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Prethermalization in Aperiodically kicked Many-Body Dynamics
No full textDriven many-body systems typically experience heating due to the lack of energy conservation. Heating may be suppressed for time-periodic drives, but little is known for less regular drive protocols. In this work, we investigate the heating dynamics in aperiodically kicked systems, specifically those driven by quasi-periodic Thue-Morse or a family of random sequences with -multipolar temporal correlations. We demonstrate that multiple heating channels can be eliminated even away from the high-frequency regime. The number of eliminated channels increases with multipolar order . We illustrate this in a classical kicked rotor chain where we find a long-lived prethermal regime. When the static Hamiltonian only involves the kinetic energy, the prethermal lifetime can strongly depend on the temporal correlations of the drive, with a power-law dependence on the kick strength , for which we can account using a simple linearization argument
Reconstruction of Manifold Embeddings into Euclidean Spaces via Intrinsic Distances
No full textWe consider the problem of reconstructing an embedding of a compact connected Riemannian manifold in a Euclidean space up to an almost isometry, given the information on intrinsic distances between points from its “sufficiently large” subset. This is one of the classical manifold learning problems. It happens that the most popular methods to deal with such a problem, with a long history in data science, namely, the classical Multidimensional scaling (MDS) and the Maximum variance unfolding (MVU), actually miss the point and may provide results very far from an isometry; moreover, they may even give no bi-Lipshitz embedding. We will provide an easy variational formulation of this problem, which leads to an algorithm always providing an almost isometric embedding with the distortion of original distances as small as desired (the parameter regulating the upper bound for the desired distortion is an input parameter of this algorithm)
Robust Multiple Stopping - A Duality Approach
No full textWe develop a method to solve, theoretically and numerically, general optimal stopping problems. Our general setting allows for multiple exercise rights—that is, optimal multiple stopping—for a robust evaluation that accounts for model uncertainty with a dominated family of priors and for general reward processes driven by multidimensional jump-diffusions. Our approach relies on first establishing robust martingale dual representation results for the multiple stopping problem that satisfy appealing almost sure pathwise optimality properties. Next, we exploit these theoretical results to develop upper and lower bounds that, as we formally show, not only converge to the true solution asymptotically, but also constitute genuine prelimiting upper and lower bounds. We illustrate the applicability of our approach in a few examples and analyze the impact of model uncertainty on optimal multiple stopping strategies
Radiation Conditions for the Helmholtz Equation in a Half Plane filled by Inhomogeneous Periodic Material
No full textIn this paper we consider time-harmonic acoustic wave propagation in a half-plane filled by inhomogeneous periodic medium. If the refractive index depends on the horizontal coordinate only, we define upward and downward radiating modes by solving a one-dimensional Sturm-Liouville eigenvalue problem with a complex-valued periodic coefficient. The upward and downward radiation conditions are introduced based on a generalized Rayleigh series. Using the variational method, we then prove uniqueness and existence for the scattering of an incoming wave mode by a grating located between an upper and lower half plane with such inhomogeneous periodic media. Finally, we discuss the application of the new radiation conditions to the scattering matrix algorithm, i.e., to rigorous coupled wave analysis or Fourier modal method
An MBO Method for Modularity Optimisation based on Total Variation and Signless Total Variation
No full textIn network science, one of the significant and challenging subjects is the detection of communities. Modularity [1] is a measure of community structure that compares connectivity in the network with the expected connectivity in a graph sampled from a random null model. Its optimisation is a common approach to tackle the community detection problem. We present a new method for modularity maximisation, which is based on the observation that modularity can be expressed in terms of total variation on the graph and signless total variation on the null model. The resulting algorithm is of Merriman–Bence–Osher (MBO) type. Different from earlier methods of this type, the new method can easily accommodate different choices of the null model. Besides theoretical investigations of the method, we include in this paper numerical comparisons with other community detection methods, among which the MBO-type methods of Hu et al. [2] and Boyd et al. [3], and the Leiden algorithm [4]
