Deep Network Approximation Characterized by Number of Neurons

This paper quantitatively characterizes the approximation power of deep feed-forward neural networks (FNNs) in terms of the number of neurons. It is shown by construction that ReLU FNNs with width$\mathcal{O}$(max{$d⌊N^{1/d}⌋$,$N$+1}) and depth $\mathcal{O}(L)$ can approximate an arbitrary Hölder continuous function of order $α∈(0,1]$ on $[0,1]^d$ with a nearly tight approximation rate $\mathcal{O}(\sqrt{d}N^{−2α/d}L^{−2α/d})$ measured in $L^p$ -norm for any $N,L∈\mathbb{N}^+$ and $p∈[1,∞]$. More generally for an arbitrary continuous function $f$ on $[0,1]^d$ with a modulus of continuity $ω_f (·)$, the constructive approximation rate is $\mathcal{O}(\sqrt{d}ω_f(N^{−2α/d}L^{−2α/d}))$. We also extend our analysis to $f$ on irregular domains or those localized in an ε-neighborhood of a $d_\mathcal{M}$-dimensional smooth manifold $\mathcal{M}⊆[0,1]^d$ with $d_\mathcal{M}≪d$. Especially, in the case of an essentially low-dimensional domain, we show an approximation rate $\mathcal{O}(ω_f(\frac{ε}{1−δ}\sqrt{\frac{d}{d_δ}}+ε)+\sqrt{d}ω_f(\frac{\sqrt{d}}{1−δ\sqrt{d_δ}}N^{−2α/d_δ}L^{−2α/d_δ})$ for ReLU FNNs to approximate $f$ in the $ε$-neighborhood, where $d_δ=\mathcal{O}(d_\mathcal{M}\frac{\rm{ln}(d/δ)}{δ^2})$ for any $δ∈(0,1)$ as a relative error for a projection to approximate an isometry when projecting $\mathcal{M}$ to a $d_δ$-dimensional domain.     

Directional $\mathcal{H}^2$ Compression Algorithm: Optimisations and Application to a Discontinuous Galerkin BEM for the Helmholtz Equation

This study aimed to specialise a directional $\mathcal{H}^2 (\mathcal{D}\mathcal{H}^2)$ compression to matrices arising from the discontinuous Galerkin (DG) discretisation of the hypersingular equation in acoustics. The significant finding is an algorithm that takes a DG stiffness matrix and finds a near-optimal $\mathcal{D}\mathcal{H}^2$ approximation for low and high-frequency problems. We introduced the necessary special optimisations to make this algorithm more efficient in the case of a DG stiffness matrix. Moreover, an automatic parameter tuning strategy makes it easy to use and versatile. Numerical comparisons with a classical Boundary Element Method (BEM) show that a DG scheme combined with a $\mathcal{D}\mathcal{H}^2$ gives better computational efficiency than a classical BEM in the case of high-order finite elements and $hp$ heterogeneous meshes. The results indicate that DG is suitable for an auto-adaptive context in integral equations.     

On the Effect of Ghost Force in the Quasicontinuum Method: Dynamic Problems in One Dimension

Numerical error caused by "ghost forces" in a quasicontinuum method is studied in the context of dynamic problems. The error in the discrete W^1,∞ norm is analyzed for the time scale $\mathcal{O}$($ε$) and the time scale $\mathcal{O}$(1) with ε being the lattice spacing.     

An Efficient Finite Element Method with Exponential Mesh Refinement for the Solution of the Allen-Cahn Equation in Non-Convex Polygons

In this paper we consider the numerical solution of the Allen-Cahn type diffuse interface model in a polygonal domain. The intersection of the interface with the re-entrant corners of the polygon causes strong corner singularities in the solution. To overcome the effect of these singularities on the accuracy of the approximate solution, for the spatial discretization we develop an efficient finite element method with exponential mesh refinement in the vicinity of the singular corners, that is based on ($k$−1)-th order Lagrange elements, $k$≥2 an integer. The problem is fully discretized by employing a first-order, semi-implicit time stepping scheme with the Invariant Energy Quadratization approach in time, which is an unconditionally energy stable method. It is shown that for the error between the exact and the approximate solution, an accuracy of $\mathcal{O}$($h^k$+$τ$) is attained in the $L^2$-norm for the number of $\mathcal{O}$($h^{−2}$ln$h^{−1}$) spatial elements, where $h$ and $τ$ are the mesh and time steps, respectively. The numerical results obtained support the analysis made.     

