Abstract : Numerical methods for solving nonlinear PDEs are at the heart of many scientific applications in physics, engineering, and biology. Recent advances in developing preconditioners and fast solvers bring significant improvement to the robustness and efficiency of numerical methods for nonlinear PDEs. A variety of novel techniques have been introduced such as nonlinear preconditioning, model order reduction, multiscale methods, heterogeneous computing, and machine learning. This minisymposium is to encourage communication among experts in these fields to discuss cutting-edge topics of numerical methods for nonlinear PDEs and their applications.
Organizer(s) : Xiao-Chuan Cai, Rongliang Chen, Li Luo
[03539] A quasi-Newton method with a secant-like diagonal approximation of Jacobian for symmetric sparse nonlinear equations
Format : Talk at Waseda University
Author(s) :
Duc Quoc Huynh (National Central University)
Feng-Nan Hwang (National Central University)
Abstract : We propose and study a new variant of the quasi-Newton method with a secant-like diagonal approximation of Jacobian (QN-SDAJ) for solving sparse symmetric nonlinear equations (SSNEs). Such problems appear in various scientific computing applications, such as finding critical points that satisfy the first-order necessary condition of unconstrained optimization problems and numerical semilinear elliptic partial differential equations. The advantages of the proposed method are conceptually simple and easy to implement. We establish the global convergence of the proposed method in conjunction with a nonmonotone line search technique under some appropriate assumptions. Several numerical experiments for some benchmark problems demonstrate the efficiency of QN-SDAJ, which outperforms the alternatives, including exact Newton, nonlinear conjugate gradient, and Broyden--Fletcher--Goldfarb--Shanno (BFGS) methods. In addition, the proposed method can also be used as an effective nonlinear preconditioner to enhance the robustness and speed up the convergence of BFGS, especially for test cases with large dimensions.
[03281] Energy stable schemes for gradient flows based on the DVD method
Format : Talk at Waseda University
Author(s) :
Jizu Huang (Academy of mathematics and systems science, Chinese academy sciences)
Abstract : In this talk, we propose a new framework to construct energy stable scheme for gradient flows based on the discrete variational derivative method. Combined with the Runge--Kutta process, we can build an arbitrary high-order and unconditionally energy stable scheme based on the discrete variational derivative method. The new energy stable scheme is implicit and leads to a large sparse nonlinear algebraic system at each time step, which can be efficiently solved by using an inexact Newton type algorithm. To avoid solving nonlinear algebraic systems, we then present a relaxed discrete variational derivative method, which can construct linear unconditionally second-order energy stable schemes. Several numerical simulations are performed to investigate the efficiency, stability, and accuracy of the newly proposed schemes.
[03859] Efficient Schwarz Preconditioning Techniques for Nonlinear Problems Using FROSch
Format : Talk at Waseda University
Author(s) :
Alexander Heinlein (Delft University of Technology (TU Delft))
Axel Klawonn (University of Cologne)
Mauro Perego (Sandia National Laboratories)
Sivasankaran Rajamanickam (Sandia National Laboratories)
Lea Saßmannshausen (University of Cologne)
Ichitaro Yamazaki (Sandia National Laboratories)
Abstract : FROSch (Fast and Robust Overlapping Schwarz) is a framework for parallel Schwarz domain decomposition preconditioners in Trilinos. Due an algebraic approach, meaning that the preconditioners can be constructed from a fully assembled matrix, FROSch is applicable to a wide range of problems. This talk is focused on the application to nonlinear problems, including computational fluid dynamics and land ice simulations. Techniques for improving the efficiency and the use of GPU architectures are discussed.
[02038] Robustness and Adaptivity of Iterative Solvers
Format : Talk at Waseda University
Author(s) :
Chensong Zhang (AMSS)
Abstract : Linear systems arising from coupled PDEs in multiphysics applications could cause robustness problems for iterative solution methods. Solving large-scale linear algebraic systems in an efficient and robust manner is a dream for many computational scientists who work on practical engineering applications. In this talk, we review some old and new techniques for improving the robustness of iterative solvers for large-scale sparse linear equations. In particular, we will discuss methods based on machine learning to select solver components automatically to improve overall simulation performance. Based on this algorithm selection model, a self-adaptive procedure can be derived to improve the robustness of iterative solvers.
[02272] Fully implicit multi-physics solver for advanced fission nuclear power plant
Format : Talk at Waseda University
Author(s) :
Han Zhang (Tsinghua University)
Abstract : The fission nuclear reactor power plant is a multi-physics, multi-scale and multi-component coupling system, resulting in a nonlinear partial differential equation system. Fully-implicit methods, such as the Jacobian-free Newton-Krylov method and the Newton-Krylov method, are promising choices for effectively solving such complex nonlinear systems due to their super-linear convergence rate. This talk focuses on the development of fully-implicit solution method for the advanced nuclear reactor power plant, as well as its engineering application.
