Event(s) on November 2022

• Wednesday, 9th November, 2022

  Title:	Finite Expression Method for Solving High-Dimensional PDEs
  Speaker:	Haizhao Yang, University of Maryland College Park
  Time/Place:	10:00  -  11:00 Zoom, (Meeting ID: 997 2337 7319)
  Abstract:	Designing efficient and accurate numerical solvers for high-dimensional partial differential equations (PDE) remains a challenging and essential topic in computational science and engineering, mainly due to the ``curse of dimensionality" in designing numerical schemes that scale in dimension. This talk introduces a new methodology that seeks an approximate PDE solution in the space of functions with finitely many analytic expressions and, hence, this methodology is named the finite expression method (FEX). It is proved in approximation theory that FEX can avoid the curse of dimensionality. As a proof of concept, a deep reinforcement learning method is proposed to implement FEX for various high-dimensional PDEs in different dimensions, achieving high and even machine accuracy with a memory complexity polynomial in dimension and an amenable time complexity. An approximate solution with finite analytic expressions also provides interpretable insights into the ground truth PDE solution, which can further help to advance the understanding of physical systems and design postprocessing techniques for a refined solution.

  
  

• Friday, 25th November, 2022

  Title:	An efficient unconditionally stable method for computing Dirichlet partitions in arbitrary domains
  Speaker:	Dr. Wang Dong, The Chinese University of Hong Kong (Shenzhen)
  Time/Place:	16:00  -  17:00 Zoom, Meeting ID: 959 1528 4941
  Abstract:	A Dirichlet k-partition of a domain is a collection of k pairwise disjoint open subsets such that the sum of their first Laplace--Dirichlet eigenvalues is minimal. In this talk, we propose a new relaxation of the problem by introducing auxiliary indicator functions of domains and develop a simple and efficient diffusion generated method to compute Dirichlet k-partitions for arbitrary domains. The method only alternates three steps: 1. convolution, 2. thresholding, and 3. projection. The method is simple, easy to implement, insensitive to initial guesses and can be effectively applied to arbitrary domains without any special discretization. At each iteration, the computational complexity is linear in the discretization of the computational domain. Moreover, we theoretically prove the energy decaying property of the method. Experiments are performed to show the accuracy of approximation, efficiency and unconditional stability of the algorithm. We apply the proposed algorithms on both 2- and 3-dimensional flat tori, triangle, square, pentagon, hexagon, disk, three-fold star, five-fold star, cube, ball, and tetrahedron domains to compute Dirichlet k-partitions for different k to show the effectiveness of the proposed method. Compared to previous work with reported computational time, the proposed method achieves hundreds of times acceleration.

  
  

• Wednesday, 30th November, 2022

  Title:	A parallel preconditioner for the all-at-once linear system from evolutionary PDEs with Crank-Nicolson discretization in time
  Speaker:	Prof. Xian-Ming Gu, School of Mathematics, Southwestern University of Finance and Economics
  Time/Place:	16:00  -  17:00 Zoom, Meeting ID: 999 7518 5920
  Abstract:	The Crank-Nicolson (CN) method is a fashionable time integrator for evolutionary partial differential equations (PDEs) arisen in many areas of applied mathematics, however since the solution at any time depends on the solution at previous time steps, thus the CN method will be inherently difficult to parallelize. In this talk, we consider a parallel approach for the solution of evolutionary PDEs with the CN scheme. Using an all at-once approach, we can solve for all time steps simultaneously using a parallelizable over time preconditioner within a standard iterative method. Due to the diagonalization of the proposed preconditioner, we can minutely prove that most eigenvalues of preconditioned matrices are equal to 1 and the others will locate in the interval [1/(1 + α), 1/(1 - α)], where 0 < α < 1 is a free parameter in the preconditioner. Meanwhile, the efficient and parallel implementation of this proposed preconditioner is described in details. Finally, we will verify our theoretical findings via numerical experiments.

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