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.     

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.     

The Hyperbolic Schrödinger Equation and the Quantum Lattice Boltzmann Approximation

The quantum lattice Boltzmann (qlB) algorithm solves the 1D Dirac equations and has been used to solve approximately the classical (i.e., non-relativistic) Schrödinger equation. We point out that the qlB method actually approximates the hyperbolic version of the non-relativistic Schrödinger equation, whose solution is thus obtained at the price of an additional small error. Such an error is of order of $(ω_c\tau)^{−1},$ where $ω_c:=\frac{mc^2}{h}$ is the Compton frequency, $ħ$ being the reduced Planck constant, $m$ the rest mass of the electrons, $c$ the speed of light, and $\tau$ a chosen reference time (i.e., 1 s), and hence it vanishes in the non-relativistic limit $c → +∞.$ This asymptotic result comes from a singular perturbation process which does not require any boundary layer and, consequently, the approximation holds uniformly, which fact is relevant in view of numerical approximations. We also discuss this occurrence more generally, for some classes of linear singularly perturbed partial differential equations.     

A Kernel-Independent Treecode Based on Barycentric Lagrange Interpolation

A kernel-independent treecode (KITC) is presented for fast summation of particle interactions. The method employs barycentric Lagrange interpolation at Chebyshev points to approximate well-separated particle-cluster interactions. The KITC requires only kernel evaluations, is suitable for non-oscillatory kernels, and relies on the scale-invariance property of barycentric Lagrange interpolation. For a given level of accuracy, the treecode reduces the operation count for pairwise interactions from $\mathcal{O}$($N^2$) to $\mathcal{O}$($N$log$N$), where $N$ is the number of particles in the system. The algorithm is demonstrated for systems of regularized Stokeslets and rotlets in 3D, and numerical results show the treecode performance in terms of error, CPU time, and memory consumption. The KITC is a relatively simple algorithm with low memory consumption, and this enables a straightforward OpenMP parallelization.     

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.     

A Fast Solver for an $\mathcal{H}_1$ Regularized PDE-Constrained Optimization Problem

In this paper we consider PDE-constrained optimization problems which incorporate an $\mathcal{H}_1$ regularization control term. We focus on a time-dependent PDE, and consider both distributed and boundary control. The problems we consider include bound constraints on the state, and we use a Moreau-Yosida penalty function to handle this. We propose Krylov solvers and Schur complement preconditioning strategies for the different problems and illustrate their performance with numerical examples.     

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.     

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.     

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.     

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.     

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.     

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 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.     

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.     

Optimal Error Estimates of Compact Finite Difference Discretizations for the Schrödinger-Poisson System

We study compact finite difference methods for the Schrödinger-Poisson equation in a bounded domain and establish their optimal error estimates under proper regularity assumptions on wave function $ψ$ and external potential $V(x)$. The Crank-Nicolson compact finite difference method and the semi-implicit compact finite difference method are both of order $\mathcal{O}$($h^4$+$τ^2$) in discrete $l^2$, $H^1$ and $l^∞$ norms with mesh size $h$ and time step $τ$. For the errors of compact finite difference approximation to the second derivative and Poisson potential are nonlocal, thus besides the standard energy method and mathematical induction method, the key technique in analysis is to estimate the nonlocal approximation errors in discrete $l^∞$ and $H^1$ norm by discrete maximum principle of elliptic equation and properties of some related matrix. Also some useful inequalities are established in this paper. Finally, extensive numerical results are reported to support our error estimates of the numerical methods.     

A Well-Conditioned Hypersingular Boundary Element Method for Electrostatic Potentials in the Presence of Inhomogeneities within Layered Media

In this paper, we will present a high-order, well-conditioned boundary element method (BEM) based on Müller's hypersingular second kind integral equation formulation to accurately compute electrostatic potentials in the presence of inhomogeneity embedded within layered media. We consider two types of inhomogeneities: the first one is a simple model of an ion channel which consists of a finite height cylindrical cavity embedded in a layered electrolytes/membrane environment, and the second one is a Janus particle made of two different semi-spherical dielectric materials. Both types of inhomogeneities have relevant applications in biology and colloidal material, respectively. The proposed BEM gives$\mathcal{O}$(1) condition numbers, allowing fast convergence of iterative solvers compared to previous work using first kind of integral equations. We also show that the second order basis converges faster and is more accurate than the first order basis for the BEM.     

