Algorithm efficiency comes from problem information
January 4 2018 in Differential Equations, Julia, Mathematics, Programming | Tags: | Author: Christopher Rackauckas
This is a high level post about algorithms (especially mathematical, scientific, and data analysis algorithms) which I hope can help people who are not researchers or numerical software developers better understand how to choose and evaluate algorithms. When scientists and programmers think about efficiency of algorithms, they tend to think about high level ideas like temporary arrays, choice of language, and parallelism. Or they tend to think about low level ideas like pointer indirection, cache efficiency, and SIMD vectorization. However, while these features matter, most decent implementations of the same algorithm tend to get quite close in efficiency to each other (probably <3x if everyone is using a performant enough programming language or constructs). If you're sitting down thinking "what is the next algorithm that I am going to invent that will be truly a step beyond what existed before?", ... READ MORE
Solving Systems of Stochastic PDEs and using GPUs in Julia
December 11 2017 in CUDA, Differential Equations, Julia, Mathematics, Stochastics | Tags: | Author: Christopher Rackauckas
What I want to describe in this post is how to solve stochastic PDEs in Julia using GPU parallelism. I will go from start to finish, describing how to use the type-genericness of the DifferentialEquations.jl library in order to write a code that uses within-method GPU-parallelism on the system of PDEs. This is mostly a proof of concept: the most efficient integrators for this problem are not compatible with GPU parallelism yet, and the GPU parallelism isn’t fully efficient yet. However, I thought it would be nice to show an early progress report showing that it works and what needs to be fixed in Base Julia and various libraries for us to get the full efficiency.
Edit May 2019
As of DifferentialEquations.jl v6.4.0, this is no longer a proof of concept. The whole library, including implicit solvers with GMRES, etc., and … READ MORE
DifferentialEquations.jl 3.0 and a Roadmap for 4.0
October 30 2017 in Differential Equations, Julia, Mathematics, Programming | Tags: | Author: Christopher Rackauckas
I am pleased to announce the release of DifferentialEquations.jl 3.0. In the last DiffEq blog post I described the current state of JuliaDiffEq and DifferentialEquations.jl along with the trajectory that we hoped to take. We identified (at that time) current shortcomings of the software and our plans to remedy them. I also recently did a survey of differential equation suites in order to understand where we stand and see where we need to improve. These research efforts were used to put together a list of goals that were systematically achieved during 3.0. What I would like to do this time around is give a broad overview of what we have released in the 3.0 timeframe, the goals that we have achieved, and the goals that we are putting off (for next Google Summer of Code?). And then, … READ MORE
A Comparison Between Differential Equation Solver Suites In MATLAB, R, Julia, Python, C, Mathematica, Maple, and Fortran
September 26 2017 in C, Differential Equations, Julia, Mathematica, Mathematics, MATLAB, Programming, R | Tags: | Author: Christopher Rackauckas
Many times a scientist is choosing a programming language or a software for a specific purpose. For the field of scientific computing, the methods for solving differential equations are one of the important areas. What I would like to do is take the time to compare and contrast between the most popular offerings. This is a good way to reflect upon what’s available and find out where there is room for improvement. I hope that by giving you the details for how each suite was put together (and the “why”, as gathered from software publications) you can come to your own conclusion as to which suites are right for you.
(Full disclosure, I am the lead developer of DifferentialEquations.jl. You will see at the end that DifferentialEquations.jl does offer pretty much everything from the other suite combined, but that’s no accident: … READ MORE
Video Introduction to DifferentialEquations.jl
July 7 2017 in Differential Equations, Julia, Mathematics, Programming, Stochastics | Tags: | Author: Christopher Rackauckas
Videos can be much easier to follow than text (though they usually have fewer details!). So, here’s a video introduction to DifferentialEquations.jl from JuliaCon. In this talk I walk through the major features of DifferentialEquations.jl by walking through the the tutorials in the documentation, highlighting usage details and explaining how to properly think about the code. I hope this helps make it easier to adopt DifferentialEquations.jl!
