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Lattice Boltzmann in micro- and nano- flow simulations
This paper was presented at the 2nd Micro and Nano Flows Conference (MNF2009), which was held at Brunel University, West London, UK. The conference was organised by Brunel University and supported by the Institution of Mechanical Engineers, IPEM, the Italian Union of Thermofluid dynamics, the Process Intensification Network, HEXAG - the Heat Exchange Action Group and the Institute of Mathematics and its Applications.One of the fundamental difficulties in micro- and nano-flow simulations is that the
validity’s of the continuum assumption and the hydro-dynamic equations start to become questionable in this flow regime. The lower-level kinetic/molecular alternatives are often either prohibitively expensive for practical purposes or poorly justified from a fundamental perspective. The lattice
Boltzmann (LB) method, which originated from a simplistic Boolean kinetic model, is recently shown to converge asymptotically to the continuum Boltzmann-BGK equation and therefore offers a theoretically sound and computationally effective approach for micro- and nano-flow simulations. In addition, its kinetic nature allows certain microscopic physics to be modeled at the macroscopic level, leading to a highly efficient model for multiphase flows with phase transitions. With the inherent computational advantages of a lattice model, e.g., the algorithm simplicity and parallelizability, the
ease of handling complex geometry and so on, the LB method has found many applications in various areas of Computational Fluid Dynamics (CFD) and matured to the extend of commercial applications. In this talk, I shall give an introduction to the LB method with the emphasis given to the theoretical
justifications for its applications in micro- and nano-flow simulations. Some recent examples will also be reported
Sensitivity analysis of a coupled hydrodynamic-vegetation model using the effectively subsampled quadratures method (ESQM v5.2)
Abstract. Coastal hydrodynamics can be greatly affected by the presence of submerged aquatic vegetation. The effect of vegetation has been incorporated into the Coupled Ocean–Atmosphere–Wave–Sediment Transport (COAWST) modeling system. The vegetation implementation includes the plant-induced three-dimensional drag, in-canopy wave-induced streaming, and the production of turbulent kinetic energy by the presence of vegetation. In this study, we evaluate the sensitivity of the flow and wave dynamics to vegetation parameters using Sobol' indices and a least squares polynomial approach referred to as the Effective Quadratures method. This method reduces the number of simulations needed for evaluating Sobol' indices and provides a robust, practical, and efficient approach for the parameter sensitivity analysis. The evaluation of Sobol' indices shows that kinetic energy, turbulent kinetic energy, and water level changes are affected by plant stem density, height, and, to a lesser degree, diameter. Wave dissipation is mostly dependent on the variation in plant stem density. Performing sensitivity analyses for the vegetation module in COAWST provides guidance to optimize efforts and reduce exploration of parameter space for future observational and modeling work.
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Finite size scaling for quantum criticality using the finite-element method
Finite size scaling for the Schr\"{o}dinger equation is a systematic approach
to calculate the quantum critical parameters for a given Hamiltonian. This
approach has been shown to give very accurate results for critical parameters
by using a systematic expansion with global basis-type functions. Recently, the
finite element method was shown to be a powerful numerical method for ab initio
electronic structure calculations with a variable real-space resolution. In
this work, we demonstrate how to obtain quantum critical parameters by
combining the finite element method (FEM) with finite size scaling (FSS) using
different ab initio approximations and exact formulations. The critical
parameters could be atomic nuclear charges, internuclear distances, electron
density, disorder, lattice structure, and external fields for stability of
atomic, molecular systems and quantum phase transitions of extended systems. To
illustrate the effectiveness of this approach we provide detailed calculations
of applying FEM to approximate solutions for the two-electron atom with varying
nuclear charge; these include Hartree-Fock, density functional theory under the
local density approximation, and an "exact"' formulation using FEM. We then use
the FSS approach to determine its critical nuclear charge for stability; here,
the size of the system is related to the number of elements used in the
calculations. Results prove to be in good agreement with previous Slater-basis
set calculations and demonstrate that it is possible to combine finite size
scaling with the finite-element method by using ab initio calculations to
obtain quantum critical parameters. The combined approach provides a promising
first-principles approach to describe quantum phase transitions for materials
and extended systems.Comment: 15 pages, 19 figures, revision based on suggestions by referee,
accepted in Phys. Rev.
Reduced Order and Surrogate Models for Gravitational Waves
We present an introduction to some of the state of the art in reduced order
and surrogate modeling in gravitational wave (GW) science. Approaches that we
cover include Principal Component Analysis, Proper Orthogonal Decomposition,
the Reduced Basis approach, the Empirical Interpolation Method, Reduced Order
Quadratures, and Compressed Likelihood evaluations. We divide the review into
three parts: representation/compression of known data, predictive models, and
data analysis. The targeted audience is that one of practitioners in GW
science, a field in which building predictive models and data analysis tools
that are both accurate and fast to evaluate, especially when dealing with large
amounts of data and intensive computations, are necessary yet can be
challenging. As such, practical presentations and, sometimes, heuristic
approaches are here preferred over rigor when the latter is not available. This
review aims to be self-contained, within reasonable page limits, with little
previous knowledge (at the undergraduate level) requirements in mathematics,
scientific computing, and other disciplines. Emphasis is placed on optimality,
as well as the curse of dimensionality and approaches that might have the
promise of beating it. We also review most of the state of the art of GW
surrogates. Some numerical algorithms, conditioning details, scalability,
parallelization and other practical points are discussed. The approaches
presented are to large extent non-intrusive and data-driven and can therefore
be applicable to other disciplines. We close with open challenges in high
dimension surrogates, which are not unique to GW science.Comment: Invited article for Living Reviews in Relativity. 93 page
Polynomial approximation of non-Gaussian unitaries by counting one photon at a time
In quantum computation with continous-variable systems, quantum advantage can
only be achieved if some non-Gaussian resource is available. Yet, non-Gaussian
unitary evolutions and measurements suited for computation are challenging to
realize in the lab. We propose and analyze two methods to apply a polynomial
approximation of any unitary operator diagonal in the amplitude quadrature
representation, including non-Gaussian operators, to an unknown input state.
Our protocols use as a primary non-Gaussian resource a single-photon counter.
We use the fidelity of the transformation with the target one on Fock and
coherent states to assess the quality of the approximate gate.Comment: 11 pages, 7 figure
Quadrature Strategies for Constructing Polynomial Approximations
Finding suitable points for multivariate polynomial interpolation and
approximation is a challenging task. Yet, despite this challenge, there has
been tremendous research dedicated to this singular cause. In this paper, we
begin by reviewing classical methods for finding suitable quadrature points for
polynomial approximation in both the univariate and multivariate setting. Then,
we categorize recent advances into those that propose a new sampling approach
and those centered on an optimization strategy. The sampling approaches yield a
favorable discretization of the domain, while the optimization methods pick a
subset of the discretized samples that minimize certain objectives. While not
all strategies follow this two-stage approach, most do. Sampling techniques
covered include subsampling quadratures, Christoffel, induced and Monte Carlo
methods. Optimization methods discussed range from linear programming ideas and
Newton's method to greedy procedures from numerical linear algebra. Our
exposition is aided by examples that implement some of the aforementioned
strategies
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