3 research outputs found
A Deep Learning algorithm to accelerate Algebraic Multigrid methods in Finite Element solvers of 3D elliptic PDEs
Algebraic multigrid (AMG) methods are among the most efficient solvers for
linear systems of equations and they are widely used for the solution of
problems stemming from the discretization of Partial Differential Equations
(PDEs). The most severe limitation of AMG methods is the dependence on
parameters that require to be fine-tuned. In particular, the strong threshold
parameter is the most relevant since it stands at the basis of the construction
of successively coarser grids needed by the AMG methods. We introduce a novel
Deep Learning algorithm that minimizes the computational cost of the AMG method
when used as a finite element solver. We show that our algorithm requires
minimal changes to any existing code. The proposed Artificial Neural Network
(ANN) tunes the value of the strong threshold parameter by interpreting the
sparse matrix of the linear system as a black-and-white image and exploiting a
pooling operator to transform it into a small multi-channel image. We
experimentally prove that the pooling successfully reduces the computational
cost of processing a large sparse matrix and preserves the features needed for
the regression task at hand. We train the proposed algorithm on a large dataset
containing problems with a highly heterogeneous diffusion coefficient defined
in different three-dimensional geometries and discretized with unstructured
grids and linear elasticity problems with a highly heterogeneous Young's
modulus. When tested on problems with coefficients or geometries not present in
the training dataset, our approach reduces the computational time by up to 30%
Preconditioned NonSymmetric/Symmetric Discontinuous Galerkin Method for Elliptic Problem with Reconstructed Discontinuous Approximation
In this paper, we propose and analyze an efficient preconditioning method for
the elliptic problem based on the reconstructed discontinuous approximation
method. We reconstruct a high-order piecewise polynomial space that arbitrary
order can be achieved with one degree of freedom per element. This space can be
directly used with the symmetric/nonsymmetric interior penalty discontinuous
Galerkin method. Compared with the standard DG method, we can enjoy the
advantage on the efficiency of the approximation. Besides, we establish an norm
equivalence result between the reconstructed high-order space and the piecewise
constant space. This property further allows us to construct an optimal
preconditioner from the piecewise constant space. The upper bound of the
condition number to the preconditioned symmetric/nonsymmetric system is shown
to be independent of the mesh size. Numerical experiments are provided to
demonstrate the validity of the theory and the efficiency of the proposed
method
Algebraic multigrid schemes for high-order nodal discontinuous galerkin methods
We present algebraic multigrid (AMG) methods for the efficient solution of the linear system of equations stemming from high-order discontinuous Galerkin (DG) discretizations of second-order elliptic problems. For DG methods, standard multigrid approaches cannot be employed because of redundancy of the degrees of freedom associated to the same grid point. We present new aggregation procedures and test them in extensive two-dimensional numerical experiments to demonstrate that the proposed AMG method is uniformly convergent with respect to all of the discretization parameters, namely the mesh-size and the polynomial approximation degree