Random Batch Algorithms for Quantum Monte Carlo Simulations

Random batch algorithms are constructed for quantum Monte Carlo simulations. The main objective is to alleviate the computational cost associated with the calculations of two-body interactions, including the pairwise interactions in the potential energy, and the two-body terms in the Jastrow factor. In the framework of variational Monte Carlo methods, the random batch algorithm is constructed based on the over-damped Langevin dynamics, so that updating the position of each particle in an $N$-particle system only requires$\mathcal{O}(1)$ operations, thus for each time step the computational cost for $N$ particles is reduced from$\mathcal{O}(N^2)$ to$\mathcal{O}(N)$. For diffusion Monte Carlo methods, the random batch algorithm uses an energy decomposition to avoid the computation of the total energy in the branching step. The effectiveness of the random batch method is demonstrated using a system of liquid $^4$He atoms interacting with a graphite surface.     

Fast One-Dimensional Convolution with General Kernels Using Sum-of-Exponential Approximation

Based on the recently-developed sum-of-exponential (SOE) approximation, in this article, we propose a fast algorithm to evaluate the one-dimensional convolution potential $φ(x)=K∗ρ=∫^1_{0}K(x−y)ρ(y)dy$ at (non)uniformly distributed target grid points {$x_i$}$^M_{i=1}$, where the kernel $K(x)$ might be singular at the origin and the source density function $ρ(x)$ is given on a source grid ${{{y_i}}}^N_{j=1}$ which can be different from the target grid. It achieves an optimal accuracy, inherited from the interpolation of the density $ρ(x)$, within $\mathcal{O}(M+N)$ operations. Using the kernel's SOE approximation $K_{ES}$, the potential is split into two integrals: the exponential convolution $φ_{ES}$=$K_{ES}∗ρ$ and the local correction integral $φ_{cor}=(K−K_{ES})∗ρ$. The exponential convolution is evaluated via the recurrence formula that is typical of the exponential function. The local correction integral is restricted to a small neighborhood of the target point where the kernel singularity is considered. Rigorous estimates of the optimal accuracy are provided. The algorithm is ideal for parallelization and favors easy extensions to complicated kernels. Extensive numerical results for different kernels are presented.     

PVFMM: A Parallel Kernel Independent FMM for Particle and Volume Potentials

We describe our implementation of a parallel fast multipole method for evaluating potentials for discrete and continuous source distributions. The first requires summation over the source points and the second requiring integration over a continuous source density. Both problems require$\mathcal{O}$($N^2$) complexity when computed directly; however, can be accelerated to $\mathcal{O}$($N$) time using FMM. In our PVFMM software library, we use kernel independent FMM and this allows us to compute potentials for a wide range of elliptic kernels. Our method is high order, adaptive and scalable. In this paper, we discuss several algorithmic improvements and performance optimizations including cache locality, vectorization, shared memory parallelism and use of coprocessors. Our distributed memory implementation uses space-filling curve for partitioning data and a hypercube communication scheme. We present convergence results for Laplace, Stokes and Helmholtz (low wavenumber) kernels for both particle and volume FMM. We measure efficiency of our method in terms of CPU cycles per unknown for different accuracies and different kernels. We also demonstrate scalability of our implementation up to several thousand processor cores on the Stampede platform at the Texas Advanced Computing Center.     

Efficient Algorithm for Many-Electron Angular Momentum and Spin Diagonalization on Atomic Subshells

We devise an efficient algorithm for the symbolic calculation of irreducible angular momentum and spin (LS) eigenspaces within the $n$-fold antisymmetrized tensor product $Λ^n$$V_u$, where n is the number of electrons and $u$ = s,p,d,··· denotes the atomic subshell. This is an essential step for dimension reduction in configuration-interaction (CI) methods applied to atomic many-electron quantum systems. The algorithm relies on the observation that each $L_z$ eigenstate with maximal eigenvalue is also an $L^2$ eigenstate (equivalently for $S_z$ and $S^2$ ), as well as the traversal of LS eigenstates using the lowering operators $L_−$ and $S_−$. Iterative application to the remaining states in $Λ^n$$V_u$ leads to an implicit simultaneous diagonalization. A detailed complexity analysis for fixed $n$ and increasing subshell number $u$ yields run time$\mathcal{O}$($u^{3n−2}$). A symbolic computer algebra implementation is available online.     

