Title: Rational approximation and its role in different branches of mathematics and applications

Speaker: Nadezda Sukhorukova (Swinburne University of Technology)

Date and Time: Wed Nov 3, 17:00 AEST (Register here for remote connection via Zoom)

Abstract:

Rational approximation is a powerful function approximation tool. Rational approximation is approximation by a ratio of two polynomials, whose coefficients are subject to optimisation. Numerical methods for rational approximation have been developed independently in different branches of mathematics. In this talk, I will present the interconnections between different numerical methods developed to rational approximation. Most of them can be extended to the case of the so called generalised rational approximation where the approximation is a ration of two linear forms and the basis functions are not limited to monomials. Finally, I am going to talk about real-life applications for rational and generalised rational approximation.queness.

Title: Sharp and strong minima for robust recovery

Speaker: Nghia Tran (Oakland University)

Date and Time: Wed Oct 27, 11:00 AEST (Register here for remote connection via Zoom)

Abstract:

In this talk, we show the important roles of sharp minima and strong minima for robust recovery. We also obtain several characterizations of sharp minima for convex regularized optimization problems. Our characterizations are quantitative and verifiable especially for the case of decomposable norm regularized problems including sparsity, group-sparsity, and low-rank convex problems. For group-sparsity optimization problems, we show that a unique solution is a strong solution and obtain quantitative characterizations for solution uniqueness.

Title: Tweaking Ramanujan’s Approximation of n!

Speaker: Sidney Morris (Federation University, and La Trobe University)

Date and Time: Wed Oct 20, 17:00 AEST (Register here for remote connection via Zoom)

Abstract:

In 1730 James Stirling, building on the work of Abraham de Moivre, published what is known as Stirling’s approximation of n!. He gave a good formula which is asymptotic to n!. Since then hundreds of papers have given alternative proofs of his result and improved upon it, including notably by Burside, Gosper, and Mortici. However Srinivasa Ramanujan gave a remarkably better asymptotic formula. Hirschhorn and Villarino gave a nice proof of Ramanujan’s result and an error estimate for the approximation. 

This century there have been several improvements of Stirling’s formula including by Nemes, Windschitl, and Chen. In this presentation it is shown 

(i) how all these asymptotic results can be easily verified; 

(ii) how Hirschhorn and Villarino’s argument allows a tweaking of Ramanujan’s result to give a better approximation; 

(iii) that a new asymptotic formula can be obtained by further tweaking of Ramanujan’s result;

(iv) that Chen’s asymptotic formula is better than the others mentioned here, and the new asymptotic formula is comparable with Chen’s.

Title: A product space reformulation with reduced dimension

Speaker: Rubén Campoy (University of Valencia)

Date and Time: Wed Oct 13, 17:00 AEST (Register here for remote connection via Zoom)

Abstract:

The product space reformulation is a powerful trick when tackling monotone inclusions defined by finitely many operators with splitting algorithms. This technique constructs an equivalent two-operator problem, embedded in a product Hilbert space, that preserves computational tractability. Each operator in the original problem requires one dimension in the product space. In this talk, we propose a new reformulation with a reduction on the dimension of the outcoming product Hilbert space. We shall discuss the case of not necessarily convex feasibility problems. As an application, we obtain a new parallel variant of the Douglas-Rachford algorithm with a reduction in the number of variables. The computational advantage is illustrated through some numerical experiments.

Title: Alternating projections with applications to Gerchberg-Saxton error reduction

Speaker: Dominikus Noll (Institut de Mathématiques de Toulouse)

Date and Time: Wed Oct 6, 17:00 AEST (Register here for remote connection via Zoom)

Abstract:

We discuss alternating projections between closed non-convex sets A, B in R^n and obtain criteria for convergence when A, B do not intersect transversally. The infeasible case, A∩B=∅, is also addressed, and here we expect convergence toward a gap between A, B. For sub-analytic sets A, B sub-linear convergence rates depending on the Lojasiewicz exponent of the distance function can be computed. We then present applications to the Gerchberg-Saxton error reduction algorithm, to Cadzow’s denoising algorithm, and to instances of the Gaussian EM-algorithm.