Date/Time : 06 Mar 2024 15:00
Venue : S17 #05-12 (SR4)
Speaker : Akiko Takeda, University of Tokyo

Random projection techniques based on the Johnson-Lindenstrauss lemma are used for randomly aggregating the constraints or variables of optimization problems while approximately preserving their optimal values, which leads to smaller-scale optimization problems. In this talk, we show a few applications of random matrix techniques for constructing random subspace algorithms that iteratively solve smaller-scale subproblems and discuss their convergence speed. This talk is based on joint works with Ryota Nozawa, and Pierre-Louis Poirion.

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Date/Time : 06 Mar 2024 15:40
Venue : S17 #05-12 (SR4)
Speaker : Pierre-Louis Poirion, University of Tokyo

In this talk, we present a randomized subspace regularized Newton method for a non-convex function. We show that our method has global convergence under appropriate assumptions, and its convergence rate is the same as that of the full regularized Newton method. Furthermore, we can obtain a local linear convergence rate, under some additional assumptions, and prove that this rate is the best we can hope, under local strong convexity sumptions, when using random subspace. Finally, we will prove that under rank deficiency conditions, one can obtain super-linear convergence in the subspace setting.

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Date/Time : 06 Mar 2024 15:30
Venue : S17 #04-05 (SR2)
Speaker : Viet-Anh Nguyen, Universite de Lille

Let $X$ be a compact normal complex space of dimension $n$ and $L$ be a holomorphic line bundle on $X$. Suppose that $\Sigma=(\Sigma_1,\ldots,\Sigma_\ell)$ is an $\ell$-tuple of distinct irreducible proper analytic subsets of $X$, $\tau=(\tau_1,\ldots,\tau_\ell)$ is an $\ell$-tuple of positive real numbers, and let $H^0_0(X,L^p)$ be the space of holomorphic sections of $L^p:=L^{\otimes p}$ that vanish to order at least $\tau_jp$ along $\Sigma_j$, $1\leq j\leq\ell$. If $Y\subset X$ is an irreducible analytic subset of dimension $m$, we consider the space $H^0_0 (X|Y, L^p)$ of holomorphic sections of $L^p|_Y$ that extend to global holomorphic sections in $H^0_0(X,L^p)$. Assuming that the triplet $(L,\Sigma,\tau)$ is big in the sense that $\dim H^0_0(X,L^p)\sim p^n$, we give a general condition on $Y$ to ensure that $\dim H^0_0(X|Y,L^p)\sim p^m$. When $L$ is endowed with a continuous Hermitian metric, we show that the Fubini-Study currents of the spaces $H^0_0(X|Y,L^p)$ converge to a certain equilibrium current on $Y$. We apply this to the study of the equidistribution of zeros in $Y$ of random holomorphic sections in $H^0_0(X|Y,L^p)$ as $p\to\infty$.

This is a joint-work with Dan Coman and George Marinescu.

Keywords:
(Partial) Bergman kernel function, big line bundle, singular Hermitian metric, holomorphic section, big cohomology class.

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Date/Time : 06 Mar 2024 16:30
Venue : S17 #05-11 (SR5)
Speaker : Li Yifei, National University of Singapore

Designing a numerical scheme that can preserve the geometric structure for anisotropic surface diffusion with an arbitrary anisotropic surface energy is a long-standing problem. In this talk, we propose and analyze a structure-preserving parametric finite element methods (SP-PFEM) for the evolution of a closed curve in 2D, which preserve two geometric structures – area conservation and energy dissipation – at the full-discretized level, The SP-PFEM innovates with a novel surface energy matrix and the Cahn-Hoffman ξ-vector, leading to a new geometric identity for dealing with the weighted mean curvature. This new identity allows our SP-PFEM to be easily extended to various geometric flows with anisotropic effects. Extensive numerical results demonstrate its efficiency, stability, and success in other geometric flows.

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Date/Time : 06 Mar 2024 17:00
Venue : S17 #04-05 (SR2)
Speaker : Rupert Hoelzl, Universitaet der Bundeswehr, Munich

A left-computable number x is called regainingly approximable if there is a computable increasing sequence (x_n)_{n in N} of rational numbers converging to x such that x-x_n < 2^-n for infinitely many n in N; and it is called nearly computable if there is such an (x_n)_n such that for every computable increasing function s:N -> N the sequence (x_{s(n+1)}-x_{s(n)}){n in N} converges computably to 0.

