Suboptimal Convergence Research Articles

High order VEM on curved domains

We deal with the virtual element method (VEM) for solving the Poisson equation on a domain \Omega with curved boundary. Given a polygonal approximation \Omega_h of the domain \Omega , the standard order m VEM [6], for m increasing, leads to a suboptimal convergence rate. We adapt the approach of [14] to VEM and we prove that an optimal convergence rate can be achieved by using a suitable correction depending on high order normal derivatives of the discrete solution at the boundary edges of \Omega_h , which, to retain computability, is evaluated after applying the projector \Pi^{\nabla} onto the space of polynomials. Numerical experiments confirm the theory.

Performance of a Deep Learning Algorithm for Automated Segmentation and Quantification of Traumatic Pelvic Hematomas on CT.

The volume of pelvic hematoma at CT has been shown to be the strongest independent predictor of major arterial injury requiring angioembolization in trauma victims with pelvic fractures, and also correlates with transfusion requirement and mortality. Measurement of pelvic hematomas (unopacified extraperitoneal blood accumulated from time of injury) using semi-automated seeded region growing is time-consuming and requires trained experts, precluding routine measurement at the point of care. Pelvic hematomas are markedly variable in shape and location, have irregular ill-defined margins, have low contrast with respect to viscera and muscle, and reside within anatomically distorted pelvises. Furthermore, pelvic hematomas occupy a small proportion of the entire volume of a chest, abdomen, and pelvis (C/A/P) trauma CT. The challenges are many, and no automated methods for segmentation and volumetric analysis have been described to date. Traditional approaches using fully convolutional networks result in coarse segmentations and class imbalance with suboptimal convergence. In this study, we implement a modified coarse-to-fine deep learning approach-the Recurrent Saliency Transformation Network (RSTN) for pelvic hematoma volume segmentation. RSTN previously yielded excellent results in pancreas segmentation, where low contrast with adjacent structures, small target volume, variable location, and fine contours are also problematic. We have curated a unique single-institution corpus of 253 C/A/P admission trauma CT studies in patients with bleeding pelvic fractures with manually labeled pelvic hematomas. We hypothesized that RSTN would result in sufficiently high Dice similarity coefficients to facilitate accurate and objective volumetric measurements for outcome prediction (arterial injury requiring angioembolization). Cases were separated into five combinations of training and test sets in an 80/20 split and fivefold cross-validation was performed. Dice scores in the test set were 0.71 (SD ± 0.10) using RSTN, compared to 0.49 (SD ± 0.16) using a baseline Deep Learning Tool Kit (DLTK) reference 3D U-Net architecture. Mean inference segmentation time for RSTN was 0.90min (± 0.26). Pearson correlation between predicted and manual labels was 0.95 with p< 0.0001. Measurement bias was within 10mL. AUC of hematoma volumes for predicting need for angioembolization was 0.81 (predicted) versus 0.80 (manual). Qualitatively, predicted labels closely followed hematoma contours and avoided muscle and displaced viscera. Further work will involve validation using a federated dataset and incorporation into a predictive model using multiple segmented features.

Interior penalty discontinuous Galerkin FEMs for a gradient beam and CNTs

Carbon nanotubes (CNTs) are widely employed to modern engineering applications. The structural beam element is an acceptable model that has been used to simulate the CNTs' response. At the same time the gradient elasticity theory, taking into account the size effect phenomena, can be considered a viable option to study the mechanical response of CNTs through a generalized gradient beam model. Given that the constitutive equations of this theory produce a boundary layer, the hp-version interior penalty discontinuous Galerkin finite element methods (IPDGFEMs) are developed in this work for the solution of a static, tensile or compressive, gradient elastic beam in macro- and micro-structure (CNT), respectively. An a priori error analysis of the aforementioned methods is performed and numerical simulations that verify the analytical findings are carried out. The consistency, the stability and the convergence of the IPDGFEMs are proved. A priori error estimates established also indicate an optimal convergence in the meshsize h, yet a slightly suboptimal convergence in the polynomial degree p.

Spatial Pyramid-Enhanced NetVLAD With Weighted Triplet Loss for Place Recognition.

