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A variational principle for determining unstable periodic orbits of flows as well as unstable spatio-temporally periodic solutions of extended systems is proposed and implemented. An initial loop approximating a periodic solution is evolved…
A key issue in dimension reduction of dissipative dynamical systems with spectral gaps is the identification of slow invariant manifolds. We present theoretical and numerical results for a variational approach to the problem of computing…
Chaotic dynamics in systems ranging from low-dimensional nonlinear differential equations to high-dimensional spatio-temporal systems including fluid turbulence is supported by non-chaotic, exactly recurring time-periodic solutions of the…
In Hamiltonian systems subjected to periodic perturbations the stable and unstable manifolds of the unstable periodic orbits provide the dynamical "skeleton" that drives the mixing process and bounds the chaotic regions of the phase space.…
An algorithm for detecting unstable periodic orbits in chaotic systems [Phys. Rev. E, 60 (1999), pp. 6172-6175] which combines the set of stabilising transformations proposed by Schmelcher and Diakonos [Phys. Rev. Lett., 78 (1997), pp.…
We propose an algorithmic framework, that employs active subspace techniques, for scalable global optimization of functions with low effective dimension (also referred to as low-rank functions). This proposal replaces the original…
An algorithm for detecting periodic orbits in chaotic systems [Phys. Rev. E, 60 (1999), pp.~6172--6175], which combines the set of stabilising transformations proposed by Schmelcher and Diakonos [Phys. Rev. Lett., 78 (1997), pp.~4733--4736]…
We undertake a systematic exploration of recurrent patterns in a 1-dimensional Kuramoto-Sivashinsky system. For a small, but already rather turbulent system, the long-time dynamics takes place on a low-dimensional invariant manifold. A set…
We propose a general strategy for reduced order modeling of systems that display highly nonlinear oscillations. By considering a continuous family of forced periodic orbits defined in relation to a stable fixed point and subsequently…
We present a method to detect the unstable periodic orbits of a multidimensional chaotic dynamical system. Our approach allows us to locate in an efficient way the unstable cycles of, in principle, arbitrary length with a high accuracy.…
The time evolution of quantum many-body systems is one of the most promising applications for near-term quantum computers. However, the utility of current quantum devices is strongly hampered by the proliferation of hardware errors. The…
Unstable periodic orbits (UPOs) are the non-chaotic, dynamical building blocks of spatio-temporal chaos, motivating a first-principles based theory for turbulence ever since the discovery of deterministic chaos. Despite their key role in…
Recurrence in the phase space of complex systems is a well-studied phenomenon, which has provided deep insights into the nonlinear dynamics of such systems. For dissipative systems, characteristics based on recurrence plots have recently…
An analysis of high-dimensional data can offer a detailed description of a system but is often challenged by the curse of dimensionality. General dimensionality reduction techniques can alleviate such difficulty by extracting a few…
We present a new adaptive method for electronic structure calculations based on novel fast algorithms for reduction of multivariate mixtures. In our calculations, spatial orbitals are maintained as Gaussian mixtures whose terms are selected…
We study the dynamical properties of a broad class of high-dimensional random dynamical systems exhibiting chaotic as well as fixed point and periodic attractors. We consider cases in which attractors can co-exists in some regions of the…
In recent years, to improve the evolutionary algorithms used to solve optimization problems involving a large number of decision variables, many attempts have been made to simplify the problem solution space of a given problem for the…
We consider model order reduction of parameterized Hamiltonian systems describing nondissipative phenomena, like wave-type and transport dominated problems. The development of reduced basis methods for such models is challenged by two main…
One of the challenges in optimization of high dimensional problems is finding appropriate solutions in a way that are as close as possible to the global optima. In this regard, one of the most common phenomena that occurs is the curse of…
This paper can be seen as an attempt of rethinking the {\em Extra-Gradient Philosophy} for solving Variational Inequality Problems. We show that the properly defined {\em Reduced Gradients} can be used instead for finding approximate…