Scipedia: Open Access Repository of the ExaQUte project: Publications

A Parallel Dynamic Asynchronous Framework for Uncertainty Quantification by Hierarchical Monte Carlo Algorithms

Riccardo Tosi — Tue, 14 Sep 2021 08:49:02 +0200

The necessity of dealing with uncertainties is growing in many different fields of science and engineering. Due to the constant development of computational capabilities, current solvers must satisfy both statistical accuracy and computational efficiency. The aim of this work is to introduce an asynchronous framework for Monte Carlo and Multilevel Monte Carlo methods to achieve such a result. The proposed approach presents the same reliability of state of the art techniques, and aims at improving the computational efficiency by adding a new level of parallelism with respect to existing algorithms: between batches, where each batch owns its hierarchy and is independent from the others. Two different numerical problems are considered and solved in a supercomputer to show the behavior of the proposed approach.

Analysis of stochastic gradient methods for PDE-constrained optimal control problems with uncertain parameters

Quentin Ayoul-Guilmard — Mon, 28 Jun 2021 12:39:02 +0200

A generic finite element framework on parallel tree-based adaptive meshes

Cecilia Soriano — Fri, 21 Feb 2020 11:43:02 +0100

We present highly scalable parallel distributed-memory algorithms and associated data structures for a generic finite element framework that supports h-adaptivity on computational domains represented as multiple connected adaptive trees—forest-of-trees—, thus providing multi-scale resolution on problems governed by partial differential equations.The framework is grounded on a rich representation of the adaptive mesh suitable for generic finite elements that is built on top of a low-level, light-weight forest-oftrees data structure handled by a specialized, highly parallel adaptive meshing engine. Along the way, we have identified the requirements that the forest-of-trees layer must fulfill to be coupled into our framework. Essentially, it must be able to describe neighboring relationships between cells in the adapted mesh (apart from hierarchical relationships) across the lower-dimensional objects at the boundary of the cells. Atop this two-layered mesh representation, we build the rest of data structures required for the numerical integration and assembly of the discrete system of linear equations.We consider algorithms that are suitable for both subassembled and fully-assembled distributed data layouts of linear system matrices. The proposed framework has been implemented within the FEMPAR scientific software library, using p4est as a practical forest-of-octrees demonstrator. A comprehensive strong scaling study of this implementation when applied to Poisson and Maxwell problems reveals remarkable scalability up to 32.2K CPU cores and 482.2M degrees of freedom. Besides, the implementation in FEMPAR of the proposed approach is up to 2.6 and 3.4 times faster than the state-of-the-art deal.II finite element software in the h-adaptive approximation of a Poisson problem with firstand second-order Lagrangian finite elements, respectively (excluding the linear solver step from the comparison).

Embedded multilevel Monte Carlo for uncertainty quantification in random domains

Cecilia Soriano — Fri, 21 Feb 2020 11:36:03 +0100

The multilevel Monte Carlo (MLMC) method has proven to be an effective variance-reduction statistical method for Uncertainty quantification in PDE models. It combines approximations at different levels of accuracy using a hierarchy of meshes in a similar way as multigrid. The generation of body-fitted mesh hierarchies is only possible for simple geometries. On top of that, MLMC for random domains involves the generation of a mesh for every sample. Instead, here we consider the use of embedded methods which make use of simple background meshes of an artificial domain (a bounding-box) for which it is easy to define a mesh hierarchy, thus eliminating the need of body-fitted unstructured meshes, but can produce ill-conditioned discrete problems. To avoid this complication, we consider the recent aggregated finite element method (AgFEM). In particular, we design an embedded MLMC framework for (geometrically and topologically) random domains implicitly defined through a random level-set function, which makes use of a set of hierarchical background meshes and the AgFEM. Performance predictions from existing theory are verified statistically in three numerical experiments, namely the solution of the Poisson equation on a circular domain of random radius, the solution of the Poisson equation on a topologically identical but more complex domain, and the solution of a heat-transfer problem in a domain that has geometric and topological uncertainties. Finally, the use of AgFE is statistically demonstrated to be crucial for complex and uncertain geometries in terms of robustness and computational cost. Date: November 28, 2019.

