178,281 research outputs found
SFOPDES: A stepwise tutorial for teaching Partial Differential Equations using a CAS
Partial Differential Equations (PDE) are one of the most difficult topics that Engineering and
Sciences students have to study in the different Math subjects in their degree.
In this talk we introduce SFOPDES (Stepwise First Order Partial Differential Equations
Solver) aimed to be used as a tutorial for helping both the teacher and the students in the
teaching and learning process of PDE.
The type of problems that SFOPDES solves can be grouped in the following three blocks:
1. Pfaff Differential Equations, which consists on finding the general solution for:
P(x; y; z) dx + Q(x; y; z) dy + R(x; y; z) dz = 0
(a) General method.
(b) Particular cases:
i. Separable equations.
ii. Exact Pfaff equations.
iii. One-separated variable equations.
2. Quasi-linear Partial Differential Equations, which consists on finding the general
solution for: P(x; y; x) p + Q(x; y; z) q = R(x; y; z)
(a) General method.
(b) Particular solution which contents a given curve.
3. Using Lagrange-Charpit Method for finding a complete integral for a given general
first order partial differential equation: F(x; y; z; p; q) = 0.
(a) General method.
(b) Particular cases:
i. F(p; q) = 0
ii. g1(x; p) = g2(y; q)
iii. z = px + qy + g(p; q)Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech
Fine-grained bit-flip protection for relaxation methods
[EN] Resilience is considered a challenging under-addressed issue that the high performance computing community (HPC) will have to face in order to produce reliable Exascale systems by the beginning of the next decade. As part of a push toward a resilient HPC ecosystem, in this paper we propose an error-resilient iterative solver for sparse linear systems based on stationary component-wise relaxation methods. Starting from a plain implementation of the Jacobi iteration, our approach introduces a low-cost component-wise technique that detects bit-flips, rejecting some component updates, and turning the initial synchronized solver into an asynchronous iteration. Our experimental study with sparse incomplete factorizations from a collection of real-world applications, and a practical GPU implementation, exposes the convergence delay incurred by the fault-tolerant implementation and its practical performance.This material is based upon work supported in part by the U.S. Department of Energy (Award Number DE-SC-0010042) and NVIDIA. E. S. Quintana-Orti was supported by project CICYT TIN2014-53495-R of MINECO and FEDER.Anzt, H.; Dongarra, J.; Quintana Ortí, ES. (2019). Fine-grained bit-flip protection for relaxation methods. Journal of Computational Science. 36:1-11. https://doi.org/10.1016/j.jocs.2016.11.013S11136Chow, E., & Patel, A. (2015). Fine-Grained Parallel Incomplete LU Factorization. SIAM Journal on Scientific Computing, 37(2), C169-C193. doi:10.1137/140968896Karpuzcu, U. R., Kim, N. S., & Torrellas, J. (2013). Coping with Parametric Variation at Near-Threshold Voltages. IEEE Micro, 33(4), 6-14. doi:10.1109/mm.2013.71Bronevetsky, G., & de Supinski, B. (2008). Soft error vulnerability of iterative linear algebra methods. Proceedings of the 22nd annual international conference on Supercomputing - ICS ’08. doi:10.1145/1375527.1375552Sao, P., & Vuduc, R. (2013). Self-stabilizing iterative solvers. Proceedings of the Workshop on Latest Advances in Scalable Algorithms for Large-Scale Systems - ScalA ’13. doi:10.1145/2530268.2530272Calhoun, J., Snir, M., Olson, L., & Garzaran, M. (2015). Understanding the Propagation of Error Due to a Silent Data Corruption in a Sparse Matrix Vector Multiply. 2015 IEEE International Conference on Cluster Computing. doi:10.1109/cluster.2015.101Chazan, D., & Miranker, W. (1969). Chaotic relaxation. Linear Algebra and its Applications, 2(2), 199-222. doi:10.1016/0024-3795(69)90028-7Frommer, A., & Szyld, D. B. (2000). On asynchronous iterations. Journal of Computational and Applied Mathematics, 123(1-2), 201-216. doi:10.1016/s0377-0427(00)00409-xDuff, I. S., & Meurant, G. A. (1989). The effect of ordering on preconditioned conjugate gradients. BIT, 29(4), 635-657. doi:10.1007/bf01932738Aliaga, J. I., Barreda, M., Dolz, M. F., Martín, A. F., Mayo, R., & Quintana-Ortí, E. S. (2014). Assessing the impact of the CPU power-saving modes on the task-parallel solution of sparse linear systems. Cluster Computing, 17(4), 1335-1348. doi:10.1007/s10586-014-0402-
