5,506 research outputs found
Koopman Operator and its Approximations for Systems with Symmetries
Nonlinear dynamical systems with symmetries exhibit a rich variety of
behaviors, including complex attractor-basin portraits and enhanced and
suppressed bifurcations. Symmetry arguments provide a way to study these
collective behaviors and to simplify their analysis. The Koopman operator is an
infinite dimensional linear operator that fully captures a system's nonlinear
dynamics through the linear evolution of functions of the state space.
Importantly, in contrast with local linearization, it preserves a system's
global nonlinear features. We demonstrate how the presence of symmetries
affects the Koopman operator structure and its spectral properties. In fact, we
show that symmetry considerations can also simplify finding the Koopman
operator approximations using the extended and kernel dynamic mode
decomposition methods (EDMD and kernel DMD). Specifically, representation
theory allows us to demonstrate that an isotypic component basis induces block
diagonal structure in operator approximations, revealing hidden organization.
Practically, if the data is symmetric, the EDMD and kernel DMD methods can be
modified to give more efficient computation of the Koopman operator
approximation and its eigenvalues, eigenfunctions, and eigenmodes. Rounding out
the development, we discuss the effect of measurement noise
Multireference Alignment using Semidefinite Programming
The multireference alignment problem consists of estimating a signal from
multiple noisy shifted observations. Inspired by existing Unique-Games
approximation algorithms, we provide a semidefinite program (SDP) based
relaxation which approximates the maximum likelihood estimator (MLE) for the
multireference alignment problem. Although we show that the MLE problem is
Unique-Games hard to approximate within any constant, we observe that our
poly-time approximation algorithm for the MLE appears to perform quite well in
typical instances, outperforming existing methods. In an attempt to explain
this behavior we provide stability guarantees for our SDP under a random noise
model on the observations. This case is more challenging to analyze than
traditional semi-random instances of Unique-Games: the noise model is on
vertices of a graph and translates into dependent noise on the edges.
Interestingly, we show that if certain positivity constraints in the SDP are
dropped, its solution becomes equivalent to performing phase correlation, a
popular method used for pairwise alignment in imaging applications. Finally, we
show how symmetry reduction techniques from matrix representation theory can
simplify the analysis and computation of the SDP, greatly decreasing its
computational cost
5 Post-processing methods for passivity enforcement
Many physical systems are passive (or dissipative): they are unable to generate energy on their own, but they can store energy in some form while exchanging power with the surrounding environment. This chapter describes the most prominent approaches for ensuring that Reduced Order Models are passive, so that their math- ematical representation satisfies an appropriate dissipativity condition. The main focus is on Linear and Time-Invariant (LTI) systems in state-space form. Different conditions for testing passivity of a given LTI model are discussed, including Linear Matrix Inequalities (LMIs), Frequency-Domain Inequalities, and spectral conditions on associated Hamiltonian matrices. Then we describe common approaches for perturbing a given non-passive system to enforce its passivity. Various examples from electronic applications are used to demonstrate both theory and algorithm performance
MRRR-based Eigensolvers for Multi-core Processors and Supercomputers
The real symmetric tridiagonal eigenproblem is of outstanding importance in
numerical computations; it arises frequently as part of eigensolvers for
standard and generalized dense Hermitian eigenproblems that are based on a
reduction to tridiagonal form. For its solution, the algorithm of Multiple
Relatively Robust Representations (MRRR or MR3 in short) - introduced in the
late 1990s - is among the fastest methods. To compute k eigenpairs of a real
n-by-n tridiagonal T, MRRR only requires O(kn) arithmetic operations; in
contrast, all the other practical methods require O(k^2 n) or O(n^3) operations
in the worst case. This thesis centers around the performance and accuracy of
MRRR.Comment: PhD thesi
Mixed-Precision Numerical Linear Algebra Algorithms: Integer Arithmetic Based LU Factorization and Iterative Refinement for Hermitian Eigenvalue Problem
Mixed-precision algorithms are a class of algorithms that uses low precision in part of the algorithm in order to save time and energy with less accurate computation and communication. These algorithms usually utilize iterative refinement processes to improve the approximate solution obtained from low precision to the accuracy we desire from doing all the computation in high precision. Due to the demand of deep learning applications, there are hardware developments offering different low-precision formats including half precision (FP16), Bfloat16 and integer operations for quantized integers, which uses integers with a shared scalar to represent a set of equally spaced numbers. As new hardware architectures focus on bringing performance in these formats, the mixed-precision algorithms have more potential leverage on them and outmatch traditional fixed-precision algorithms. This dissertation consists of two articles. In the first article, we adapt one of the most fundamental algorithms in numerical linear algebra---LU factorization with partial pivoting--- to use integer arithmetic. With the goal of obtaining a low accuracy factorization as the preconditioner of generalized minimal residual (GMRES) to solve systems of linear equations, the LU factorization is adapted to use two different fixed-point formats for matrices L and U. A left-looking variant is also proposed for matrices with unbounded column growth. Finally, GMRES iterative refinement has shown that it can work on matrices with condition numbers up to 10000 with the algorithm that uses int16 as input and int32 accumulator for the update step. The second article targets symmetric and Hermitian eigenvalue problems. In this section we revisit the SICE algorithm from Dongarra et al. By applying the Sherman-Morrison formula on the diagonally-shifted tridiagonal systems, we propose an updated SICE-SM algorithm. By incorporating the latest two-stage algorithms from the PLASMA and MAGMA software libraries for numerical linear algebra, we achieved up to 3.6x speedup using the mixed-precision eigensolver with the blocked SICE-SM algorithm for iterative refinement when compared with full double complex precision solvers for the cases with a portion of eigenvalues and eigenvectors requested
Financial Risk Measurement for Financial Risk Management
Current practice largely follows restrictive approaches to market risk measurement, such as historical simulation or RiskMetrics. In contrast, we propose flexible methods that exploit recent developments in financial econometrics and are likely to produce more accurate risk assessments, treating both portfolio-level and asset-level analysis. Asset-level analysis is particularly challenging because the demands of real-world risk management in financial institutions - in particular, real-time risk tracking in very high-dimensional situations - impose strict limits on model complexity. Hence we stress powerful yet parsimonious models that are easily estimated. In addition, we emphasize the need for deeper understanding of the links between market risk and macroeconomic fundamentals, focusing primarily on links among equity return volatilities, real growth, and real growth volatilities. Throughout, we strive not only to deepen our scientific understanding of market risk, but also cross-fertilize the academic and practitioner communities, promoting improved market risk measurement technologies that draw on the best of both.Market risk, volatility, GARCH
Line-Graph Lattices: Euclidean and Non-Euclidean Flat Bands, and Implementations in Circuit Quantum Electrodynamics
Materials science and the study of the electronic properties of solids are a
major field of interest in both physics and engineering. The starting point for
all such calculations is single-electron, or non-interacting, band structure
calculations, and in the limit of strong on-site confinement this can be
reduced to graph-like tight-binding models. In this context, both
mathematicians and physicists have developed largely independent methods for
solving these models. In this paper we will combine and present results from
both fields. In particular, we will discuss a class of lattices which can be
realized as line graphs of other lattices, both in Euclidean and hyperbolic
space. These lattices display highly unusual features including flat bands and
localized eigenstates of compact support. We will use the methods of both
fields to show how these properties arise and systems for classifying the
phenomenology of these lattices, as well as criteria for maximizing the gaps.
Furthermore, we will present a particular hardware implementation using
superconducting coplanar waveguide resonators that can realize a wide variety
of these lattices in both non-interacting and interacting form
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