1,013 research outputs found
Uncertainty quantification in coastal aquifers using the multilevel Monte Carlo method
We consider a class of density-driven flow problems. We are particularly
interested in the problem of the salinization of coastal aquifers. We consider
the Henry saltwater intrusion problem with uncertain porosity, permeability,
and recharge parameters as a test case. The reason for the presence of
uncertainties is the lack of knowledge, inaccurate measurements, and inability
to measure parameters at each spatial or time location. This problem is
nonlinear and time-dependent. The solution is the salt mass fraction, which is
uncertain and changes in time. Uncertainties in porosity, permeability,
recharge, and mass fraction are modeled using random fields. This work
investigates the applicability of the well-known multilevel Monte Carlo (MLMC)
method for such problems. The MLMC method can reduce the total computational
and storage costs. Moreover, the MLMC method runs multiple scenarios on
different spatial and time meshes and then estimates the mean value of the mass
fraction. The parallelization is performed in both the physical space and
stochastic space. To solve every deterministic scenario, we run the parallel
multigrid solver ug4 in a black-box fashion. We use the solution obtained from
the quasi-Monte Carlo method as a reference solution.Comment: 24 pages, 3 tables, 11 figure
Resolução da equação da onda utilizando métodos multigrid espaço-tempo
Orientador: Prof. Dr. Marcio Augusto Villela PintoCoorientador: Prof. Dr. Sebastião Romero FrancoTese (doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Métodos Numéricos em Engenharia. Defesa : Curitiba, 10/04/2023Inclui referências: p. 109-117Resumo: Neste trabalho apresenta-se a avaliação de diferentes formas de solução para problemas modelados pela equação da onda, para os casos 1D e 2D. Utiliza-se para discretização espacial, o Método das Diferenças Finitas ponderado por um parâmetro n em diferentes estágios de tempo, para obter-se um esquema de solução implícito. Com isso, propõe-se a utilização de diferentes varreduras no tempo, a fim de gerar métodos robustos e eficientes, desde a clássica Time-Stepping, até outra varredura que envolve simultaneamente o espaço e o tempo, como Waveform Relaxation. Neste trabalho, combina-se o método dos Subdomínios com a estratégia Waveform Relaxation para reduzir as fortes oscilações que ocorrem o início do processo iterativo. Obtém-se excelentes resultados ao aplicar o método Multigrid para esta classe de problemas, já que, melhora-se muito os fatores de convergência calculados a partir das soluções aproximadas do sistema de equações resultante das discretizações. Na verificação das metodologias propostas e suas características, apresentamse simulações de propagação de ondas envolvendo problemas uni e bidimensionais, onde analisa-se os erros de discretização, ordens efetiva e aparente, fator de convergência, ordens de complexidade e tempo computacional.Abstract: In this thesis presents the evaluation of different forms of solution for probelms modeled by the wave equation, for the 1D and 2D cases. The Finite Difference Method is used for the spatial discretization, weighted by a parameter n at different time steps, in order to obtain an implicit solution. With this, it is proposed the use of different sweeps in time, in order to generate robust and efficient methods, from the classical Time-Stepping, to other less usual sweep as Waveform Relaxation. In this work, the Subdomains method is combined with the Waveform Relaxation strategy to reduce the strong oscillations that occur early in the iterative process. Excellent results are obtained when applying the Multigrid method for this class of problems, since the convergence factors calculated from the approximate solutions of the system of equations resulting from the discretizations are greatly improved. In the verification of the proposed methodologies and their respective advantages, simulations of wave propagation involving one- and two-dimensional problems are presented, where the discretization errors, effective and apparent orders, convergence factor, complexity orders and computational time are analyzed
Algebraic Temporal Blocking for Sparse Iterative Solvers on Multi-Core CPUs
Sparse linear iterative solvers are essential for many large-scale
simulations. Much of the runtime of these solvers is often spent in the
implicit evaluation of matrix polynomials via a sequence of sparse
matrix-vector products. A variety of approaches has been proposed to make these
polynomial evaluations explicit (i.e., fix the coefficients), e.g., polynomial
preconditioners or s-step Krylov methods. Furthermore, it is nowadays a popular
practice to approximate triangular solves by a matrix polynomial to increase
parallelism. Such algorithms allow to evaluate the polynomial using a so-called
matrix power kernel (MPK), which computes the product between a power of a
sparse matrix A and a dense vector x, or a related operation. Recently we have
shown that using the level-based formulation of sparse matrix-vector
multiplications in the Recursive Algebraic Coloring Engine (RACE) framework we
can perform temporal cache blocking of MPK to increase its performance. In this
work, we demonstrate the application of this cache-blocking optimization in
sparse iterative solvers.
