628 research outputs found
Function spaces and functional frameworks
The goal is to provide an overview about function spaces, and more generally speaking functional frameworks that include metric spacs, normed spaces, inner product spaces, and convex sets for variational inequalities. Throughout, the implication to algorithms and practical applications is made and sometimes illustrated with numerical simulations from my own work
Iterative solution to the biharmonic equation in mixed form discretized by the Hybrid High-Order method
We consider the solution to the biharmonic equation in mixed form discretized
by the Hybrid High-Order (HHO) methods. The two resulting second-order elliptic
problems can be decoupled via the introduction of a new unknown, corresponding
to the boundary value of the solution of the first Laplacian problem. This
technique yields a global linear problem that can be solved iteratively via a
Krylov-type method. More precisely, at each iteration of the scheme, two
second-order elliptic problems have to be solved, and a normal derivative on
the boundary has to be computed. In this work, we specialize this scheme for
the HHO discretization. To this aim, an explicit technique to compute the
discrete normal derivative of an HHO solution of a Laplacian problem is
proposed. Moreover, we show that the resulting discrete scheme is well-posed.
Finally, a new preconditioner is designed to speed up the convergence of the
Krylov method. Numerical experiments assessing the performance of the proposed
iterative algorithm on both two- and three-dimensional test cases are
presented
LFA-tuned matrix-free multigrid method for the elastic Helmholtz equation
We present an efficient matrix-free geometric multigrid method for the
elastic Helmholtz equation, and a suitable discretization. Many discretization
methods had been considered in the literature for the Helmholtz equations, as
well as many solvers and preconditioners, some of which are adapted for the
elastic version of the equation. However, there is very little work considering
the reciprocity of discretization and a solver. In this work, we aim to bridge
this gap. By choosing an appropriate stencil for re-discretization of the
equation on the coarse grid, we develop a multigrid method that can be easily
implemented as matrix-free, relying on stencils rather than sparse matrices.
This is crucial for efficient implementation on modern hardware. Using two-grid
local Fourier analysis, we validate the compatibility of our discretization
with our solver, and tune a choice of weights for the stencil for which the
convergence rate of the multigrid cycle is optimal. It results in a scalable
multigrid preconditioner that can tackle large real-world 3D scenarios.Comment: 20 page
A full approximation scheme multilevel method for nonlinear variational inequalities
We present the full approximation scheme constraint decomposition (FASCD)
multilevel method for solving variational inequalities (VIs). FASCD is a common
extension of both the full approximation scheme (FAS) multigrid technique for
nonlinear partial differential equations, due to A.~Brandt, and the constraint
decomposition (CD) method introduced by X.-C.~Tai for VIs arising in
optimization. We extend the CD idea by exploiting the telescoping nature of
certain function space subset decompositions arising from multilevel mesh
hierarchies. When a reduced-space (active set) Newton method is applied as a
smoother, with work proportional to the number of unknowns on a given mesh
level, FASCD V-cycles exhibit nearly mesh-independent convergence rates, and
full multigrid cycles are optimal solvers. The example problems include
differential operators which are symmetric linear, nonsymmetric linear, and
nonlinear, in unilateral and bilateral VI problems.Comment: 25 pages, 9 figure
Learning Mesh Motion Techniques with Application to Fluid-Structure Interaction
Mesh degeneration is a bottleneck for fluid-structure interaction (FSI)
simulations and for shape optimization via the method of mappings. In both
cases, an appropriate mesh motion technique is required. The choice is
typically based on heuristics, e.g., the solution operators of partial
differential equations (PDE), such as the Laplace or biharmonic equation.
Especially the latter, which shows good numerical performance for large
displacements, is expensive. Moreover, from a continuous perspective, choosing
the mesh motion technique is to a certain extent arbitrary and has no influence
on the physically relevant quantities. Therefore, we consider approaches
inspired by machine learning. We present a hybrid PDE-NN approach, where the
neural network (NN) serves as parameterization of a coefficient in a second
order nonlinear PDE. We ensure existence of solutions for the nonlinear PDE by
the choice of the neural network architecture. Moreover, we present an approach
where a neural network corrects the harmonic extension such that the boundary
displacement is not changed. In order to avoid technical difficulties in
coupling finite element and machine learning software, we work with a splitting
of the monolithic FSI system into three smaller subsystems. This allows to
solve the mesh motion equation in a separate step. We assess the quality of the
learned mesh motion technique by applying it to a FSI benchmark problem
BDDC preconditioners for virtual element approximations of the three-dimensional Stokes equations
The Virtual Element Method (VEM) is a novel family of numerical methods for
approximating partial differential equations on very general polygonal or
polyhedral computational grids. This work aims to propose a Balancing Domain
Decomposition by Constraints (BDDC) preconditioner that allows using the
conjugate gradient method to compute the solution of the saddle-point linear
systems arising from the VEM discretization of the three-dimensional Stokes
equations. We prove the scalability and quasi-optimality of the algorithm and
confirm the theoretical findings with parallel computations. Numerical results
with adaptively generated coarse spaces confirm the method's robustness in the
presence of large jumps in the viscosity and with high-order VEM
discretizations
Exploiting spatial symmetries for solving Poisson's equation
