569 research outputs found
A Recommendation System for Preconditioned Iterative Solvers
Solving linear systems of equations is an integral part of most scientific simulations. In
recent years, there has been a considerable interest in large scale scientific simulation of
complex physical processes. Iterative solvers are usually preferred for solving linear systems
of such magnitude due to their lower computational requirements. Currently, computational
scientists have access to a multitude of iterative solver options available as "plug-and-
play" components in various problem solving environments. Choosing the right solver
configuration from the available choices is critical for ensuring convergence and achieving
good performance, especially for large complex matrices. However, identifying the
"best" preconditioned iterative solver and parameters is challenging even for an expert due
to issues such as the lack of a unified theoretical model, complexity of the solver configuration
space, and multiple selection criteria. Therefore, it is desirable to have principled
practitioner-centric strategies for identifying solver configuration(s) for solving large linear
systems.
The current dissertation presents a general practitioner-centric framework for (a) problem
independent retrospective analysis, and (b) problem-specific predictive modeling of
performance data. Our retrospective performance analysis methodology introduces new
metrics such as area under performance-profile curve and conditional variance-based finetuning
score that facilitate a robust comparative performance evaluation as well as parameter
sensitivity analysis. We present results using this analysis approach on a number of popular
preconditioned iterative solvers available in packages such as PETSc, Trilinos, Hypre, ILUPACK, and WSMP. The predictive modeling of performance data is an integral part
of our multi-stage approach for solver recommendation. The key novelty of our approach
lies in our modular learning based formulation that comprises of three sub problems: (a)
solvability modeling, (b) performance modeling, and (c) performance optimization, which
provides the flexibility to effectively target challenges such as software failure and multiobjective
optimization. Our choice of a "solver trial" instance space represented in terms
of the characteristics of the corresponding "linear system", "solver configuration" and their
interactions, leads to a scalable and elegant formulation. Empirical evaluation of our approach
on performance datasets associated with fairly large groups of solver configurations
demonstrates that one can obtain high quality recommendations that are close to the ideal
choices
Statistical and Machine Learning Techniques Applied to Algorithm Selection for Solving Sparse Linear Systems
There are many applications and problems in science and engineering that require large-scale numerical simulations and computations. The issue of choosing an appropriate method to solve these problems is very common, however it is not a trivial one, principally because this decision is most of the times too hard for humans to make, or certain degree of expertise and knowledge in the particular discipline, or in mathematics, are required. Thus, the development of a methodology that can facilitate or automate this process and helps to understand the problem, would be of great interest and help. The proposal is to utilize various statistically based machine-learning and data mining techniques to analyze and automate the process of choosing an appropriate numerical algorithm for solving a specific set of problems (sparse linear systems) based on their individual properties
Quantum Circulant Preconditioner for Linear System of Equations
We consider the quantum linear solver for with the circulant
preconditioner . The main technique is the singular value estimation (SVE)
introduced in [I. Kerenidis and A. Prakash, Quantum recommendation system, in
ITCS 2017]. However, some modifications of SVE should be made to solve the
preconditioned linear system . Moreover, different from
the preconditioned linear system considered in [B. D. Clader, B. C. Jacobs, C.
R. Sprouse, Preconditioned quantum linear system algorithm, Phys. Rev. Lett.,
2013], the circulant preconditioner is easy to construct and can be directly
applied to general dense non-Hermitian cases. The time complexity depends on
the condition numbers of and , as well as the Frobenius norm
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