By integrating the RACE library into the Trilinos framework, we demonstrate
the speedups achieved in preconditioned) s-step GMRES, polynomial
preconditioners, and algebraic multigrid (AMG). For MPK-dominated algorithms we
achieve speedups of up to 3x on modern multi-core compute nodes. For algorithms
with moderate contributions from subspace orthogonalization, the gain reduces
significantly, which is often caused by the insufficient quality of the
orthogonalization routines. Finally, we showcase the application of
RACE-accelerated solvers in a real-world wind turbine simulation (Nalu-Wind)
and highlight the new opportunities and perspectives opened up by RACE as a
cache-blocking technique for MPK-enabled sparse solvers.Comment: 25 pages, 11 figures, 3 table
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
Prognostic and health management of critical aircraft systems and components: an overview
This article belongs to the Special Issue Feature Papers in Fault Diagnosis & Sensors 2023Prognostic and health management (PHM) plays a vital role in ensuring the safety and reliability of aircraft systems. The process entails the proactive surveillance and evaluation of the state and functional effectiveness of crucial subsystems. The principal aim of PHM is to predict the remaining useful life (RUL) of subsystems and proactively mitigate future breakdowns in order to minimize consequences. The achievement of this objective is helped by employing predictive modeling techniques and doing real-time data analysis. The incorporation of prognostic methodologies is of utmost importance in the execution of condition-based maintenance (CBM), a strategic approach that emphasizes the prioritization of repairing components that have experienced quantifiable damage. Multiple methodologies are employed to support the advancement of prognostics for aviation systems, encompassing physics-based modeling, data-driven techniques, and hybrid prognosis. These methodologies enable the prediction and mitigation of failures by identifying relevant health indicators. Despite the promising outcomes in the aviation sector pertaining to the implementation of PHM, there exists a deficiency in the research concerning the efficient integration of hybrid PHM applications. The primary aim of this paper is to provide a thorough analysis of the current state of research advancements in prognostics for aircraft systems, with a specific focus on prominent algorithms and their practical applications and challenges. The paper concludes by providing a detailed analysis of prospective directions for future research within the field.European Union funding: 95568
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
Training Manual on Advanced Analytical Tools for Social Science Research Vol.1
Applying appropriate analytical techniques form the backbone of any research endeavor in agriculture,
fisheries, and allied sciences. Without proper knowledge of applying statistical/econometric tools,
software, and derivation of inferences from the same, it would not be possible to gather relevant
interpretations of the investigation. Hence, the importance of well-designed data collection protocol,
analysis, and interpretation cannot be underestimated. Such inferences form the basis of sound policy
planning and resource management. Technology advancements and the development of analytical
software have made the data analysis process less laborious. A basic understanding of the application
of advanced analytical tools and their interpretation increases the productivity and efficiency of social
science researchers engaged in agriculture/animal/fisheries science research. Hence the Winter School
on Advanced Analytical Tools for Social Science Research is designed to enhance the analytical skills
of social science researchers from NARES by allowing them to familiarize with advanced analytical
procedures and their practical applications.
This Winter School is a step towards familiarizing recent analytical techniques in social science to
derive quality research outputs. The course is designed to acquaint the participants with areas such as
exploratory data analysis, sampling techniques, data classificatory techniques, non–parametric
methods, econometric analysis, and time series modeling, etc. Lectures on GIS/Spatial modeling,
scaling techniques, data mining and big data analytics, machine learning techniques, and ecosystem
evaluation have also been touched upon. The course is more practical-oriented, with a greater emphasis
on interpreting the results. It employs a combination of lectures and exercises using statistical software
Geometric multigrid method for solving Poisson's equation on octree grids with irregular boundaries
A method is presented to include irregular domain boundaries in a geometric multigrid solver. Dirichlet boundary conditions can be imposed on an irregular boundary defined by a level set function. Our implementation employs quadtree/octree grids with adaptive refinement, a cell-centered discretization and pointwise smoothing. Boundary locations are determined at a subgrid resolution by performing line searches. For grid blocks near the interface, custom operator stencils are stored that take the interface into account. For grid block away from boundaries, a standard second-order accurate discretization is used. The convergence properties, robustness and computational cost of the method are illustrated with several test cases. New version program summary: Program Title: Afivo CPC Library link to program files: https://doi.org/10.17632/5y43rjdmxd.2 Developer's repository link: https://github.com/MD-CWI/afivo Licensing provisions: GPLv3 Programming language: Fortran Journal reference of previous version: Comput. Phys. Commun. 233 (2018) 156–166. https://doi.org/10.1016/j.cpc.2018.06.018 Does the new version supersede the previous version?: Yes. Reasons for the new version: Add support for internal boundaries in the geometric multigrid solver. Summary of revisions: The geometric multigrid solver was generalized in several ways: a coarse grid solver from the Hypre library is used, operator stencils are now stored per grid block, and methods for including boundaries via a level set function were added. Nature of problem: The goal is to solve Poisson's equation in the presence of irregular boundaries that are not aligned with the computational grid. It is assumed these irregular boundaries are defined by a level set function, and that a Dirichlet type boundary condition is applied. The main applications are 2D and 3D simulations with octree-based adaptive mesh refinement, in which the mesh frequently changes but the irregular boundaries do not. Solution method: A geometric multigrid method compatible with octree grids is developed, using a cell-centered discretization and point-wise smoothing. Near irregular boundaries, custom operator stencils are stored. Line searches are performed to locate interfaces with sub-grid resolution. To increase the methods robustness, this line search is modified on coarse grids if boundaries are otherwise not resolved. The multigrid solver uses OpenMP parallelization
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