(1) "On the ideas of Multigrid, Domain Decomposition, and RAS" (2) "Simple Basic Preconditioning ideas for some Kernel methods" (3) "Two-scale Neural Networks for optimal control of linear convection-dominated equations"

Prelegent(ci)

(1) Yi Yu (2) Blanca Avuso De Dios (3) Marcus Sarkis

Afiliacja

(1) Guangxi University, China. (2) Universita degli Studi Milano-Bicocca, Italy. (3) Worcester Polytechnic Institute, USA.

Język referatu

angielski

Termin

11 czerwca 2026 10:15

Pokój

p. 4070

Seminarium

Seminarium Zakładu Analizy Numerycznej

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(1) Classical preconditioners for linear systems often fall into two main categories: multigrid method (successive subspace correction, SSC) and domain decomposition method (parallel subspace correction, PSC). In this talk, we focus on domain decomposition methods and discuss Restricted Additive Schwarz (RAS) and its transpose form ASH (Additive Schwarz with Harmonic extension). We show the difference with the traditional Additive Schwarz Method (ASM) and demonstrate by numerical examples that "RAS is superior to ASM in terms of both iteration counts and CPU time, as well as communication cost when implemented on distributed memory computers." [1] We then discuss the fundamental concepts and key ideas behind RAS, and show that these ideas can be naturally extended to nonlinear preconditioners and emerging applications in neural networks.
[1] X.C. Cai and M. Sarkis, A restricted additive Schwarc preconditioner for general sparse linear systems, SISC, 1999

(2) Kernel methods provide an elegant and often effective framework to design algorithms for non-linear, non-parametric learning. However, at least in their basic form, the resulting schemes become unfeasible when dealing with large datasets due to computational requirements in terms of time and especially memory. Overcoming these drawbacks  has prompt an upsurge of research in the design of several computational strategies for large scale kernel methods. To alleviate the memory limitation some kind of Randomization is typically used. In this talk we consider the solution to a Kernel Ridge Regression (KRR) problem. We explore how some simple ideas from Domain Decomposition/Subspace Correction methods can be combined with existing solvers to alleviate the memory bottleneck and improve the efficiency. Extensive numerical results show the viability of the proposed solvers. The talk is based on the current collaboration with G. Meanti, L. Rosasco and G. Vitale from University of Genova, and M. Rando (now at INRIA).

(3) We propose a two-scale neural network method for optimal control problems governed by convection-dominated convection–diffusion–reaction equations. Building on two-scale architectures developed for singularly perturbed forward problems, we augment the spatial input with suitably rescaled features that become increasingly important as the diffusion coefficient becomes small. The approach employs separate neural networks for the state and adjoint state variables of the optimality system, reflecting the fact that these quantities develop sharp layers in different parts of the domain due to opposite convection fields. By choosing different center points for the two networks, the architecture naturally aligns with the layer location of each variable. We present two formulations of the method, one based on the first-order optimality conditions and another using penalization of the PDE constraint and combine them with a successive training strategy that gradually decreases the diffusion coefficient toward its target value. Numerical experiments on benchmark problems illustrate the effectiveness and behavior of the proposed approach. This is a joint work with Sijing Liu (WPI) , Zhongqiang Zhang (WPI) and Yi Zhang (The University of North Carolina at Greensboro, USA)

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