Report
Teaser, summary, work performed and final results
Periodic Reporting for period 1 - CHiPS (CHallenges in Preservation of Structure)
Teaser
It is difficult to overestimate the importance of differential equations in modern society. They are literally everywhere. Differential equations have been used to describe weather and climate phenomena. And the transportation of pollution in the atmosphere, and plankton and...
Summary
It is difficult to overestimate the importance of differential equations in modern society. They are literally everywhere.
Differential equations have been used to describe weather and climate phenomena. And the transportation of pollution in the atmosphere, and plankton and nutrients in the sea. They’ve been used to model the spread of diseases. And the movement of planets and satellites.
Unsurprisingly, the computation of numerical solutions to differential equations has been an important branch of mathematics for decades.
When modelling complex physical phenomena in a computer, geometry is often essential, as geometry - and symmetry - is the mathematical manifestation of the underlying laws of physics.
Yet, this fact has historically been ignored when creating numerical approximations to differential equations. The main quest has been to come up with all-purpose algorithms.
But lately, focus has shifted to instead considering special classes of differential equations and purpose-built algorithms.
These novel numerical methods are not only quantitatively accurate, but also qualitatively accurate as they preserve the key structural properties of the particular problem under study.
Such methods are called structure preserving integrators.
This approach can lead to both better and faster numerical approximations.
Work performed
The CHiPS project addresses fundamental questions within structure preserving algorithms.
The project deals with mathematical work with direct practical applications; in image processing and shape analysis, particle dynamics in turbulent fluids, equations with highly oscillatory solutions with applications in chemistry, problems in Big Data and Bayesian statistics, in mechanics and control.
The philosophy of the CHiPS project is simple: All physical laws have a geometric foundation.
Our aim is to create new methods for solving differential equations that conserve these geometric properties and invariants.
The goal is to obtain faster computational methods and accurate numerical solutions and the number of actual applications is abound.
Here is a sample of the results achieved so far (there are more results and more details in the technical report):
- better image denoising using novel optimization methods
- new algorithms to compute splines, usable for quantum computing
- extension of algorithms to simulate ferromagnetic fluids
- software to aid devising better stochastic integrators
- software for image denoising using a new regulariser
As for dissemination, apart from participation to, and organisation of, numerous conferences, we have made a pedagogical video to explain our methods and objectives.
Final results
One of the new, major challenges of now is how to analyze truly large amounts of data. Big Data.
Big Data analysis cannot be done without statistics, but we need a completely new type of statistics to truly solve the computational challenges.
The interplay of geometry, statistics and computational geometric methods, is important when working with big data.
Our work on shape analysis and its use in computer animation demonstrate that geometric insight is very valuable for understanding and manipulating certain type of data.
We have also obtained numerical methods for image processing which solve optimisation problems replacing them by differential equations.
In our work on Schrödinger equations and highly oscillatory problems we propose methods that can impact future research on quantum computers.
While highly applicable, the approach has some very theoretical aspects to it. There is more structure to the method than you might think. More abstract mathematics than one can see. It is interesting and fascinating in its own right.
These new challenging applications also bring new mathematical challenges and theoretical questions along with them, that we are trying to untangle. In doing so we will get a deeper understanding of the practical problems, possible resulting in even better solutions.
But mathematics does not end when computation begins.
The CHiPS project is only really successful if the results are helpful to others outside the mathematics community. An important goal for the project is to make sure these innovative approaches are implemented in usable software tools.
The goal of the CHiPS project is twofold. We want to develop new mathematics and fast methods. And we want these methods to be used.
Website & more info
More info: https://www.ntnu.edu/imf/chips.