Polynomial Test Functions Research Articles

Variational Physics Informed Neural Networks: the Role of Quadratures and Test Functions

In this work we analyze how quadrature rules of different precisions and piecewise polynomial test functions of different degrees affect the convergence rate of Variational Physics Informed Neural Networks (VPINN) with respect to mesh refinement, while solving elliptic boundary-value problems. Using a Petrov-Galerkin framework relying on an inf-sup condition, we derive an a priori error estimate in the energy norm between the exact solution and a suitable high-order piecewise interpolant of a computed neural network. Numerical experiments confirm the theoretical predictions and highlight the importance of the inf-sup condition. Our results suggest, somehow counterintuitively, that for smooth solutions the best strategy to achieve a high decay rate of the error consists in choosing test functions of the lowest polynomial degree, while using quadrature formulas of suitably high precision.

Heavy-Traffic Analysis of Queueing Systems with No Complete Resource Pooling

We study the heavy-traffic limit of the generalized switch operating under MaxWeight, without assuming that the complete resource pooling condition is satisfied and allowing for correlated arrivals. The main contribution of this paper is the steady-state mean of linear combinations of queue lengths in heavy traffic. We showcase the generality of our result by presenting various stochastic networks as corollaries, each of which is a contribution by itself. In particular, we study the input-queued switch with correlated arrivals, and we show that, if the state space collapses to a full-dimensional subspace, the correlation among the arrival processes does not matter in heavy traffic. We exemplify this last case with a parallel-server system, an [Formula: see text]-system, and an ad hoc wireless network. Whereas these results are obtained using the drift method, we additionally present a negative result showing a limitation of the drift method. We show that it is not possible to obtain the individual queue lengths using the drift method with polynomial test functions. We do this by presenting an alternate view of the drift method in terms of a system of linear equations, and we use this system of equations to obtain bounds on arbitrary linear combinations of the queue lengths.

Moment and polynomial bounds for ruin-related quantities in risk theory

We establish a novel numerical quantification method based upon mathematical programming for the ruin-related quantities on collective risk models with fairly general features, such as reflecting, non-homogeneous and multidimensional dynamics with non-smooth coefficients. Rather than seeking approximations with a great deal of computing effort as in the existing numerical approximation methods, the proposed method aims to instantly provide deterministic upper and lower bounds for such quantities as linear combinations of moments or in the explicit polynomial form, without random number generation or sophisticated numerical algorithms. Given that closed-form solutions are rarely available especially in finite-time problems, the upper and lower bounds have great potential to be useful information that can complement approximate solutions obtained from existing numerical methods. We present numerical results throughout to justify the theoretical developments and convergence results so as to illustrate the effectiveness of the proposed method with low computational complexity. We also demonstrate further improvements of the bounds by scaling the problem domain, employing piecewise polynomial test functions and the exponential tempering of the polynomial bases.

Global fluctuations for Multiple Orthogonal Polynomial Ensembles

We study the fluctuations of linear statistics with polynomial test functions for Multiple Orthogonal Polynomial Ensembles. Multiple Orthogonal Polynomial Ensembles form an important class of determinantal point processes that include random matrix models such as the GUE with external source, complex Wishart matrices, multi-matrix models and others.Our analysis is based on the recurrence matrix for the multiple orthogonal polynomials, that is constructed out of the nearest neighbor recurrences. If the coefficients for the nearest neighbor recurrences have limits, then we show that the right-limit of this recurrence matrix is a matrix that can be viewed as representation of a Toeplitz operator with respect to a non-standard basis. This will allow us to prove Central Limit Theorems for linear statistics of Multiple Orthogonal Polynomial Ensembles. A particular novelty is the use of the Baker–Campbell–Hausdorff formula to prove that the higher cumulants of the linear statistics converge to zero.We illustrate the main results by discussing Central Limit Theorems for the Gaussian Unitary Ensembles with external source, complex Wishart matrices and specializations of the Schur measure related to multiple Charlier, multiple Krawtchouk and multiple Meixner polynomials.

Quantitative normal approximation of linear statistics of $\beta$-ensembles

We present a new approach, inspired by Stein’s method, to prove a central limit theorem (CLT) for linear statistics of $\beta$-ensembles in the one-cut regime. Compared with the previous proofs, our result requires less regularity on the potential and provides a rate of convergence in the quadratic Kantorovich or Wasserstein-2 distance. The rate depends both on the regularity of the potential and the test functions, and we prove that it is optimal in the case of the Gaussian Unitary Ensemble (GUE) for certain polynomial test functions. The method relies on a general normal approximation result of independent interest which is valid for a large class of Gibbs-type distributions. In the context of $\beta$-ensembles, this leads to a multi-dimensional CLT for a sequence of linear statistics which are approximate eigenfunctions of the infinitesimal generator of Dyson Brownian motion once the various error terms are controlled using the rigidity results of Bourgade, Erdős and Yau.

Revisiting generalized FEM: a Petrov\u2013Galerkin enrichment based FEM interpolation for Helmholtz problem

A concept of Petrov–Galerkin enrichment which is appropriate for highly accurate and stable interpolation of variational solutions is introduced. In the finite element context, the setting refers to standard trial functions for the solution, while the test space will be enriched. The FEM interpolation procedure that we propose will be justified by local wavelets with vanishing moments based on Gegenbauer polynomials. For the reference Helmholtz equation, the continuous piecewise polynomial test functions are enriched using dispersion analysis on uniform meshes in 2d and 3d. From a-priori and a-posteriori numerical analysis it follows that the Petrov–Galerkin based enrichment approximates the exact interpolate solution of the Helmholtz equation with at least seventh order of accuracy.

