Proper Orthogonal Decomposition (POD), combined with the Method of Snapshots and Galerkin projection, is a popular method for the model order reduction of nonlinear PDEs. The POD requires the left singular vectors from the singular value decomposition (SVD) of an n-by-m "snapshot matrix" S, each column of which represents the computed state of the system at a given time. However, the direct computation of this decomposition can be computationally expensive, particularly for snapshot matrices that are too large to fit in memory. Hierarchical Approximate POD (HAPOD) (Himpe 2016) is a recent method for the approximate truncated SVD that requires only a single pass over S, is easily parallelizable, and can be computationally cheaper than direct SVD, all while guaranteeing the requested accuracy for the resulting basis. This method processes the columns of S in blocks based on a predefined rooted tree of processors, concatenating the outputs from each stage to form the inputs for the next. However, depending on the selected parameter values and the properties of S, the performance of HAPOD may be no better than that of direct SVD. In this work, we numerically explore the parameter values and snapshot matrix properties for which HAPOD is computationally advantageous over the full SVD and compare its performance to that of a parallelized incremental SVD method (Brand 2002, Brand 2003, and Arrighi2015). In particular, in addition to the two major processor tree structures detailed in the initial publication of HAPOD (Himpe2016), we explore the viability of a new structure designed with an MPI implementation in mind. / Master of Science / Singular Value Decomposition (SVD) provides a way to represent numeric data that breaks the data up into its most important components, as well as measuring how significant each part is. This decomposition is widely used to assist in finding patterns in data and making decisions accordingly, or to obtain simple, yet accurate, representations of complex physical processes. Examples of useful data to decompose include the velocity of water flowing past an obstacle in a river, a large collection of images, or user ratings for a large number of movies. However, computing the SVD directly can be computationally expensive, and usually requires repeated access to the entire dataset. As these data sets can be very large, up to hundreds of gigabytes or even several terabytes, storing all of the data in memory at once may be infeasible. Thus, repeated access to the entire dataset requires that the files be read repeatedly from the hard disk, which can make the required computations exceptionally slow. Fortunately, for many applications, only the most important parts of the data are needed, and the rest can be discarded. As a result, several methods have surfaced that can pick out the most important parts of the data while accessing the original data only once, piece by piece, and can be much faster than computing the SVD directly. In addition, the recent bottleneck in individual computer processor speeds has motivated a need for methods that can efficiently run on a large number of processors in parallel. Hierarchical Approximate POD (HAPOD) [1] is a recently-developed method that can efficiently pick out the most important parts of the data while only accessing the original data once, and which is very easy to run in parallel. However, depending on a user-defined algorithm parameter (weight), HAPOD may return more information than is needed to satisfy the requested accuracy, which determines how much data can be discarded. It turns out that the input weights that result in less extra data also result in slower computations and the eventual need for more data to be stored in memory at once. This thesis explores how to choose this input weight to best balance the amount of extra information used with the speed of the method, and also explores how the properties of the data, such as the size of the data or the distribution of levels of significance of each part, impact the effectiveness of HAPOD.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/81938 |
| Date | 25 January 2018 |
| Creators | Beach, Benjamin Josiah |
| Contributors | Mathematics, Gugercin, Serkan, Borggaard, Jeffrey T., Embree, Mark P. |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Detected Language | English |
| Type | Thesis |
| Format | ETD, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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