- Title Information
- Title
- Multigrid and Domain Decomposition Methods in Fault-Prone Environments
- Name:
Personal
- Name Part
- Glusa, Christian Alexander
- Role
- Role Term:
Text
- creator
- Name:
Personal
- Name Part
- Ainsworth, Mark
- Role
- Role Term:
Text
- Advisor
- Name:
Personal
- Name Part
- Karniadakis, George
- Role
- Role Term:
Text
- Reader
- Name:
Personal
- Name Part
- Guzman, Johnny
- Role
- Role Term:
Text
- Reader
- Name:
Corporate
- Name Part
- Brown University. Department of Applied Mathematics
- Role
- Role Term:
Text
- sponsor
- Origin Information
- Copyright Date
- 2017
- Physical Description
- Extent
- xxiii, 198 p.
- digitalOrigin
- born digital
- Note:
thesis
- Thesis (Ph. D.)--Brown University, 2017
- Genre (aat)
- theses
- Abstract
- Many scientific simulations rely on the solution of very large linear systems of equations, and multigrid and domain decomposition methods are widely used solvers. It is unclear whether or not existing implementations of these methods will continue to perform in the same way on the next generation of supercomputers.
In the first part, we study the impact of faults on the convergence of iterative solvers. We describe the different categories of faults and how they may be modeled using random matrices. Theoretical results concerning products of random matrices are given. We apply this novel theoretical framework to a two grid method, and extend the obtained convergence bounds to the multi-level case. It will transpire that multigrid in the presence of a fixed fault level will degenerate as the problem size grows. By protecting one of its steps, the prolongation operation, fault resilience is achieved. Practical details concerning the detection and mitigation of faults are discussed in order to leverage the theoretical results to obtain a fault-tolerant multigrid implementation. Finally, we extend the framework to cover faults in overlapping domain decomposition solvers.
In the second part, we turn our attention to the efficient solution of non-local and fractional equations using iterative linear solvers. Non-local equations can be used to describe physical phenomena more accurately than classical local models. We introduce archetypes of elliptic and parabolic non-local equations, and state results concerning the regularity of their solution. We give details concerning finite element approximations and derive a priori and a posteriori error bounds. We then turn to the question of efficient implementation. We obtain sparse representations of the non-local operators by exploiting the low rank of off-diagonal blocks, and demonstrate that the assemble-solve-estimate-refine cycle has quasi-linear complexity. We give numerical examples, involving fractional Poisson and heat equations and reaction-diffusion systems.
- Subject
- Topic
- Numerical Anlaysis
- Subject
- Topic
- Scientific Computation
- Subject (fast)
(authorityURI="http://id.worldcat.org/fast", valueURI="http://id.worldcat.org/fast/00921988")
- Topic
- Fault-tolerant computing
- Subject (fast)
(authorityURI="http://id.worldcat.org/fast", valueURI="http://id.worldcat.org/fast/01028886")
- Topic
- Multigrid methods (Numerical analysis)
- Subject
- Topic
- Domain Decomposition methods
- Subject
- Topic
- Fractional Laplacian
- Subject
- Topic
- Non-Local Equations
- Subject (fast)
(authorityURI="http://id.worldcat.org/fast", valueURI="http://id.worldcat.org/fast/00980827")
- Topic
- Iterative methods (Mathematics)
- Subject (fast)
(authorityURI="http://id.worldcat.org/fast", valueURI="http://id.worldcat.org/fast/00956032")
- Topic
- High performance computing
- Language
- Language Term (ISO639-2B)
- English
- Record Information
- Record Content Source (marcorg)
- RPB
- Record Creation Date
(encoding="iso8601")
- 20170616
- Identifier:
DOI
- 10.7301/Z0X63KCM
- Access Condition:
rights statement
(href="http://rightsstatements.org/vocab/InC/1.0/")
- In Copyright
- Access Condition:
restriction on access
- Collection is open for research.
- Type of Resource (primo)
- dissertations