Dr. Sigal Gottlieb

Research experience

Ph.D. Applied Mathematics, Brown University

Research projects

Some past and present research projects include:

Runge-Kutta schemes.

Spectral methods for time-dependent problems.

Explicit and implicit error inhibiting schemes with post-processing

Potential research projects for students

We’ve all experienced rounding errors when doing homework: you round each variable to 2 decimal places, multiply them together and before you know it, your solution is totally wrong. Rounding errors have been a big problem in many numerical computations.

These have led to real-world tragedies, as in the death  of 28 US service people in 1991, when roundoff errors caused the Patriot missile to malfunction.

Many mathematical problems are hard to solve explicitly, and so we approximate their solution using numerical methods.

Computing an accurate approximate solution is often done iteratively: we make a guess x0, then use the equations to get  a better guess x1, etc. This, for example, is how we often approximate the solution of a linear system of equations A x = b (where x is unknown)

when it is too large to solve exactly. This type of iterative approach is also used when approximating the solution of  ordinary and partial differential equations. The problem with this iterative type of approach is that it allows the rounding errors to build up! 

A solution to this is to use more precision in the computation (this means saving more decimal place information): but this makes the code take longer to run!

A smart approach mixes high and low precision to use low precision computationally time-consuming parts of the computation and correcting these with high precision computationally fast steps.

There are three types of questions we ask in this research:

1. (Theory) How do we design a mixed (low/high) precision approach that will speed up the computation while keeping it accurate?

2. (Practice) How do we implement such an approach and see what really happens on a computer?

3. (One step further) This type of approach is not only efficient in terms of time but in terms of energy consumption as well! How can we design such “green” mixed precision iterative methods?

Suggestions welcome!