We propose an end-to-end place recognition model based on a novel deep neural network. First, we propose to exploit the spatial pyramid structure of the images to enhance the vector of locally aggregated descriptors (VLAD) such that the enhanced VLAD features can reflect the structural information of the images. To encode this feature extraction into the deep learning method, we build a spatial pyramid-enhanced VLAD (SPE-VLAD) layer. Next, we impose weight constraints on the terms of the traditional triplet loss (T-loss) function such that the weighted T-loss (WT-loss) function avoids the suboptimal convergence of the learning process. The loss function can work well under weakly supervised scenarios in that it determines the semantically positive and negative samples of each query through not only the GPS tags but also the Euclidean distance between the image representations. The SPE-VLAD layer and the WT-loss layer are integrated with the VGG-16 network or ResNet-18 network to form a novel end-to-end deep neural network that can be easily trained via the standard backpropagation method. We conduct experiments on three benchmark data sets, and the results demonstrate that the proposed model defeats the state-of-the-art deep learning approaches applied to place recognition.

Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws on moving simplex meshes

In Klingenberg, Schn\"ucke and Xia (Math. Comp. 86 (2017), 1203-1232) an arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method to solve conservation laws has been developed and analyzed. In this paper, the ALE-DG method will be extended to several dimensions. The method will be designed for simplex meshes. This will ensure that the method satisfies the geometric conservation law, if the accuracy of the time integrator is not less than the value of the spatial dimension. For the semi-discrete method the L2-stability will be proven. Furthermore, an error estimate which provides the suboptimal (k+1/2) convergence with respect to the L-infinity-norm will be presented, when an arbitrary monotone flux is used and for each cell the approximating functions are given by polynomials of degree $k$. The two dimensional fully-discrete explicit method will be combined with the bound preserving limiter developed by Zhang, Xia and Shu in (J. Sci. Comput. 50 (2012), 29-62). This limiter does not affect the high order accuracy of a numerical method. Then, for the ALE-DG method revised by the limiter the validity of a discrete maximum principle will be proven. The numerical stability, robustness and accuracy of the method will be shown by a variety of two dimensional computational experiments on moving triangular meshes.

Complete expected improvement converges to an optimal budget allocation

AbstractThe ranking and selection problem is a well-known mathematical framework for the formal study of optimal information collection. Expected improvement (EI) is a leading algorithmic approach to this problem; the practical benefits of EI have repeatedly been demonstrated in the literature, especially in the widely studied setting of Gaussian sampling distributions. However, it was recently proved that some of the most well-known EI-type methods achieve suboptimal convergence rates. We investigate a recently proposed variant of EI (known as ‘complete EI’) and prove that, with some minor modifications, it can be made to converge to the rate-optimal static budget allocation without requiring any tuning.

A mixed DG method and an HDG method for incompressible magnetohydrodynamics

Abstract In this paper we propose and analyze a mixed discontinuous Galerkin (DG) method and an hybridizable DG (HDG) method for the stationary magnetohydrodynamics (MHD) equations with two types of boundary (or constraint) conditions. The mixed DG method is based on a recent work proposed by Houston et al. (2009, A mixed DG method for linearized incompressible magnetohydrodynamics. J. Sci. Comput., 40, 281–314) for the linearized MHD. With two novel discrete Sobolev embedding type estimates for the discontinuous polynomials, we provide a priori error estimates for the method on the nonlinear MHD equations. In the smooth case we have optimal convergence rate for the velocity, magnetic field and pressure in the energy norm; the Lagrange multiplier only has suboptimal convergence order. With the minimal regularity assumption on the exact solution, the approximation is optimal for all unknowns. To the best of our knowledge, these are the first a priori error estimates for DG methods for the nonlinear MHD equations. In addition, we also propose and analyze the first divergence-free HDG method for the problem with several unique features comparing with the mixed DG method.

A simple finite element method for elliptic bulk problems with embedded surfaces

In this paper, we develop a simple finite element method for simulation of embedded layers of high permeability in a matrix of lower permeability using a basic model of Darcy flow in embedded cracks. The cracks are allowed to cut through the mesh in arbitrary fashion and we take the flow in the crack into account by superposition. The fact that we use continuous elements leads to suboptimal convergence due to the loss of regularity across the crack. We therefore refine the mesh in the vicinity of the crack in order to recover optimal order convergence in terms of the global mesh parameter. The proper degree of refinement is determined based on an a priori error estimate and can thus be performed before the actual finite element computation is started. Numerical examples showing this effect and confirming the theoretical results are provided. The approach is easy to implement and beneficial for rapid assessment of the effect of crack orientation and may for example be used in an optimization loop.

