Research vision

My central interest is the design of numerical methods for variational problems by passing through a continuous dynamical system:

\[ \text{variational problem} \quad\longrightarrow\quad \text{flow} \quad\longrightarrow\quad \text{discretization} \quad\longrightarrow\quad \text{algorithm}. \]

This perspective is useful because it provides an abstract framework for understanding families of algorithms. A closely related dual to this is the Bayesian formulation of variational problems and the relationship between the aforementioned flows and Monte-Carlo sampling.

Projects

Unconditionally stable gradient and momentum flow solvers

Classical linear solvers can be interpreted as discretizations or approximations of continuous residual flows. The aim is to design finite-time spectral filters and Krylov approximations with strong stability properties.

Duration: Oct 2025 to July 2026

Topics: linear systems, Krylov methods, exponential integrators, spectral filters.

Inertial PDEs for image inpainting

Second-order-in-time variants of dissipative imaging PDEs may improve basin escape and information propagation while retaining stability through suitable semigroup discretizations.

Duration: Jan 2026 to July 2026

Topics: Cahn–Hilliard, image inpainting, exponential integrators, attractors.

Nonlinear regression as a dissipative dynamical system

Nonlinear regression can be studied through gradient flows in function space, enabling questions about well-posedness, attractors, discretization, and convergence.

Duration: July 2026 to Oct 2026

Topics: RKHS, Hilbert-space flows, global attractors, Łojasiewicz inequalities.

Boundary-aware kernel interpolation

Kernel interpolation on symmetric domains can incorporate exact boundary conditions using group actions, reflections, and symmetry-aware kernels.

Duration: July 2026 to Nov 2026

Topics: RBF interpolation, RKHS theory, Coxeter groups, Approximation theory.