1,166 research outputs found
Spectral Sparsification for Communication-Efficient Collaborative Rotation and Translation Estimation
We propose fast and communication-efficient optimization algorithms for
multi-robot rotation averaging and translation estimation problems that arise
from collaborative simultaneous localization and mapping (SLAM),
structure-from-motion (SfM), and camera network localization applications. Our
methods are based on theoretical relations between the Hessians of the
underlying Riemannian optimization problems and the Laplacians of suitably
weighted graphs. We leverage these results to design a collaborative solver in
which robots coordinate with a central server to perform approximate
second-order optimization, by solving a Laplacian system at each iteration.
Crucially, our algorithms permit robots to employ spectral sparsification to
sparsify intermediate dense matrices before communication, and hence provide a
mechanism to trade off accuracy with communication efficiency with provable
guarantees. We perform rigorous theoretical analysis of our methods and prove
that they enjoy (local) linear rate of convergence. Furthermore, we show that
our methods can be combined with graduated non-convexity to achieve
outlier-robust estimation. Extensive experiments on real-world SLAM and SfM
scenarios demonstrate the superior convergence rate and communication
efficiency of our methods.Comment: Revised extended technical report (37 pages, 15 figures, 6 tables
A distributed-memory parallel algorithm for discretized integral equations using Julia
Boundary value problems involving elliptic PDEs such as the Laplace and the
Helmholtz equations are ubiquitous in physics and engineering. Many such
problems have alternative formulations as integral equations that are
mathematically more tractable than their PDE counterparts. However, the
integral equation formulation poses a challenge in solving the dense linear
systems that arise upon discretization. In cases where iterative methods
converge rapidly, existing methods that draw on fast summation schemes such as
the Fast Multipole Method are highly efficient and well established. More
recently, linear complexity direct solvers that sidestep convergence issues by
directly computing an invertible factorization have been developed. However,
storage and compute costs are high, which limits their ability to solve
large-scale problems in practice. In this work, we introduce a
distributed-memory parallel algorithm based on an existing direct solver named
``strong recursive skeletonization factorization.'' The analysis of its
parallel scalability applies generally to a class of existing methods that
exploit the so-called strong admissibility. Specifically, we apply low-rank
compression to certain off-diagonal matrix blocks in a way that minimizes data
movement. Given a compression tolerance, our method constructs an approximate
factorization of a discretized integral operator (dense matrix), which can be
used to solve linear systems efficiently in parallel. Compared to iterative
algorithms, our method is particularly suitable for problems involving
ill-conditioned matrices or multiple right-hand sides. Large-scale numerical
experiments are presented to demonstrate the performance of our implementation
using the Julia language
Finite difference method in prolate spheroidal coordinates for freely suspended spheroidal particles in linear flows of viscous and viscoelastic fluids
A finite difference scheme is used to develop a numerical method to solve the
flow of an unbounded viscoelastic fluid with zero to moderate inertia around a
prolate spheroidal particle. The equations are written in prolate spheroidal
coordinates, and the shape of the particle is exactly resolved as one of the
coordinate surfaces representing the inner boundary of the computational
domain. As the prolate spheroidal grid is naturally clustered near the particle
surface, good resolution is obtained in the regions where the gradients of
relevant flow variables are most significant. This coordinate system also
allows large domain sizes with a reasonable number of mesh points to simulate
unbounded fluid around a particle. Changing the aspect ratio of the inner
computational boundary enables simulations of different particle shapes ranging
from a sphere to a slender fiber. Numerical studies of the latter particle
shape allow testing of slender body theories. The mass and momentum equations
are solved with a Schur complement approach allowing us to solve the zero
inertia case necessary to isolate the viscoelastic effects. The singularities
associated with the coordinate system are overcome using L'Hopital's rule. A
straightforward imposition of conditions representing a time-varying
combination of linear flows on the outer boundary allows us to study various
flows with the same computational domain geometry. {For the special but
important case of zero fluid and particle inertia we obtain a novel formulation
that satisfies the force- and torque-free constraint in an iteration-free
manner.} The numerical method is demonstrated for various flows of Newtonian
and viscoelastic fluids around spheres and spheroids (including those with
large aspect ratio). Good agreement is demonstrated with existing theoretical
and numerical results.Comment: 32 pages, 12 figures. Accepted at Journal of Computational Physic
High-Order Mixed Finite Element Variable Eddington Factor Methods
We apply high-order mixed finite element discretization techniques and their
associated preconditioned iterative solvers to the Variable Eddington Factor
(VEF) equations in two spatial dimensions. The mixed finite element VEF
discretizations are coupled to a high-order Discontinuous Galerkin (DG)
discretization of the Discrete Ordinates transport equation to form effective
linear transport algorithms that are compatible with high-order (curved)
meshes. This combination of VEF and transport discretizations is motivated by