This paper presents a strategy to accelerate virtually any Poisson solver by taking advantage of s spatial reflection symmetries. More precisely, we have proved the existence of an inexpensive block diagonalisation that transforms the original Poisson equation into a set of 2s fully decoupled subsystems then solved concurrently. This block diagonalisation is identical regardless of the mesh connectivity (structured or unstructured) and the geometric complexity of the problem, therefore applying to a wide range of academic and industrial configurations. In fact, it simplifies the task of discretising complex geometries since it only requires meshing a portion of the domain that is then mirrored implicitly by the symmetriesâ hyperplanes. Thus, the resulting meshes naturally inherit the exploited symmetries, and their memory footprint becomes 2s times smaller. Thanks to the subsystemsâ better spectral properties, iterative solvers converge significantly faster. Additionally, imposing an adequate grid pointsâ ordering allows reducing the operatorsâ footprint and replacing the standard sparse matrix-vector products with the sparse matrixmatrix product, a higher arithmetic intensity kernel. As a result, matrix multiplications are accelerated, and massive simulations become more affordable. Finally, we include numerical experiments based on a turbulent flow simulation and making state-of-theart solvers exploit a varying number of symmetries. On the one hand, algebraic multigrid and preconditioned Krylov subspace methods require up to 23% and 72% fewer iterations, resulting in up to 1.7x and 5.6x overall speedups, respectively. On the other, sparse direct solversâ memory footprint, setup and solution costs are reduced by up to 48%, 58% and 46%, respectively.This work has been financially supported by two competitive R+D projects: RETOtwin (PDC2021-120970-I00), given by MCIN/AEI/10.13039/501100011033 and European Union Next GenerationEU/PRTR, and FusionCAT (001-P-001722), given by Generalitat de Catalunya RIS3CAT-FEDER. Ădel Alsalti-Baldellou has also been supported by the predoctoral grants DIN2018-010061 and 2019-DI-90, given by MCIN/AEI/10.13039/501100011033 and the Catalan Agency for Management of University and Research Grants (AGAUR), respectively.Peer ReviewedPostprint (published version
Optimization and coarse-grid selection for algebraic multigrid
Multigrid methods are often the most efficient approaches for solving the very
large linear systems that arise from discretized PDEs and other problems. Algebraic
multigrid (AMG) methods are used when the discretization lacks the structure needed
to enable more efficient geometric multigrid techniques. AMG methods rely in part
on heuristic graph algorithms to achieve their performance. Reduction-based AMG
(AMGr) algorithms attempt to formalize these heuristics.
The main focus of this thesis is to develop eâ”ective algebraic multigrid methods.
A key step in all AMG approaches is the choice of the coarse/fine partitioning, aiming
to balance the convergence of the iteration with its cost. In past work (MacLachlan
and Saad, A greedy strategy for coarse-grid selection, SISC 2007), a constrained
combinatorial optimization problem was used to define the âbestâ coarse grid within
the setting of two-level reduction-based AMG and was shown to be NP-complete. In
the first part of the thesis, a new coarsening algorithm based on simulated annealing
has been developed to solve this problem. The new coarsening algorithm gives better
results than the greedy algorithm developed previously.
The goal of the second part of the thesis is to improve the classical AMGr method.
Convergence factor bounds do not hold when AMGr algorithms are applied to matrices
that are not diagonally dominant. In this part of our research, we present
modifications to the classical AMGr algorithm that improve its performance on such
matrices. For non-diagonally dominant matrices, we find that strength of connection
plays a vital role in the performance of AMGr. To generalize the diagonal
approximations of AFF used in classical AMGr, we use a sparse approximate inverse
(SPAI) method, with nonzero pattern determined by strong connections, to define
the AMGr-style interpolation operator, coupled with rescaling based on relaxed vectors.
We present numerical results demonstrating the robustness of this approach for
non-diagonally dominant systems.
In the third part of this research, we have developed an improved deterministic
coarsening algorithm that generalizes an existing technique known as Lloydâs algorithm.
The improved algorithm provides better control of the number of clusters than
classical approaches and attempts to provide more âcompactâ groupings
An extension of the approximate component mode synthesis method to the heterogeneous Helmholtz equation
In this work we propose and analyze an extension of the approximate component
mode synthesis (ACMS) method to the heterogeneous Helmholtz equation. The ACMS
method has originally been introduced by Hetmaniuk and Lehoucq as a multiscale
method to solve elliptic partial differential equations. The ACMS method uses a
domain decomposition to separate the numerical approximation by splitting the
variational problem into two independent parts: local Helmholtz problems and a
global interface problem. While the former are naturally local and decoupled
such that they can be easily solved in parallel, the latter requires the
construction of suitable local basis functions relying on local eigenmodes and
suitable extensions. We carry out a full error analysis of this approach
focusing on the case where the domain decomposition is kept fixed, but the
number of eigenfunctions is increased. The theoretical results in this work are
supported by numerical experiments verifying algebraic convergence for the
method. In certain, practically relevant cases, even exponential convergence
for the local Helmholtz problems can be achieved without oversampling
Modeling of cardiac fibers as oriented liquid crystals
In this work we propose a mathematical model that describes the orientation
of ventricular cardiac fibers. These fibers are commonly computed as the
normalized gradient of certain harmonic potentials, so our work consisted in
finding the equations that such a vector field satisfies, considering the
unitary norm constraint. The resulting equations belong to the Frank-Oseen
theory of nematic liquid crystals, which yield a bulk of mathematical
properties to the cardiac fibers, such as the characterization of
singularities. The numerical methods available in literature are
computationally expensive and not sufficiently robust for the complex
geometries obtained from the human heart, so we also propose a preconditioned
projected gradient descent scheme that circumvents these difficulties in the
tested scenarios. The resulting model further confirms recent experimental
observations of liquid crystal behavior of soft tissue, and provides an
accurate mathematical description of such behavior
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