Tensor computations with dimensionality manipulations
Loading...
No Access Until
Permanent Link(s)
Collections
Other Titles
Authors
Abstract
Methodologies that ensure the compressibility of tensors are introduced. Bounds on the storage costs with respect to various tensor formats are derived. A new algorithm combining data-sparse tensor formats and factored alternating direction implicit method is designed to solve Sylvester tensor equations, and incorporated in a fast spectral Poisson equation solver on cubes with optimal complexity. New parallelizable algorithms for computing the tensor-train decomposition of tensors in original format, streaming data, Tucker format, and that satisfy algebraic relations, are proposed. Based on the input format, the algorithms involve deterministic or probabilistic aspects, and all have guarantees of accuracy. Scaling analysis and numerical experiments are provided to demonstrate computational and storage efficiency. An ultraspherical spectral method is developed for fractional partial differential equations via the Caffarelli--Silvestre extension on disk and rectangular domains. A parallel domain decomposition solver is designed for multi-core performance of non-smooth functions. The discretized equation is solved via direct tensor equation solvers, and numerical performance is shown with a fractional PDE constrained optimization problem. Linear systems in electron correlation calculation from computational chemistry are converted into tensor equations to reduce computing and storage costs. Several algorithms are developed to exploit the sparsity and data-sparsity of chemical structures. Numerical results indicate that tensor equation solvers are competitive over traditional linear system solvers with both canonical and localized orbital bases formulations. The quantized tensor-train format of tensors is introduced to approximate analytic functions via Chebyshev polynomial expansions. Analysis of different types of singularities is carried out, leading to theoretical guarantees of coefficient storage compressibility.
Journal / Series
Volume & Issue
Description
200 pages
Sponsorship
Date Issued
2022-05
Publisher
Keywords
Approximation theory; Numerical analysis; Numerical linear algebra; Scientific computing
Location
Effective Date
Expiration Date
Sector
Employer
Union
Union Local
NAICS
Number of Workers
Committee Chair
Townsend, Alex John
Committee Co-Chair
Committee Member
Udell, Madeleine Richards
Bindel, David S.
Bindel, David S.
Degree Discipline
Applied Mathematics
Degree Name
Ph. D., Applied Mathematics
Degree Level
Doctor of Philosophy
Related Version
Related DOI
Related To
Related Part
Based on Related Item
Has Other Format(s)
Part of Related Item
Related To
Related Publication(s)
Link(s) to Related Publication(s)
References
Link(s) to Reference(s)
Previously Published As
Government Document
ISBN
ISMN
ISSN
Other Identifiers
Rights
Rights URI
Types
dissertation or thesis