University of Coimbra

We consider degenerate fully nonlinear equations, whose degeneracy rate depends on the gradient of solutions. We work under a Dini-continuity condition on the degeneracy term and prove that solutions are continuously differentiable. Then we frame this class of equations in the context of a free transmission problem. Here, we discuss the existence of solutions and establish a result on interior regularity. We conclude the talk by discussing a boundary regularity estimate; of particular interest is the case of point-wise regularity at the intersection of the fixed and the free boundaries. This is based on joint work with David Stolnicki.

Direct-search methods are a class of popular derivative-free

algorithms characterized by evaluating the objective function

using a step size and a number of (polling) directions.

When applied to the minimization of smooth functions, the

polling directions are typically taken from positive spanning sets

which in turn must have at least n+1 vectors in an n-dimensional variable space.

In addition, to ensure the global convergence of these algorithms,

the positive spanning sets used throughout the iterations

must be uniformly non-degenerate in the sense of having a positive

(cosine) measure bounded away from zero.

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However, recent numerical results indicated that randomly generating

the polling directions without imposing the positive spanning property

can improve the performance of these methods, especially when the number

of directions is chosen considerably less than n+1.

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In this talk, we analyze direct-search algorithms when the polling

directions are probabilistic descent, meaning that with a certain

probability at least one of them is of descent type. Such a framework

enjoys almost-sure global convergence. More interestingly, we will show

a global decaying rate of $1/\sqrt{k}$ for the gradient size, with

overwhelmingly high probability, matching the corresponding rate for

the deterministic versions of the gradient method or of direct search.

Our analysis helps to understand numerical behavior and the choice of

the number of polling directions.

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This is joint work with Clément Royer, Serge Gratton, and Zaikun Zhang.

Using the Borel-Schur algebra, we construct explicit characteristic-free resolutions for Weyl modules for the general linear group. These resolutions provide an answer to the problem, posed in the 80's by A. Akin and D. A. Buchsbaum, of constructing finite explicit and universal resolutions of Weyl modules by direct sums of divided powers. Next we apply the Schur functor to these resolutions and prove a conjecture of Boltje and Hartmann on resolutions of co-Specht modules. This is joint work with I. Yudin.

The advent of multicore processors and other technologies like Graphical Processing Units (GPU) will considerably influence future research in High Performance Computing.

To take advantage of these architectures in dense linear algebra operations, new algorithms are

proposed that use finer granularity and minimize synchronization points.

After presenting some of these algorithms, we address the issue of pivoting and investigate randomization techniques to avoid pivoting in some cases.

In the particular case of GPUs, we show how linear algebra operations can be enhanced using

hybrid CPU-GPU calculations and mixed precision algorithms.