19 research outputs found
On the computation of geometric features of spectra of linear operators on Hilbert spaces
Computing spectra is a central problem in computational mathematics with an
abundance of applications throughout the sciences. However, in many
applications gaining an approximation of the spectrum is not enough. Often it
is vital to determine geometric features of spectra such as Lebesgue measure,
capacity or fractal dimensions, different types of spectral radii and numerical
ranges, or to detect essential spectral gaps and the corresponding failure of
the finite section method. Despite new results on computing spectra and the
substantial interest in these geometric problems, there remain no general
methods able to compute such geometric features of spectra of
infinite-dimensional operators. We provide the first algorithms for the
computation of many of these longstanding problems (including the above). As
demonstrated with computational examples, the new algorithms yield a library of
new methods. Recent progress in computational spectral problems in infinite
dimensions has led to the Solvability Complexity Index (SCI) hierarchy, which
classifies the difficulty of computational problems. These results reveal that
infinite-dimensional spectral problems yield an intricate infinite
classification theory determining which spectral problems can be solved and
with which type of algorithm. This is very much related to S. Smale's
comprehensive program on the foundations of computational mathematics initiated
in the 1980s. We classify the computation of geometric features of spectra in
the SCI hierarchy, allowing us to precisely determine the boundaries of what
computers can achieve and prove that our algorithms are optimal. We also
provide a new universal technique for establishing lower bounds in the SCI
hierarchy, which both greatly simplifies previous SCI arguments and allows new,
formerly unattainable, classifications
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Pseudoergodic operators and periodic boundary conditions
There is an increasing literature on random non self-adjoint infinite matrices with motivations ranging from condensed matter physics to neural networks. Many of these operators fall into the class of ``pseudoergodic'' operators, which allows the elimination of probabilistic arguments when studying spectral properties. Parallel to this is the increased awareness that spectral properties of non self-adjoint operators, in particular stability, may be better captured via the notion of pseudospectra as opposed to spectra. Although it is well known that the finite section method applied to these matrices does not converge to the spectrum, it is often found in practice that the pseudospectrum behaves better with appropriate boundary conditions. We make this precise by giving a simple proof that the finite section method with periodic boundary conditions converges to the pseudospectrum of the full operator. Our results hold in any dimension (not just for banded bi-infinite matrices) and can be considered as a generalisation of the well known classical result for banded Laurent operators and their circulant approximations. Furthermore, we numerically demonstrate a convergent algorithm for the pseudospectrum including cases where periodic boundary conditions converge faster than the method of uneven sections. Finally we show that the result carries over to pseudoergodic operators acting on spaces for .This work was supported by EPSRC grant EP/L016516/1
Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems
Koopman operators are infinite-dimensional operators that globally linearize
nonlinear dynamical systems, making their spectral information useful for
understanding dynamics. However, Koopman operators can have continuous spectra
and infinite-dimensional invariant subspaces, making computing their spectral
information a considerable challenge. This paper describes data-driven
algorithms with rigorous convergence guarantees for computing spectral
information of Koopman operators from trajectory data. We introduce residual
dynamic mode decomposition (ResDMD), which provides the first scheme for
computing the spectra and pseudospectra of general Koopman operators from
snapshot data without spectral pollution. Using the resolvent operator and
ResDMD, we also compute smoothed approximations of spectral measures associated
with measure-preserving dynamical systems. We prove explicit convergence
theorems for our algorithms, which can achieve high-order convergence even for
chaotic systems, when computing the density of the continuous spectrum and the
discrete spectrum. We demonstrate our algorithms on the tent map, Gauss
iterated map, nonlinear pendulum, double pendulum, Lorenz system, and an
-dimensional extended Lorenz system. Finally, we provide kernelized
variants of our algorithms for dynamical systems with a high-dimensional
state-space. This allows us to compute the spectral measure associated with the
dynamics of a protein molecule that has a 20,046-dimensional state-space, and
compute nonlinear Koopman modes with error bounds for turbulent flow past
aerofoils with Reynolds number that has a 295,122-dimensional
state-space
The foundations of spectral computations via the Solvability Complexity Index hierarchy: Part I
The problem of computing spectra of operators is arguably one of the most
investigated areas of computational mathematics. Recent progress and the
current paper reveal that, unlike the finite-dimensional case,
infinite-dimensional problems yield a highly intricate infinite classification
theory determining which spectral problems can be solved and with which type of
algorithms. Classifying spectral problems and providing optimal algorithms is
uncharted territory in the foundations of computational mathematics. This paper
is the first of a two-part series establishing the foundations of computational
spectral theory through the Solvability Complexity Index (SCI) hierarchy and
has three purposes. First, we establish answers to many longstanding open
questions on the existence of algorithms. We show that for large classes of
partial differential operators on unbounded domains, spectra can be computed
with error control from point sampling operator coefficients. Further results
include computing spectra of operators on graphs with error control, the
spectral gap problem, spectral classifications, and discrete spectra,
multiplicities and eigenspaces. Second, these classifications determine which
types of problems can be used in computer-assisted proofs. The theory for this
is virtually non-existent, and we provide some of the first results in this
infinite classification theory. Third, our proofs are constructive, yielding a
library of new algorithms and techniques that handle problems that before were
out of reach. We show several examples on contemporary problems in the physical
sciences. Our approach is closely related to Smale's program on the foundations
of computational mathematics initiated in the 1980s, as many spectral problems
can only be computed via several limits, a phenomenon shared with the
foundations of polynomial root finding with rational maps, as proved by
McMullen
Kernel Density Estimation with Linked Boundary Conditions
Kernel density estimation on a finite interval poses an outstanding challenge
because of the well-recognized bias at the boundaries of the interval.
