Infinite Dimensional Theory Research Articles

State-Space Modeling of Viscous Unsteady Aerodynamic Loads

In this paper, we summarize our recently developed viscous unsteady theory, which couples potential flow with the triple-deck boundary-layer theory. This approach provides a viscous extension of potential-flow unsteady aerodynamics. As such, a Reynolds-number-dependent transfer function is determined for unsteady lift. We then use the Wiener–Hammerstein structure to develop a finite-dimensional approximation of such an infinite-dimensional theory, presenting it in a state-space model. This novel nonlinear state-space model of viscous unsteady aerodynamic loads is expected to serve aerodynamicists better than the classical Theodorsen’s model because it captures viscous effects (that is, Reynolds number dependence) as well as nonlinearity and additional lag in the lift dynamics; it also allows simulation of arbitrary time-varying airfoil motions (not necessarily harmonic). Moreover, being in a state-space form makes it quite convenient for simulation and coupling with structural dynamics to perform aeroelasticity, flight dynamics analysis, and control design. We then develop a linearization of such a model, which enables analytical results. Subsequently, we derive an analytical representation of the viscous lift frequency response function: an explicit function of both the frequency and Reynolds number. We also develop a state-space model of the linearized response. We finally simulate the nonlinear and linear models to a nonharmonic small-amplitude pitching maneuver at a Reynolds number of 100,000 and compare the resulting lift and pitching moment with those obtained from potential flow; this is in reference to relatively higher-fidelity computations of the unsteady Reynolds-averaged Navier–Stokes equations.

Maximum a posteriori probability estimates in infinite-dimensional Bayesian inverse problems

A demanding challenge in Bayesian inversion is to efficiently characterize the posterior distribution. This task is problematic especially in high-dimensional non-Gaussian problems, where the structure of the posterior can be very chaotic and difficult to analyse. Current inverse problem literature often approaches the problem by considering suitable point estimators for the task. Typically the choice is made between the maximum a posteriori (MAP) or the conditional mean (CM) estimate. The benefits of either choice are not well-understood from the perspective of infinite-dimensional theory. Most importantly, there exists no general scheme regarding how to connect the topological description of a MAP estimate to a variational problem. The recent results by Dashti and others (Dashti et al 2013 Inverse Problems 29 095017) resolve this issue for nonlinear inverse problems in Gaussian framework. In this work we improve the current understanding by introducing a novel concept called the weak MAP (wMAP) estimate. We show that any MAP estimate in the sense of Dashti et al (2013 Inverse Problems 29 095017) is a wMAP estimate and, moreover, how the wMAP estimate connects to a variational formulation in general infinite-dimensional non-Gaussian problems. The variational formulation enables to study many properties of the infinite-dimensional MAP estimate that were earlier impossible to study. In a recent work by the authors (Burger and Lucka 2014 Maximum a posteriori estimates in linear inverse problems with logconcave priors are proper bayes estimators preprint) the MAP estimator was studied in the context of the Bayes cost method. Using Bregman distances, proper convex Bayes cost functions were introduced for which the MAP estimator is the Bayes estimator. Here, we generalize these results to the infinite-dimensional setting. Moreover, we discuss the implications of our results for some examples of prior models such as the Besov prior and hierarchical prior.

The Simplicity of Robust Direct Adaptive Control with Disturbance Rejection for Linear Infinite-Dimensional Systems: An Introduction

Our purpose in this paper is to provide an introduction to direct adaptive control in infinite dimensions and to prove the remarkable simplicity of infinite dimensional adaptive control even in infinite dimensions. However, we will show that the mathematical foundations are much more stringent in infinite dimensional theory. Consequently, this paper will emphasize some important fundamentals and will not be comprehensive. Much more can be done, but here we hope to provide a starting point for further exploration. Given a linear continuous-time infinite-dimensional plant on a Hilbert space and disturbances of known and unknown waveform, we show that there exists a very simple stabilizing direct adaptive control law with certain disturbance rejection and robustness properties. The closed loop system is shown to be exponentially convergent to a neighborhood with radius proportional to bounds on the size of the disturbance. The plant is described by a closed densely defined linear operator that generates a continuous semigroup of bounded operators on the Hilbert space of states. To illustrate the use of the control law, we apply the result to a partial differential equation example of an unstable diffusion system, but other applications are possible.