Convergence of $$\text{ dG(1) }$$ dG(1) in elastoplastic evolution

The discontinuous Galerkin (dG) methodology provides a hierarchy of time discretization schemes for evolutionary problems such as elastoplasticity with the Prandtl-Reus flow rule. A dG time discretization has been proposed for a variational inequality in the context of rate-independent inelastic material behaviour in Alberty and Carstensen in (CMME 191:4949–4968, 2002) with the help of duality in convex analysis to justify certain jump terms. This paper establishes the first a priori error analysis for the dG(1) scheme with discontinuous piecewise linear polynomials in the temporal and lowest-order finite elements for the spatial discretization. Compared to a generalized mid-point rule, the dG(1) formulation distributes the action of the material law in the form of the variational inequality in time and so it introduces an error in the material law. This may result in a suboptimal convergence rate for the dG(1) scheme and this paper shows that the stress error in the $$L^\infty (L^2)$$ norm is merely $$O(h+k^{3/2})$$ based on a seemingly sharp error analysis. The numerical investigation for a benchmark problem with known analytic solution provides empirical evidence of a higher convergence rate of the dG(1) scheme compared to dG(0).

An effective framework for asynchronous incremental graph processing

Although many graph processing systems have been proposed, graphs in the real-world are often dynamic. It is important to keep the results of graph computation up-to-date. Incremental computation is demonstrated to be an efficient solution to update calculated results. Recently, many incremental graph processing systems have been proposed to handle dynamic graphs in an asynchronous way and are able to achieve better performance than those processed in a synchronous way. However, these solutions still suffer from suboptimal convergence speed due to their slow propagation of important vertex state (important to convergence speed) and poor locality. In order to solve these problems, we propose a novel graph processing framework. It introduces a dynamic partition method to gather the important vertices for high locality, and then uses a priority-based scheduling algorithm to assign them with a higher priority for an effective processing order. By such means, it is able to reduce the number of updates and increase the locality, thereby reducing the convergence time. Experimental results show that our method reduces the number of updates by 30%, and reduces the total execution time by 35%, compared with state-of-the-art systems.

Uniform boundary estimates in homogenization of higher-order elliptic systems

This paper focuses on uniform boundary estimates in homogenization of a family of higher-order elliptic operators $$\mathcal {L}_\varepsilon $$ , with rapidly oscillating periodic coefficients. We derive uniform boundary $$C^{m-1,\lambda } (0\!<\!\lambda \!<\!1)$$ and $$ W^{m,p}$$ estimates in $$C^1$$ domains, as well as uniform boundary $$C^{m-1,1}$$ estimate in $$C^{1,\theta } (0\!<\!\theta \!<\!1)$$ domains without the symmetry assumption on the operator. The proof, motivated by the works “Armstrong and Smart in Ann Sci Ec Norm Super (4) 49(2):423–481 (2016) and Shen in Anal PDE 8(7):1565–1601 (2015),” is based on a suboptimal convergence rate in $$H^{m-1}(\Omega )$$ . Compared to “Kenig et al. in Arch Ration Mech Anal 203(3):1009–1036 (2012) and Shen (2015),” the convergence rate obtained here does not require the symmetry assumption on the operator, nor additional assumptions on the regularity of $$u_0$$ (the solution to the homogenized problem), and thus might be of some independent interests even for second-order elliptic systems.

Discontinuous Galerkin methods for Maxwell’s equations in Drude metamaterials on unstructured meshes

In this follow-up work, we extend the discontinuous Galerkin (DG) methods previously developed on rectangular meshes (Li et al., 2017) to triangular meshes. The DG schemes in Li et al. (2017) are both optimally convergent and energy conserving. However, as we shall see in the numerical results section, the DG schemes on triangular meshes only have suboptimal convergence rate. We prove the energy conservation and an error estimate for the semi-discrete schemes. The stability of the fully discrete scheme is proved and its error estimate is stated. We present extensive numerical results with convergence consistent of our error estimate, and simulations of wave propagation in Drude metamaterials to demonstrate the flexibility of triangular meshes.

