Solving differential equations fast, precisely, and reliably

Introduction

Being a language of nature, differential equations are ubiquitous in science and technology. Solving them is a fundamental computational task with a long and rich history. Applications usually require approximate solutions, which can be computed using numerical methods such as Runge-Kutta schemes. Alternatively, one may search for symbolic solutions, which have the advantage of presenting the solutions in an exact and more intelligible way. However, such solutions do not always exist and may be hard to compute.

Proposal Objectives

The present proposal aims at making the resolution of differential equations both faster and more reliable. We will undertake a systematic analysis of the cost to compute both numeric and symbolic solutions, as a function of:

1. The required precision
2. Special properties of the equation and its solutions
3. Hardware specifics of the computer

This includes the cost to certify approximate numeric solutions, e.g., through the computation of provable error bounds.

Methodology

In order to compute symbolic solutions more efficiently, we will develop a new theory that relies on two techniques from computer algebra that were improved significantly in the past decade:

• Numerical homotopy continuation
• Sparse interpolation

Implementation and Validation

Theoretical progress on the above problems will be accompanied by open source implementations. For this purpose, we will also implement several high-performance libraries of independent interest:

• Non-conventional medium precision arithmetic
• Reliable homotopy continuation
• Sparse interpolation
• Faster-than-just-in-time compilation

Altogether, these implementations will validate the correctness and efficiency of our approach. They should also allow us to tackle problems from applications that are currently out of reach.