Fixed-Functionals of three-dimensional Quantum Einstein Gravity

We study the non-perturbative renormalization group flow of f(R)-gravity in three-dimensional Asymptotically Safe Quantum Einstein Gravity. Within the conformally reduced approximation, we derive an exact partial differential equation governing the RG-scale dependence of the function f(R). This equation is shown to possess two isolated and one continuous one-parameter family of scale-independent, regular solutions which constitute the natural generalization of RG fixed points to the realm of infinite-dimensional theory spaces. All solutions are bounded from below and give rise to positive definite kinetic terms. Moreover, they admit either one or two UV-relevant deformations, indicating that the corresponding UV-critical hypersurfaces remain finite dimensional despite the inclusion of an infinite number of coupling constants. The impact of our findings on the gravitational Asymptotic Safety program and its connection to new massive gravity is briefly discussed.

On the Cauchy-Kowalevski Theorem

After a short review of the basic properties of analytic functions, we apply the infinite-dimensional theory to get a simple proof of the Cauchy-Kowalevski theorem, in an infinite-dimensional version which seems to be new.

Conformal sector of quantum Einstein gravity in the local potential approximation: Non-Gaussian fixed point and a phase of unbroken diffeomorphism invariance

We explore the nonperturbative renormalization group flow of quantum Einstein gravity (QEG) on an infinite dimensional theory space. We consider ``conformally reduced'' gravity where only fluctuations of the conformal factor are quantized and employ the local potential approximation for its effective average action. The requirement of ``background independence'' in quantum gravity entails a partial differential equation governing the scale dependence of the potential for the conformal factor which differs significantly from that of a scalar matter field. In the infinite dimensional space of potential functions we find a Gaussian as well as a non-Gaussian fixed point which provides further evidence for the viability of the asymptotic safety scenario. The analog of the invariant cubic in the curvature which spoils perturbative renormalizability is seen to be unproblematic for the asymptotic safety of the conformally reduced theory. The scaling fields and dimensions of both fixed points are obtained explicitly and possible implications for the predictivity of the theory are discussed. Spacetime manifolds with ${R}^{d}$ as well as ${S}^{d}$ topology are considered. Solving the flow equation for the potential numerically we obtain examples of renormalization group trajectories inside the ultraviolet critical surface of the non-Gaussian fixed point. The quantum theories based upon some of them show a phase transition from the familiar (low energy) phase of gravity with spontaneously broken diffeomorphism invariance to a new phase of unbroken diffeomorphism invariance; the latter phase is characterized by a vanishing expectation value of the metric.

SOME FAMILIES OF q-VECTOR FIELDS ON PATH SPACES

Some families of H-valued vector fields with calculable Lie brackets are given. These provide examples of vector fields on path spaces with a divergence and we show that versions of Bismut type formulae for forms on a compact Riemannian manifold arise as projections of the infinite dimensional theory.

Differential Galois theory of infinite dimension

This paper is the second part of our work on differential Galois theory as we promised in [U3]. Differential Galois theory has a long history since Lie tried to apply the idea of Abel and Galois to differential equations in the 19th century (cf. [U3], Introduction). When we consider Galois theory of differential equation, we have to separate the finite dimensional theory from the infinite dimensional theory. As Kolchin theory shows, the first is constructed on a rigorous foundation. The latter, however, seems inachieved despite of several important contributions of Drach, Vessiot,…. We propose in this paper a differential Galois theory of infinite dimension in a rigorous and transparent framework. We explain the idea of the classical authors by one of the simplest examples and point out the problems.

Ioffe's normal cone and the foundations of welfare economics: The infinite dimensional theory

We establish the relevance of Ioffe's normal cone for basic theorems of welfare economics in the context of a commodity space formalized as a vactor lattice and endowed with a locally convex linear topology in which the positive cone is closed.

Short-distance behavior of renornalizable quantum field theories

We introduce particular coordinates to the infinite-dimensional theory space (consisting of all possible hamiltonians) of euclidean quantum field theory such that all but a finite number of coordinates vanish for renormalizable theories. With this choice we can analyze the structure of the renormalization group equations by counting scaling dimensions of the coordinates. Following Wegner's idea we combine the renormalization group with the operator product expansion to determine all possible non-analytic dependence on the renormalized coupling constants in the short-distance expansions of the n-point correlation functions.