the use of high-order mixed finite element methods in hydrodynamics
calculations at the Lawrence Livermore National Laboratory. Due to the
mathematical structure of the VEF equations, the standard Raviart Thomas (RT)
mixed finite elements cannot be used to approximate the vector variable in the
VEF equations. Instead, we investigate three alternatives based on the use of
continuous finite elements for each vector component, a non-conforming RT
approach where DG-like techniques are used, and a hybridized RT method. We
present numerical results that demonstrate high-order accuracy, compatibility
with curved meshes, and robust and efficient convergence in iteratively solving
the coupled transport-VEF system and in the preconditioned linear solvers used
to invert the discretized VEF equations
Fast boundary element methods for the simulation of wave phenomena
This thesis is concerned with the efficient implementation of boundary element methods (BEM) for their application in wave problems. BEM present a particularly useful tool, since they reduce the dimension of the problems by one, resulting in much fewer unknowns. However, this comes at the cost of dense system matrices, whose entries require the integration of singular kernel functions over pairs of boundary elements. Because calculating these four-dimensional integrals by cubature rules is expensive, a novel approach based on singularity cancellation and analytical integration is proposed. In this way, the dimension of the integrals is reduced and closed formulae are obtained for the most challenging cases. This allows for the accurate calculation of the matrix entries while requiring less computational work compared with conventional numerical integration. Furthermore, a new algorithm based on hierarchical low-rank approximation is presented, which compresses the dense matrices and improves the complexity of the method. The idea is to collect the matrices corresponding to different time steps in a third-order tensor and to approximate individual sub-blocks by a combination of analytic and algebraic low-rank techniques. By exploiting the low-rank structure in several ways, the method scales almost linearly in the number of spatial degrees of freedom and number of time steps. The superior performance of the new method is demonstrated in numerical examples.Diese Arbeit befasst sich mit der effizienten Implementierung von Randelementmethoden (REM) fĂŒr ihre Anwendung auf Wellenprobleme. REM stellen ein besonders nĂŒtzliches Werkzeug dar, da sie die Dimension der Probleme um eins reduzieren, was zu weit weniger Unbekannten fĂŒhrt. Allerdings ist dies mit vollbesetzten Matrizen verbunden, deren EintrĂ€ge die Integration singulĂ€rer Kernfunktionen ĂŒber Paare von Randelementen erfordern. Da die Berechnung dieser vierdimensionalen Integrale durch Kubaturformeln aufwendig ist, wird ein neuer Ansatz basierend auf Regularisierung und analytischer Integration verfolgt. Auf diese Weise reduziert sich die Dimension der Integrale und es ergeben sich geschlossene Formeln fĂŒr die schwierigsten FĂ€lle. Dies ermöglicht die genaue Berechnung der MatrixeintrĂ€ge mit geringerem Rechenaufwand als konventionelle numerische Integration. AuĂerdem wird ein neuer Algorithmus beruhend auf hierarchischer Niedrigrangapproximation prĂ€sentiert, der die Matrizen komprimiert und die KomplexitĂ€t der Methode verbessert. Die Idee ist, die Matrizen der verschiedenen Zeitpunkte in einem Tensor dritter Ordnung zu sammeln und einzelne Teilblöcke durch eine Kombination von analytischen und algebraischen Niedrigrangverfahren zu approximieren. Durch Ausnutzung der Niedrigrangstruktur skaliert die Methode fast linear mit der Anzahl der rĂ€umlichen Freiheitsgrade und der Anzahl der Zeitschritte. Die ĂŒberlegene Leistung der neuen Methode wird anhand numerischer Beispiele aufgezeigt
Robust Methods for Multiscale Coarse Approximations of Diffusion Models in Perforated Domains
For the Poisson equation posed in a domain containing a large number of
polygonal perforations, we propose a low-dimensional coarse approximation space
based on a coarse polygonal partitioning of the domain. Similarly to other
multiscale numerical methods, this coarse space is spanned by locally discrete
harmonic basis functions. Along the subdomain boundaries, the basis functions
are piecewise polynomial. The main contribution of this article is an error
estimate regarding the H1-projection over the coarse space which depends only
on the regularity of the solution over the edges of the coarse partitioning.
For a specific edge refinement procedure, the error analysis establishes
superconvergence of the method even if the true solution has a low general
regularity. Combined with domain decomposition (DD) methods, the coarse space
leads to an efficient two-level iterative linear solver which reaches the
fine-scale finite element error in few iterations. It also bodes well as a
preconditioner for Krylov methods and provides scalability with respect to the
number of subdomains. Numerical experiments showcase the increased precision of
the coarse approximation as well as the efficiency and scalability of the
coarse space as a component of a DD algorithm.Comment: 32 pages, 14 figures, submitted to Journal of Computational Physic
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
Computational modelling and optimal control of interacting particle systems: connecting dynamic density functional theory and PDE-constrained optimization
Processes that can be described by systems of interacting particles are ubiquitous in nature, society, and industry, ranging from animal flocking, the spread of diseases, and formation of opinions to nano-filtration, brewing, and printing. In real-world applications it is often relevant to not only model a process of interest, but to also optimize it in order to achieve a desired outcome with minimal resources, such as time, money, or energy.
