Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena

Introduction

Modern numerical algorithms and high-performance computing systems are capable of simulating physical systems far more efficiently than those from just a few years ago. However, this recent increase in computational power comes with an insatiable demand for complexity and scale: scientists and artists now wish to model textiles at the level of individual fibers and simulate oceans down to the smallest ripple.

Challenges in Current Approaches

The typical strategy of analytically deriving integrators and hand-tuning parameters will inevitably fail to cope with the increasing non-linearity and complexity of these problems. This shift in problem complexity necessitates entirely novel strategies for discovering numerical algorithms.

Proposed Research Directions

We will redefine the state of the art in numerical simulation and animation by combining three different approaches:

1. Development of Analytical Tools: We will develop analytical tools customized for large scales; we will derive optimal numerical algorithms by viewing physics through the lens of computational complexity and by observing limiting behaviors as problems increase in size.

2. Physical Insights through Data: Next, we will gain physical insights by simulating huge numbers of smaller simulations and generating large data sets. By discovering trends in this data, we will faithfully approximate aggregate behaviors in systems that are far too complex for analytical techniques to penetrate.

3. Constrained Optimization for Algorithm Derivation: Finally, by framing the derivation of numerical algorithms as a constrained optimization problem, we will be able to deliver provably optimal code for a given piece of hardware and precisely control accuracy/efficiency trade-offs.

Expected Outcomes

The combination of these research directions will enable efficient simulations of massively complicated systems that are currently unfeasible to compute. Due to the timely and groundbreaking nature of these proposed directions, we also expect to develop entirely unprecedented methods for physics simulation and discover a number of theoretical insights and scientific advances along the way.