6 research outputs found
Koopman Spectral Linearization vs. Carleman Linearization: A Computational Comparison Study
Nonlinearity presents a significant challenge in problems involving dynamical
systems, prompting the exploration of various linearization techniques,
including the well-known Carleman Linearization. In this paper, we introduce
the Koopman Spectral Linearization method tailored for nonlinear autonomous
dynamical systems. This innovative linearization approach harnesses the
Chebyshev differentiation matrix and the Koopman Operator to yield a lifted
linear system. It holds the promise of serving as an alternative approach that
can be employed in scenarios where Carleman linearization is traditionally
applied. Numerical experiments demonstrate the effectiveness of this
linearization approach for several commonly used nonlinear dynamical systems.Comment: 17 pages, 7 figure
Carleman Linearization of Nonlinear Systems and Its Finite-Section Approximations
The Carleman linearization is one of the mainstream approaches to lift a
finite-dimensional nonlinear dynamical system into an infinite-dimensional
linear system with the promise of providing accurate approximations of the
original nonlinear system over larger regions around the equilibrium for longer
time horizons with respect to the conventional first-order linearization
approach. Finite-section approximations of the lifted system has been widely
used to study dynamical and control properties of the original nonlinear
system. In this context, some of the outstanding problems are to determine
under what conditions, as the finite-section order (i.e., truncation length)
increases, the trajectory of the resulting approximate linear system from the
finite-section scheme converges to that of the original nonlinear system and
whether the time interval over which the convergence happens can be quantified
explicitly. In this paper, we provide explicit error bounds for the
finite-section approximation and prove that the convergence is indeed
exponential with respect to the finite-section order. For a class of nonlinear
systems, it is shown that one can achieve exponential convergence over the
entire time horizon up to infinity. Our results are practically plausible as
our proposed error bound estimates can be used to compute proper truncation
lengths for a given application, e.g., determining proper sampling period for
model predictive control and reachability analysis for safety verifications. We
validate our theoretical findings through several illustrative simulations.Comment: 25 Pages, 10 figure
Potential quantum advantage for simulation of fluid dynamics
Numerical simulation of turbulent fluid dynamics needs to either parameterize
turbulence-which introduces large uncertainties-or explicitly resolve the
smallest scales-which is prohibitively expensive. Here we provide evidence
through analytic bounds and numerical studies that a potential quantum
exponential speedup can be achieved to simulate the Navier-Stokes equations
governing turbulence using quantum computing. Specifically, we provide a
formulation of the lattice Boltzmann equation for which we give evidence that
low-order Carleman linearization is much more accurate than previously believed
for these systems and that for computationally interesting examples. This is
achieved via a combination of reformulating the nonlinearity and accurately
linearizing the dynamical equations, effectively trading nonlinearity for
additional degrees of freedom that add negligible expense in the quantum
solver. Based on this we apply a quantum algorithm for simulating the
Carleman-linerized lattice Boltzmann equation and provide evidence that its
cost scales logarithmically with system size, compared to polynomial scaling in
the best known classical algorithms. This work suggests that an exponential
quantum advantage may exist for simulating fluid dynamics, paving the way for
simulating nonlinear multiscale transport phenomena in a wide range of
disciplines using quantum computing
Computer Aided Verification
This open access two-volume set LNCS 13371 and 13372 constitutes the refereed proceedings of the 34rd International Conference on Computer Aided Verification, CAV 2022, which was held in Haifa, Israel, in August 2022. The 40 full papers presented together with 9 tool papers and 2 case studies were carefully reviewed and selected from 209 submissions. The papers were organized in the following topical sections: Part I: Invited papers; formal methods for probabilistic programs; formal methods for neural networks; software Verification and model checking; hyperproperties and security; formal methods for hardware, cyber-physical, and hybrid systems. Part II: Probabilistic techniques; automata and logic; deductive verification and decision procedures; machine learning; synthesis and concurrency. This is an open access book