Numerical Solution of 3D Poisson-Nernst-Planck Equations Coupled with Classical Density Functional Theory for Modeling Ion and Electron Transport in a Confined Environment

We have developed efficient numerical algorithms for solving 3D steady-state Poisson-Nernst-Planck (PNP) equations with excess chemical potentials described by the classical density functional theory (cDFT). The coupled PNP equations are discretized by a finite difference scheme and solved iteratively using the Gummel method with relaxation. The Nernst-Planck equations are transformed into Laplace equations through the Slotboom transformation. Then, the algebraic multigrid method is applied to efficiently solve the Poisson equation and the transformed Nernst-Planck equations. A novel strategy for calculating excess chemical potentials through fast Fourier transforms is proposed, which reduces computational complexity from $\mathcal{O}$($N^2$) to $\mathcal{O}$($NlogN$), where $N$ is the number of grid points. Integrals involving the Dirac delta function are evaluated directly by coordinate transformation, which yields more accurate results compared to applying numerical quadrature to an approximated delta function. Numerical results for ion and electron transport in solid electrolyte for lithium-ion (Li-ion) batteries are shown to be in good agreement with the experimental data and the results from previous studies.     

Solving Maxwell's Equation in Meta-Materials by a CG-DG Method

In this paper, an approach combining the DG method in space with CG method in time (CG-DG method) is developed to solve time-dependent Maxwell's equations when meta-materials are involved. Both the unconditional $L^2$-stability and error estimate of order $\mathcal{O}$($τ^ {r+1}$+$h^{k+\frac{1}{2}}$) are obtained when polynomials of degree at most r is used for the temporal discretization and at most k for the spatial discretization. Numerical results in 3D are given to validate the theoretical results.     

Laparoscopic surgery in children is associated with an intraoperative hypermetabolic response

Background Laparoscopic surgery is thought to be associated with a reduced metabolic response compared to open surgery. Oxygen consumption ( ) and energy metabolism during laparoscopic surgery have not been characterized in children. Methods We measured respiratory gas exchange intraoperatively in children undergoing 19 open and 20 laparoscopic procedures. Premature infants and patients with metabolic, renal, and cardiac abnormalities were excluded. Anesthesia was standardized. Unheated carbon dioxide was used for insufflation. was measured by indirect calorimetry. Core temperature was measured using an esophageal temperature probe. Results We found a steady increase in during laparoscopy. The increase in was more marked in younger children and was associated with a significant rise in core temperature. Open surgery was not associated with significant changes in core temperature or . Conclusions Laparoscopy in children is associated with an intraoperative hypermetabolic response characterized by increased oxygen consumption and core temperature. These changes are more marked in younger children. M. C. McHoney and L. Corizia contributed equally to the study, analysis, and writing of the paper     

Finite Neuron Method and Convergence Analysis

We study a family of $H^m$-conforming piecewise polynomials based on the artificial neural network, referred to as the finite neuron method (FNM), for numerical solution of $2m$-th-order partial differential equations in$\mathbb{R}^d$ for any $m,d≥1$ and then provide convergence analysis for this method. Given a general domain Ω$⊂\mathbb{R}^d$ and a partition$\mathcal{T}_h$ of Ω, it is still an open problem in general how to construct a conforming finite element subspace of $H^m$(Ω) that has adequate approximation properties. By using techniques from artificial neural networks, we construct a family of $H^m$-conforming functions consisting of piecewise polynomials of degree $k$ for any $k≥m$ and we further obtain the error estimate when they are applied to solve the elliptic boundary value problem of any order in any dimension. For example, the error estimates that $‖u−u_N‖_{H^m(\rm{Ω})}=\mathcal{O}(N^{−\frac{1}{2}−\frac{1}{d}})$ is obtained for the error between the exact solution $u$ and the finite neuron approximation $u_N$. A discussion is also provided on the difference and relationship between the finite neuron method and finite element methods (FEM). For example, for the finite neuron method, the underlying finite element grids are not given a priori and the discrete solution can be obtained by only solving a non-linear and non-convex optimization problem. Despite the many desirable theoretical properties of the finite neuron method analyzed in the paper, its practical value requires further investigation as the aforementioned underlying non-linear and non-convex optimization problem can be expensive and challenging to solve. For completeness and the convenience of the reader, some basic known results and their proofs are introduced.     

