We present a novel method to perform numerical integration over curved polyhedra enclosed by high-order parametric surfaces. Such a polyhedron is first decomposed into a set of triangular and/or rectangular pyramids, whose certain faces correspond to the given parametric surfaces. Each pyramid serves as an integration cell with a geometric mapping from a standard parent domain (e.g., a unit cube), where the tensor-product Gauss quadrature is adopted. As no constraint is imposed on the decomposition, certain resulting pyramids may intersect with themselves, and thus their geometric mappings may present negative Jacobian values. We call such cells the folded cells and refer to the corresponding decomposition as a folded decomposition. We show that folded cells do not cause any issues in practice as they are only used to numerically compute certain integrals of interest. The same idea can be applied to planar curved polygons as well. We demonstrate both theoretically and numerically that folded cells can retain the same accuracy as the cells with strictly positive Jacobians. On the other hand, folded cells allow for a much easier and much more flexible decomposition for general curved polyhedra, on which one can robustly compute integrals. In the end, we show that folded cells can flexibly and robustly accommodate real-world complex geometries by presenting several examples in the context of immersed isogeometric analysis, where involved sharp features can be well respected in generating integration cells.
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Integration：Integration, the VLSI Journal。
Explanation：集成，VLSI雜志。
Publisher：Elsevier。
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In this study, novel physics-informed neural network (PINN) methods for coupling neighboring support points and automatic differentiation (AD) through Taylor series expansion are proposed to allow efficient training with improved accuracy. The computation of differential operators required for PINNs loss evaluation at collocation points are conventionally obtained via AD. Although AD has the advantage of being able to compute the exact gradients at any point, such PINNs can only achieve high accuracies with large numbers of collocation points, otherwise they are prone to optimizing towards unphysical solution. To make PINN training fast, the dual ideas of using numerical differentiation (ND)-inspired method and coupling it with AD are employed to define the loss function. The ND-based formulation for training loss can strongly link neighboring collocation points to enable efficient training in sparse sample regimes, but its accuracy is restricted by the interpolation scheme. The proposed coupled-automatic-numerical differentiation framework, labeled as can-PINN, unifies the advantages of AD and ND, providing more robust and efficient training than AD-based PINNs, while further improving accuracy by up to 1-2 orders of magnitude relative to ND-based PINNs. For a proof-of-concept demonstration of this can-scheme to fluid dynamic problems, two numerical-inspired instantiations of can-PINN schemes for the convection and pressure gradient terms were derived to solve the incompressible Navier-Stokes (N-S) equations. The superior performance of can-PINNs is demonstrated on several challenging problems, including the flow mixing phenomena, lid driven flow in a cavity, and channel flow over a backward facing step. The results reveal that for challenging problems like these, can-PINNs can consistently achieve very good accuracy whereas conventional AD-based PINNs fail.
This paper deals with unconstrained optimization problems based on numerical analysis of ordinary differential equations (ODEs). Although it has been known for a long time that there is a relation between optimization methods and discretization of ODEs, research in this direction has recently been gaining attention. In recent studies, the dissipation laws of ODEs have often played an important role. By contrast, in the context of numerical analysis, a technique called geometric numerical integration, which explores discretization to maintain geometrical properties such as the dissipation law, is actively studied. However, in research investigating the relationship between optimization and ODEs, techniques of geometric numerical integration have not been sufficiently investigated. In this paper, we show that a recent geometric numerical integration technique for gradient flow reads a new step-size criterion for the steepest descent method. Consequently, owing to the discrete dissipation law, convergence rates can be proved in a form similar to the discussion in ODEs. Although the proposed method is a variant of the existing steepest descent method, it is suggested that various analyses of the optimization methods via ODEs can be performed in the same way after discretization using geometric numerical integration.
In numerical simulations of complex flows with discontinuities, it is necessary to use nonlinear schemes. The spectrum of the scheme used have a significant impact on the resolution and stability of the computation. Based on the approximate dispersion relation method, we combine the corresponding spectral property with the dispersion relation preservation proposed by De and Eswaran (J. Comput. Phys. 218 (2006) 398-416) and propose a quasi-linear dispersion relation preservation (QL-DRP) analysis method, through which the group velocity of the nonlinear scheme can be determined. In particular, we derive the group velocity property when a high-order Runge-Kutta scheme is used and compare the performance of different time schemes with QL-DRP. The rationality of the QL-DRP method is verified by a numerical simulation and the discrete Fourier transform method. To further evaluate the performance of a nonlinear scheme in finding the group velocity, new hyperbolic equations are designed. The validity of QL-DRP and the group velocity preservation of several schemes are investigated using two examples of the equation for one-dimensional wave propagation and the new hyperbolic equations. The results show that the QL-DRP method integrated with high-order time schemes can determine the group velocity for nonlinear schemes and evaluate their performance reasonably and efficiently.
