199 research outputs found
Massively Parallel Implementation of FETI-2LM Methods for the Simulation of the Sparse Receiving Array Evolution of the GRAVES Radar System for Space Surveillance and Tracking
International audienceThis paper presents the massively parallel implementation of the FETI-2LM techniques (Finite Element Tearing and Interconnecting with two Lagrange Multipliers) in the FACTOPO code to solve large-scale sparse receiving array evolutions of the GRAVES bistatic radar in the VHF band. There are four main ingredients in the proposed work and methodology: 1) the implementation of a total field weak formulation of FETI-2LM algorithms for multi-sources modeling using an efficient block Krylov recycling strategy for the calculation of the hundreds of antenna embedded radiation patterns of the sparse array; 2) the implementation of a meshing strategy consisting in generating the sparse array by populating a regular periodic grid; 3) the implementation of the proposed methodology on massively parallel clusters; 4) the investigation of performances of the optimized GRAVES elementary antenna in the VHF band, followed by the demonstration of the expected gain performances even with stronger interferences due to the densification of the array. These simulations of the GRAVES sparse array requiring the resolution of sparse linear systems with 10.47 billion unknowns have been made possible thanks to recent developments of the FETI-2LM domain decomposition method and the use of 13, 692 Intel Xeon Broadwell E5-2680v4 computing cores. A total of 3.8 million cumulated hours have been invested in the interest of the augmentation of the antennas. Until now FETI-2LM methods have been successfully implemented for antenna and diffraction electromagnetic simulations in the S-C-X-Ku and Ka bands and to the best to our knowledge this is the first time that they have been used in the VHF band
Parallel preconditioners for high order discretizations arising from full system modeling for brain microwave imaging
This paper combines the use of high order finite element methods with
parallel preconditioners of domain decomposition type for solving
electromagnetic problems arising from brain microwave imaging. The numerical
algorithms involved in such complex imaging systems are computationally
expensive since they require solving the direct problem of Maxwell's equations
several times. Moreover, wave propagation problems in the high frequency regime
are challenging because a sufficiently high number of unknowns is required to
accurately represent the solution. In order to use these algorithms in practice
for brain stroke diagnosis, running time should be reasonable. The method
presented in this paper, coupling high order finite elements and parallel
preconditioners, makes it possible to reduce the overall computational cost and
simulation time while maintaining accuracy
A Fast Algorithm for Parabolic PDE-based Inverse Problems Based on Laplace Transforms and Flexible Krylov Solvers
We consider the problem of estimating parameters in large-scale weakly
nonlinear inverse problems for which the underlying governing equations is a
linear, time-dependent, parabolic partial differential equation. A major
challenge in solving these inverse problems using Newton-type methods is the
computational cost associated with solving the forward problem and with
repeated construction of the Jacobian, which represents the sensitivity of the
measurements to the unknown parameters. Forming the Jacobian can be
prohibitively expensive because it requires repeated solutions of the forward
and adjoint time-dependent parabolic partial differential equations
corresponding to multiple sources and receivers. We propose an efficient method
based on a Laplace transform-based exponential time integrator combined with a
flexible Krylov subspace approach to solve the resulting shifted systems of
equations efficiently. Our proposed solver speeds up the computation of the
forward and adjoint problems, thus yielding significant speedup in total
inversion time. We consider an application from Transient Hydraulic Tomography
(THT), which is an imaging technique to estimate hydraulic parameters related
to the subsurface from pressure measurements obtained by a series of pumping
tests. The algorithms discussed are applied to a synthetic example taken from
THT to demonstrate the resulting computational gains of this proposed method
Subspace Recycling for Sequences of Shifted Systems with Applications in Image Recovery
For many applications involving a sequence of linear systems with slowly
changing system matrices, subspace recycling, which exploits relationships
among systems and reuses search space information, can achieve huge gains in
iterations across the total number of linear system solves in the sequence.
However, for general (i.e., non-identity) shifted systems with the shift value
varying over a wide range, the properties of the linear systems vary widely as
well, which makes recycling less effective. If such a sequence of systems is
embedded in a nonlinear iteration, the problem is compounded, and special
approaches are needed to use recycling effectively.
In this paper, we develop new, more efficient, Krylov subspace recycling
approaches for large-scale image reconstruction and restoration techniques that
employ a nonlinear iteration to compute a suitable regularization matrix. For
each new regularization matrix, we need to solve regularized linear systems,
, for a sequence of regularization parameters,
, to find the optimally regularized solution that, in turn, will
be used to update the regularization matrix.
In this paper, we analyze system and solution characteristics to choose
appropriate techniques to solve each system rapidly. Specifically, we use an
inner-outer recycling approach with a larger, principal recycle space for each
nonlinear step and smaller recycle spaces for each shift. We propose an
efficient way to obtain good initial guesses from the principle recycle space
and smaller shift-specific recycle spaces that lead to fast convergence. Our
method is substantially reduces the total number of matrix-vector products that
would arise in a naive approach. Our approach is more generally applicable to
sequences of shifted systems where the matrices in the sum are positive
semi-definite
Numerical aspects of Casimir energy computation in acoustic scattering
Computing the Casimir force and energy between objects is a classical problem
of quantum theory going back to the 1940s. Several different approaches have
been developed in the literature often based on different physical principles.
Most notably a representation of the Casimir energy in terms of determinants of
boundary layer operators makes it accessible to a numerical approach. In this
paper, we first give an overview of the various methods and discuss the
connection to the Krein-spectral shift function and computational aspects. We
propose variants of Krylov subspace methods for the computation of the Casimir
energy for large-scale problems and demonstrate Casimir computations for
several complex configurations. This allows for Casimir energy calculation for
large-scale practical problems and significantly speeds up the computations in
that case
Block Iterative Methods and Recycling for Improved Scalability of Linear Solvers
International audienceContemporary large-scale Partial Differential Equation (PDE) simulations usually require the solution of large and sparse linear systems. Moreover, it is often needed to solve these linear systems with different or multiple Right-Hand Sides (RHSs). In this paper, various strategies will be presented to extend the scalability of existing linear solvers using appropriate recycling strategies or block methodsâi.e., by treating multiple right-hand sides simultaneously. The scalability of this work is assessed by performing simulations on up to 8,192 cores for solving linear systems arising from various physical phenomena modeled by Poisson's equation, the system of linear elasticity, or Maxwell's equation. This work is shipped as part of on open-source software, readily available and usable in any C, C++, or Python code. In particular, some simulations are performed on top of a well-established library, PETSc, and it is shown how our approaches can be used to decrease time to solution down by 30%
Block Iterative Methods and Recycling for Improved Scalability of Linear Solvers
International audienceContemporary large-scale Partial Differential Equation (PDE) simulations usually require the solution of large and sparse linear systems. Moreover, it is often needed to solve these linear systems with different or multiple Right-Hand Sides (RHSs). In this paper, various strategies will be presented to extend the scalability of existing linear solvers using appropriate recycling strategies or block methodsâi.e., by treating multiple right-hand sides simultaneously. The scalability of this work is assessed by performing simulations on up to 8,192 cores for solving linear systems arising from various physical phenomena modeled by Poisson's equation, the system of linear elasticity, or Maxwell's equation. This work is shipped as part of on open-source software, readily available and usable in any C, C++, or Python code. In particular, some simulations are performed on top of a well-established library, PETSc, and it is shown how our approaches can be used to decrease time to solution down by 30%
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