Space-Time Discontinuous Galerkin Method for Maxwell's Equations

A fully discrete discontinuous Galerkin method is introduced for solving time-dependent Maxwell's equations. Distinguished from the Runge-Kutta discontinuous Galerkin method (RKDG) and the finite element time domain method (FETD), in our scheme, discontinuous Galerkin methods are used to discretize not only the spatial domain but also the temporal domain. The proposed numerical scheme is proved to be unconditionally stable, and a convergent rate $\mathcal{O}((∆t)^{r+1}+h^{k+1/2})$ is established under the $L^2$ -norm when polynomials of degree at most $r$ and $k$ are used for temporal and spatial approximation, respectively. Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction. An ultra-convergence of order $(∆t)^{2r+1}$ in time step is observed numerically for the numerical fluxes w.r.t. temporal variable at the grid points.     

A Comparison of Higher-Order Weak Numerical Schemes for Stopped Stochastic Differential Equations

We review, implement, and compare numerical integration schemes for spatially bounded diffusions stopped at the boundary which possess a convergence rate of the discretization error with respect to the time step $h$ higher than $\mathcal{O}$$(√h)$. We address specific implementation issues of the most general-purpose of such schemes. They have been coded into a single Matlab program and compared, according to their accuracy and computational cost, on a wide range of problems in up to R⁴⁸. The paper is self-contained and the code will be made freely downloadable.     

Correlation Functions,Universal Ratios and Goldstone Mode Singularities in $n$-Vector Models

Correlation functions in the $\mathcal{O}$$(n)$ models below the critical temperature are considered. Based on Monte Carlo (MC) data, we confirm the fact stated earlier by Engels and Vogt, that the transverse two-plane correlation function of the $\mathcal{O}$$(4)$ model for lattice sizes about $L=120$ and small external fields $h$ is very well described by a Gaussian approximation. However, we show that fits of not lower quality are provided by certain non-Gaussian approximation. We have also tested larger lattice sizes, up to $L=512$. The Fourier-transformed transverse and longitudinal two-point correlation functions have Goldstone mode singularities in the thermodynamic limit at $k→0$ and $h=+0$, i.e., $G_⊥$(k)$≃ak^{−λ_⊥}$ and$G_{||}($k$)≃bk^{−λ_{||}}$, respectively. Here $a$ and $b$ are the amplitudes, $k$=|k| is the magnitude of the wave vector k. The exponents $λ_⊥$, $λ_{||}$ and the ratio $bM^2/a^2$, where $M$ is the spontaneous magnetization, are universal according to the GFD (grouping of Feynman diagrams) approach. Here we find that the universality follows also from the standard (Gaussian) theory, yielding $bM^2/a^2$=$(n−1)/16$. Our MC estimates of this ratio are $0.06±0.01$ for $n=2$, $0.17±0.01$ for $n=4$ and $0.498±0.010$ for $n=10$. According to these and our earlier MC results, the asymptotic behavior and Goldstone mode singularities are not exactly described by the standard theory. This is expected from the GFD theory. We have found appropriate analytic approximations for $G_⊥$(k) and $G_{||}$(k), well fitting the simulation data for small $k$. We have used them to test the Patashinski-Pokrovski relation and have found that it holds approximately.     