Adaptive thresholding for large volatility matrix estimation based on high-frequency financial data

Universal thresholding methods have been developed to estimate the large sparse integrated volatility matrix of underlying assets based on high-frequency financial data. Since the integrated volatility matrix often has entries with a wide range of variability, universal thresholding estimators do not take the varying entries into consideration and may have unsatisfactory performances. This paper investigates adaptive thresholding estimation of the large integrated volatility matrix. We first construct an estimator for the asymptotic variance of the pre-averaging realized volatility estimator and then use the two estimators to develop an adaptive thresholding estimator of the large volatility matrix. It is shown that the adaptive thresholding estimator can achieve the optimal rate of convergence over the class of the sparse integrated volatility matrix when both the number of assets and sample size are allowed to go to infinity, while the universal thresholding estimator can achieve only the sub-optimal convergence rate. Also we discuss how to harness the adaptive thresholding scheme in the approximate factor model. The simulation study is conducted to check the finite sample performance of the adaptive thresholding estimators.

A linear smoothed higher-order CS-FEM for the analysis of notched laminated composites

Higher-order elements with highly accurate solutions are attractive for stress analysis and stress concentration problems. However, the distorted eight-node serendipity quadrilateral element is known to yield inaccurate results and sub-optimal convergence rate. In this paper, we present a higher-order CS-FEM to alleviate the effect of distorted mesh and guarantee the quality of solutions by employing a linear smoothing technique over eight-node quadratic serendipity elements. The modified strain matrix is computed by the divergence theorem between the nodal shape functions and their derivatives using Taylor's expansion of the weak form. The proposed method eliminates the need for isoparametric mapping and numerical studies demonstrate that the proposed method is insensitive to mesh distortion. The improved accuracy and superior convergence rates are numerically demonstrated with a few benchmark problems. The analysis of the stress concentration around cutouts also proves that the present method has good performance for the laminated composites.

Local discontinuous Galerkin methods for parabolic interface problems with homogeneous and non-homogeneous jump conditions

In this paper, we present the local discontinuous Galerkin (LDG) methods for solving second order parabolic interface problems in two-dimensional convex polygonal domains with homogeneous and non-homogeneous jump conditions, respectively. We start analyzing from the homogeneous problems. After giving the semi discrete LDG method, we derive its stability and a prior error estimate. Then, after a series of derivation, we obtain the LDG method for the non-homogeneous problems, which takes the same formulation as the homogeneous case with a special choice of numerical fluxes that incorporate the jump conditions across interface. So its analysis can be easily obtained within the framework developed for the homogeneous case. It is an advantage of the LDG method that it provides a natural framework to enforce the discontinuities in both the potential and the flux weakly in the discrete formulation provided that the triangulation of the domain is fitted to the interface. Numerical experiments are conducted to confirm our theoretical analysis with the help of the explicit temporal discretization and show that the LDG methods have the optimal and suboptimal convergence rate in the L2 norms for the potential and the flux, respectively. In addition, they also indicate that the LDG methods possess the same convergence properties in the L∞ norms. What is more, to avoid the strict time-step restriction of explicit schemes, the implicit integration factor (IIF) method is employed, which not only relaxes the time step to Δt=O(hmin) but also remains the important property of LDG method that the computation can proceed element by element and avoid solving a global system of algebraic equations as the standard implicit schemes do. We provide numerical examples to illustrate that the numerical scheme combined LDG method with IIF method indeed reduces the computational cost greatly.

An Arbitrary Lagrangian–Eulerian Local Discontinuous Galerkin Method for Hamilton–Jacobi Equations

In this paper, an arbitrary Lagrangian–Eulerian local discontinuous Galerkin (ALE-LDG) method for Hamilton–Jacobi equations will be developed, analyzed and numerically tested. This method is based on the time-dependent approximation space defined on the moving mesh. A priori error estimates will be stated with respect to the $$\mathrm {L}^{\infty }\left( 0,T;\mathrm {L}^{2}\left( \Omega \right) \right) $$ -norm. In particular, the optimal ( $$k+1$$ ) convergence in one dimension and the suboptimal ( $$k+\frac{1}{2}$$ ) convergence in two dimensions will be proven for the semi-discrete method, when a local Lax–Friedrichs flux and piecewise polynomials of degree k on the reference cell are used. Furthermore, the validity of the geometric conservation law will be proven for the fully-discrete method. Also, the link between the piecewise constant ALE-LDG method and the monotone scheme, which converges to the unique viscosity solution, will be shown. The capability of the method will be demonstrated by a variety of one and two dimensional numerical examples with convex and noneconvex Hamiltonian.

