Direct numerical simulation of complex viscoelastic flows via fast lattice-Boltzmann solution of the Fokker–Planck equation

Micro–macro simulations of polymeric solutions rely on the coupling between macroscopic conservation equations for the fluid flow and stochastic differential equations for kinetic viscoelastic models at the microscopic scale. In the present work we introduce a novel micro–macro numerical approach, where the macroscopic equations are solved by a finite-volume method and the microscopic equation by a lattice-Boltzmann one. The kinetic model is given by molecular analogy with a finitely extensible non-linear elastic (FENE) dumbbell and is deterministically solved through an equivalent Fokker–Planck equation. The key features of the proposed approach are: (i) a proper scaling and coupling between the micro lattice-Boltzmann solution and the macro finite-volume one; (ii) a fast microscopic solver thanks to an implementation for Graphic Processing Unit (GPU) and the local adaptivity of the lattice-Boltzmann mesh; (iii) an operator-splitting algorithm for the convection of the macroscopic viscoelastic stresses instead of the whole probability density of the dumbbell configuration. This latter feature allows the application of the proposed method to non-homogeneous flow conditions with low memory-storage requirements. The model optimization is achieved through an extensive analysis of the lattice-Boltzmann solution, which finally provides control on the numerical error and on the computational time. The resulting micro–macro model is validated against the benchmark problem of a viscoelastic flow past a confined cylinder and the results obtained confirm the validity of the approach

Certified lattice reduction

Quadratic form reduction and lattice reduction are fundamental tools in computational number theory and in computer science, especially in cryptography. The celebrated Lenstra-Lenstra-Lov\'asz reduction algorithm (so-called LLL) has been improved in many ways through the past decades and remains one of the central methods used for reducing integral lattice basis. In particular, its floating-point variants-where the rational arithmetic required by Gram-Schmidt orthogonalization is replaced by floating-point arithmetic-are now the fastest known. However, the systematic study of the reduction theory of real quadratic forms or, more generally, of real lattices is not widely represented in the literature. When the problem arises, the lattice is usually replaced by an integral approximation of (a multiple of) the original lattice, which is then reduced. While practically useful and proven in some special cases, this method doesn't offer any guarantee of success in general. In this work, we present an adaptive-precision version of a generalized LLL algorithm that covers this case in all generality. In particular, we replace floating-point arithmetic by Interval Arithmetic to certify the behavior of the algorithm. We conclude by giving a typical application of the result in algebraic number theory for the reduction of ideal lattices in number fields.Comment: 23 page

Decoding by Sampling: A Randomized Lattice Algorithm for Bounded Distance Decoding

Despite its reduced complexity, lattice reduction-aided decoding exhibits a widening gap to maximum-likelihood (ML) performance as the dimension increases. To improve its performance, this paper presents randomized lattice decoding based on Klein's sampling technique, which is a randomized version of Babai's nearest plane algorithm (i.e., successive interference cancelation (SIC)). To find the closest lattice point, Klein's algorithm is used to sample some lattice points and the closest among those samples is chosen. Lattice reduction increases the probability of finding the closest lattice point, and only needs to be run once during pre-processing. Further, the sampling can operate very efficiently in parallel. The technical contribution of this paper is two-fold: we analyze and optimize the decoding radius of sampling decoding resulting in better error performance than Klein's original algorithm, and propose a very efficient implementation of random rounding. Of particular interest is that a fixed gain in the decoding radius compared to Babai's decoding can be achieved at polynomial complexity. The proposed decoder is useful for moderate dimensions where sphere decoding becomes computationally intensive, while lattice reduction-aided decoding starts to suffer considerable loss. Simulation results demonstrate near-ML performance is achieved by a moderate number of samples, even if the dimension is as high as 32

CO adsorption on metal surfaces: a hybrid functional study with plane wave basis set

We present a detailed study of the adsorption of CO on Cu, Rh, and Pt (111) surfaces in top and hollow sites. The study has been performed using the local density approximation, the gradient corrected functional PBE, and the hybrid Hartree-Fock density functionals PBE0 and HSE03 within the framework of generalized Kohn-Sham density functional theory using a plane-wave basis set. As expected, the LDA and GGA functionals show a tendency to favor the hollow sites, at variance with experimental findings that give the top site as the most stable adsorption site. The PBE0 and HSE03 functionals reduce this tendency. In fact, they predict the correct adsorption site for Cu and Rh but fail for Pt. But even in this case, the hybrid functional destabilizes the hollow site by 50 meV compared to the PBE functional. The results of the total energy calculations are presented along with an analysis of the projected density of states.Comment: 32 pages, 6 tables, 3 figures. (Re)Submitted to Phys. Rev. B; LDA results added in the tables; minor changes in the tex

Numerical Ricci-flat metrics on K3

We develop numerical algorithms for solving the Einstein equation on Calabi-Yau manifolds at arbitrary values of their complex structure and Kahler parameters. We show that Kahler geometry can be exploited for significant gains in computational efficiency. As a proof of principle, we apply our methods to a one-parameter family of K3 surfaces constructed as blow-ups of the T^4/Z_2 orbifold with many discrete symmetries. High-resolution metrics may be obtained on a time scale of days using a desktop computer. We compute various geometric and spectral quantities from our numerical metrics. Using similar resources we expect our methods to practically extend to Calabi-Yau three-folds with a high degree of discrete symmetry, although we expect the general three-fold to remain a challenge due to memory requirements.Comment: 38 pages, 10 figures; program code and animations of figures downloadable from http://schwinger.harvard.edu/~wiseman/K3/ ; v2 minor corrections, references adde