DifferentialEquations.jl 2.0: State of the Ecosystem
May 8 2017 in CUDA, Differential Equations, HPC, Julia, Mathematics, Stochastics, Uncategorized | Tags: | Author: Christopher Rackauckas
In this blog post I want to summarize what we have accomplished with DifferentialEquations’ 2.0 release and detail where we are going next. I want to put the design changes and development work into a larger context so that way everyone can better understand what has been achieved, and better understand how we are planning to tackle our next challenges.
If you find this project interesting and would like to support our work, please star our Github repository. Thanks!
Now let’s get started.
DifferentialEquations.jl 1.0: The Core
Before we start talking about 2.0, let’s understand first what 1.0 was all about. DifferentialEquations.jl 1.0 was about answering a single question: how can we put the wide array of differential equations into one simple and efficient interface. The result of this was the common interface explained in the first blog post. Essentially, we created … READ MORE
Some Fun With Julia Types: Symbolic Expressions in the ODE Solver
May 4 2017 in Differential Equations, Julia, Mathematics, Uncategorized | Tags: | Author: Christopher Rackauckas
In Julia, you can naturally write generic algorithms which work on any type which has specific “actions”. For example, an “AbstractArray” is a type which has a specific set of functions implemented. This means that in any generically-written algorithm that wants an array, you can give it an AbstractArray and it will “just work”. This kind of abstraction makes it easy to write a simple algorithm and then use that same exact code for other purposes. For example, distributed computing can be done by just passing in a DistributedArray, and the algorithm can be accomplished on the GPU by using a GPUArrays. Because Julia’s functions will auto-specialize on the types you give it, Julia automatically makes efficient versions specifically for the types you pass in which, at compile-time, strips away the costs of the abstraction.
This means … READ MORE
Building a Web App in Julia: DifferentialEquations.jl Online
January 17 2017 in Differential Equations, Julia, Mathematics | Tags: | Author: Christopher Rackauckas
Web apps are all the rage because of accessibility. However, there are usually problems with trying to modify some existing software to be a web app: specifically user interface and performance. In order for a web application to perform well, it must be able to have a clean user interface with a “smart” backend which can automatically accommodate to the user’s needs, but also give back a nice looking response in under a second. This has typically limited the types of applications which can become responsive web applications.
Scientific software is one domain where user interface has been obstructive at best, and always in need of better performance. However, the programming language Julia has allowed us to build both an easy to use and highly performant ecosystem of numerical differential equation solvers over the last 8 months. Thus we had … READ MORE
6 Months of DifferentialEquations.jl: Where We Are and Where We Are Going
December 15 2016 in Differential Equations, Julia, Mathematics, Programming, Stochastics, Uncategorized | Tags: DifferentialEquations.jl, julia | Author: Christopher Rackauckas
So around 6 months ago, DifferentialEquations.jl was first registered. It was at first made to be a library which can solve “some” types of differential equations, and that “some” didn’t even include ordinary differential equations. The focus was mostly fast algorithms for stochastic differential equations and partial differential equations.
Needless to say, Julia makes you too productive. Ambitions grew. By the first release announcement, much had already changed. Not only were there ordinary differential equation solvers, there were many. But the key difference was a change in focus. Instead of just looking to give a production-quality library of fast methods, a major goal of DifferentialEquations.jl became to unify the various existing packages of Julia to give one user-friendly interface.
Since that release announcement, we have made enormous progress. At this point, I believe we have both the most expansive and flexible … READ MORE
Introducing DifferentialEquations.jl
August 1 2016 in Differential Equations, FEM, Julia, Programming, Stochastics | Tags: DifferentialEquations.jl, julia | Author: Christopher Rackauckas
Edit: This post is very old. See this post for more up-to-date information.
Differential equations are ubiquitous throughout mathematics and the sciences. In fact, I myself have studied various forms of differential equations stemming from fields including biology, chemistry, economics, and climatology. What was interesting is that, although many different people are using differential equations for many different things, pretty much everyone wants the same thing: to quickly solve differential equations in their various forms, and make some pretty plots to describe what happened.
The goal of DifferentialEquations.jl is to do exactly that: to make it easy solve differential equations with the latest and greatest algorithms, and put out a pretty plot. The core idea behind DifferentialEquations.jl is that, while it is easy to describe a differential equation, they have such diverse behavior that experts have spent over a century compiling … READ MORE