Pressure-Robust Error Analysis for Raviart-Thomas Enriched Scott-Vogelius Pairs
No full textRecent work shows that it is possible to enrich the Scott--Vogelius finite element pair by cer- tain Raviart--Thomas functions to obtain an inf-sup stable and divergence-free method on general shape-regular meshes. A skew-symmetric consistency term was suggested for avoiding an ad- ditional stabilization term for higher order elements, but no L2 (Ω) error estimate was shown for the Stokes equations. This note closes this gap. In addition, the optimal choice of the stabilization parameter is studied numerically
An Ergodic and Isotropic Zero-Conductance Model with Arbitrarily Strong Local Connectivity
No full textWe exhibit a percolating ergodic and isotropic lattice model in all but at least two dimensions that has zero effective conductivity in all spatial directions and for all non-trivial choices of the connectivity parameter. The model is based on the so-called randomly stretched lattice where we additionally elongate layers containing few open edges
A Weighted Hybridizable Discontinuous Galerkin Method for Drift-Diffusion Problems
No full textIn this work we propose a weighted hybridizable discontinuous Galerkin method (W-HDG) for drift-diffusion problems. By using specific exponential weights when computing the L2 product in each cell of the discretization, we are able to replicate the behavior of the Slotboom change of variables, and eliminate the drift term from the local matrix contributions. We show that the proposed numerical scheme is well-posed, and numerically validates that it has the same properties of classical HDG methods, including optimal convergence, and superconvergence of postprocessed solutions. For polynomial degree zero, dimension one, and vanishing HDG stabilization parameter, W-HDG coincides with the Scharfetter-Gummel stabilized finite volume scheme (i.e., it produces the same system matrix). The use of local exponential weights generalizes the Scharfetter-Gummel stabilization (the state-of-the-art for Finite Volume discretization of transport-dominated problems) to arbitrary high-order approximations
Cardiac Dynamics of a Human Ventricular Tissue Model with Focus on Early Afterdepolarizations
No full textThis study aims to investigate the computationally complex cardiac dynamics of the well-known human ventricular model of Ten Tusscher and Panfilov from 2006. The corresponding physical system of the cellular model is described by a set of nonlinear differential equations containing various system parameters. If one or a set of specific system parameters crosses a certain threshold, the system is forced to change dynamics, which might result in dangerous cardiac dynamics and can be precursors to cardiac death. To perform an efficient numerical analysis, the original model is revised and simplified such that the modified models perfectly match the trajectory (time-dependent cardiac potential) of the original model. Moreover, it is demonstrated that the reduced models have the same dynamics. Furthermore, using the lowest dimensional model, it is shown by means of bifurcation analysis that combinations of reduced slow and rapid potassium currents and an enhanced calcium current may lead to early afterdepolarizations, which are pathological voltage oscillations during the repolarization or plateau phase of cardiac action potentials and are considered potential precursors to cardiac arrhythmia. Finally, to outline synchronization effects and pattern formation on a larger scale (macro scale), a two-dimensional epicardial monodomain equation is studied
Hessian Barrier Algorithms for Non-Convex Conic Optimization
No full textA key problem in mathematical imaging, signal processing and computational statistics is the minimization of non-convex objective functions that may be non-differentiable at the relative boundary of the feasible set. This paper proposes a new family of first- and second-order interior-point methods for non-convex optimization problems with linear and conic constraints, combining logarithmically homogeneous barriers with quadratic and cubic regularization respectively. Our approach is based on a potential-reduction mechanism and, under the Lipschitz continuity of the corresponding derivative with respect to the local barrier-induced norm, attains a suitably defined class of approximate first- or second-order KKT points with worst-case iteration complexity (first-order) and (second-order), respectively. Based on these findings, we develop new path-following schemes attaining the same complexity, modulo adjusting constants. These complexity bounds are known to be optimal in the unconstrained case, and our work shows that they are upper bounds in the case with complicated constraints as well. To the best of our knowledge, this work is the first which achieves these worst-case complexity bounds under such weak conditions for general conic constrained non-convex optimization problems