My research is focused on numerical methods for initial boundary value problems, and also for the optimization of continuum based models arising in material science. Scientifically, my work is aimed at characterizing how non-convex and non-local effects lead to the formation of self-assembled structures, as well as the emergence of new length scales that arise in pattern formation. As a general theme, my work is often centered on identifying an underlying analytical structure, which may then be used in a numerical setting.

I have also been involved in educational projects including the development of undergraduate tutorial videos for the MIT OpenCourseWare (OCW) project for linear algebra (18.06sc), and differential equations (18.03sc). I have also participated as a judge for the New York math fair.

- Numerical methods/analysis for partial differential equations.
- Numerical methods for global non-convex/non-local energy minimization.
- Application areas: self-assembly in many particle systems, computations for fluids.

- Time stepping for stiff PDEs
- Embedded boundary and spectral methods
- Infinite dimensional conic programming.
- D.C. (difference of convex) programming for infinite dimensional problems.
- Gradient flows and phase field models.
- Incompressible Navier-Stokes equations.

My current research involves two separate numerical areas: (i) numerical methods for partial differential equations, and (ii) numerical methods for applied optimization. My focus is on applications to material self-assembly, and computational fluid dynamics.

Much of my current research is aimed at minimizing an energy functionals arising from material models. Scientifically, the long term goal is to quantify how anisotropic forces, or the inclusion of multiple species can affect the emergent length scales observed in the self-assembly of nano-structures. Mathematically, we have been developing conic programming, and DC programming techniques to attack these difficult optimization problems, and gain insight into the underlying energy landscapes.

With regards to numerical PDEs, my interests lie in developing fast and accurate methods that improve upon current fluid dynamics solvers. Currently my interests have focused on overcoming the stability limitations that arise in well-known IMEX (implicit-explicit) numerical time integrators, and also in resolving accuracy limitations (referred to as order reduction) in Runge-Kutta schemes.

- Mahdi Bandegi, D. Shirokoff. Approximate global minimizers to pairwise interaction problems via convex relaxation, 2016.
- R. R. Rosales, B. Seibold, D. Shirokoff, D. Zhou. Unconditional stability for Multistep ImEx schemes – Theory, 2017.
- D. Zhou, B. Seibold, D. Shirokoff, P. Chidyagwai, R. Rosales. Meshfree finite differences for a vector Poisson and pressure Poisson equation with electric boundary conditions.
*Meshfree methods for Partial Differential Equations VII Lecture Notes in Computational Science and Engineering*, Vol. 100 Griebel, M. and Schweitzer, M.A. (ed.), Springer, Switzerland, 223—246, 2015. - X. Cheng, D. Li, D. Shirokoff, B. Wetton, On the spectral gap of a square distance matrix,
*J. Stat. Phys.*, 166(3):1029—1035, 2017. - D. Shirokoff, R. Chiksi, J-C. Nave, Sufficient conditions for global minimality of metastable states in a class of non-convex functionals: a simple approach via quadratic lower bounds,
*J. Nonlinear Sci.*, 25(3):539-582, 2015. - R. Galagusz, D. Shirokoff, J-C. Nave, A Fourier penalty method for solving the time-dependent Maxwell’s equations in domains with curved boundaries,
*J. Comput. Phys.*, 306:167—198, 2016. - D. Shirokoff, J-C. Nave, A Sharp-interface active penalty method for the incompressible Navier-Stokes equations,
*J. Sci. Comput.*, 62(1):53—77, 2015. - D. Shirokoff, R. R. Rosales. An efficient method for the incompressible Navier-Stokes equations on irregular domains with no-slip boundary conditions, high order up to the boundary,
*J. Comput. Phys.*, 230: 8619—8646, 2011.