Novel Randomized Algorithms for QR with Column Pivoting, and Their Implementations in RandLAPACK
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Melnichenko, Maksim↗
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Randomized Projection for Rank-Revealing Matrix Factorizations and Low-Rank Approximations
Rank-revealing matrix decompositions provide an essential tool in spectral analysis of matrices, including the Singular Value Decomposition (SVD) and related low-rank approximation techniques. QR with Column Pivoting (QRCP) is usually suitable for these purposes, but it can be much slower than the unpivoted QR algorithm. For large matrices, the difference in performance is due to increased communication between the processor and slow memory, which QRCP needs in order to choose pivots during decomposition. Our main algorithm, Randomized QR with Column Pivoting (RQRCP), uses randomized projection to make pivot decisions from a much smaller sample matrix, which we can construct to reside in a faster level of memory than the original matrix. This technique may be understood as trading vastly reduced communication for a controlled increase in uncertainty during the decision process. Furthermore, for rank-revealing purposes, the selection mechanism in RQRCP produces results that are the same quality as the standard algorithm, but with performance near that of unpivoted QR (often an order of magnitude faster for large matrices). Additionally, we also propose two formulas that facilitate further performance improvements. The first efficiently updates sample matrices to avoid computing new randomized projections. The second avoids large trailing updates during the decomposition in truncated low-rank approximations. Our truncated version of RQRCP also provides a key initial step in our truncated SVD approximation, TUXV. These advances open up a new performance domain for large matrix factorizations that will support efficient problem-solving techniques for challenging applications in science, engineering, and data analysis.
97 MATHEMATICS AND COMPUTING↗
Applications of truncated QR methods to sinusoidal frequency estimation
Three truncated QR methods are proposed for sinusoidal frequency estimation: (1) truncated QR without column pivoting (TQR), (2) truncated QR with preordered columns, and (3) truncated QR with column pivoting. It is demonstrated that the benefit of truncated SVD for high frequency resolution is achievable under the truncated QR approach with much lower computational cost. Other attractive features of the proposed methods include the ease of updating, which is difficult for the SVD method, and numerical stability. TQR methods thus offer efficient ways to identify sinusoidals closely clustered in frequencies under stationary and nonstationary conditions.
Hsieh, S. F.↗
Computing rank‐revealing factorizations of matrices stored out‐of‐core
This paper describes efficient algorithms for computing rank-revealing factorizations of matrices that are too large to fit in main memory (RAM), and must instead be stored on slow external memory devices such as disks (out-of-core or out-of-memory). Traditional algorithms for computing rank-revealing factorizations (such as the column pivoted QR factorization and the singular value decomposition) are very communication intensive as they require many vector-vector and matrix-vector operations, which become prohibitively expensive when data is not in RAM. Randomization allows to reformulate new methods so that large contiguous blocks of the matrix are processed in bulk. The paper describes two distinct methods. The first is a blocked version of column pivoted Householder QR, organized as a “left-looking” method to minimize the number of the expensive write operations. The second method results employs a UTV factorization. It is organized as an algorithm-by-blocks to overlap computations and I/O operations. As it incorporates power iterations, it is much better at revealing the numerical rank. Numerical experiments on several computers demonstrate that the new algorithms are almost as fast when processing data stored on slow memory devices as traditional algorithms are for data stored in RAM.
97 MATHEMATICS AND COMPUTING↗
Algorithm For Solution Of Subset-Regression Problems
Reliable and flexible algorithm for solution of subset-regression problem performs QR decomposition with new column-pivoting strategy, enables selection of subset directly from originally defined regression parameters. This feature, in combination with number of extensions, makes algorithm very flexible for use in analysis of subset-regression problems in which parameters have physical meanings. Also extended to enable joint processing of columns contaminated by noise with those free of noise, without using scaling techniques.
