On sparse representations for piecewise smooth signals
Abstract
It is well known that piecewise smooth signals are approximately sparse in a wavelet basis. However, other sparse representations are possible, such as the discrete gradient basis. It turns out that signals drawn from a random piecewise constant model have sparser representations in the discrete gradient basis than in Haar wavelets (with high probability). I will talk about this result and its implications, and also show some numerical experiments in which the use of the gradient basis improves compressive signal reconstruction.
Carleman Estimates and Unique Continuation for Fractional Schroedinger Equations
Abstract
equations and discuss how these imply the strong unique continuation
principle even in the presence of rough potentials. Moreover, I show how
they can be used to derive quantitative unique continuation results in
the setting of compact manifolds. These quantitative estimates can then
be exploited to deduce upper bounds on the Hausdorff dimension of nodal
domains (of eigenfunctions to the investigated Dirichlet-to-Neumann maps).
Fast evaluation of the inverse Poisson CDF
Abstract
One general approach to random number generation is to take a uniformly distributed (0,1) random variable and then invert the cumulative distribution function (CDF) to generate samples from another distribution. This talk follows this approach, approximating the inverse CDF for the Poisson distribution in a way which is particularly efficient for vector execution on NVIDIA GPUs.
Towards the compatibility of Geometric Langlands with the extended Whittaker model
Abstract
Let $G$ be a connected reductive group and $X$ a smooth complete curve, both defined over an algebraically closed field of characteristic zero. Let $Bun_G$ denote the stack of $G$-bundles on $X$. In analogy with the classical theory of Whittaker coefficients for automorphic functions, we construct a “Fourier transform” functor, called $coeff_{G}$, from the DG category of D-modules on $Bun_G$ to a certain DG category $Wh(G, ext)$, called the extended Whittaker category. Combined with work in progress by other mathematicians and the speaker, this construction allows to formulate the compatibility of the Langlands duality functor $$\mathbb{L}_G : \operatorname{IndCoh}_{N}(LocSys_{\check{G}} ) \to D(Bun_G)$$ with the Whittaker model. For $G = GL_n$ and $G = PGL_n$, we prove that $coeff_G$ is fully faithful. This result guarantees that, for those groups, $\mathbb{L}_G$ is unique (if it exists) and necessarily fully faithful.
Fast and backward stable computation of roots of polynomials
Abstract
A stable algorithm to compute the roots of polynomials is presented. The roots are found by computing the eigenvalues of the associated companion matrix by Francis's implicitly-shifted $QR$ algorithm. A companion matrix is an upper Hessenberg matrix that is unitary-plus-rank-one, that is, it is the sum of a unitary matrix and a rank-one matrix. These properties are preserved by iterations of Francis's algorithm, and it is these properties that are exploited here. The matrix is represented as a product of $3n-1$ Givens rotators plus the rank-one part, so only $O(n)$ storage space is required. In fact, the information about the rank-one part is also encoded in the rotators, so it is not necessary to store the rank-one part explicitly. Francis's algorithm implemented on this representation requires only $O(n)$ flops per iteration and thus $O(n^{2})$ flops overall. The algorithm is described, backward stability is proved under certain conditions on the polynomial coefficients, and an extensive set of numerical experiments is presented. The algorithm is shown to be about as accurate as the (slow) Francis $QR$ algorithm applied to the companion matrix without exploiting the structure. It is faster than other fast methods that have been proposed, and its accuracy is comparable or better.
The convergence of stationary iterations with indefinite splitting
Abstract
The relationship of diagonal dominance ideas to the convergence of stationary iterations is well known. There are a multitude of situations in which such considerations can be used to guarantee convergence when the splitting matrix (the preconditioner) is positive definite. In this talk we will describe and prove sufficient conditions for convergence of a stationary iteration based on a splitting with an indefinite preconditioner. Simple examples covered by this theory coming from Optimization and Economics will be described.
This is joint work with Michael Ferris and Tom Rutherford
X-ray imaging with emitter arrays
Abstract
We investigate an X-ray imaging system that fires multiple point sources of radiation simultaneously from close proximity to a probe. Radiation traverses the probe in a non-parallel fashion, which makes it necessary to use tomosynthesis as a preliminary step to calculating a 2D shadowgraph. The system geometry requires imaging techniques that differ substantially from planar X-rays or CT tomography. We present a proof of concept of such an imaging system, along with relevant artefact removal techniques. This work is joint with Kishan Patel.
Phase transitions in bootstrap percolation
Abstract
Colouring graphs without odd holes
Abstract
Gyárfás conjectured in 1985 that if $G$ is a graph with no induced cycle of odd length at least 5, then the chromatic number of $G$ is bounded by a function of its clique number. We prove this conjecture. Joint work with Paul Seymour.