[01916] BDDC Algorithms for Oseen problems with HDG Discretizations
Format : Talk at Waseda University
Author(s) :
Xuemin Tu (University of Kansas)
Abstract : In this talk, the balancing domain decomposition by constraints methods (BDDC)
are applied to the linear system arising from the Oseen equation with the hybridizable discontinuous
Galerkin (HDG) discretization.
The original system is reduced to a subdomain interface problem which
is asymmetric indefinite but can be positive definite in a special subspace.
Edge/face average constraints can ensure all BDDC preconditioned GMRES
iterates stay in this special subspace. Some additional edge/face constraints
are used to improve the convergence. When the viscosity
is large and the subdomain size is small enough, the number of iterations is independent of the number of subdomains and
depends only slightly on the subdomain problem size. When
the viscosity is small, the convergence can deteriorate.
[01407] Nonlinear Preconditioning Strategies Based on Residual Learning for PDEs
Format : Talk at Waseda University
Author(s) :
Li Luo (University of Macau)
Xiao-Chuan Cai (University of Macau)
Abstract : We present nonlinearly preconditioned inexact Newton methods for solving highly nonlinear system of algebraic equations from the discretization of PDEs. From a large number of numerical experiments, we observe that when the inexact Newton stagnates or fails to converge, the space of residuals often contains a subspace that is difficult to resolve by Newton iteration. We introduce a learning technique to identify this subspace and then improve the convergence.
[01782] Generalized multiscale finite element method for highly heterogeneous compressible flow
Format : Talk at Waseda University
Author(s) :
Shubin Fu (Eastern Institute for Advanced Study)
Lina Zhao (City University of Hong Kong)
Eric Chung (The Chinese University of Hong Kong)
Abstract : I will present generalized multiscale finite element method for highly heterogeneous compressible flow. We follow the major steps of the GMsFEM to construct a permeability dependent offline basis for fast coarse-grid simulation. To further increase the accuracy of the multiscale method, a residual driven online multiscale basis is added to the offline space. Rich numerical tests on typical 3D highly heterogeneous media are presented to demonstrate the impressive computational advantages of the proposed multiscale method.
[01417] Recent advances on high-performance computing algorithms for patient-specific blood flow simulations
Format : Talk at Waseda University
Author(s) :
Rongliang Chen (Shenzhen Institutes of Advanced Technology Chinese Academy of Sciences)
Abstract : Patient-specific blood flow simulations have the potential to provide quantitative predictive tools for virtual surgery, treatment planning, and risk stratification. To accurately resolve the blood flows based on the patient-specific geometry and parameters is still a big challenge because of the complex geometry and the turbulence, and it is also important to obtain the results in a short amount of computing time so that the simulation can be used in surgery planning. In this talk, we will precent some recent results of the multi-organ blood flow simulations with patient-specific geometry and parameters on a large-scale supercomputer. Several mathematical, biomechanical, and supercomputing issues will be discussed in detail. We will also report the parallel performance of the methods on a supercomputer with a large number of processors.
[02019] ENO schemes with adaptive order for solving hyperbolic conservation laws
Format : Talk at Waseda University
Author(s) :
Hua Shen (University of Electronic Science and Technology of China)
Abstract : We present a class of ENO schemes with adaptive order for solving hyperbolic conservation laws. The proposed schemes select the optimal polynomial from several candidates that are reconstructed on stencils of unequal sizes by using a novel strategy. In this way, the schemes give high-order accuracy whenever the data is smooth but avoid the Gibbs phenomenon at discontinuities.
[01460] Scalable multilevel preconditioners for hybrid-DG discretizations of nonlinear cell-by-cell cardiac models
Format : Online Talk on Zoom
Author(s) :
Ngoc Mai Monica Huynh (University of Pavia)
Abstract : We present theoretical and numerical results for a scalable and quasi-optimal BDDC preconditioner for Discontinuous Galerkin discretizations of cardiac cell-by-cell models in order to approximate the discontinuous nature of cellular networks.
The resulting discrete cell-by-cell models have discontinuous global solutions across the cell boundaries, hence the proposed BDDC preconditioner is based on appropriate dual and primal spaces with additional constraints which transfer information between cells/subdomains without influencing the overall discontinuity of the global solution.
[04011] Nonlinear FETI-DP domain decomposition methods combined with deep learning
Format : Online Talk on Zoom
Author(s) :
Axel Klawonn (University of Cologne)
Martin Lanser (University of Cologne)
Janine Weber (University of Cologne)
Abstract : In nonlinear-FETI-DP domain decomposition methods the choice of the nonlinear elimination set and of the coarse space have a huge impact on the nonlinear and linear convergence behavior. In this talk, we will show new results combining recently developed approaches for the adaptive choice of the nonlinear elimination set with adaptive coarse spaces. Additionally, we will discuss approaches to improve the computational efficiency and nonlinear convergence by enhancing Nonlinear-FETI-DP with techniques from machine learning.
