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UID:20250712T122019EDT-99256tpaVf@132.216.98.100
DTSTAMP:20250712T162019Z
DESCRIPTION:Title:\n\nCompressive Fourier collocation methods for high-dime
 nsional diffusion equations with periodic boundary conditions\n\nAbstract:
  \n\nHigh-dimensional Partial Differential Equations (PDEs) are a popular 
 mathematical modelling tool. However\, standard numerical techniques for s
 olving High-dimensional PDEs are typically affected by the curse of dimens
 ionality. In this work\, we tackle this challenge while focusing on statio
 nary diffusion equations defined over a high-dimensional domain with perio
 dic boundary conditions. Inspired by recent progress in high-dimensional s
 parse function approximation\, we propose a new method called compressive 
 Fourier collocation. Combining ideas from compressive sensing and spectral
  collocation\, our method uses Monte Carlo sampling and employs sparse rec
 overy techniques\, such as orthogonal matching pursuit and L1 minimization
 \, to approximate the Fourier coefficients on given index sets of the PDE 
 solution. We conduct a rigorous theoretical analysis showing that the appr
 oximation error of the proposed method is comparable with the best s-term 
 approximation (with respect to the Fourier basis) to the solution and miti
 gates the curse of dimensionality with respect to the number of collocatio
 n points under sufficient conditions on the regularity of the diffusion co
 efficient. We present numerical experiments that illustrate the accuracy a
 nd stability of the method for the approximation of sparse and compressibl
 e solutions. In our current work\, noticing that a bottleneck towards impr
 oving the solution accuracy is the choice of the index set\, we develop a 
 method using orthogonal matching pursuit to select the elements of the ind
 ex set adaptively. In addition\, we seek an efficient neural network model
  to solve the high-dimensional PDE\, to compare the performance of the ada
 ptive method with a deep learning-based approach.\n
DTSTART:20230213T213000Z
DTEND:20230213T223000Z
LOCATION:Room 1104\, Burnside Hall\, CA\, QC\, Montreal\, H3A 0B9\, 805 rue
  Sherbrooke Ouest
SUMMARY:Weiqi Wang  (Concordia University)
URL:/mathstat/channels/event/weiqi-wang-concordia-univ
 ersity-345859
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