Multivalued matrix functions arise in solving various kinds of matrix equations. The matrix logarithm is the prototypical example. Another example is the Lambert W function of a matrix, which is much less well known but has been attracting recent interest. A theme of the talk is the importance of choosing appropriate principal values and making sure that the correct choices of signs and branches are used, both in theory and in computation. We will give examples where incorrect results have previously been obtained.
We focus on matrix inverse trigonometric and inverse hyperbolic functions, beginning by investigating existence and characterization. Turning to the principal values, various functional identities are derived, some of which are new even in the scalar case, including a "round trip'' formula that relates acos(cos A) to A and similar formulas for the other inverse functions. Key tools used in the derivations are the matrix unwinding function and the matrix sign function.
A new inverse scaling and squaring type algorithm employing a Schur decomposition and variable-degree Padé approximation is derived for computing acos, and it is shown how it can also be used to compute asin, acosh, and asinh. In numerical experiments the algorithm is found to behave in a forward stable fashion and to be superior to computing these functions via logarithmic formulas.
In this talk I will describe the construction of a manifold called "multispace" which uses the Lagrange interpolation data of functions as coordinates and which has the jet space for functions embedded as a sub manifold.
The purpose for the construction is the simultaneous calculation of smooth and discrete invariants of Lie group actions on the base space, via a moving frame, together with their differential (smooth) and recurrence (discrete) relations, with all appropriate continuum limits guaranteed by the construction.
Two main applications have been described to date, to discrete integrable systems and to discrete variational systems. I will discuss the application to variational shallow water systems, showing how a simultaneous smooth and discrete
Noether's theorem leads to the conservation of energy and momenta in the discrete scheme, with guaranteed continuum limits to the smooth energy and momenta.
This is joint work with Gloria Mari Beffa (U Wisconsin-Madison).
In this talk, I will outline results obtained in the last fifteen years, all stemming from the original aim to include boundary conditions into the celebrated Inverse Scattering Transform, which is in essence a nonlinear Fourier transform.
The talk will revisit the Fourier transform on R, embedding it in a general way of thinking about integral transform that relies on a formulation in the complex domain (called a Riemann-Hilbert formulation) and start from this idea to describe a generalised approach, now known as the unified transform, or Fokas transform. This circle of ideas has produced unexpected and very general results for the solution of linear boundary value problems, including interface and moving boundary problems, new ideas for numerical schemes, a way to study of nonselfadjoint differential operators, as well as going some way to achieving the original aim of solving nonlinear integrable problems in domains with boundaries.
I will describe the key ideas and some of the most unexpected results.
Discontinuous Galerkin (DG) methods, as the name suggests, are a class of Galerkin finite element methods in which the condition of continuity of functions across element boundaries is relaxed. DG methods have become a powerful tool in the approximate solution of hyperbolic problems, the context in which they were first developed. DG methods have also been applied with much success to elliptic PDEs and systems, and it is this class of applications that will be the focus of this presentation. A simple model problem will serve as the vehicle for motivating the DG approach and comparing it to the conventional finite element method. A further application will be in the context of problems arising in solid and fluid mechanics, and in which dependence on a parameter may lead to singular behaviour of the approximate solution. It will be shown how uniformly convergent DG approaches can be constructed in such situations.
Quadrature is the term for the numerical evaluation of integrals. It's a beautiful subject because it's so accessible, yet full of conceptual surprises and challenges. This talk will review ten of these, with plenty of history and numerical demonstrations. Some are old if not well known, some are new, and two are subjects of my current research.
The talk will be about simultaneous Gaussian quadrature (introduced by Borges in 1994) for two integrals of the same function f but on two disjoint intervals. The quadrature nodes are zeros of a type II multiple orthogonal polynomial for an Angelesco system. We recall some known results for the quadrature nodes and the quadrature weights and prove some new results about the convergence of the quadrature formulas. Furthermore we give some estimates of the quadrature weights. Our results are based on a vector equilibrium problem in potential theory and weighted polynomial approximation. This is joint work with Doron Lubinsky (Georgia Institute of Technology, Atlanta GA, USA).
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