Comparison of mechanical work and metabolic energy consumption during normal gait

The validity of using mechanical measures of work to indicate the metabolic energy consumption during normal gait was examined. These mechanical measures were (a) mechanical work done on the center of mass per kilogram body mass per second (\documentclass{article}\pagestyle{empty}\begin{document}$ \dot W_{{\rm cm}} $\end{document}), calculated by integration of ground reaction forces measured by force platforms; (b) total body segmental work per kilogram body mass per second (\documentclass{article}\pagestyle{empty}\begin{document}$ \dot W_{{\rm seg}} $\end{document}), calculated from individual body segment energies measured by motion analysis; and (c) the sum of the normalized absolute moment impulses per second acting on the joints of the lower extremities (\documentclass{article}\pagestyle{empty}\begin{document}$ \dot M $\end{document}), calculated from both force and motion data. The metabolic energy consumption, determined by analysis of expired air, and the three mechanical measures of work were calculated for six normal subjects walking at five speeds. Each measure of mechanical work per second walked was highly correlated with metabolic energy consumption/kg · s (r = 0.89 for W_cm, r = 0.79 for \documentclass{article}\pagestyle{empty}\begin{document}$ \dot W_{{\rm seg}} $\end{document}, and r = 0.85 for M), but a poorer correlation was found between each measure of mechanical work per meter walked and net metabolic energy consumption/kg. m (r = 0.54 for W_cm, r = 0.28 for W_seg, and r = 0.03 for M). These mechanical parameters, particularly when measured per time, may be useful in comparing metabolic energy consumption between individuals or between different walking conditions for the same individual.     

Efficient Implementation of Smoothed Particle Hydrodynamics (SPH) with Plane Sweep Algorithm

Neighbour search (NS) is the core of any implementations of smoothed particle hydrodynamics (SPH). In this paper, we present an efficient$\mathcal{O}$($N$log$N$) neighbour search method based on the plane sweep (PW) algorithm with $N$ being the number of SPH particles. The resulting method, dubbed the PWNS method, is totally independent of grids (i.e., purely meshfree) and capable of treating variable smoothing length, arbitrary particle distribution and heterogenous kernels. Several state-of-the-art data structures and algorithms, e.g., the segment tree and the Morton code, are optimized and implemented. By simply allowing multiple lines to sweep the SPH particles simultaneously from different initial positions, a parallelization of the PWNS method with satisfactory speedup and load-balancing can be easily achieved. That is, the PWNS SPH solver has a great potential for large scale fluid dynamics simulations.     

A New Family of Nonconforming Elements with $H$(curl)-Continuity for the 3D Quad-Curl Problem

We propose and analyze a new family of nonconforming finite elements for the three-dimensional quad-curl problem. The proposed finite element spaces are subspaces of $\boldsymbol{H}$(curl), but not of $\boldsymbol{H}$(grad curl), which are different from the existing nonconforming ones [10,12,13]. The well-posedness of the discrete problem is proved and optimal error estimates in discrete $\boldsymbol{H}$(grad curl) norm, $\boldsymbol{H}$(curl) norm and $L^2$ norm are derived. Numerical experiments are provided to illustrate the good performance of the method and confirm our theoretical predictions.     

An Adaptive Finite Element Method with Hybrid Basis for Singularly Perturbed Nonlinear Eigenvalue Problems

In this paper, we propose a uniformly convergent adaptive finite element method with hybrid basis (AFEM-HB) for the discretization of singularly perturbed nonlinear eigenvalue problems under constraints with applications in Bose-Einstein condensation (BEC) and quantum chemistry. We begin with the time-independent Gross-Pitaevskii equation and show how to reformulate it into a singularly perturbed nonlinear eigenvalue problem under a constraint. Matched asymptotic approximations for the problem are reviewed to confirm the asymptotic behaviors of the solutions in the boundary/interior layer regions. By using the normalized gradient flow, we propose an adaptive finite element with hybrid basis to solve the singularly perturbed nonlinear eigenvalue problem. Our basis functions and the mesh are chosen adaptively to the small parameter ε. Extensive numerical results are reported to show the uniform convergence property of our method. We also apply the AFEM-HB to compute the ground and excited states of BEC with box/harmonic/optical lattice potential in the semiclassical regime (0<ε≪1). In addition, we give a detailed error analysis of our AFEM-HB to a simpler singularly perturbed two point boundary value problem, show that our method has a minimum uniform convergence order $\mathcal{O}$(1/$(NlnN)^\frac{2}{3}$).