Verhaegen, Michel↗
Bringing randomized algorithms to mainstream numerical linear algebra
Numerical linear algebra (NLA) underpins huge swaths of computational science and engineering. For scientists and engineers to make the most of the DOE’s computing resources, it is essential that they have access to high-performance implementations of algorithms with best-in-class scalability and reliability. Despite this, prevailing NLA libraries have little to no support for breakthrough algorithms from the field of randomized numerical linear algebra (RandNLA) that have been developed over the past twenty years. The goal of this LDRD was to break a log-jam that had prevented broad adoption of RandNLA. Our work had two thrusts. The first was to develop RandBLAS: a trustworthy and high-performance C++ library for randomized dimension reduction (an operation widely known as sketching). The second was the development of a novel randomized algorithm for computing a challenging type of matrix decomposition known as Householder QR with column pivoting (Householder QRCP). In this one-year late-start LDRD we successfully delivered RandBLAS 1.0 and new CPU and GPU codes for Householder QRCP. RandBLAS has extensive documentation at https://randblas.readthedocs.io/en/stable/. Papers on RandBLAS and and our high-performance QRCP codes are forthcoming.
97 MATHEMATICS AND COMPUTING↗
Mechanical features of the shuttle rotating service structure
With the development of the space shuttle launching facilities, it became mandatory to develop a shuttle rotating service structure to provide for the insertion and/or removal of payloads at the launch pads. The rotating service structure is a welded tubular steel space frame 189 feet high, 65 feet wide, and weighing 2100 tons. At the pivot column the structure is supported on a 30 inch diameter hemispherical bearing. At the opposite terminus the structure is supported on two truck assemblies each having eight 36 inch diameter double flanged wheels. The following features of the rotating service structure are discussed: (1) thermal expansion and contraction; (2) hurricane tie downs; (3) payload changeout room; (4) payload ground handling mechanism; (5) payload and orbiter access platforms; and (6) orbiter cargo bay access.
Crump, J. M.↗
On the reliable and flexible solution of practical subset regression problems
A new algorithm for solving subset regression problems is described. The algorithm performs a QR decomposition with a new column-pivoting strategy, which permits subset selection directly from the originally defined regression parameters. This, in combination with a number of extensions of the new technique, makes the method a very flexible tool for analyzing subset regression problems in which the parameters have a physical meaning.
Verhaegen, M. H.↗
The minimal residual QR-factorization algorithm for reliably solving subset regression problems
A new algorithm to solve test subset regression problems is described, called the minimal residual QR factorization algorithm (MRQR). This scheme performs a QR factorization with a new column pivoting strategy. Basically, this strategy is based on the change in the residual of the least squares problem. Furthermore, it is demonstrated that this basic scheme might be extended in a numerically efficient way to combine the advantages of existing numerical procedures, such as the singular value decomposition, with those of more classical statistical procedures, such as stepwise regression. This extension is presented as an advisory expert system that guides the user in solving the subset regression problem. The advantages of the new procedure are highlighted by a numerical example.
Verhaegen, M. H.↗
The use of the QR factorization in the partial realization problem
The use of the QR factorization of the Hankel matrix in solving the partial realization problem is analyzed. Straightforward use of the QR factorization results in a realization scheme that possesses all of the computational advantages of Rissanen's realization scheme. These latter properties are computational efficiency, recursiveness, use of limited computer memory, and the realization of a system triplet having a condensed structure. Moreover, this scheme is robust when the order of the system corresponds to the rank of the Hankel matrix. When this latter condition is violated, an approximate realization could be determined via the QR factorization. In this second scheme, the given Hankel matrix is approximated by a low-rank non-Hankel matrix. Furthermore, it is demonstrated that column pivoting might be incorporated in this second scheme. The results presented are derived for a single input/single output system, but this does not seem to be a restriction.
Verhaegen, M. H.↗
On the reliable and flexible solution of practical subset regression problems
A new algorithm for solving subset regression problems is described. The algorithm performs a QR decomposition with a new column-pivoting strategy, which permits subset selection directly from the originally defined regression parameters. This, in combination with a number of extensions of the new technique, makes the method a very flexible tool for analyzing subset regression problems in which the parameters have a physical meaning.
Verhaegen, M.↗
Motion systems providing three or four degrees of freedom
A motion system is provided by a platform generally parallel to a base and connected thereto by a column and powered and controlled extensible members, at least three of which are connected between distributed points around the column. In a three degree of freedom device, the column is conical, rigidly supported at its base with a universal joint at its top. The points of attachment define triangles in the base and in the platform surrounding the column with one extensible member connected between each. In the four degree of freedom version, the column is modified by making it effectively a column which is pivoted or guided at the base or contains an extensible member, preferably retains its triangular shape and its universal joint connection to the platform at its apex. For stability four powered and controlled extensible members are provided between points in the base and platform distributed around the column, a preferred pattern of arrangement being a square with the column at the center.
Chou, Richard C.↗
Selective electrified polyethylene upcycling by pore-modulated pyrolysis
Plastic waste is a increasing problem, accumulating in landfills and the environment. Pyrolysis is a promising and industrially relevant approach for transforming plastic waste into value-added chemicals. However, the selectivity and yield of traditional plastic pyrolysis are poor, with products featuring broad molar mass distributions. Here we report a highly selective, energy-efficient and catalyst-free pyrolysis method that can upcycle plastic into value-added chemicals via pore-modulated pyrolysis. Using a Joule-heated carbon column, we demonstrate the pivotal role of the reactor’s graded porous structure in decreasing the polydispersity of the reaction intermediates, enabling high product selectivity and yield. The decreasing pore size of the reactor modulates the mass transport in an apparent gating effect—preventing high-molar-mass species from exiting the reactor before sufficient pyrolysis has occurred. Using polyethylene as a model reactant, we demonstrate a high yield of 65.9 ± 5.2% and up to 80.8% selectivity toward value-added aviation fuel precursor (C8–C18 hydrocarbons) without the use of any catalysts.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
The Polarimetric Radar Scattering Properties of Oriented Aggregates
The scattering properties of aggregates are studied herein. Early aggregates (<7 monomers) of branched planar crystals and mature aggregates (up to 100 monomers) of columns are randomly generated with varying assumptions about the monomer attachment processes and the orientation behavior during collection. The resulting physical properties of the aggregates correspond well with prior in situ and retrieved sizes and shapes. Assumed azimuthally uniform orientations during collection and monomer pivoting upon attachment resulted in flatter and denser aggregates. The column aggregates had lower density and more spherical shapes than the branched planar crystal aggregates. The scattering properties were calculated using the discrete dipole approximation for a set of orientation angles and transformed to spectral coefficients representing modes of orientation angle variability. The zeroth- and second-order coefficients dominate this variability, with the zeroth-order coefficients representing the scattering properties for randomly oriented particles. The second-order coefficients for backscatter showed differences between horizontal and vertical polarization increasing with density, and these coefficients for specific differential phase increase with both mass and density. Similarly, coefficients for the copolar covariance decreased with density. Rapid changes in the contributions to the radar moments from the second-order coefficients from low to moderate density were observed, likely due to the increasing presence of horizontally aligned monomers in the aggregate structure. Differences in how differential reflectivity and correlation coefficient evolve with the orientation distribution parameters suggest that these measurements, along with specific differential phase and reflectivity, provide complementary information about aggregate sizes, shapes, and orientation distributions.
Cloud microphysics↗
Block LU factorization
Many of the currently popular 'block algorithms' are scalar algorithms in which the operations have been grouped and reordered into matrix operations. One genuine block algorithm in practical use is block LU factorization, and this has recently been shown by Demmel and Higham to be unstable in general. It is shown here that block LU factorization is stable if A is block diagonally dominant by columns. Moreover, for a general matrix the level of instability in block LU factorization can be founded in terms of the condition number kappa(A) and the growth factor for Gaussian elimination without pivoting. A consequence is that block LU factorization is stable for a matrix A that is symmetric positive definite or point diagonally dominant by rows or columns as long as A is well-conditioned.
Demmel, James W.↗
Experimental parameters for plasma wakefield acceleration in a narrow plasma column
Recent theoretical advancements propose multiple positron acceleration schemes in plasma wakefield acceleration (PWFA). One of the most promising ideas involves the creation of an electron-driven blowout wake within a finite-radius pre-ionized plasma column. This leads to the formation of an elongated region of sheath electrons at the closing of the first wake period capable of accelerating positrons while simultaneously providing a transverse focusing force. Additionally, the proposed scheme has shown to be a potential means of suppressing instabilities. We present an experimental opportunity to explore the narrow column PWFA at the Facility for Advanced Accelerator Experimental Tests II (FACET-II): the E333 experiment. As a pivotal first step towards achieving positron acceleration in PWFA, we have planned a precursor experiment utilizing the currently available single-bunch electron beam to study the physics of the narrow plasma PWFA scheme. We outline the feasible experimental parameters, including beam and ionization laser parameters, along with the required laser optics for the experiment. The primary observable signatures for this stage of the experiment are the final energy spectrum and transverse position of the electron bunch, anticipating reduced energy loss and enhanced beam guidance in the narrow plasma column compared to the nominal PWFA. Comprehensive simulations are used to detail our experimental plan.
Lee, Valentina [University of Colorado, Boulder, C↗
Algorithms for solving large sparse systems of simultaneous linear equations on vector processors
Very efficient algorithms for solving large sparse systems of simultaneous linear equations have been developed for serial processing computers. These involve a reordering of matrix rows and columns in order to obtain a near triangular pattern of nonzero elements. Then an LU factorization is developed to represent the matrix inverse in terms of a sequence of elementary Gaussian eliminations, or pivots. In this paper it is shown how these algorithms are adapted for efficient implementation on vector processors. Results obtained on the CYBER 200 Model 205 are presented for a series of large test problems which show the comparative advantages of the triangularization and vector processing algorithms.
David, R. E.↗
Fox Trails
1. This software utilizes python pandas to pull data from P6 databases or XER files. The software transforms the datasets into multiple main tables by joining, filtering, iteratively flattening hierarchical structured data, and pivoting datasets to give simple flat output tables. The activity table includes all of the information related to an activity including activity codes, global, EPS, and project codes, UDFs, and WBS information as separate columns. This includes the code id, code value and sequence number for all levels in hierarchical codes. The resource table is similar to the activity table and includes all of the information related to resources on activities including UPFs and resource codes. The resource time phased table takes the resource information and time phases it for the budget, forecast, late, and actual dates/units/costs that closely matches P6's user interface's values as it implements the resource curve and calendars. The wbs table contains the WBS structure broken out by levels and includes UDFs, codes, and notebook topics. The final P6 data table is the relationships table which simply contains the relationships. 2. When a user updates the tool with data (via giving it P6 project names with database username/password information or XER files) the system creates the data in #1, then creates a networkx graph with the activity data imbedded in the node data and the relationships added as edges. Each edge also has it's float calculated (working time distance between the predecessor and successor) and attached to the edge. Activities are also tagged as a potential start of a path based on their constraints, constraint dates, remaining start date, and activity status. When a user enters an activity ID into the UI, it runs a shortest path calculation on the network graph between each node tagged as potential start to the entered activity id based on the float tagged on the edge. Each path returned by the algorithm contains all of the nodes on the path in order, as well as the total float of the edges that make the path. This data is then collected and returned to the user in the form of a gantt chart with groupings for each path that includes the total float for each group. 3. Similar to 2, if the user passes through a reference dataset each activity set in the path is checked to see if it had a path in the reference dataset, if that path was the primary path between the start and end activities, and what has changed regarding logic and durations. These changes are color coded and summarized before sent to the user to be displayed by the UI for simple discovery. 4. Utilizing the data from #1, the user can submit desired grouping code(s) and filters to the system. The system will then pull the activities, resources, and relationships and create a gantt chart based on the groupings sent and filtered based on the filters sent. 5. The system will produce a gantt chart in a similar method to #4, but allows interactivity with the data. As the user interacts with the gantt chart, the software captures the changes and stores it with the user making the change so that project controls and implement those changes in P6.
Fox, Ben↗