A Second Order Godunov Method for Multidimensional Relativistic Magnetohydrodynamics
We describe a new Godunov algorithm for relativistic magnetohydrodynamics
(RMHD) that combines a simple, unsplit second order accurate integrator with
the constrained transport (CT) method for enforcing the solenoidal constraint
on the magnetic field. A variety of approximate Riemann solvers are implemented
to compute the fluxes of the conserved variables. The methods are tested with a
comprehensive suite of multidimensional problems. These tests have helped us
develop a hierarchy of correction steps that are applied when the integration
algorithm predicts unphysical states due to errors in the fluxes, or errors in
the inversion between conserved and primitive variables. Although used
exceedingly rarely, these corrections dramatically improve the stability of the
algorithm. We present preliminary results from the application of these
algorithms to two problems in RMHD: the propagation of supersonic magnetized
jets, and the amplification of magnetic field by turbulence driven by the
relativistic Kelvin-Helmholtz instability (KHI). Both of these applications
reveal important differences between the results computed with Riemann solvers
that adopt different approximations for the fluxes. For example, we show that
use of Riemann solvers which include both contact and rotational
discontinuities can increase the strength of the magnetic field within the
cocoon by a factor of ten in simulations of RMHD jets, and can increase the
spectral resolution of three-dimensional RMHD turbulence driven by the KHI by a
factor of 2. This increase in accuracy far outweighs the associated increase in
computational cost. Our RMHD scheme is publicly available as part of the Athena
code.Comment: 75 pages, 28 figures, accepted for publication in ApJS. Version with
high resolution figures available from
http://jila.colorado.edu/~krb3u/Athena_SR/rmhd_method_paper.pd
Bridging the computational gap between mesoscopic and continuum modeling of red blood cells for fully resolved blood flow
We present a computational framework for the simulation of blood flow with
fully resolved red blood cells (RBCs) using a modular approach that consists of
a lattice Boltzmann solver for the blood plasma, a novel finite element based
solver for the deformable bodies and an immersed boundary method for the
fluid-solid interaction. For the RBCs, we propose a nodal projective FEM
(npFEM) solver which has theoretical advantages over the more commonly used
mass-spring systems (mesoscopic modeling), such as an unconditional stability,
versatile material expressivity, and one set of parameters to fully describe
the behavior of the body at any mesh resolution. At the same time, the method
is substantially faster than other FEM solvers proposed in this field, and has
an efficiency that is comparable to the one of mesoscopic models. At its core,
the solver uses specially defined potential energies, and builds upon them a
fast iterative procedure based on quasi-Newton techniques. For a known
material, our solver has only one free parameter that demands tuning, related
to the body viscoelasticity. In contrast, state-of-the-art solvers for
deformable bodies have more free parameters, and the calibration of the models
demands special assumptions regarding the mesh topology, which restrict their
generality and mesh independence. We propose as well a modification to the
potential energy proposed by Skalak et al. 1973 for the red blood cell
membrane, which enhances the strain hardening behavior at higher deformations.
Our viscoelastic model for the red blood cell, while simple enough and
applicable to any kind of solver as a post-convergence step, can capture
accurately the characteristic recovery time and tank-treading frequencies. The
framework is validated using experimental data, and it proves to be scalable
for multiple deformable bodies
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