On the Construction of Well-Conditioned Hierarchical Bases for $\mathcal{H}(div)$-Conforming $\mathbb{R}^n$ Simplicial Elements

Hierarchical bases of arbitrary order for $\mathcal{H}(div)$-conforming triangular and tetrahedral elements are constructed with the goal of improving the conditioning of the mass and stiffness matrices. For the basis with the triangular element, it is found numerically that the conditioning is acceptable up to the approximation of order four, and is better than a corresponding basis in the dissertation by Sabine Zaglmayr [High Order Finite Element Methods for Electromagnetic Field Computation, Johannes Kepler Universität, Linz, 2006]. The sparsity of the mass matrices from the newly constructed basis and from the one by Zaglmayr is similar for approximations up to order four. The stiffness matrix with the new basis is much sparser than that with the basis by Zaglmayr for approximations up to order four. For the tetrahedral element, it is identified numerically that the conditioning is acceptable only up to the approximation of order three. Compared with the newly constructed basis for the triangular element, the sparsity of the mass matrices from the basis for the tetrahedral element is relatively sparser.     

A Fast Time Splitting Finite Difference Approach to Gross–Pitaevskii Equations

We propose an idea to solve the Gross–Pitaevskii equation for dark structures inside an infinite constant background density $ρ_∞$=${|ψ_∞|}^2$, without the introduction of artificial boundary conditions. We map the unbounded physical domain $\mathbb{R}^3$ into the bounded domain ${(−1,1)}^3$ and discretize the rescaled equation by equispaced 4th-order finite differences. This results in a free boundary approach, which can be solved in time by the Strang splitting method. The linear part is solved by a new, fast approximation of the action of the matrix exponential at machine precision accuracy, while the nonlinear part can be solved exactly. Numerical results confirm existing ones based on the Fourier pseudospectral method and point out some weaknesses of the latter such as the need of a quite large computational domain, and thus a consequent critical computational effort, in order to provide reliable time evolution of the vortical structures, of their reconnections, and of integral quantities like mass, energy, and momentum. The free boundary approach reproduces them correctly, also in finite subdomains, at low computational cost. We show the versatility of this method by carrying out one- and three-dimensional simulations and by using it also in the case of Bose–Einstein condensates, for which $ψ$→0 as the spatial variables tend to infinity.     

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/$(NlnN)^\frac{2}{3}$).     

A proposed method for approximate estimates of the ion-activity products of calcium oxalate and calcium phosphate in spot-urine samples or in urine samples collected during less well defined periods of time

The purpose of this study is to derive approximate estimates of ion-activity products of CaOx (AP_CaOx) and CaP (AP_CaP) useful for spot urine or other less well defined short-term urine collections. In accordance with previously applied and described principles for estimating ion-activity products, the intention was to derive simplified estimates of AP(CaOx)_CONC and AP(CaP)_CONC by using urine concentrations of the most important determinants of AP_CaOx and AP_CaP. A comparison thus was made between estimates derived in that manner and AP(CaOx) index and AP(CaP) index obtained from calculations based on analysis of 24 h urine samples. The best fit between estimates of AP(CaOx) index in 24 h urine and AP(CaOx) index_CONC was obtained with the following formula (r = 0.99; p = 0.0000): ${\text{AP(CaOx)index}}_{\text{CONC}} = \frac{{2.09 \times c{Ca}^{0.84} \times c{Ox} }}{{c{Cit}^{0.22} \times c{Mg}^{0.12}}}. $ The corresponding formula for AP (CaP) index_CONC was derived from comparison with the corresponding calculations of AP(CaP) index in 24 h urine (r = 0.91; p = 0.0000): ${\text{AP(CaP)index}}_{\text{CONC}} = \frac{{0.432 \times c{Ca}^{1.07} \times c{P}^{0.70} \times ({pH} - 4.5)^{6.8}}}{{c{Cit}^{0.20} }}. $ The proposed simplified formulas enable calculation of approximate estimates of AP_CaOx and AP_CaP in spot-urine samples or any kind of urine collection for which duration of the collection period is less well known and by using the concentrations (c) of the variables in mmol/L in the two formulas.