A modified Finite Element formulation for the imposition of the slip boundary condition over embedded volumeless geometries

Cecilia Soriano — Fri, 21 Feb 2020 11:33:02 +0100

This work describes a novel formulation for the simulation of Navier-Stokes problems including embedded objects. The new proposal is based on the use of a modified finite element space, which replaces the standard one within the elements intersected by the immersed geometry. The modified space is able to exactly reproduce the jumps happening at the embedded boundary while preserving the conformity across the faces intersected by the embedded object. The paper focuses particularly on the imposition of a slip boundary condition on the surface of the embedded geometry, proposing a new technique for the application of such constraint. The new proposal is carefully benchmarked using the results of a body fitted technique and of an alternative embedded approach. Potential applications of interest are also presented.

Computational modeling of the fluid flow in type B aortic dissection using a modified finite element embedded formulation

Cecilia Soriano — Fri, 21 Feb 2020 11:25:16 +0100

This work explores the use of an embedded computational fluid dynamics method to study the type B aortic dissection. The use of the proposed technique makes it possible to easily test different intimal flap configurations without any need of remeshing. To validate the presented methodology, we take as reference test case an in vitro experiment present in the literature. This experiment, which considers several intimal flap tear configurations (number, size and location), mimics the blood flow in a real type B aortic dissection. We prove the correctness and suitability of the presented approach by comparing the pressure values and waveform. The obtained results exhibit a remarkable similarity with the experimental reference data. Complementary, we present a feasible surgical application of the presented computer method. The aim is to help the clinicians in the decision making before the type B aortic dissection surgical fenestration. The capabilities of the proposed technique are exploited to efficiently create artificial reentry tear configurations. We highlight that only the radius and center of the reentry tear need to be specified by the clinicians, without any need to modify neither the model geometry nor the mesh. The obtained computational surgical fenestration results are in line with the medical observations in similar clinical studies.

Parallel unstructured mesh adaptation using iterative remeshing and repartitioning

Cecilia Soriano — Fri, 21 Feb 2020 11:04:35 +0100

Mesh adaptation has proven to be a powerful tool for increasing the accuracy ofnumerical simulations whenever the solution exhibits strong non-uniform features over the com-putational domain. Sequential remeshing currently represents a bottleneck for parallel solvers. Tothis aim, we present a parallel remeshing algorithm that enables the reuse of existing sequential remeshing libraries, a non-intrusive linkage with third-party solvers, and an efficient exploitationof distributed parallel environments. The numerical procedure is implemented in the open sourceParMmg software package.

Balancing domain decomposition by constraints associated with subobjects

Cecilia Soriano — Fri, 21 Feb 2020 10:55:02 +0100

A simple variant of the BDDC preconditioner in which constraints are imposed on a selected set of subobjects (subdomain subedges, subfaces and vertices between pairs of subedges) is presented. We are able to show that the condition number of the preconditioner is bounded by C 1 + log(L/h)2, where C is a constant, and h and L are the characteristic sizes of the mesh and the subobjects, respectively. As L can be chosen almost freely, the condition number can theoretically be as small as O(1). We will discuss the pros and cons of the preconditioner and its application to heterogeneous problems. Numerical results on supercomputers are provided.

Distributed-memory parallelization of the aggregated unfitted finite element method

Cecilia Soriano — Fri, 21 Feb 2020 10:47:03 +0100

The aggregated unfitted finite element method (AgFEM) is a methodology recently introduced in order to address conditioning and stability problems associated with embedded, unfitted, or extended finite element methods. The method is based on removal of basis functions associated with badly cut cells by introducing carefully designed constraints, which results in well-posed systems of linear algebraic equations, while preserving the optimal approximation order of the underlying finite element spaces. The specific goal of this work is to present the implementation and performance of the method on distributed-memory platforms aiming at the efficient solution of large-scale problems. In particular, we show that, by considering AgFEM, the resulting systems of linear algebraic equations can be effectively solved using standard algebraic multigrid preconditioners. This is in contrast with previous works that consider highly customized preconditioners in order to allow one the usage of iterative solvers in combination with unfitted techniques. Another novelty with respect to the methods available in the literature is the problem sizes that can be handled with the proposed approach. While most of previous references discussing linear solvers for unfitted methods are based on serial non-scalable algorithms, we propose a parallel distributed-memory method able to efficiently solve problems at large scales. This is demonstrated by means of a weak scaling test defined on complex 3D domains up to 300M degrees of freedom and one billion cells on 16K CPU cores in the Marenostrum-IV platform. The parallel implementation of the AgFEM method is available in the large-scale finite element package FEMPAR.

