A numerical algorithm is presented to solve a benchmark problem proposed by Hemker. The algorithm incorporates asymptotic information into the design of appropriate piecewise-uniform Shishkin meshes. Moreover, different co-ordinate systems are utilized due to the different geometries and associated layer structures that are involved in this problem. Numerical results are presented to demonstrate the effectiveness of the proposed numerical algorithm.
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Benchmark datasets for visual recognition assume that data is uniformly distributed, while real-world datasets obey long-tailed distribution. Current approaches handle the long-tailed problem to transform the long-tailed dataset to uniform distribution by re-sampling or re-weighting strategies. These approaches emphasize the tail classes but ignore the hard examples in head classes, which result in performance degradation. In this paper, we propose a novel gradient harmonized mechanism with category-wise adaptive precision to decouple the difficulty and sample size imbalance in the long-tailed problem, which are correspondingly solved via intra- and inter-category balance strategies. Specifically, intra-category balance focuses on the hard examples in each category to optimize the decision boundary, while inter-category balance aims to correct the shift of decision boundary by taking each category as a unit. Extensive experiments demonstrate that the proposed method consistently outperforms other approaches on all the datasets.
The Schrijver graph $S(n,k)$ is defined for integers $n$ and $k$ with $n \geq 2k$ as the graph whose vertices are all the $k$-subsets of $\{1,2,\ldots,n\}$ that do not include two consecutive elements modulo $n$, where two such sets are adjacent if they are disjoint. A result of Schrijver asserts that the chromatic number of $S(n,k)$ is $n-2k+2$ (Nieuw Arch. Wiskd., 1978). In the computational Schrijver problem, we are given an access to a coloring of the vertices of $S(n,k)$ with $n-2k+1$ colors, and the goal is to find a monochromatic edge. The Schrijver problem is known to be complete in the complexity class $\mathsf{PPA}$. We prove that it can be solved by a randomized algorithm with running time $n^{O(1)} \cdot k^{O(k)}$, hence it is fixed-parameter tractable with respect to the parameter $k$.
We introduce a filtering technique for Discontinuous Galerkin approximations of hyperbolic problems. Following an approach already proposed for the Hamilton-Jacobi equations by other authors, we aim at reducing the spurious oscillations that arise in presence of discontinuities when high order spatial discretizations are employed. This goal is achieved using a filter function that keeps the high order scheme when the solution is regular and switches to a monotone low order approximation if it is not. The method has been implemented in the framework of the $deal.II$ numerical library, whose mesh adaptation capabilities are also used to reduce the region in which the low order approximation is used. A number of numerical experiments demonstrate the potential of the proposed filtering technique.
In this work, we introduce a novel approach to formulating an artificial viscosity for shock capturing in nonlinear hyperbolic systems by utilizing the property that the solutions of hyperbolic conservation laws are not reversible in time in the vicinity of shocks. The proposed approach does not require any additional governing equations or a priori knowledge of the hyperbolic system in question, is independent of the mesh and approximation order, and requires the use of only one tunable parameter. The primary novelty is that the resulting artificial viscosity is unique for each component of the conservation law which is advantageous for systems in which some components exhibit discontinuities while others do not. The efficacy of the method is shown in numerical experiments of multi-dimensional hyperbolic conservation laws such as nonlinear transport, Euler equations, and ideal magnetohydrodynamics using a high-order discontinuous spectral element method on unstructured grids.
We describe a numerical algorithm for approximating the equilibrium-reduced density matrix and the effective (mean force) Hamiltonian for a set of system spins coupled strongly to a set of bath spins when the total system (system+bath) is held in canonical thermal equilibrium by weak coupling with a "super-bath". Our approach is a generalization of now standard typicality algorithms for computing the quantum expectation value of observables of bare quantum systems via trace estimators and Krylov subspace methods. In particular, our algorithm makes use of the fact that the reduced system density, when the bath is measured in a given random state, tends to concentrate about the corresponding thermodynamic averaged reduced system density. Theoretical error analysis and numerical experiments are given to validate the accuracy of our algorithm. Further numerical experiments demonstrate the potential of our approach for applications including the study of quantum phase transitions and entanglement entropy for long-range interaction systems.
