Geometric Numerical Integration
Symmetries in a physical system lead to conservation laws such as conservation of mass, momentum, angular momentum and energy.
This animation shows the solution of the Schrödinger equation with radially symmetric initial conditions. The radial symmetry of the system results in conservation of angular momentum. Do the numerical methods we use for simulating this system conserve the angular momentum?
Geometric Numerical Integration is about the design and analysis of methods that can conserve some of these properties exactly or approximately.
Areas: Lie algebras/groups, exponential splittings, convergence analysis.
Approximating the Matrix Exponential
The solution to a linear PDE
u' = A u
such as the Schrödinger equation can be expressed explicitly in terms of the exponential of A,
u(t) = exp(t A) u(0).
We may approximate the exponential by truncating the Taylor expansion. These are polynomial approximations. The first such truncation is the Forward Euler method, which suffers from stability issues. Krylov subspace methods are among the most effective polynomial approximations but require extremely small time steps when matrix norm becomes large. The case of Coloumb potential shown in the animation is one such case.
I am interested in the design of exponential splittings and Krylov methods that can allow us to take much larger steps and lead to efficient approximation of the exponential.
Areas: Exponential splittings, Krylov methods, numerical optimization
Control of Quantum Systems
Controlling a process at the atomic and molecular scale requires an ability to manipulate the electromagnetic field effectively. In the case of spins, whether they are used for building a quantum computer or in the design of a better NMR (or MRI), we can exert control by manipulating the magnetic field. In femtochemistry and atomic physics, one uses lasers to exert control on the wavefunction.
This animation shows a wavefunction (orange) that would normally have stayed trapped in the central well of the potential (blue). Under the influence of the laser, it starts moving around.
This project is about designing optimization approaches and faster solvers that could be used for designing optimal control pulses.
Areas: optimal control, numerical optimization, stochasticity
I may also have other projects not listed here.