32 research outputs found
Quantum Computing for Fusion Energy Science Applications
This is a review of recent research exploring and extending present-day
quantum computing capabilities for fusion energy science applications. We begin
with a brief tutorial on both ideal and open quantum dynamics, universal
quantum computation, and quantum algorithms. Then, we explore the topic of
using quantum computers to simulate both linear and nonlinear dynamics in
greater detail. Because quantum computers can only efficiently perform linear
operations on the quantum state, it is challenging to perform nonlinear
operations that are generically required to describe the nonlinear differential
equations of interest. In this work, we extend previous results on embedding
nonlinear systems within linear systems by explicitly deriving the connection
between the Koopman evolution operator, the Perron-Frobenius evolution
operator, and the Koopman-von Neumann evolution (KvN) operator. We also
explicitly derive the connection between the Koopman and Carleman approaches to
embedding. Extension of the KvN framework to the complex-analytic setting
relevant to Carleman embedding, and the proof that different choices of complex
analytic reproducing kernel Hilbert spaces depend on the choice of Hilbert
space metric are covered in the appendices. Finally, we conclude with a review
of recent quantum hardware implementations of algorithms on present-day quantum
hardware platforms that may one day be accelerated through Hamiltonian
simulation. We discuss the simulation of toy models of wave-particle
interactions through the simulation of quantum maps and of wave-wave
interactions important in nonlinear plasma dynamics.Comment: 42 pages; 12 figures; invited paper at the 2021-2022 International
Sherwood Fusion Theory Conferenc
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OKID as a general approach to linear and bilinear system identification
This work advances the understanding of the complex world of system identification, i.e. the set of techniques to find mathematical models of dynamical systems from measured input-output data, and exploits well-established approaches for linear systems to address nonlinear system identification problems.
We focus on observer/Kalman filter identification (OKID), a method for simultaneous identification of a linear state-space model and the associated Kalman filter from noisy input-output measurements.
OKID, developed at NASA, resulted in a very successful algorithm known as OKID/ERA (OKID followed by eigensystem realization algorithm). We show how ERA is not the only method to complete the OKID process, developing novel algorithms based on the preliminary estimation of the Kalman filter output residuals.
The new algorithms do not only show potential for better performance, they also cast light on OKID, explicitly establishing the Kalman filter as central to linear system identification in the presence of noise, paralleling its role in signal estimation and filtering. The Kalman filter embedded in the OKID core equation is capable of converting the original problem, affected by random noise, into a purely deterministic problem.
The new interpretation leads to the extension of OKID to output-only system identification, providing a new tool for applications in structural health monitoring, and raises OKID to the level of a unified approach for input-output and output-only linear system identification. Any algorithm for linear system identification formulated in the absence of noise can now optimally handle noisy data via a preliminary step consisting in solving the OKID core equation.
The OKID framework developed for linear system identification is then extended to bilinear systems, which are of interest because several natural phenomena are inherently bilinear and also because high-order bilinear models are universal approximators for a wide class of nonlinear systems.
The formulation of an optimal bilinear observer for bilinear state-space models, similar to the Kalman filter in the linear case, leads to the development of an extension of OKID to bilinear system identification. This is the first application of OKID to nonlinear problems, not only because bilinear systems are themselves nonlinear, but also because one can think of bilinear OKID as a technique to find bilinear approximations of nonlinear systems.
Furthermore, the same strategy adopted in this work could be used to extend OKID directly to other classes of nonlinear models
State Estimation for Distributed Systems with Stochastic and Set-membership Uncertainties
State estimation techniques for centralized, distributed, and decentralized systems are studied. An easy-to-implement state estimation concept is introduced that generalizes and combines basic principles of Kalman filter theory and ellipsoidal calculus. By means of this method, stochastic and set-membership uncertainties can be taken into consideration simultaneously. Different solutions for implementing these estimation algorithms in distributed networked systems are presented
Nonlinear robust H∞ control.
