Mathematics

The Bhatnagar-Gross-Krook (BGK) single-relaxation-time collision model for the Boltzmann equation serves as the foundation of the lattice BGK (LBGK) method developed in recent years. The description of the collision as a uniform relaxation process of the distribution function towards its equilibrium is, in many scenarios, simplistic. Based on a previous series of papers, we present a collision model formulated as independent relaxations of the irreducible components of the Hermit coefficients in the reference frame moving with the fluid. These components, corresponding to the irreducible representation of the rotation group, are the minimum tensor components that can be separately relaxed without violating rotation symmetry. For the 2nd, 3rd and 4th moments respectively, two, two and three independent relaxation rates can exist, giving rise to the shear and bulk viscosity, thermal diffusivity and some high-order relaxation process not explicitly manifested in the Navier-Stokes-Fourier equations. Using the binomial transform, the Hermite coefficients are evaluated in the absolute frame to avoid the numerical dissipation introduced by interpolation. Extensive numerical verification is also provided.

This paper is concerned with the convergence analysis of two-level hierarchical basis (TLHB) methods in a general setting, where the decomposition associated with two hierarchical component spaces is not required to be a direct sum. The TLHB scheme can be regarded as a combination of compatible relaxation and coarse-grid correction. Most of the previous works focus on the case of exact coarse solver, and the existing identity for the convergence factor of exact TLHB methods involves a tricky max-min problem. In this work, we present a new and purely algebraic analysis of TLHB methods, which gives a succinct identity for the convergence factor of exact TLHB methods. The new identity can be conveniently utilized to derive an optimal interpolation and analyze the influence of coarse space on the convergence factor. Moreover, we establish two-sided bounds for the convergence factor of TLHB methods with inexact coarse solver, which extend the existing TLHB theory.

This paper introduces a new approximation scheme for solving high-dimensional semilinear partial differential equations (PDEs) and backward stochastic differential equations (BSDEs). First, we decompose a target semilinear PDE (BSDE) into two parts, namely "dominant" linear and "small" nonlinear PDEs. Then, we employ a Deep BSDE solver with a new control variate method to solve those PDEs, where approximations based on an asymptotic expansion technique are effectively applied to the linear part and also used as control variates for the nonlinear part. Moreover, our theoretical result indicates that errors of the proposed method become much smaller than those of the original Deep BSDE solver. Finally, we show numerical experiments to demonstrate the validity of our method, which is consistent with the theoretical result in this paper.

This paper proposes and analyzes a new operator splitting method for stochastic Maxwell equations driven by additive noise, which not only decomposes the original multi-dimensional system into some local one-dimensional subsystems, but also separates the deterministic and stochastic parts. This method is numerically efficient, and preserves the symplecticity, the multi-symplecticity as well as the growth rate of the averaged energy. A detailed H 2 -regularity analysis of stochastic Maxwell equations is obtained, which is a crucial prerequisite of the error analysis. Under the regularity assumptions of the initial data and the noise, the convergence order one in mean square sense of the operator splitting method is established.

We provide a new upper bound for sampling numbers ( g n ) n∈N associated to the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants C,c>0 (which are specified in the paper) such that g 2 n ≤ Clog(n) n ∑ k≥⌊cn⌋ σ 2 k ,n≥2, where ( σ k ) k∈N is the sequence of singular numbers (approximation numbers) of the Hilbert-Schmidt embedding Id:H(K)→ L 2 (D, ϱ D ) . The algorithm which realizes the bound is a least squares algorithm based on a specific set of sampling nodes. These are constructed out of a random draw in combination with a down-sampling procedure coming from the celebrated proof of Weaver's conjecture, which was shown to be equivalent to the Kadison-Singer problem. Our result is non-constructive since we only show the existence of a linear sampling operator realizing the above bound. The general result can for instance be applied to the well-known situation of H s mix ( T d ) in L 2 ( T d ) with s>1/2 . We obtain the asymptotic bound g n ≤ C s,d n −s log(n ) (d−1)s+1/2 , which improves on very recent results by shortening the gap between upper and lower bound to log(n) − − − − − √ .

We apply the nonconforming discretisation of Wu and Xu (2019) to the tri-Helmholtz equation on the plane where the source term is a functional evaluating the test function on a one-dimensional mesh-aligned embedded curve. We present error analysis for the convergence of the discretisation and linear convergence as a function of mesh size is recovered almost everywhere away from the embedded curve which aligns with classic regularity theory.

A variety of techniques have been developed for the approximation of non-periodic functions. In particular, there are approximation techniques based on rank- 1 lattices and transformed rank- 1 lattices, including methods that use sampling sets consisting of Chebyshev- and tent-transformed nodes. We compare these methods with a parameterized transformed Fourier system that yields similar ??2 -approximation errors.

In this work we propose a novel block preconditioner, labelled Explicit Decoupling Factor Approximation (EDFA), to accelerate the convergence of Krylov subspace solvers used to address the sequence of non-symmetric systems of linear equations originating from flow simulations in porous media. The flow model is discretized blending the Mixed Hybrid Finite Element (MHFE) method for Darcy's equation with the Finite Volume (FV) scheme for the mass conservation. The EDFA preconditioner is characterized by two features: the exploitation of the system matrix decoupling factors to recast the Schur complement and their inexact fully-parallel computation by means of restriction operators. We introduce two adaptive techniques aimed at building the restriction operators according to the properties of the system at hand. The proposed block preconditioner has been tested through an extensive experimentation on both synthetic and real-case applications, pointing out its robustness and computational efficiency.

Here we introduce a new reconstruction technique for two-dimensional Bragg Scattering Tomography (BST), based on the Radon transform models of [arXiv preprint, arXiv:2004.10961 (2020)]. Our method uses a combination of ideas from multibang control and microlocal analysis to construct an objective function which can regularize the BST artifacts; specifically the boundary artifacts due to sharp cutoff in sinogram space (as observed in [arXiv preprint, arXiv:2007.00208 (2020)]), and artifacts arising from approximations made in constructing the model used for inversion. We then test our algorithm in a variety of Monte Carlo (MC) simulated examples of practical interest in airport baggage screening and threat detection. The data used in our studies is generated with a novel Monte-Carlo code presented here. The model, which is available from the authors upon request, captures both the Bragg scatter effects described by BST as well as beam attenuation and Compton scatter.

In this article, we develop and analyse a new spectral method to solve the semi-classical Schrödinger equation based on the Gaussian wave-packet transform (GWPT) and Hagedorn's semi-classical wave-packets (HWP). The GWPT equivalently recasts the highly oscillatory wave equation as a much less oscillatory one (the w equation) coupled with a set of ordinary differential equations governing the dynamics of the so-called GWPT parameters. The Hamiltonian of the w equation consists of a quadratic part and a small non-quadratic perturbation, which is of order O( ε ??) , where ε?? is the rescaled Planck's constant. By expanding the solution of the w equation as a superposition of Hagedorn's wave-packets, we construct a spectral method while the O( ε ??) perturbation part is treated by the Galerkin approximation. This numerical implementation of the GWPT avoids imposing artificial boundary conditions and facilitates rigorous numerical analysis. For arbitrary dimensional cases, we establish how the error of solving the semi-classical Schrödinger equation with the GWPT is determined by the errors of solving the w equation and the GWPT parameters. We prove that this scheme has the spectral convergence with respect to the number of Hagedorn's wave-packets in one dimension. Extensive numerical tests are provided to demonstrate the properties of the proposed method.