Compensator design for stability enhancement with collocated controllers

A continuum model rather than a finite element model is used. The optimal compensator design is formulated as a stochastic regulator problem and is shown to be solvable by the general infinite-dimensional theory developed by the author despite the lack of exponential stabilizability. Infinite-dimensional steady-state Riccati equations characterizing the feedback control gain and the Kalman filter gain operators can be solved explicitly. The associated performance indexes including the mean-square control-effort are calculated in closed form. As a first approximation, the compensator transfer function can be realized as a bank of bandpass filters in parallel centered at the undamped mode frequencies. Numerical calculations for the gain and bandwidths for a typical configuration are presented. The performance of the compensator is evaluated when in fact in the true model there is no actuator noise. The theoretical problem involved is to show that the infinite-dimensional stochastic process is asymptotically stationary. It is possible to calculate the steady-state covariance in closed form and thereby calculate performance indexes of interest explicitly, facilitating the choice of optimal design parameters. >

Integration of almost-periodic functions and infinite-dimensional theory of diophantine approximations

Integration of almost-periodic functions and infinite-dimensional theory of diophantine approximations

Disturbance decoupling and invariant subspaces for delay systems

In this paper we consider the disturbance decoupling problem for distributed parameter systems, with special attention to the case of delay systems. We present several examples which illustrate the difficulties of the infinite dimensional theory for the case of general distributed parameter systems and for the case of delay systems. In this last case we single out a class of subspaces whose invariant properties are easily characterized and which seems to be interesting from the point of view of the applications.

Some observations on the spectra of positive operators on finite-dimensional C∗-algebras

The classical theorems of O. Perron and G. Frobenius about spectral properties of matrices with positive entries have been studied and generalized by various authors. In the book of A. Berman and R.J. Plemmons [3] the finite-dimensional aspects of this theory are described, whereas in the two monographs of H.H. Schaefer [20, 21] the infinite-dimensional theory is developed. Our purpose is to continue these extensions to finite-dimensional C ∗-algebras, obtaining a complete description of the spectrum of positive irreducible operators on such ordered vector spaces.

Input–Output Description of Roomy Systems

In the classical dynamic systems theory, precise information about a system can be deduced from its input–output map. In fact, for minimal systems, a complete pole-zero theory can be constructed using polynomial coprime factorization techniques, together with algebraic properties of the state space module. In this paper, an infinite-dimensional theory in the same style is presented whereby the input–output maps are assumed to exhibit some energy conservation properties and whereby these maps are ascertained to belong to a class of systems with nontrivial nullspace, called “roomy” systems. As a result, a coprime factorization theory can be deduced based not on properties of polynomials but of analytical functions, a complete polar description of the system can be given and a zero description for a somewhat more restricted class. The mathematical tools used lean heavily on Helson and Lowdenslaeger’s invariant subspace theory of Hardy spaces which quite naturally comes into play through the Bochner–Chandrase...

Statistical mechanics of the melting transition in lattice models of polymers

Five two-dimensional lattice models, four with rotational isomeric and excluded volume interactions and one with cross links, are used to discuss the theory of the melting transition in polymers. The models have been chosen because they are isomorphic to exactly solvable six vertex and dimer models. The orders of the thermodynamic transitions are extremely varied from model to model, including first-order, 3/2 order and infinite order transitions. These models are used to test and reveal the shortcomings of the Flory–Huggins approximate theory, which is most aptly described as an infinite dimensional theory.

Position Operators and Localizability of Quantum Systems Described by Finite- and Infinite-Dimensional Wave Equations

The principles of quantum mechanics impose certain restrictions on acceptable position operators. The probability density as derived from the position operator should agree with the fourth component of the current vector. The various relativistic wave equations are reexamined from the point of view of this consistency test. These and the group theoretical considerations lead to the conclusion that only a Dirac particle can be considered to be elementary. The position operator of the infinite-dimensional theories is investigated and the advantages of theories using the unitary representations of the Lorentz group are discussed.