Mathematically, the dynamics of interacting particle systems can be described using Dynamic Density Functional Theory (DDFT). The resulting models are nonlinear, nonlocal partial differential equations (PDEs) that include convolution integral terms. Such terms also enter the naturally arising no-flux boundary conditions. Due to the nonlocal, nonlinear nature of such problems they are challenging both to analyse and solve numerically.
In order to optimize processes that are modelled by PDEs, one can apply tools from PDE-constrained optimization. The aim here is to drive a quantity of interest towards a target state by varying a control variable. This is constrained by a PDE describing the process of interest, in which the control enters as a model parameter. Such problems can be tackled by deriving and solving the (first-order) optimality system, which couples the PDE model with a second PDE and an algebraic equation. Solving such a system numerically is challenging, since large matrices arise in its discretization, for which efficient solution strategies have to be found. Most work in PDE-constrained optimization addresses problems in which the control is applied linearly, and which are constrained by local, often linear PDEs, since introducing nonlinearity significantly increases the complexity in both the analysis and numerical solution
of the optimization problem.
However, in order to optimize real-world processes described by nonlinear, nonlocal DDFT models, one has to develop an optimal control framework for such models. The aim is to drive the particles to some desired distribution by applying control either linearly, through a particle source, or bilinearly, though an advective field. The optimization process is constrained by the DDFT model that describes how the particles move under the influence of advection, diffusion, external forces, and particleâparticle interactions. In order to tackle this, the (first-order) optimality system is derived, which, since it involves nonlinear (integro-)PDEs that are coupled nonlocally in space and time, is significantly harder than in the standard case. Novel numerical methods are developed, effectively combining pseudospectral methods and iterative solvers, to efficiently and accurately solve such a system.
In a next step this framework is extended so that it can capture and optimize industrially relevant processes, such as brewing and nano-filtration. In order to do so, extensions to both the DDFT model and the numerical method are made. Firstly, since industrial processes often involve tubes, funnels, channels, or tanks of various shapes, the PDE model itself, as well as the optimization problem, need to be solved on complicated domains. This is achieved by developing a novel spectral element approach that is compatible with both the PDE solver and the optimal control framework. Secondly, many industrial processes, such as nano-filtration, involve more than one type of particle. Therefore, the DDFT model is extended to describe multiple particle species. Finally, depending on the application of interest, additional physical effects need to be included in the model. In this thesis, to model sedimentation processes in brewing, the model is modified to capture volume exclusion effects
Reduced Order Modeling based Inexact FETI-DP solver for lattice structures
This paper addresses the overwhelming computational resources needed with
standard numerical approaches to simulate architected materials. Those
multiscale heterogeneous lattice structures gain intensive interest in
conjunction with the improvement of additive manufacturing as they offer, among
many others, excellent stiffness-to-weight ratios. We develop here a dedicated
HPC solver that benefits from the specific nature of the underlying problem in
order to drastically reduce the computational costs (memory and time) for the
full fine-scale analysis of lattice structures. Our purpose is to take
advantage of the natural domain decomposition into cells and, even more
importantly, of the geometrical and mechanical similarities among cells. Our
solver consists in a so-called inexact FETI-DP method where the local,
cell-wise operators and solutions are approximated with reduced order modeling
techniques. Instead of considering independently every cell, we end up with
only few principal local problems to solve and make use of the corresponding
principal cell-wise operators to approximate all the others. It results in a
scalable algorithm that saves numerous local factorizations. Our solver is
applied for the isogeometric analysis of lattices built by spline composition,
which offers the opportunity to compute the reduced basis with macro-scale
data, thereby making our method also multiscale and matrix-free. The solver is
tested against various 2D and 3D analyses. It shows major gains with respect to
black-box solvers; in particular, problems of several millions of degrees of
freedom can be solved with a simple computer within few minutes.Comment: 30 pages, 12 figures, 2 table
Parallel-in-Time Solver for the All-at-Once Runge--Kutta Discretization
In this article, we derive fast and robust parallel-in-time preconditioned
iterative methods for the all-at-once linear systems arising upon
discretization of time-dependent PDEs. The discretization we employ is based on
a Runge--Kutta method in time, for which the development of parallel solvers is
an emerging research area in the literature of numerical methods for
time-dependent PDEs. By making use of classical theory of block matrices, one
is able to derive a preconditioner for the systems considered. The block
structure of the preconditioner allows for parallelism in the time variable, as
long as one is able to provide an optimal solver for the system of the stages
of the method. We thus propose a preconditioner for the latter system based on
a singular value decomposition (SVD) of the (real) Runge--Kutta matrix
. Supposing is invertible,
we prove that the spectrum of the system for the stages preconditioned by our
SVD-based preconditioner is contained within the right-half of the unit circle,
under suitable assumptions on the matrix (the assumptions are well
posed due to the polar decomposition of ). We show the
numerical efficiency of our SVD-based preconditioner by solving the system of
the stages arising from the discretization of the heat equation and the Stokes
equations, with sequential time-stepping. Finally, we provide numerical results
of the all-at-once approach for both problems, showing the speed-up achieved on
a parallel architecture
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