On Nesterov acceleration for Landweber iteration of linear ill-posed problems

Abstract In this paper we deal with Nesterov acceleration and show that it speeds up Landweber iteration when applied to linear ill-posed problems. It is proven that, if the exact solution x † ∈ ℛ ⁢ ( ( T * ⁢ T ) μ ) {x^{\dagger}\in{\cal R}((T^{*}T)^{\mu})} , then optimal convergence rates are obtained if μ ≤ 1 2 {\mu\leq\frac{1}{2}} and if the iteration is terminated according to an a priori stopping rule. If μ &gt; 1 2 {\mu&gt;\frac{1}{2}} or if the iteration is terminated according to the discrepancy principle, only suboptimal convergence rates can be guaranteed. Nevertheless, the number of iterations for Nesterov acceleration is always much smaller if the dimension of the problem is large. Numerical results verify the theoretical ones.

Three Natural Computation methods for joint channel estimation and symbol detection in multiuser communications

This paper studies three of the most important optimization algorithms belonging to Natural Computation (NC): genetic algorithm (GA), tabu search (TS) and simulated quenching (SQ). A concise overview of these methods, including their fundamentals, drawbacks and comparison, is described in the first half of the paper. Our work is particularized and focused on a specific application: joint channel estimation and symbol detection in a Direct-Sequence/Code-Division Multiple-Access (DS/CDMA) multiuser communications scenario; therefore, its channel model is described and the three methods are explained and particularized for solving this. Important issues such as suboptimal convergence, cycling search or control of the population diversity have deserved special attention. Several numerical simulations analyze the performance of these three methods, showing, as well, comparative results with well-known classical algorithms such as the Minimum Mean Square Error estimator (MMSE), the Matched Filter (MF) or Radial Basis Function (RBF)-based detection schemes. As a consequence, the three proposed methods would allow transmission at higher data rates over channels under more severe fading and interference conditions. Simulations show that our proposals require less computational load in most cases. For instance, the proposed GA saves about 73% of time with respect to the standard GA. Besides, when the number of active users doubles from 10 to 20, the complexity of the proposed GA increases by a factor of 8.33, in contrast to 32 for the optimum maximum likelihood detector. The load of TS and SQ is around 15–25% higher than that of the proposed GA.

Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws: Analysis and application in one dimension

In this paper, we develop and analyze an arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method with a time-dependent approximation space for one dimensional conservation laws, which satisfies the geometric conservation law. For the semi-discrete ALE-DG method, when applied to nonlinear scalar conservation laws, a cell entropy inequality, L2 stability and error estimates are proven. More precisely, we prove the sub-optimal (k + 12) convergence for monotone fluxes, and optimal (k + 1) convergence for an upwind flux, when a piecewise P k polynomial approximation space is used. For the fully-discrete ALE-DG method, the geometric conservation law and the local maximum principle are proven. Moreover we state conditions for slope limiters, which ensure total variation stability of the method. Numerical examples show the capability of the method.

Stochastic MAC-layer interference model for opportunistic spectrum access: A weighted graphical game approach

This article investigates the problem of distributed channel selection in opportunistic spectrum access networks from a perspective of interference minimization. The traditional physical (PHY)-layer interference model is for information theoretic analysis. When practical multiple access mechanisms are considered, the recently developed binary medium access control (MAC)-layer interference model in the previous work is more useful, in which the experienced interference of a user is defined as the number of competing users. However, the binary model is not accurate in mathematics analysis with poor achievable performance. Therefore, we propose a real-valued one called stochastic MAC-layer interference model, where the utility of a player is defined as a function of the aggregate weight of the stochastic interference of competing neighbors. Then, the distributed channel selection problem in the stochastic MAC-layer interference model is formulated as a weighted stochastic MAC-layer interference minimization game and we proved that the game is an exact potential game which exists one pure strategy Nash equilibrium point at least. By using the proposed stochastic learning-automata based uncoupled algorithm with heterogeneous learning parameter (SLA-H), we can achieve suboptimal convergence averagely and this result can be verified in the simulation. Moreover, the simulated results also prove that the proposed stochastic model can achieve higher throughput performance and faster convergence behavior than the binary one.