Brittany D. Hamfeldt
|Hours:||By Appointment (Summer 2018)|
My research focuses on the development of numerical methods for solving nonlinear partial differential equations. One of the guiding principles in my work is the idea that good numerical methods should be instructed by a solid theoretical understanding of the underlying equation, which ensures that methods perform correctly in even the most challenging settings. A second principle in my work is the idea that numerical methods should be influenced by applications in order to ensure that these new mathematical developments can contribute to real world problems.
A particular focus of my work is the solution of fully nonlinear elliptic equations and related applications to optimal mass transportation. I have introduced new formulations of these problems, which led to the first PDE based methods for optimal mass transportation. These methods have enabled the development of new techniques for solving geophysical inverse problems and for designing lenses to reshape beams of light. Recently, I have introduced a new framework for solving a large class of fully nonlinear elliptic equations on unstructured meshes. I have also developed new techniques for proving that the numerical methods compute the correct weak solution of the equation.
- PhD, Applied Mathematics, Simon Fraser University, 2012
- MSc, Applied Mathematics, Simon Fraser University, 2009
- BSc, Mathematics, Trinity Western University, 2007
- Optimal mass transportation
- Fully nonlinear partial differential equations
- Viscosity solutions
- Finite difference methods
- Adaptive methods
- Seismic full waveform inversion
- Reflector and refractor design
Much of my current research is focused on fully nonlinear elliptic partial differential equations, with problems ranging from theoretical results in PDE analysis to the implementation of cutting-edge numerical methods to the use of these methods in advancing scientific applications.
One of my current goals is to develop efficient, convergent numerical methods for approximating nonlinear PDEs on unstructured meshes or point clouds. The approximation of nonlinear equations on non-uniform meshes or complicated domains is needed to make progress on current problems arising in optics, geophysics, meteorology, and many other applications. In realistic settings, these equations are often equipped with non-standard or non-local boundary conditions. I am pursuing the reformulation of several such equations, with the goal of producing well-posed problems that can be solved numerically.
- Brittany D. Froese, Adam M. Oberman, and Tiago Salvador. Numerical methods for the 2-Hessian elliptic partial differential equation. IMA J. Numer. Anal., 2016.
- Zexin Feng, Brittany D. Froese, Chih-Yu Huang, Donglin Ma, and Rongguang Liang. Creating unconventional geometric beams with large depth of field using double freeform-surface optics. Appl. Optics, 54(20):6277-6281, 2015.
- Björn Engquist, Brittany D. Froese, and Yen-Hsi Richard Tsai. Fast sweeping methods for hyperbolic systems of conservation laws at steady state II. J. Comput. Phys., 286:70-86, 2015.
- Björn Engquist and Brittany D. Froese. Application of the Wasserstein metric to seismic signals. Comm. Math. Sci., 12(5):979-988, 2014.
- Jean-David Benamou, Brittany D. Froese, and Adam M. Oberman. Numerical solution of the optimal transportation problem using the Monge-Ampère equation. J. Comput. Phys., 260:107-126, 2014.
- Brittany D. Froese and Adam M. Oberman. Convergent filtered schemes for the Monge-Ampère partial differential equation. SIAM J. Numer. Anal., 51(1):423-444, 2013.
- Brittany D. Froese. A numerical method for the elliptic Monge-Ampère equation with transport boundary conditions. SIAM J. Sci. Comput., 34(3):A1432-A1459, 2012.