Preconditioning the solution of the time- dependent neutron diffusion equation by recycling Krylov subspaces

[EN] Spectral preconditioners are based on the fact that the convergence rate of Krylov subspace methods is improved if the eigenvalues of smallest magnitude of the system matrix are `removed'. In this paper, two preconditioning strategies are studied to solve a set of linear systems associated with the numerical integration of the time dependent neutron di usion equation. Both strategies can be implemented using the matrix-vector product as the main operation and succeed at reducing the total number of iterations needed to solve the set of systems.This work has been partially supported by the Spanish Ministerio de Educacion y Ciencia under projects MTM2010-18674 and ENE2011-22823.González Pintor, S.; Ginestar Peiro, D.; Verdú Martín, GJ. (2014). Preconditioning the solution of the time- dependent neutron diffusion equation by recycling Krylov subspaces. International Journal of Computer Mathematics. 91(1):42-52. https://doi.org/10.1080/00207160.2013.771181S425291

An Sn Application of Homotopy Continuation in Neutral Particle Transport

The objective of this dissertation is to investigate the usefulness of homotopy continuation applied in the context of neutral particle transport where traditional methods of acceleration degrade. This occurs in higher dimensional heterogeneous problems [51]. We focus on utilizing homotopy continuation as a means of providing a better initial guess for difficult problems. We investigate various homotopy formulations for two primary diffcult problems: a thick-diffusive fixed internal source, and a k-eigenvalue problem with high dominance ratio. We also investigate the usefulness of homotopy continuation for computationally intensive problems with 30-energy groups. We find that homotopy continuation exhibits usefulness in specific problem formulations. In the thick-diffusive problem it shows benefit when there is a strong internal source in thin materials. In the k-eigenvalue problem, homotopy continuation provides an improvement in convergence speed for fixed point iteration methods in high dominance ratio problems. We also show that one of our imbeddings successfully stabilizes the nonlinear formulation of the k-eigenvalue problem with a high dominance ratio

Contribution to the study of efficient iterative methods for the numerical solution of partial differential equations

Multigrid and domain decomposition methods provide efficient algorithms for the numerical solution of partial differential equations arising in the modelling of many applications in Computational Science and Engineering. This manuscript covers certain aspects of modern iterative solution methods for the solution of large-scale problems issued from the discretization of partial differential equations. More specifically, we focus on geometric multigrid methods, non-overlapping substructuring methods and flexible Krylov subspace methods with a particular emphasis on their combination. Firstly, the combination of multigrid and Krylov subspace methods is investigated on a linear partial differential equation modelling wave propagation in heterogeneous media. Secondly, we focus on non-overlapping domain decomposition methods for a specific finite element discretization known as the hp finite element, where unrefinement/refinement is allowed both by decreasing/increasing the step size h or by decreasing/increasing the polynomial degree p of the approximation on each element. Results on condition number bounds for the domain decomposition preconditioned operators are given and illustrated by numerical results on academic problems in two and three dimensions. Thirdly, we review recent advances related to a class of Krylov subspace methods allowing variable preconditioning. We examine in detail flexible Krylov subspace methods including augmentation and/or spectral deflation, where deflation aims at capturing approximate invariant subspace information. We also present flexible Krylov subspace methods for the solution of linear systems with multiple right-hand sides given simultaneously. The efficiency of the numerical methods is demonstrated on challenging applications in seismics requiring the solution of huge linear systems of equations with multiple right-hand sides on parallel distributed memory computers. Finally, we expose current and future prospectives towards the design of efficient algorithms on extreme scale machines for the solution of problems coming from the discretization of partial differential equations

Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference

The 6th ECCOMAS Young Investigators Conference YIC2021 will take place from July 7th through 9th, 2021 at Universitat Politècnica de València, Spain. The main objective is to bring together in a relaxed environment young students, researchers and professors from all areas related with computational science and engineering, as in the previous YIC conferences series organized under the auspices of the European Community on Computational Methods in Applied Sciences (ECCOMAS). Participation of senior scientists sharing their knowledge and experience is thus critical for this event.YIC 2021 is organized at Universitat Politécnica de València by the Sociedad Española de Métodos Numéricos en Ingeniería (SEMNI) and the Sociedad Española de Matemática Aplicada (SEMA). It is promoted by the ECCOMAS.The main goal of the YIC 2021 conference is to provide a forum for presenting and discussing the current state-of-the-art achievements on Computational Methods and Applied Sciences,including theoretical models, numerical methods, algorithmic strategies and challenging engineering applications.Nadal Soriano, E.; Rodrigo Cardiel, C.; Martínez Casas, J. (2022). Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. https://doi.org/10.4995/YIC2021.2021.15320EDITORIA

Linear Diffusion Acceleration for Neutron Transport Problems

Nuclear engineers are interested in solutions of the Neutron Transport Equation (NTE), with the goal of improving the safety and efficiency of reactors and critical nuclear systems. Complex simulations are used to obtain detailed solutions of the NTE, and can require immense computational resources to execute. A variety of methods have been developed to ease the computational burden of simulating full-scale, whole-core reactor problems. Among these is transport acceleration, which improves the convergence rate of iterative transport calculations. In addition to the use of acceleration methods, certain approximations are often made when solving the NTE. The 2D/1D approximation is used to generate a 3D solution of the NTE by iteratively solving coupled 2D radial and 1D axial equations. This method is one of the foundational techniques used in the neutronics code MPACT. Also, the Transport-Corrected P0 (TCP0) approximation for neutron scattering is often used in reactor analysis codes to simplify higher-order scattering physics. Unfortunately, both of these approximations allow for non-positive flux solutions of the NTE. More importantly, some spatial discretizations of the NTE also permit negative solutions. Under certain conditions, this can cause instability for nonlinear acceleration methods such as Coarse Mesh Finite Difference (CMFD). In this thesis, we propose a novel acceleration scheme called Linear Diffusion Acceleration (LDA) that does not possess the nonlinearities present in CMFD. This thesis work presents LDA as an alternative acceleration scheme to CMFD. As the name suggests, the LDA method is linear with respect to the scalar flux. Therefore, LDA is not susceptible to the same nonlinear modes of numerical failure as CMFD. In addition, LDA is shown to possess similar convergence properties as CMFD for practical problems that have no negative scalar fluxes. Transport acceleration with LDA allows for the use of some of the aforementioned approximations, in which the positivity of the scalar flux is not guaranteed. Fourier analysis of CMFD and LDA is performed to compare the theoretical convergence rates of the two methods for simple, spatially-heterogeneous problems. In addition, simple and practical case studies are presented in which CMFD fails due to nonlinearity. For these cases, LDA is shown to retain stability. Certain other advantages of LDA, which are a consequence of its mathematical structure, are also discussed.PHDNuclear Engineering & Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169898/1/zdodson_1.pd

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal