105 research outputs found
A rational deferred correction approach to parabolic optimal control problems
The accurate and efficient solution of time-dependent PDE-constrained optimization problems is a challenging task, in large part due to the very high dimension of the matrix systems that need to be solved. We devise a new deferred correction method for coupled systems of time-dependent PDEs, allowing one to iteratively improve the accuracy of low-order time stepping schemes. We consider two variants of our method, a splitting and a coupling version, and analyze their convergence properties. We then test our approach on a number of PDE-constrained optimization problems. We obtain solution accuracies far superior to that achieved when solving a single discretized problem, in particular in cases where the accuracy is limited by the time discretization. Our approach allows for the direct reuse of existing solvers for the resulting matrix systems, as well as state-of-the-art preconditioning strategies
Near-optimal perfectly matched layers for indefinite Helmholtz problems
A new construction of an absorbing boundary condition for indefinite
Helmholtz problems on unbounded domains is presented. This construction is
based on a near-best uniform rational interpolant of the inverse square root
function on the union of a negative and positive real interval, designed with
the help of a classical result by Zolotarev. Using Krein's interpretation of a
Stieltjes continued fraction, this interpolant can be converted into a
three-term finite difference discretization of a perfectly matched layer (PML)
which converges exponentially fast in the number of grid points. The
convergence rate is asymptotically optimal for both propagative and evanescent
wave modes. Several numerical experiments and illustrations are included.Comment: Accepted for publication in SIAM Review. To appear 201
Krylov Subspace Recycling With Randomized Sketching For Matrix Functions
A Krylov subspace recycling method for the efficient evaluation of a sequence
of matrix functions acting on a set of vectors is developed. The method
improves over the recycling methods presented in [Burke et al.,
arXiv:2209.14163, 2022] in that it uses a closed-form expression for the
augmented FOM approximants and hence circumvents the use of numerical
quadrature. We further extend our method to use randomized sketching in order
to avoid the arithmetic cost of orthogonalizing a full Krylov basis, offering
an attractive solution to the fact that recycling algorithms built from shifted
augmented FOM cannot easily be restarted. The efficacy of the proposed
algorithms is demonstrated with numerical experiments.Comment: 19 pages, 5 figure
A sketch-and-select Arnoldi process
A sketch-and-select Arnoldi process to generate a well-conditioned basis of a
Krylov space is proposed. At each iteration the procedure utilizes randomized
sketching to select a limited number of previously computed basis vectors to
project out of the current basis vector. The computational cost grows linearly
with the dimension of the Krylov basis. The subset selection problem for the
projection step is approximately solved with a number of heuristic algorithms
and greedy methods used in statistical learning and compressive sensing
Fast and exact fixed-radius neighbor search based on sorting
Fixed-radius near neighbor search is a fundamental data operation that
retrieves all data points within a user-specified distance to a query point.
There are efficient algorithms that can provide fast approximate query
responses, but they often have a very compute-intensive indexing phase and
require careful parameter tuning. Therefore, exact brute force and tree-based
search methods are still widely used. Here we propose a new fixed-radius near
neighbor search method, called SNN, that significantly improves over brute
force and tree-based methods in terms of index and query time, provably returns
exact results, and requires no parameter tuning. SNN exploits a sorting of the
data points by their first principal component to prune the query search space.
Further speedup is gained from an efficient implementation using high-level
Basic Linear Algebra Subprograms (BLAS). We provide theoretical analysis of our
method and demonstrate its practical performance when used stand-alone and when
applied within the DBSCAN clustering algorithm.Comment: arXiv admin note: text overlap with arXiv:2202.0145
Fast and explainable clustering based on sorting
We introduce a fast and explainable clustering method called CLASSIX. It
consists of two phases, namely a greedy aggregation phase of the sorted data
into groups of nearby data points, followed by the merging of groups into
clusters. The algorithm is controlled by two scalar parameters, namely a
distance parameter for the aggregation and another parameter controlling the
minimal cluster size. Extensive experiments are conducted to give a
comprehensive evaluation of the clustering performance on synthetic and
real-world datasets, with various cluster shapes and low to high feature
dimensionality. Our experiments demonstrate that CLASSIX competes with
state-of-the-art clustering algorithms. The algorithm has linear space
complexity and achieves near linear time complexity on a wide range of
problems. Its inherent simplicity allows for the generation of intuitive
explanations of the computed clusters
Randomized sketching of nonlinear eigenvalue problems
Rational approximation is a powerful tool to obtain accurate surrogates for
nonlinear functions that are easy to evaluate and linearize. The interpolatory
adaptive Antoulas--Anderson (AAA) method is one approach to construct such
approximants numerically. For large-scale vector- and matrix-valued functions,
however, the direct application of the set-valued variant of AAA becomes
inefficient. We propose and analyze a new sketching approach for such functions
called sketchAAA that, with high probability, leads to much better approximants
than previously suggested approaches while retaining efficiency. The sketching
approach works in a black-box fashion where only evaluations of the nonlinear
function at sampling points are needed. Numerical tests with nonlinear
eigenvalue problems illustrate the efficacy of our approach, with speedups
above 200 for sampling large-scale black-box functions without sacrificing on
accuracy.Comment: 15 page
Different Ambidextrous Learning Architectures and the Role of HRM Systems
During the past decade ambidexterity has emerged as the central research stream in organization science to investigate how organizations manage to remain successful over time. By using the lens of organizational learning, ambidexterity can be defined as the simultaneous pursuit of exploration and exploitation. However, the link between ambidexterity and the human resource management of a firm is still a blind spot on the ambidexterity research map. To shed light on this issue, we show how different ambidextrous learning architectures can be created and maintained by the means of consistent HRM systems. By doing so, we show how HRM systems as specific bundles of HRM practices facilitate ambidextrous learning. Thereby we emphasize the challenge of creating and sustaining the horizontal and vertical fit of an HRM system with regard to different ambidextrous designs.Ambidexterity; Exploration; Exploitation; Organizational Learning; HRM; Strategic Human Resource Management
ABBA: adaptive Brownian bridge-based symbolic aggregation of time series
From Springer Nature via Jisc Publications RouterHistory: received 2019-05-29, accepted 2020-05-14, registration 2020-05-14, pub-electronic 2020-06-03, online 2020-06-03, pub-print 2020-07Publication status: PublishedFunder: Engineering and Physical Sciences Research Council; doi: http://dx.doi.org/10.13039/50110000026; Grant(s): EP/N509565/1Abstract: A new symbolic representation of time series, called ABBA, is introduced. It is based on an adaptive polygonal chain approximation of the time series into a sequence of tuples, followed by a mean-based clustering to obtain the symbolic representation. We show that the reconstruction error of this representation can be modelled as a random walk with pinned start and end points, a so-called Brownian bridge. This insight allows us to make ABBA essentially parameter-free, except for the approximation tolerance which must be chosen. Extensive comparisons with the SAX and 1d-SAX representations are included in the form of performance profiles, showing that ABBA is often able to better preserve the essential shape information of time series compared to other approaches, in particular when time warping measures are used. Advantages and applications of ABBA are discussed, including its in-built differencing property and use for anomaly detection, and Python implementations provided
- …