In this paper we present results on asymptotic characteristics of multivariate function classes in the uniform norm. Our main interest is the approximation of functions with mixed smoothness parameter not larger than $1/2$. Our focus will be on the behavior of the best $m$-term trigonometric approximation as well as the decay of Kolmogorov and entropy numbers in the uniform norm. It turns out that these quantities share a few fundamental abstract properties like their behavior under real interpolation, such that they can be treated simultaneously. We start with proving estimates on finite rank convolution operators with range in a step hyperbolic cross. These results imply bounds for the corresponding function space embeddings by a well-known decomposition technique. The decay of Kolmogorov numbers have direct implications for the problem of sampling recovery in $L_2$ in situations where recent results in the literature are not applicable since the corresponding approximation numbers are not square summable.
A method is presented for the evaluation of integrals on tetrahedra where the integrand has a singularity at one vertex. The approach uses a transformation to spherical polar coordinates which explicitly eliminates the singularity and facilitates the evaluation of integration limits. The method can also be implemented in an adaptive form which gives convergence to a required tolerance. Results from the method are compared to the output from an exact analytical method and show high accuracy. In particular, when the adaptive algorithm is used, highly accurate results are found for poorly conditioned tetrahedra which normally present difficulties for numerical quadrature techniques.
Cancelable biometric schemes aim at generating secure biometric templates by combining user specific tokens, such as password, stored secret or salt, along with biometric data. This type of transformation is constructed as a composition of a biometric transformation with a feature extraction algorithm. The security requirements of cancelable biometric schemes concern the irreversibility, unlinkability and revocability of templates, without losing in accuracy of comparison. While several schemes were recently attacked regarding these requirements, full reversibility of such a composition in order to produce colliding biometric characteristics, and specifically presentation attacks, were never demonstrated to the best of our knowledge. In this paper, we formalize these attacks for a traditional cancelable scheme with the help of integer linear programming (ILP) and quadratically constrained quadratic programming (QCQP). Solving these optimization problems allows an adversary to slightly alter its fingerprint image in order to impersonate any individual. Moreover, in an even more severe scenario, it is possible to simultaneously impersonate several individuals.
Let $K$ be a finite simplicial, cubical, delta or CW complex. The persistence map $\mathrm{PH}$ takes a filter $f:K \rightarrow \mathbb{R}$ as input and returns the barcodes $\mathrm{PH}(f)$ of the associated sublevel set persistent homology modules. We address the inverse problem: given a target barcode $D$, computing the fiber $\mathrm{PH}^{-1}(D)$. For this, we use the fact that $\mathrm{PH}^{-1}(D)$ decomposes as complex of polyhedra when $K$ is a simplicial complex, and we generalise this result to arbitrary based chain complexes. We then design and implement a depth first search algorithm that recovers the polyhedra forming the fiber $\mathrm{PH}^{-1}(D)$. As an application, we solve a corpus of 120 sample problems, providing a first insight into the statistical structure of these fibers, for general CW complexes.
In this paper we propose a modified Lie-type spectral splitting approximation where the external potential is of quadratic type. It is proved that we can approximate the solution to a nonlinear Schroedinger equation by solving the linear problem and treating the nonlinear term separately, with a rigorous estimate of the remainder term. Furthermore, we show by means of numerical experiments that such a modified approximation is more efficient than the standard one.
We introduce the continuized Nesterov acceleration, a close variant of Nesterov acceleration whose variables are indexed by a continuous time parameter. The two variables continuously mix following a linear ordinary differential equation and take gradient steps at random times. This continuized variant benefits from the best of the continuous and the discrete frameworks: as a continuous process, one can use differential calculus to analyze convergence and obtain analytical expressions for the parameters; and a discretization of the continuized process can be computed exactly with convergence rates similar to those of Nesterov original acceleration. We show that the discretization has the same structure as Nesterov acceleration, but with random parameters. We provide continuized Nesterov acceleration under deterministic as well as stochastic gradients, with either additive or multiplicative noise. Finally, using our continuized framework and expressing the gossip averaging problem as the stochastic minimization of a certain energy function, we provide the first rigorous acceleration of asynchronous gossip algorithms.
The autoregressive process is one of the fundamental and most important models that analyze a time series. Theoretical results and practical tools for fitting an autoregressive process with i.i.d. innovations are well-established. However, when the innovations are white noise but not i.i.d., those tools fail to generate a consistent confidence interval for the autoregressive coefficients. Focus on an autoregressive process with \textit{dependent} and \textit{non-stationary} innovations, this paper provides a consistent result and a Gaussian approximation theorem for the Yule-Walker estimator. Moreover, it introduces the second order wild bootstrap that constructs a consistent confidence interval for the estimator. Numerical experiments confirm the validity of the proposed algorithm with different kinds of white noise innovations. Meanwhile, the classical method(e.g., AR(Sieve) bootstrap) fails to generate a correct confidence interval when the innovations are dependent. According to Kreiss et al. \cite{10.1214/11-AOS900} and the Wold decomposition, assuming a real-life time series satisfies an autoregressive process is reasonable. However, innovations in that process are more likely to be white noises instead of i.i.d.. Therefore, our method should provide a practical tool that handles real-life problems.
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