Second-order flows for computing the ground states of rotating Bose-Einstein condensates

• URL: http://arxiv.org/abs/2205.00805v2
• Date: Fri, 6 Jan 2023 15:13:31 GMT
• Title: Second-order flows for computing the ground states of rotating Bose-Einstein condensates
• Authors: Haifan Chen, Guozhi Dong, Wei Liu, Ziqing Xie
• Abstract summary: Some artificial evolutionary differential equations involving second-order time derivatives are considered to be first-order. The proposed artificial dynamics are novel second-order hyperbolic partial differential equations with dissipation. New algorithms are superior to the state-of-the-art numerical methods based on the gradient flow.
• Score: 5.252966797394752
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Second-order flows in this paper refer to some artificial evolutionary differential equations involving second-order time derivatives distinguished from gradient flows which are considered to be first-order flows. This is a popular topic due to the recent advances of inertial dynamics with damping in convex optimization. Mathematically, the ground state of a rotating Bose-Einstein condensate (BEC) can be modeled as a minimizer of the Gross-Pitaevskii energy functional with angular momentum rotational term under the normalization constraint. We introduce two types of second-order flows as energy minimization strategies for this constrained non-convex optimization problem, in order to approach the ground state. The proposed artificial dynamics are novel second-order nonlinear hyperbolic partial differential equations with dissipation. Several numerical discretization schemes are discussed, including explicit and semi-implicit methods for temporal discretization, combined with a Fourier pseudospectral method for spatial discretization. These provide us a series of efficient and robust algorithms for computing the ground states of rotating BECs. Particularly, the newly developed algorithms turn out to be superior to the state-of-the-art numerical methods based on the gradient flow. In comparison with the gradient flow type approaches: When explicit temporal discretization strategies are adopted, the proposed methods allow for larger stable time step sizes; While for semi-implicit discretization, using the same step size, a much smaller number of iterations are needed for the proposed methods to reach the stopping criterion, and every time step encounters almost the same computational complexity. Rich and detailed numerical examples are documented for verification and comparison.

Related papers

• Distributed Optimization via Energy Conservation Laws in Dilated Coordinates [5.35599092568615]
  This paper introduces an energy conservation approach for analyzing continuous-time dynamical systems in dilated coordinates. convergence rates can be explicitly expressed in terms of the inverse time-dilation factor. Its accelerated convergence behavior is benchmarked against various state-of-the-art distributed optimization algorithms on practical, large-scale problems.
  arXiv  Detail & Related papers  (2024-09-28T08:02:43Z)
• Quantum Algorithm for Solving the Advection Equation using Hamiltonian Simulation [0.0]
  One-dimensional advection can be simulated directly since the central finite difference operator for first-order derivatives is anti-Hermitian. A single copy of the initial quantum state is required and the circuit depth grows linearly with the required number of time steps.
  arXiv  Detail & Related papers  (2023-12-15T13:39:27Z)
• Accelerated First-Order Optimization under Nonlinear Constraints [73.2273449996098]
  We exploit between first-order algorithms for constrained optimization and non-smooth systems to design a new class of accelerated first-order algorithms. An important property of these algorithms is that constraints are expressed in terms of velocities instead of sparse variables.
  arXiv  Detail & Related papers  (2023-02-01T08:50:48Z)
• Fast Computation of Optimal Transport via Entropy-Regularized Extragradient Methods [75.34939761152587]
  Efficient computation of the optimal transport distance between two distributions serves as an algorithm that empowers various applications. This paper develops a scalable first-order optimization-based method that computes optimal transport to within $varepsilon$ additive accuracy.
  arXiv  Detail & Related papers  (2023-01-30T15:46:39Z)
• On optimization of coherent and incoherent controls for two-level quantum systems [77.34726150561087]
  This article considers some control problems for closed and open two-level quantum systems. The closed system's dynamics is governed by the Schr"odinger equation with coherent control. The open system's dynamics is governed by the Gorini-Kossakowski-Sudarshan-Lindblad master equation.
  arXiv  Detail & Related papers  (2022-05-05T09:08:03Z)
• Computing the action ground state for the rotating nonlinear Schr\"odinger equation [6.6772808699409705]
  We consider the computations of the action ground state for a rotating nonlinear Schr"odinger equation. In the focusing case, we identify an equivalent formulation of the problem which simplifies the constraint. In the defocusing case, we prove that the ground state can be obtained by the unconstrained minimization.
  arXiv  Detail & Related papers  (2022-03-12T09:15:27Z)
• An application of the splitting-up method for the computation of a neural network representation for the solution for the filtering equations [68.8204255655161]
  Filtering equations play a central role in many real-life applications, including numerical weather prediction, finance and engineering. One of the classical approaches to approximate the solution of the filtering equations is to use a PDE inspired method, called the splitting-up method. We combine this method with a neural network representation to produce an approximation of the unnormalised conditional distribution of the signal process.
  arXiv  Detail & Related papers  (2022-01-10T11:01:36Z)
• A Two-Time-Scale Stochastic Optimization Framework with Applications in Control and Reinforcement Learning [13.908826484332282]
  We study a new two-time-scale gradient method for solving optimization problems. Our first contribution is to characterize the finite-time complexity of the proposed two-time-scale gradient algorithm. We apply our framework to gradient-based policy evaluation algorithms in reinforcement learning.
  arXiv  Detail & Related papers  (2021-09-29T23:15:23Z)
• Single-Timescale Stochastic Nonconvex-Concave Optimization for Smooth Nonlinear TD Learning [145.54544979467872]
  We propose two single-timescale single-loop algorithms that require only one data point each step. Our results are expressed in a form of simultaneous primal and dual side convergence.
  arXiv  Detail & Related papers  (2020-08-23T20:36:49Z)
• Convergence and sample complexity of gradient methods for the model-free linear quadratic regulator problem [27.09339991866556]
  We show that ODE searches for optimal control for an unknown computation system by directly searching over the corresponding space of controllers. We take a step towards demystifying the performance and efficiency of such methods by focusing on the gradient-flow dynamics set of stabilizing feedback gains and a similar result holds for the forward disctization of the ODE.
  arXiv  Detail & Related papers  (2019-12-26T16:56:59Z)
• A Near-Optimal Gradient Flow for Learning Neural Energy-Based Models [93.24030378630175]
  We propose a novel numerical scheme to optimize the gradient flows for learning energy-based models (EBMs) We derive a second-order Wasserstein gradient flow of the global relative entropy from Fokker-Planck equation. Compared with existing schemes, Wasserstein gradient flow is a smoother and near-optimal numerical scheme to approximate real data densities.
  arXiv  Detail & Related papers  (2019-10-31T02:26:20Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.