Motivated by an application in cancer research, we consider a boundary
constraint linking the values of the unknown target density function at the
boundaries. We provide a kernel density estimator (KDE) that successfully
incorporates this linked boundary condition, leading to a non-self-adjoint
diffusion process and expansions in non-separable generalized eigenfunctions.
The solution is rigorously analyzed through an integral representation given by
the unified transform (or Fokas method). The new KDE possesses many desirable
properties, such as consistency, asymptotically negligible bias at the
boundaries, and an increased rate of approximation, as measured by the AMISE.
We apply our method to the motivating example in biology and provide numerical
experiments with synthetic data, including comparisons with state-of-the-art
KDEs (which currently cannot handle linked boundary constraints). Results
suggest that the new method is fast and accurate. Furthermore, we demonstrate
how to build statistical estimators of the boundary conditions satisfied by the
target function without apriori knowledge. Our analysis can also be extended to
more general boundary conditions that may be encountered in applications
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How to Compute Spectra with Error Control.
Computing the spectra of operators is a fundamental problem in the sciences, with wide-ranging applications in condensed-matter physics, quantum mechanics and chemistry, statistical mechanics, etc. While there are algorithms that in certain cases converge to the spectrum, no general procedure is known that (a) always converges, (b) provides bounds on the errors of approximation, and (c) provides approximate eigenvectors. This may lead to incorrect simulations. It has been an open problem since the 1950s to decide whether such reliable methods exist at all. We affirmatively resolve this question, and the algorithms provided are optimal, realizing the boundary of what digital computers can achieve. Moreover, they are easy to implement and parallelize, offer fundamental speed-ups, and allow problems that before, regardless of computing power, were out of reach. Results are demonstrated on difficult problems such as the spectra of quasicrystals and non-Hermitian phase transitions in optics.This work was supported by Engineering and Physical Sciences Research Council Grants No. EP/L016516/1, No. EP/R008272/1, No. EP/N014588/1, and No. EP/ L003457/1, as well as a Royal Society University Research Fellowship
Beyond expectations: Residual Dynamic Mode Decomposition and Variance for Stochastic Dynamical Systems
Koopman operators linearize nonlinear dynamical systems, making their
spectral information of crucial interest. Numerous algorithms have been
developed to approximate these spectral properties, and Dynamic Mode
Decomposition (DMD) stands out as the poster child of projection-based methods.
Although the Koopman operator itself is linear, the fact that it acts in an
infinite-dimensional space of observables poses various challenges. These
include spurious modes, essential spectra, and the verification of Koopman mode
decompositions. While recent work has addressed these challenges for
deterministic systems, there remains a notable gap in verified DMD methods
tailored for stochastic systems, where the Koopman operator measures the
expectation of observables. We show that it is necessary to go beyond
expectations to address these issues. By incorporating variance into the
Koopman framework, we address these challenges. Through an additional DMD-type
matrix, we approximate the sum of a squared residual and a variance term, each
of which can be approximated individually using batched snapshot data. This
allows verified computation of the spectral properties of stochastic Koopman
operators, controlling the projection error. We also introduce the concept of
variance-pseudospectra to gauge statistical coherency. Finally, we present a
suite of convergence results for the spectral quantities of stochastic Koopman
operators. Our study concludes with practical applications using both simulated
and experimental data. In neural recordings from awake mice, we demonstrate how
variance-pseudospectra can reveal physiologically significant information
unavailable to standard expectation-based dynamical models
On the infinite-dimensional QR algorithm
Abstract: Spectral computations of infinite-dimensional operators are notoriously difficult, yet ubiquitous in the sciences. Indeed, despite more than half a century of research, it is still unknown which classes of operators allow for the computation of spectra and eigenvectors with convergence rates and error control. Recent progress in classifying the difficulty of spectral problems into complexity hierarchies has revealed that the most difficult spectral problems are so hard that one needs three limits in the computation, and no convergence rates nor error control is possible. This begs the question: which classes of operators allow for computations with convergence rates and error control? In this paper, we address this basic question, and the algorithm used is an infinite-dimensional version of the QR algorithm. Indeed, we generalise the QR algorithm to infinite-dimensional operators. We prove that not only is the algorithm executable on a finite machine, but one can also recover the extremal parts of the spectrum and corresponding eigenvectors, with convergence rates and error control. This allows for new classification results in the hierarchy of computational problems that existing algorithms have not been able to capture. The algorithm and convergence theorems are demonstrated on a wealth of examples with comparisons to standard approaches (that are notorious for providing false solutions). We also find that in some cases the IQR algorithm performs better than predicted by theory and make conjectures for future study