We study the numerical approximation by space-time finite element methods of a multi-physics system coupling hyperbolic elastodynamics with parabolic transport and modelling poro- and thermoelasticity. The equations are rewritten as a first-order system in time. Discretizations by continuous Galerkin methods in space and time with inf-sup stable pairs of finite elements for the spatial approximation of the unknowns are investigated. Optimal order error estimates of energy-type are proven. Superconvergence at the time nodes is addressed briefly. The error analysis can be extended to discontinuous and enriched Galerkin space discretizations. The error estimates are confirmed by numerical experiments.
The Koopman operator is beneficial for analyzing nonlinear and stochastic dynamics; it is linear but infinite-dimensional, and it governs the evolution of observables. The extended dynamic mode decomposition (EDMD) is one of the famous methods in the Koopman operator approach. The EDMD employs a data set of snapshot pairs and a specific dictionary to evaluate an approximation for the Koopman operator, i.e., the Koopman matrix. In this study, we focus on stochastic differential equations, and a method to obtain the Koopman matrix is proposed. The proposed method does not need any data set, which employs the original system equations to evaluate some of the targeted elements of the Koopman matrix. The proposed method comprises combinatorics, an approximation of the resolvent, and extrapolations. Comparisons with the EDMD are performed for a noisy van der Pol system. The proposed method yields reasonable results even in cases wherein the EDMD exhibits a slow convergence behavior.
In this paper, we consider the constrained energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with discontinuous Galerkin (DG) coupling for the linear elasticity equations in highly heterogeneous and high contrast media. We will introduce the construction of a DG version of the CEM-GMsFEM, such as auxiliary basis functions and offline basis functions. The DG version of the method offers some advantages such as flexibility in coarse grid construction and sparsity of resulting discrete systems. Moreover, to our best knowledge, this is the first time where the proof of the convergence of the CEM-GMsFEM in the DG form is given. Some numerical examples will be presented to illustrate the performance of the method.
One of the most important problems in system identification and statistics is how to estimate the unknown parameters of a given model. Optimization methods and specialized procedures, such as Empirical Minimization (EM) can be used in case the likelihood function can be computed. For situations where one can only simulate from a parametric model, but the likelihood is difficult or impossible to evaluate, a technique known as the Two-Stage (TS) Approach can be applied to obtain reliable parametric estimates. Unfortunately, there is currently a lack of theoretical justification for TS. In this paper, we propose a statistical decision-theoretical derivation of TS, which leads to Bayesian and Minimax estimators. We also show how to apply the TS approach on models for independent and identically distributed samples, by computing quantiles of the data as a first step, and using a linear function as the second stage. The proposed method is illustrated via numerical simulations.
The numerical solution of singular eigenvalue problems is complicated by the fact that small perturbations of the coefficients may have an arbitrarily bad effect on eigenvalue accuracy. However, it has been known for a long time that such perturbations are exceptional and standard eigenvalue solvers, such as the QZ algorithm, tend to yield good accuracy despite the inevitable presence of roundoff error. Recently, Lotz and Noferini quantified this phenomenon by introducing the concept of $\delta$-weak eigenvalue condition numbers. In this work, we consider singular quadratic eigenvalue problems and two popular linearizations. Our results show that a correctly chosen linearization increases $\delta$-weak eigenvalue condition numbers only marginally, justifying the use of these linearizations in numerical solvers also in the singular case. We propose a very simple but often effective algorithm for computing well-conditioned eigenvalues of a singular quadratic eigenvalue problems by adding small random perturbations to the coefficients. We prove that the eigenvalue condition number is, with high probability, a reliable criterion for detecting and excluding spurious eigenvalues created from the singular part.
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