In this talk we study the relationship between both concepts by constructing on the one hand a non-computable number that is both regainingly approximable and nearly computable, and on the other hand a left-computable number that is nearly computable but not regainingly approximable; it then easily follows that the two notions are incomparable with non-trivial intersection. With this relationship clarified, we then hold the keys to answering an open question of Merkle and Titov: they studied speedable numbers, that is, left-computable numbers whose approximations can be sped up in a certain sense, and asked whether, among the left-computable numbers, being Martin-Loef random is equivalent to being non-speedable. As we show that the concepts of speedable and regainingly approximable numbers are equivalent within the nearly computable numbers, our second construction provides a negative answer.

This is joint work with Philip Janicki.

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Date/Time : 28 Feb 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Jiequn Han, Flatiron Institute, USA

Partial differential equations (PDEs) are among the most ubiquitous tools in modeling natural phenomena, and various numerical methods have been developed for decades to solve PDE problems. While deep learning has introduced new techniques to the field, the limited accuracy of deep neural networks hinders their application to scientific problems that require high precision.

This talk will present a warm-start approach that combines the strengths of deep neural networks and classical numerical solvers. The approach uses neural networks to provide an initial guess, enabling classical numerical solvers to achieve a good solution more efficiently. We will demonstrate the advantages of the proposed method through two examples. In the first example, we will demonstrate how the initial guess provided by neural networks helps identify the basin of attraction of the true solution in the inverse scattering problem. In the second example, we will show how the initial guess provided by neural networks leads to faster convergence in solving the Reynolds-averaged Navier-Stokes equations. In both examples, the combination of the classical PDE solver and the neural network outperforms either approach alone. The potential of this approach will be discussed, along with the new challenges that must be tackled to solve challenging scientific problems using this new paradigm

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Date/Time : 28 Feb 2024 16:30
Venue : S17 #05-12 (SR4)
Speaker : Guo Ruchi, The Chinese University of Hong Kong

In physics and mathematics, a large class of PDE systems can be formulated as energy minimization problems subject to specific constraints. Lagrange multipliers are widely used for solving such problems, which however leads to min-max optimization problems known as saddle point systems. The development of fast solvers for saddle point systems, especially those nonlinear ones, presents considerable challenges in the sense that

(i) one has to consider the preconditioning in the primal and dual directions collectively;

(ii) the preconditioners have to evolve in iterations to accommodate the nonlinearity;

(iii) all the subsystems and subproblems cannot be solved exactly due to the high computational cost involved.

In this work, we develop an efficient transformed primal-dual (TPD) algorithm, and this is done by modifying the original gradient flow from the perspective of matrix factorization. We prove the global optimal linear convergence in terms of the condition numbers measured by variable metrics. To demonstrate the effectiveness of our approach, we apply the algorithm to a Darcy-Forchheimer model and a nonlinear ferromagnetic model, showing it is much more efficient than some traditional methods.

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Date/Time : 27 Feb 2024 14:00
Venue : S17 #05-12 (SR4)
Speaker : Shilin Yu, Xiamen University

Unipotent representations are believed to be the building blocks of unitary dual of semisimple/reductive Lie groups. In their study of special unipotent representations for complex semisimple groups, Barbasch and Vogan constructed a duality map between the nilpotent orbits of $G$ and that of its Langlands dual group $G^\vee$ (also discovered by Lusztig and Spaltenstein), which allows them to describe the unipotent ideals and representations of $G$ in terms of $G^\vee$. Later Sommers and Achar extended this duality map by considering pairs $(\mathbb{O}, C)$ consisting of a nilpotent orbit $\mathbb{O}$ and a conjugacy class $C$ in its Lusztig canonical quotient group.

In the joint work with Lucas Mason-Brown and Dmytro Matvieievskyi (arXiv:2309.14853), we have extended the Sommers’ duality map in another direction by considering covers of nilpotent orbits, which turns out to be equivalent to Achar’s duality map (in a nontrivial way). This extended duality map allows us to describe the (generalized) unipotent ideals and bimodules attached to nilpotent covers, proposed previously by Ivan Losev and my other two coauthors, in terms of the Langlands dual group.

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Date/Time : 22 Feb 2024 14:00
Venue : S17 #05-11 (SR5)
Speaker : Sebastian Pokutta, Zuse Institute Berlin (ZIB) and TU Berlin

Conditional Gradient methods are an important class of methods to minimize (non-)smooth convex functions over (combinatorial) polytopes. Recently these methods received a lot of attention as they allow for structured optimization and hence learning, incorporating the underlying polyhedral structure into solutions. In this talk I will give a broad overview of these methods, their applications, as well as present some recent results both in traditional optimization and learning as well as in deep learning.

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