Dear GMS,
I hope you are doing well.
We will have a seminar on Monday, March 23, between 11:30 and 12:30 at 225 St. Paul's.
Our speaker will be @Gabriel Hamm<mailto:hammg2@myumanitoba.ca>. You can find more information about his talk below.
Title: Fast algorithms for Sobolev orthogonal polynomials
Abstract: Sobolev orthogonal polynomials are those polynomials orthogonal with respect to an inner product including derivatives. They have a beautiful and rich theoretical history. We propose fast algorithms for Sobolev orthogonal polynomials by careful consideration of the properties of the Sobolev–Gram matrix. We begin with a description of a matrix equation for the Sobolev–Gram matrix with a number of terms proportional to the order of the Sobolev inner product. We describe conditions on the vectorial measure that cause the Sobolev–Gram matrix to be banded, leading to linear complexity Cholesky factorization. Next, we convert the problem of including Dirac measures into that of a finite-rank perturbation of a known Cholesky factorization, which can also be performed in linear complexity. Finally, in the case of conversion to Chebyshev polynomials, we harness the power of randomized numerical linear algebra to solve the connection problem in O(n log^{O(1)}n) flops.
See you all in the seminar!
GMS Website<https://sites.google.com/view/umgradmathsociety/> / Instagram<https://www.instagram.com/umgradmathsociety/>
Sincerely,
Berkant<https://cnnk.xyz>
GMS Executive
Dear GMS,
I hope this email finds you well.
We will have a seminar on March 9, Monday (tomorrow) between 11:30 and 12:30 at 225 St. Paul's.
Our speaker will be @Gabriel Ogulu<mailto:Gabriel.Ogulu@umanitoba.ca>. You can find more information about his talk below.
Title: Matrix Analysis of the Ultraspherical Spectral Method
Abstract: Spectral methods are prized in scientific computing for their exponential convergence rates, yet classical collocation approaches suffer from severe algebraic bottle necks: they generate dense differentiation matrices with condition numbers scaling as O(N4). This research project investigates the Ultraspherical Spectral Method, a structured approach that overcomes these limitations by utilizing a hierarchy of orthogonal polynomial bases to represent differentiation as a sparse, banded operation. We perform a rigorous matrix analysis of the infinite-dimensional operators arising from this discretization, specifically constructing the banded differentiation (D(2)), multiplication (X), and basis conversion (R(2)) matrices. A key focus is the solution of ”almost-banded” linear systems structures arising from the imposition of boundary conditions on banded operators. We implement a stable QR factorization algorithm utilizing Givens rotations to solve these systems while minimizing fill-in. Numerical experiments performed in Julia validate the theoretical claims on two canonical problems: the Airy boundary value problem and the Harmonic Oscillator eigenvalue problem. Our benchmarks confirm that the ultraspherical solver achieves O(N) computational complexity, a significant improvement over the O(N3) scaling of dense spectral methods, while maintaining bounded or slowly growing condition numbers. This work demonstrates that exploiting structured linear algebra is essential for unlocking the full potential of spectral discretizations in modern differential equation solvers.
See you all in the seminar!
GMS Website<https://sites.google.com/view/umgradmathsociety/> / Instagram<https://www.instagram.com/umgradmathsociety/>
Sincerely,
Berkant<https://cnnk.xyz>
GMS Executive
