** Speaker:** Tim Davis

Department of Computer Science and Engineering, TAMU

__Title:__Sparse Matrix Algorithms : Combinatorics + Numerical Methods + Applications

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

__Location:__ Blocker 117

__Abstract:__

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.

(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

__Title:__Maximum norm a posteriori error estimates for parabolic partial differential equations

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

__Location:__ Blocker 220

__Abstract:__

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.

References:

[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.

References:

[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

__Title:__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

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