A new theory is proposed for the full-information finite and infinite horizontime
robust H∞ control that is equivalently effective for the regulation and/or tracking
problems of the general class of time-varying nonlinear systems under the presence of
exogenous disturbance inputs. The theory employs the sequence of linear-quadratic and
time-varying approximations, that were recently introduced in the optimal control
framework, to transform the nonlinear H∞ control problem into a sequence of linearquadratic
robust H∞ control problems by using well-known results from the existing
Riccati-based theory of the maturing classical linear robust control. The proposed
method, as in the optimal control case, requires solving an approximating sequence of
Riccati equations (ASRE), to find linear time-varying feedback controllers for such
disturbed nonlinear systems while employing classical methods. Under very mild
conditions of local Lipschitz continuity, these iterative sequences of solutions are
known to converge to the unique viscosity solution of the Hamilton-lacobi-Bellman
partial differential equation of the original nonlinear optimal control problem in the
weak form (Cimen, 2003); and should hold for the robust control problems herein. The
theory is analytically illustrated by directly applying it to some sophisticated nonlinear
dynamical models of practical real-world applications. Under a r -iteration sense, such
a theory gives the control engineer and designer more transparent control requirements
to be incorporated a priori to fine-tune between robustness and optimality needs. It is
believed, however, that the automatic state-regulation robust ASRE feedback control
systems and techniques provided in this thesis yield very effective control actions in
theory, in view of its computational simplicity and its validation by means of classical
numerical techniques, and can straightforwardly be implemented in practice as the
feedback controller is constrained to be linear with respect to its inputs
Selected Topics in Gravity, Field Theory and Quantum Mechanics
Quantum field theory has achieved some extraordinary successes over the past sixty years; however, it retains a set of challenging problems. It is not yet able to describe gravity in a mathematically consistent manner. CP violation remains unexplained. Grand unified theories have been eliminated by experiment, and a viable unification model has yet to replace them. Even the highly successful quantum chromodynamics, despite significant computational achievements, struggles to provide theoretical insight into the low-energy regime of quark physics, where the nature and structure of hadrons are determined. The only proposal for resolving the fine-tuning problem, low-energy supersymmetry, has been eliminated by results from the LHC. Since mathematics is the true and proper language for quantitative physical models, we expect new mathematical constructions to provide insight into physical phenomena and fresh approaches for building physical theories
Constructive Methods of Invariant Manifolds for Kinetic Problems
We present the Constructive Methods of Invariant Manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The physical problem of reduced description is studied in a most general form as a problem of constructing the slow invariant manifold. The invariance conditions are formulated as the differential equation for a manifold immersed in the phase space (the invariance equation). The equation of motion for immersed manifolds is obtained (the film extension of the dynamics). Invariant manifolds are fixed points for this equation, and slow invariant manifolds are Lyapunov stable fixed points, thus slowness is presented as stability. A collection of methods for construction of slow invariant manifolds is presented, in particular, the Newton method subject to incomplete linearization is the analogue of KAM methods for dissipative systems.
The systematic use of thermodynamics structures and of the quasi--chemical representation allow to construct approximations which are in concordance with physical restrictions.
We systematically consider a discrete analogue of the slow (stable) positively invariant manifolds for dissipative systems, invariant grids. Dynamic and static postprocessing procedures give us the opportunity to estimate the accuracy of obtained approximations, and to improve this accuracy significantly.
The following examples of applications are presented: Nonperturbative deviation of physically consistent hydrodynamics from the Boltzmann equation and from the reversible dynamics, for Knudsen numbers Kn~1; construction of the moment equations for nonequilibrium media and their dynamical correction (instead of extension of list of variables) to gain more accuracy in description of highly nonequilibrium flows; determination of molecules dimension (as diameters of equivalent hard spheres) from experimental viscosity data; invariant grids for a two-dimensional catalytic reaction and a four-dimensional oxidation reaction (six species, two balances); universal continuous media description of dilute polymeric solution; the limits of macroscopic description for polymer molecules, etc