Stephen D. Shank

Department of Mathematics

Massachussetts Institute of Technology
77 Massachussetts Avenue
Cambridge, MA 02139
sshank@mit.edu
Curriculum Vitae

Research interests

My current research interests are in the fields of scientific computing and numerical analysis. Specifically, I have worked on the following topics in numerical linear algbera:
• Iterative solution of linear systems: Krylov subspace methods for large/sparse linear systems, augmentation strategies, algebraic Schwarz methods, preconditioning KKT systems
• Low-rank approximation: large-scale Sylvester and Lyapunov equations and their generalizations, subspace projection strategies, extended and rational Krylov subspace methods for matrix equations, low-rank Krylov subspace methods
• Wave propagation (acoustic, frequency domain): iterative solution of the Helmholtz equation, shifted Laplacian preconditioning, optimal control of the Helmholtz equation
My research aspirations are to further investigate existing low-rank solution techniques for matrix equations arising in PDE-constrained optimization and linear systems that arise in the numerical solution of stochastic PDEs. I am also interested in working on electromagnetic waves, as well as multigrid and high performance computing.

Publications

David Fritzsche, Andreas Frommer, Stephen D. Shank, and Daniel B. Szyld.
Overlapping blocks by growing a partition with applications to preconditioning.
SIAM Journal on Scientific Computing, SIAM, 2013, Vol. 35(1), pp. A453-A473

Stephen D. Shank and Valeria Simoncini.
Krylov Subspace Methods for Large-Scale Constrained Sylvester Equations.
SIAM Journal on Matrix Analysis and Applications, SIAM, 2013, Vol. 34(4), pp. 1448-1463

Selected talks

Overlapping blocks by growing a partition with applications to preconditioning.
SIAM Conference on Applied Linear Algebra, 2012. Universitat Politècnica de València, Valencia, Spain. 6/19/12.

KKT preconditioners for indefinite PDE systems: The Helmholtz case.
Mid-Atlantic Numerical Analysis Day 2013. Temple University, Philadelphia, PA. 11/22/13.

Other activities

Participant in the 2013 Gene Golub SIAM Summer School at Fudan University in Shanghai, China.
Co-founder of Kanga Technologies, LLC.
Co-creator of KangaStock app, written in PHP, Python and SQL. Current used by a corrugated plant to take inventory.
Co-creator of The SAT Game, with a companion app on Apple's app store. Written in PHP, Python and SQL.

The above image

 The image above arises from the PDE-constrained optimization problem \begin{aligned} \min_{u,f} \frac{1}{2}\| u - \hat{u} \|_{L_2(\Omega_\text{ROI})}^2 & + \beta \| f \|_{L_2(\Omega)}^2 \\ \mbox{ such that } -\Delta u - \omega^2 u & = f \mbox{ in } \Omega \\ \frac{\partial u}{\partial \eta} - i \omega u & = 0 \mbox{ on } \partial \Omega \\ \mbox{supp}(f) & \subset \Omega_\text{CAA} \end{aligned} so that $$u$$ and $$f$$ are constrained to solve a partial differential equation, while minimizing a given objective function. The interpretation is as follows: we have a computational domain $$\Omega$$ and a type of physics we seek to model, in this case acoustic waves modeled by the Helmholtz equation with first-order Sommerfeld boundary conditions. Our goal is to create some desirable effect in our state by choosing our source function appropriately. Our unknowns $$u$$ and $$f$$ are guaranteed to solve this PDE by construction, but in constrast to the typical forward problem, $$f$$ is not known beforehand. The computed source $$f$$ is given from the minimization problem, but this source is only allowed to be nonzero in some other subregion $$\Omega_\text{CAA} \subset \Omega$$, called a control allowable area. We hope our computed state $$u$$ will be close to $$\hat{u}$$ in $$\Omega_\text{ROI}$$, called a region of interest, where $$\hat{u}$$ is some desired state we seek to create in this region. The objective function achieves this by bring $$u$$ as close to $$\hat{u}$$ as possible (in the $$L_2$$ sense) while adding a penalty to the norm of $$f$$ which regularizes the problem. The constraint guarantees that we solve the PDE corresponding whatever physics we are trying to model. Here we sought to create a plane wave with a nearby source.