971 research outputs found
Spectra and leading directions for differential-algebraic equations
The state of the art in the spectral theory of linear time-varying differential-algebraic equations (DAEs) is surveyed. To characterize the asymptotic behavior and the growth rate of solutions, basic spectral notions such as Lyapunov- and Bohl exponents, and Sacker-Sell spectra are discussed. For DAEs in strangeness-free form, the results extend those for ordinary differential equations, but only under additional conditions. This has consequences concerning the boundedness of solutions of inhomogeneous equations. Also, linear subspaces of leading directions are characterized, which are associated with spectral intervals and which generalize eigenvectors and invariant subspaces as they are used in the linear time-invariant setting
Singular perturbation of polynomial potentials in the complex domain with applications to PT-symmetric families
In the first part of the paper, we discuss eigenvalue problems of the form
-w"+Pw=Ew with complex potential P and zero boundary conditions at infinity on
two rays in the complex plane. We give sufficient conditions for continuity of
the spectrum when the leading coefficient of P tends to 0. In the second part,
we apply these results to the study of topology and geometry of the real
spectral loci of PT-symmetric families with P of degree 3 and 4, and prove
several related results on the location of zeros of their eigenfunctions.Comment: The main result on singular perturbation is substantially improved,
generalized, and the proof is simplified. 37 pages, 16 figure
Geometric methods on low-rank matrix and tensor manifolds
In this chapter we present numerical methods for low-rank matrix and tensor problems that explicitly make use of the geometry of rank constrained matrix and tensor spaces. We focus on two types of problems: The first are optimization problems, like matrix and tensor completion, solving linear systems and eigenvalue problems. Such problems can be solved by numerical optimization for manifolds, called Riemannian optimization methods. We will explain the basic elements of differential geometry in order to apply such methods efficiently to rank constrained matrix and tensor spaces. The second type of problem is ordinary differential equations, defined on matrix and tensor spaces. We show how their solution can be approximated by the dynamical low-rank principle, and discuss several numerical integrators that rely in an essential way on geometric properties that are characteristic to sets of low rank matrices and tensors
Spectral methods for CFD
One of the objectives of these notes is to provide a basic introduction to spectral methods with a particular emphasis on applications to computational fluid dynamics. Another objective is to summarize some of the most important developments in spectral methods in the last two years. The fundamentals of spectral methods for simple problems will be covered in depth, and the essential elements of several fluid dynamical applications will be sketched
Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation
In this paper we consider the spectral and nonlinear stability of periodic
traveling wave solutions of a generalized Kuramoto-Sivashinsky equation. In
particular, we resolve the long-standing question of nonlinear modulational
stability by demonstrating that spectrally stable waves are nonlinearly stable
when subject to small localized (integrable) perturbations. Our analysis is
based upon detailed estimates of the linearized solution operator, which are
complicated by the fact that the (necessarily essential) spectrum of the
associated linearization intersects the imaginary axis at the origin. We carry
out a numerical Evans function study of the spectral problem and find bands of
spectrally stable periodic traveling waves, in close agreement with previous
numerical studies of Frisch-She-Thual, Bar-Nepomnyashchy,
Chang-Demekhin-Kopelevich, and others carried out by other techniques. We also
compare predictions of the associated Whitham modulation equations, which
formally describe the dynamics of weak large scale perturbations of a periodic
wave train, with numerical time evolution studies, demonstrating their
effectiveness at a practical level. For the reader's convenience, we include in
an appendix the corresponding treatment of the Swift-Hohenberg equation, a
nonconservative counterpart of the generalized Kuramoto-Sivashinsky equation
for which the nonlinear stability analysis is considerably simpler, together
with numerical Evans function analyses extending spectral stability analyses of
Mielke and Schneider.Comment: 78 pages, 11 figure
Tensor Analysis and Fusion of Multimodal Brain Images
Current high-throughput data acquisition technologies probe dynamical systems
with different imaging modalities, generating massive data sets at different
spatial and temporal resolutions posing challenging problems in multimodal data
fusion. A case in point is the attempt to parse out the brain structures and
networks that underpin human cognitive processes by analysis of different
neuroimaging modalities (functional MRI, EEG, NIRS etc.). We emphasize that the
multimodal, multi-scale nature of neuroimaging data is well reflected by a
multi-way (tensor) structure where the underlying processes can be summarized
by a relatively small number of components or "atoms". We introduce
Markov-Penrose diagrams - an integration of Bayesian DAG and tensor network
notation in order to analyze these models. These diagrams not only clarify
matrix and tensor EEG and fMRI time/frequency analysis and inverse problems,
but also help understand multimodal fusion via Multiway Partial Least Squares
and Coupled Matrix-Tensor Factorization. We show here, for the first time, that
Granger causal analysis of brain networks is a tensor regression problem, thus
allowing the atomic decomposition of brain networks. Analysis of EEG and fMRI
recordings shows the potential of the methods and suggests their use in other
scientific domains.Comment: 23 pages, 15 figures, submitted to Proceedings of the IEE
Derivation of the Zakharov equations
This paper continues the study of the validity of the Zakharov model
describing Langmuir turbulence. We give an existence theorem for a class of
singular quasilinear equations. This theorem is valid for well-prepared initial
data. We apply this result to the Euler-Maxwell equations describing
laser-plasma interactions, to obtain, in a high-frequency limit, an asymptotic
estimate that describes solutions of the Euler-Maxwell equations in terms of
WKB approximate solutions which leading terms are solutions of the Zakharov
equations. Because of transparency properties of the Euler-Maxwell equations,
this study is led in a supercritical (highly nonlinear) regime. In such a
regime, resonances between plasma waves, electromagnetric waves and acoustic
waves could create instabilities in small time. The key of this work is the
control of these resonances. The proof involves the techniques of geometric
optics of Joly, M\'etivier and Rauch, recent results of Lannes on norms of
pseudodifferential operators, and a semiclassical, paradifferential calculus
Robust stability of differential-algebraic equations
This paper presents a survey of recent results on the robust stability analysis and the distance to instability for linear time-invariant and time-varying differential-algebraic equations (DAEs). Different stability concepts such as exponential and asymptotic stability are studied and their robustness is analyzed under general as well as restricted sets of real or complex perturbations. Formulas for the distances are presented whenever these are available and the continuity of the distances in terms of the data is discussed. Some open problems and challenges are indicated
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