Stiffness is a phenomena in differential equations which requires implicit methods in order to be efficiently solved. However, implicit methods are inherently more costly and thus inefficient when they are not needed. This is an issue because many problems are not always stiff, and instead switch from stiff and nonstiff modes. The purpose of this project would be to develop functionality for detecting stiffness during integration and testing algorithms for automatic switching between appropriate algorithms. These would not only be more efficient on a large class of problems, but also decrease the cognative burden on the user by being efficient for a large class of algorithms and likely become the new default methods.
Recommended Skills: Background knowledge in numerical methods for solving differential equations. The student is expected to already be familiar with the concept of stiffness in ODEs, but not necessarily an expert.
Expected Results: Implementation of a trait-based engine for algorithm-specific stiffness metrics, and implementations of new algorithms for switching strategies.
Mentors: Chris Rackauckas
Julia needs to have a full set of ordinary differential equations (ODE) and algebraic differential equation (DAE) solvers, as they are vital for numeric programming. There are many advantages to having a native Julia implementation, including the ability to use Julia-defined types (for things like arbitrary precision) and composability with other packages. A library of methods can be built for the common interface, seamlessly integrating with the other available methods. Possible families of methods to implement are:
Recommended Skills: Background knowledge in numerical analysis, numerical linear algebra, and the ability to write fast code.
Expected Results: Contributions of production-quality ODE/DAE solver methods.
Mentors: Chris Rackauckas
Molecular dynamics simulations are large N-body problems which predict the properties of materials. While in theory anyone with an ODE solver can write the ODE to be solved, in practice these problems can be very large and thus very difficult to specify. The purpose of this project is to expand on the tooling of DiffEqPhysics.jl to build methods for these types of simulations. Extensions to force-field simulators which allow for constant temperature or use Lennard-Jones potentials, along with tools for easily calculating system properties like temperature and pressure, would make the Julia ecosystem much friendlier to these forms of modeling.
Recommended Skills: Background knowledge in physics.
Expected Results: Tools for performing molecular dynamics simulations and examples which show the usage of such tools.
Mentors: Chris Rackauckas, David Sanders
Global Sensitivity Analysis is a popular tool to assess the affect that parameters have on a differential equation model. A good introduction can be found in this thesis. Global Sensitivity Analysis tools can be much more efficient than Local Sensitivity Analysis tools, and give a better view of how parameters affect the model in a more general sense. Julia currently has an implementation Local Sensitivity Analysis, but there is no method for Global Sensitivity Analysis. The goal of this project would be to implement methods like the Morris method in DiffEqSensitivity.jl which can be used with any differential equation solver on the common interface.
In addition, adjoint sensitivity analysis is a more efficient method than standard local sensitivity analysis when the number of parameters is large. It is the differential equations extension of “backpropagation” and is used in many other domains like parameter estimation as part of the optimization process. An introduction to the adjoint sensitivity equations can be found in this documentation.
Recommended Skills: An understanding of how to use DifferentialEquations.jl to solve equations.
Expected Results: Efficient functions for performing global and adjoint sensitivity analysis.
Mentors: Chris Rackauckas
There are two ways to approach libraires for partial differential equations (PDEs): one can build “toolkits” which enable users to discretize any PDE but require knowledge of numerical PDE methods, or one can build “full-stop” PDE solvers for specific PDEs. There are many different ways solving PDEs could be approached, and here are some ideas for potential projects:
1) Enhancement of existing tools for discretizing PDEs. The finite differencing (FDM) library DiffEqOperators.jl could be enahnced to allow non-uniform grids or composition of operators. The finite element method (FEM) library FEniCS.jl could wrap more of the FEniCS library. 2) Full stop solvers of common fluid dynamical equations, such as diffusion-advection- convection equations, or of hyperbolic PDEs such as the Hamilton-Jacobi-Bellman equations would be useful to many users. 3) Using stochastic differential equation (SDE) solvers to efficiently (and highly parallel) approximate certain PDEs. 4) Development of ODE solvers for more efficiently solving specific types of PDE discretizations. See the “Native Julia solvers for ordinary differential equations” project.
Recommended Skills: Background knowledge in numerical methods for solving differential equations. Some basic knowledge of PDEs, but mostly a willingness to learn and a strong understanding of calculus and linear algebra.
Expected Results: A production-quality PDE solver package for some common PDEs.
Mentors: Chris Rackauckas