The surrogate matrix methodology: A reference implementation for low-cost assembly in isogeometric analysis

Cecilia Soriano — Fri, 21 Feb 2020 10:41:02 +0100

A reference implementation of a new method in isogeometric analysis (IGA) is presented. It delivers lowcost variable-scale approximations (surrogates) of the matrices which IGA conventionally requires to be computed by element-scale quadrature. To generate surrogate matrices, quadrature must only be performed on a fraction of the elements in the computational domain. In this way, quadrature determines only a subset of the entries in the final matrix. The remaining matrix entries are computed by a simple B-spline interpolation procedure. We present the modifications and extensions required for a reference implementation in the open-source IGA software library GeoPDEs. The exposition is fashioned to help facilitate similar modifications in other contemporary software libraries. Method name: Surrogate matrix method for isogeometric analysis

The surrogate matrix methodology: Low-cost assembly for isogeometric analysis

Cecilia Soriano — Fri, 21 Feb 2020 10:18:03 +0100

A new methodology in isogeometric analysis (IGA) is presented. This methodology delivers low-cost variable-scale approximations (surrogates) of the matrices which IGA conventionally requires to be computed from element-scale quadrature formulas. To generate surrogate matrices, quadrature must only be performed on certain elements in the computational domain. This, in turn, determines only a subset of the entries in the final matrix.The remaining matrix entries are computed by a simple B-spline interpolation procedure. Poisson’s equation, membrane vibration, plate bending, and Stokes’ flow problems are studied. In these problems, the use of surrogate matrices has a negligible impact on solution accuracy. Because only a small fraction of the original quadrature must be performed, we are able to report beyond a fifty-fold reduction in overall assembly time in the same software. The capacity for even further speed-ups is clearly demonstrated. The implementation used here was achieved by a small number of modifications to the open-source IGA software library GeoPDEs. Similar modifications could be made to other present-day software libraries.

The surrogate matrix methodology: a priori error estimation

Cecilia Soriano — Fri, 21 Feb 2020 09:59:22 +0100

We give the first mathematically rigorous analysis of an emerging approach to finite element analysis (see, e.g., Bauer et al. [Appl. Numer. Math., 2017]), which we hereby refer to as the surrogate matrix methodology. This methodology is based on the piece-wise smooth approximation of the matrices involved in a standard finite element discretization. In particular, it relies on the projection of smooth so-called stencil functions onto high-order polynomial subspaces. The performance advantage of the surrogate matrix methodology is seen in constructions where each stencil function uniquely determines the values of a significant collection of matrix entries. Such constructions are shown to be widely achievable through the use of locally-structured meshes. Therefore, this methodology can be applied to a wide variety of physically meaningful problems, including nonlinear problems and problems with curvilinear geometries. Rigorous a priori error analysis certifies the convergence of a novel surrogate method for the variable coefficient Poisson equation. The flexibility of the methodology is also demonstrated through the construction of novel methods for linear elasticity and nonlinear diffusion problems. In numerous numerical experiments, we demonstrate the efficacy of these new methods in a matrix-free environment with geometric multigrid solvers. In our experiments, up to a twenty-fold decrease in computation time is witnessed over the classical method with an otherwise identical implementation.

AutoParallel: A Python module for automatic parallelization and distributed execution of affine loop nests

Cecilia Soriano — Fri, 21 Feb 2020 09:42:37 +0100

The last improvements in programming languages, programming models, and frameworks have focused on abstracting the users from many programming issues. Among others, recent programming frameworks include simpler syntax, automatic memory management and garbage collection, which simplifies code re-usage through library packages, and easily configurable tools for deployment. For instance, Python has risen to the top of the list of the programming languages due to the simplicity of its syntax, while still achieving a good performance even being an interpreted language. Moreover, the community has helped to develop a large number of libraries and modules, tuning them to obtain great performance.
However, there is still room for improvement when preventing users from dealing directly with distributed and parallel computing issues. This paper proposes and evaluates AutoParallel, a Python module to automatically find an appropriate task-based parallelization of affine loop nests to execute them in parallel in a distributed computing infrastructure. This parallelization can also include the building of data blocks to increase task granularity in order to achieve a good execution performance. Moreover, AutoParallel is based on sequential programming and only contains a small annotation in the form of a Python decorator so that anyone with little programming skills can scale up an application to hundreds of cores.

ExaQUte Leaflet

Cecilia Soriano — Mon, 18 Nov 2019 11:06:02 +0100

ExaQUte Leaflet

ExaQUte_4_Everyone

Cecilia Soriano — Mon, 18 Nov 2019 10:58:02 +0100

ExaQUte Description for the general public