained in the computation process: computed solution and current mesh. Such a posteriori error estimates for parabolic partial differential equations will be the subject of this talk. For classical and singularly perturbed semilinear parabolic equations, we give computable a posteriori error estimates in the maximum norm, which, in the singularly perturbed regime, hold uniformly in the small perturbation parameter. The parabolic equations are discretized in time using the backward Euler, Crank-Nicolson and discontinuous Galerkin methods. Both semidiscrete (no spatial discretization) and fully discrete cases will be addressed. The analysis invokes certain bounds for the Green's function of the parabolic operator. When dealing with the full discretizations, we also employ the elliptic reconstruction technique. Although parts of our analysis are quite technical, it will be demonstrated (using a first-order ODE example as a trivial case of a parabolic PDE) that some main ideas are quite elementary.
[1] N. Kopteva and T. Linß, Maximum norm a posteriori error estimation for parabolic problems using elliptic reconstructions, SIAM J. Numer. Anal., 51 (2013), 1494-1524.
[2] A. Demlow, O. Lakkis, and C. Makridakis, A posteriori error estimates in
the maximum norm for parabolic problems, SIAM J. Numer. Anal., 47 (2009), pp. 2157–2176.

Slides of talk

Speaker: Peter Kuchment

Department of Mathematics, TAMU


Unreasonable effectiveness of mathematics in natural science and engineering

Time / Date:   3:00 - 4:00 PM Wednesday, September 30

Location: Blocker 105


Slides of talk