Imperial College, London
This workshop proposal brings together 4 speakers on different topics in numerical analysis. The aim of the workshop is to have demonstrations of Julia packages in this area in a way to motivate the results to a “general” audience. Our speakers cover a variety of areas to give a broad demonstration of what is now possible using Julia.
ApproxFun.jl, Approximating Functions and Solving Differential Equations Sheehan Olver, Imperial College, London
ApproxFun.jl is a Julia package that makes working with functions on a computer fast and easy. Basic algebra and calculus operations are available, so that one can, for example, differentiate the function $\sin\cos^2 x$ as easy as typing sin(cos(x)^2)'. Further capabilities include solving differential equations and random number sampling. This talk will demonstrate the capabilities of ApproxFun, and how Julia’s approach to typing allows for these capabilities to be used with other types, such as Dual numbers (from DualNumbers.jl) or BigFloats.
ValidatedNumerics.jl, Rigorous Floating-point Calculations with Interval Arithmetic David P. Sanders, Universidad Nacional Autónoma de México
I will present a suite of Julia packages in the JuliaIntervals organisation which provide implementations of numerical methods that provide results that are guaranteed to be correct. These are based on interval arithmetic, i.e. defining arithmetic and elementary functions acting on intervals of real numbers.
This gives us a tool to calculate with continuous sets of real numbers, and thus rigorously bound the range of a function over a given set. Applications of this technology include finding, in a guaranteed way, all the roots of a multivariable function in a given region of space, and finding the global optimum of a function.
FastTransforms.jl, Fast Orthogonal Polynomial Transforms R. Mikael Slevinsky, University of Manitoba
FastTransforms.jl allows the user to conveniently work with orthogonal polynomials with degrees well into the millions. Transforms include conversion between Jacobi polynomial expansions, with Chebyshev, Legendre, and ultraspherical polynomial transforms as special cases. For the signal processor, all three types of nonuniform fast Fourier transforms available. As well, spherical harmonic transforms and transforms between orthogonal polynomials on the triangle allow for the efficient simulation of partial differential equations of evolution. Algorithms include methods based on asymptotic formulae to relate the transforms to a small number of fast Fourier transforms, matrix factorizations based on the Hadamard product, hierarchical matrix decompositions à la Fast Multipole Method, and the butterfly algorithm.
MatrixDepot.jl, Testing Linear Algebra Algorithms in Julia Weijian Zhang, University of Manchester
Test matrices are important for exploring the behavior of linear algebra algorithms and for measuring their performance with respect to accuracy, stability, convergence rate, speed, or robustness. We give a brief historical remark on the development of test matrices in different programming languages and then focus on the advantage of using MatrixDepot.jl for testing and exploring new algorithms in Julia. Using both contrived and real-world examples, we demonstrate the power of MatrixDepot.jl which takes advantage of many nice Julia features, such as using multiple dispatch to help provide a simple user interface and to allow matrices to be generated in any of the numeric data types supported by the language.
I am a Reader (equivalent to Assoc. Professor) in Applied Mathematics and Mathematical Physics at Imperial College, London. My research is in numerics, in particular spectral methods and complex analytical methods. I have extensive experience with programming Julia, having developed several packages (ApproxFun.jl, BandedMatrices.jl, BlockBandedMatrices.jl, etc.).
