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.     

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.     

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.     

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.     

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.     

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.     

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.     

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/$(NlnN)^\frac{2}{3}$).     

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.     

An Adaptive Combined Preconditioner with Applications in Radiation Diffusion Equations

The paper aims to develop an effective preconditioner and conduct the convergence analysis of the corresponding preconditioned GMRES for the solution of discrete problems originating from multi-group radiation diffusion equations. We firstly investigate the performances of the most widely used preconditioners (ILU(k) and AMG) and their combinations ($B_{co}$ and$\widetilde{B}_{co}$), and provide drawbacks on their feasibilities. Secondly, we reveal the underlying complementarity of ILU(k) and AMG by analyzing the features suitable for AMG using more detailed measurements on multiscale nature of matrices and the effect of ILU(k) on multiscale nature. Moreover, we present an adaptive combined preconditioner $B^α_{co}$ involving an improved ILU(0) along with its convergence constraints. Numerical results demonstrate that $B^α_{co}$-GMRES holds the best robustness and efficiency. At last, we analyze the convergence of GMRES with combined preconditioning which not only provides a persuasive support for our proposed algorithms, but also updates the existing estimation theory on condition numbers of combined preconditioned systems.     

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     

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.     

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.     

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.     

Transposition of the posterior tibial tendon in spastic equinovarus

Summary Eleven posterior tibial tendon transfers were performed in eight patients with spastic equinovarus with an average follow-up of 3 years (range years). The main goals—to eliminate the need for braces and to improve gait—were achieved in all patients. An adequate operation technique kept the well-known postoperative complications, e.g., valgus and calcaneal deformity, to a minimum.

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Zusammenfassung Die Transposition der Musc. tibialis posterior-Sehne als operative Behandlung des spastischen Equinovarus wurde elfmal durchgeführt bei acht Patienten. Die Ergebnisse wurden Jahre nach der Operation kontrolliert. Das Hauptziel: Unabhängigkeit von Orthesen and Verbesserung der Gehfähigkeit, konnte bei allen Patienten erreicht werden. Eine adequate Operationstechnik konnte die bekannten postoperativen Komplikationen, wie eine Valgus- und Hackenfuß-Deformität bis auf ein Minimum beschränken.

     

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.     

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.     

Early diastolic function during exertion influences exercise intolerance in patients with hypertrophic cardiomyopathy

Background

Hypertrophic cardiomyopathy (HCM) patients with preserved left ventricular ejection fraction (LVEF) often develop dyspnea and exercise intolerance. Diastolic dysfunction may contribute to exercise intolerance in these patients. This study aimed to clarify our hypothesis as to whether diastolic function rather than systolic function would be associated with exercise intolerance in HCM using two-dimensional (2D) speckle tracking echocardiography during exercise.

Methods

Thirty-three HCM patients (mean age 59.3 ± 15.7 years) underwent 2D speckle tracking echocardiography at rest and during submaximal semi-supine bicycle exercise. Global longitudinal strain (LS), LS rate during systole (LSRs), early diastole (LSRe), and late diastole (LSRa) were measured. The symptom-limited cardiopulmonary exercise testing was performed using a cycle ergometer for measuring the peak oxygen consumption (peak $ \dot{V}_{{{\text{O}}_{2} }} $ ).

Results

In the multivariate linear regression analysis, peak $ \dot{V}_{{{\text{O}}_{2} }} $ did not associate with strain or strain rate at rest. However, peak $ \dot{V}_{{{\text{O}}_{2} }} $ correlated with LS (β = ?0.403, p = 0.007), LSRe (β = 6.041, p = 0.001), and LSRa (β = 5.117, p = 0.021) during exercise after adjustment for age, gender, and heart rate. The first quartile peak $ \dot{V}_{{{\text{O}}_{2} }} $ (14.2 mL/min/kg) was assessed to predict exercise intolerance. The C-statistic of delta LSRe was 0.74, which was relatively greater than that of delta LS (0.70) and delta LSRa (0.58), indicating that early diastolic function rather than systolic and late diastolic function affects exercise intolerance.

Conclusions

LSRe during exercise is closely associated with the peak $ \dot{V}_{{{\text{O}}_{2} }} $ . Early diastolic function during exercise is an important determinant of exercise capacity in patients with HCM.     

A Mixed Finite Element Scheme for Biharmonic Equation with Variable Coefficient and von Kármán Equations

In this paper, a new mixed finite element scheme using element-wise stabilization is introduced for the biharmonic equation with variable coefficient on Lipschitz polyhedral domains. The proposed scheme doesn’t involve any integration along mesh interfaces. The gradient of the solution is approximated by $H$(div)-conforming $BDM_{k+1}$ element or vector valued Lagrange element with order $k+1,$ while the solution is approximated by Lagrange element with order $k+2$ for any $k≥0.$ This scheme can be easily implemented and produces symmetric and positive definite linear system. We provide a new discrete $H^2$-norm stability, which is useful not only in analysis of this scheme but also in ${\rm C}^0$ interior penalty methods and DG methods. Optimal convergences in both discrete $H^2$-norm and $L^2$-norm are derived. This scheme with its analysis is further generalized to the von Kármán equations. Finally, numerical results verifying the theoretical estimates of the proposed algorithms are also presented.     

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.