This session will take place live in L1 and online. A Teams link will be shared 30 minutes before the session begins.
This session will take place live in L1 and online. A Teams link will be shared 30 minutes before the session begins.
Zoom passcode: Basis
Giancarlo Rota once wrote of matroids that "It is as if one were to
condense all trends of present day mathematics onto a single
structure, a feat that anyone would a priori deem impossible, were it
not for the fact that matroids do exist" (Indiscrete Thoughts, 1997).
This makes matroid theory a natural hub through which ideas flow from
one field of mathematics to the next. At the end of our three-day
workshop, participants will understand the most common objects and
constructions in matroid theory to the depth suitable for exploring
many of these interesting connections. We will also pick up some
highly practical matroid tools for working through problems in
persistent homology, (optimal) cycle representatives, and other
objects of interest in TDA.
Circuits in persistent homology
Exercise: write your own persistent homology algorithm!
Zoom Passcode: Basis
Giancarlo Rota once wrote of matroids that "It is as if one were to
condense all trends of present day mathematics onto a single
structure, a feat that anyone would a priori deem impossible, were it
not for the fact that matroids do exist" (Indiscrete Thoughts, 1997).
This makes matroid theory a natural hub through which ideas flow from
one field of mathematics to the next. At the end of our three-day
workshop, participants will understand the most common objects and
constructions in matroid theory to the depth suitable for exploring
many of these interesting connections. We will also pick up some
highly practical matroid tools for working through problems in
persistent homology, (optimal) cycle representatives, and other
objects of interest in TDA.
Matroid representations, continued
Matroids in homological algebra
Zoom Passcode: Basis
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Giancarlo Rota once wrote of matroids that "It is as if one were to
condense all trends of present day mathematics onto a single
structure, a feat that anyone would a priori deem impossible, were it
not for the fact that matroids do exist" (Indiscrete Thoughts, 1997).
This makes matroid theory a natural hub through which ideas flow from
one field of mathematics to the next. At the end of our three-day
workshop, participants will understand the most common objects and
constructions in matroid theory to the depth suitable for exploring
many of these interesting connections. We will also pick up some
highly practical matroid tools for working through problems in
persistent homology, (optimal) cycle representatives, and other
objects of interest in TDA.
Definitions There are many definitions of matroids. Here's how to organize them.
Examples We work with matroids every day. Here are a few you have seen.
Important properties What's so great about a matroid?
The essential operations: deletion, contraction, and dualization
Working with matroids: matrix representations
That neural networks may be pruned to high sparsities and retain high accuracy is well established. Recent research efforts focus on pruning immediately after initialization so as to allow the computational savings afforded by sparsity to extend to the training process. In this work, we introduce a new `DCT plus Sparse' layer architecture, which maintains information propagation and trainability even with as little as 0.01% trainable kernel parameters remaining. We show that standard training of networks built with these layers, and pruned at initialization, achieves state-of-the-art accuracy for extreme sparsities on a variety of benchmark network architectures and datasets. Moreover, these results are achieved using only simple heuristics to determine the locations of the trainable parameters in the network, and thus without having to initially store or compute with the full, unpruned network, as is required by competing prune-at-initialization algorithms. Switching from standard sparse layers to DCT plus Sparse layers does not increase the storage footprint of a network and incurs only a small additional computational overhead.
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Topology optimisation finds the optimal material distribution of a fluid or solid in a domain, subject to PDE, volume, and box constraints. The optimisation problem is normally nonconvex and can support multiple local minima. In recent work [1], the authors developed an algorithm for systematically discovering multiple minima of two-dimensional problems through a combination of barrier methods, active-set strategies, and deflation. The bottleneck of the algorithm is solving the Newton systems that arise. In this talk, we will present preconditioning methods for these linear systems as they occur in the topology optimization of Stokes flow. The strategies involve a mix of block preconditioning and specialized multigrid relaxation schemes that reduce the computational work required and allow the application of the algorithm to three-dimensional problems.
[1] “Computing multiple solutions of topology optimization problems”, I. P. A. Papadopoulos, P. E. Farrell, T. M. Surowiec, 2020, https://arxiv.org/abs/2004.11797
A link for this talk will be sent to our mailing list a day or two in advance. If you are not on the list and wish to be sent a link, please contact @email.
A wide variety of fixed-point iterative methods for the solution of nonlinear operator equations in Hilbert spaces exists. In many cases, such schemes can be interpreted as iterative local linearisation methods, which can be obtained by applying a suitable preconditioning operator to the original (nonlinear) equation. Based on this observation, we will derive a unified abstract framework which recovers some prominent iterative methods. It will be shown that for strongly monotone operators this unified iteration scheme satisfies an energy contraction property. Consequently, the generated sequence converges to a solution of the original problem.
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A link for this talk will be sent to our mailing list a day or two in advance. If you are not on the list and wish to be sent a link, please contact @email.
Point cloud registration is the task of finding the transformation that aligns two data sets. We make the assumption that the data lies on a low-dimensional algebraic variety. The task is phrased as an optimization problem over the special orthogonal group of rotations. We solve this problem using Riemannian optimization algorithms and show numerical examples that illustrate the efficiency of this approach for point cloud registration.
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A link for this talk will be sent to our mailing list a day or two in advance. If you are not on the list and wish to be sent a link, please contact @email.
For high order finite elements (continuous piecewise polynomials) the conditioning of the basis is important. However, so far there seems no generally accepted concept of "a well-conditioned basis”, or a general strategy for how to obtain such representations. In this presentation, we use the $L^2$ condition number as a measure of the conditioning, and construct representations by frames such that the associated $L^2$ condition number is bounded independently of the polynomial degree. The main tools include the bubble transform, which is a stable decomposition of functions into local modes, and orthogonal polynomials on simplexes. We also include a brief discussion on potential applications in preconditioning. This is a joint work with Ragnar Winther.
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A link for this talk will be sent to our mailing list a day or two in advance. If you are not on the list and wish to be sent a link, please contact @email.