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

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The CLE_4 Conformal Loop Ensemble in a 2D simply connected domain is a random countable collection of fractal Jordan curves that satisfies a statistical conformal invariance and appears, or is conjectured to appear, as a scaling limit of interfaces in various statistical physics models in 2D, for instance in the double dimer model. The CLE_4   is also related to the 2D Gaussian free field. Given a simply connected domain D and a point z in D, we consider the CLE_4 loop that surrounds z and study the extremal distance between the loop and the boundary of the domain, and the conformal radius of the interior surrounded by the loop seen from z. Because of the confomal invariance, the joint law of this two quantities does not depend (up to a scale factor) on the choice of the domain D and the point z in D. The law of the conformal radius alone has been known since the works of Schramm, Sheffield and Wilson. We complement their result by deriving the joint law of (extremal distance, conformal radius). Both quantities can be read on the same 1D Brownian path, by tacking a last passage time and a first hitting time. This joint law, together with some distortion bounds, provides some exponents related to the CLE_4. This is a joint work with Juhan Aru and Avelio Sepulveda.

We combine the notions of graded Clifford algebras and Drinfeld algebras. This gives us a framework to study algebras with a PBW property and underlying vector space $\mathbb{C}[G] \# Cl(V) \otimes S(U) $ for $G$-modules $U$ and $V$. The class of graded Clifford-Drinfeld algebras contains the Hecke-Clifford algebras defined by Nazarov, Khongsap-Wang. We give a new example of a GCD algebra which plays a role in an Arakawa-Suzuki duality involving the Clifford algebra.

We discuss a novel backward Ito-Ventzell formula and an extension of the Aleeksev-Gröbner interpolating formula to stochastic flows. We also present some natural spectral conditions that yield direct and simple proofs of time uniform estimates of the difference between the two stochastic flows when their drift and diffusion functions are not the same, yielding what seems to be the first results of this type for this class of  anticipative models.

We illustrate the impact of these results in the context of diffusion perturbation theory, interacting diffusions and discrete time approximations.

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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