GLOBAL FLUCTUATIONS FOR LINEAR STATISTICS OF β-JACOBI ENSEMBLES

We study the global fluctuations for linear statistics of the form [Formula: see text] as n → ∞, for C1functions f, and λ1, …, λnbeing the eigenvalues of a (general) β-Jacobi ensemble. The fluctuation from the mean [Formula: see text] turns out to be given asymptotically by a Gaussian process. We compute the covariance matrix for the process and show that it is diagonalized by a shifted Chebyshev polynomial basis; in addition, we analyze the deviation from the predicted mean for polynomial test functions, and we obtain a law of large numbers.

Adaptive hybrid optimization strategy for calibration and parameter estimation of physical process models

A new adaptive hybrid optimization strategy, entitled squads, is proposed for complex inverse analysis of computationally intensive physics-based models. Typically, models are calibrated and model parameters are estimated by minimization of the discrepancy between model simulations characterizing the system and existing observations requiring a substantial number of model evaluations. Squads is designed to be computationally efficient and robust in identification of the global optimum (i.e. maximum or minimum value of an objective function). It integrates global and local optimization using Adaptive Particle Swarm Optimization (APSO) and Levenberg–Marquardt (LM) optimization using adaptive rules based on runtime performance. The global strategy (APSO) optimizes the location of a set of solutions (particles) in the parameter space. The local strategy (LM) is applied only to a subset of the particles at different stages of the optimization based on the adaptive rules. After the LM adjustment of the subset of particle positions, the updated particles are returned to APSO. Therefore, squads is a global strategy that utilizes a local optimization speedup. The advantages of coupling APSO and LM in the manner implemented in squads is demonstrated by comparisons of squads performance against Levenberg–Marquardt (LM), Particle Swarm Optimization (PSO), Adaptive Particle Swarm Optimization (APSO; i.e. TRIBES), and an existing hybrid optimization strategy (hPSO). All the strategies are tested on 2D, 5D and 10D Rosenbrock and Griewank polynomial test functions and a synthetic hydrogeologic application to identify the source of a contaminant plume in an aquifer. Tests are performed using a series of runs with random initial guesses for the estimated parameters. The performance of the strategies are compared based on their robustness, defined as the percentage of runs that identify the global optimum, and their efficiency, quantified by a statistical representation of the number of function evaluations performed prior to identification of the global optimum. Squads is observed to have better performance than the other strategies for the test functions and the hydrogeologic application when both robustness and efficiency are taken into consideration.

Analytical formulation of meshless local integral equation method

In this paper, the exact forms of integrals in the meshless local boundary integral equation method are derived and implemented for elastostatic problems. A weak form for a set of governing equations with a unit test function or polynomial test functions is transformed into local integral equations. Each node has its own support domain and is surrounded by a local integral domain with different shapes of boundaries. The meshless approximation based on the radial basis function (RBF) is employed for the implementation of displacements. A completed set of closed forms of the local boundary integrals are obtained. As there are no numerical integrations to be carried out the computational time is significantly reduced. Three examples are presented to demonstrate the application of this approach in solid mechanics.

Meshless Local Integral Equation Method with Analytical Formulation and its Application to Computational Fracture Mechanics

In this paper, the exact forms of integrals in the meshless local boundary integral equation method are presented and applications were demonstrated for fracture mechanics. A weak form for a set of governing equations with a unit test function or polynomial test functions is transformed into local integral equations. Each node has its own support domain and is surrounded by a local integral domain with different shapes of boundaries. The meshless approximation based on the Radial Basis Function (RBF) is employed for the implementation of displacements. For cracked plate, opening displacement on the crack surface can be obtained with satisfactory accuracy without special treatment for stresses singularities at crack tip.

A New Insight into the Differential Quadrature Method in Solving 2-D Elliptic PDEs

When the local differential quadrature (LDQ) has been successfully applied to solve two-dimensional problems, the global method of DQ still has a problem by requiring to solve the inversions of ill-posed matrices. Previously, when one uses (n 1)th order polynomial test functions to determine the weight- ing coefficients with n grid points, the resultant n n Vandermonde matrix is highly ill-conditioned and its inversion is hard to solve. Now we use (m 1)th order poly- nomial test functions by n grid points that the size of Vandermonde matrix is m n, of which m is much less than n. We find that the (m 1)th order polynomial test functions are accurate enough to express the solutions, and the novel method sig- nificantly improves the ill-condition of algebraic equations. Such a new DQ as being combined with FTIM (Fictitious Time Integration Method) can solve 2-D elliptic type PDEs successfully. There are some examples tested in this paper and the numerical errors are found to be very small.

The Noise in the Circular Law and the Gaussian Free Field

Fill an n × n matrix with independent complex Gaussians of variance 1/n. As n → ∞, the eigenvalues {zk} onverge to a sum of an H1-noise on the unit disk and an independent H1/2-noise on the unit circle. More precisely, for C1 functions of suitable growth, the distribution of ∑k=1n(f(zk)−Ef(zk)) converges to that of a mean-zero Gaussian with variance given by the sum of the squares of the disk H1 and the circle H1/2 norms of f. As a consequence, with pn the characteristic polynomial, it is found that log |pn| − E log |pn| tends to the planar Gaussian free field conditioned to be harmonic outside the unit disk. Further, for polynomial test functions f, we prove that the limiting covariance structure is universal for a class of models including Haar distributed unitary matrices.

A variant of the finite superelement method for computing viscous incompressible flows

A variant of the finite superelement method (FSEM) is proposed for computing viscous incompressible convection-dominated flows on a triangular unstructured mesh. To construct the high-order FSEM scheme, vector polynomial test functions (SE basis) are calculated in each grid cell by solving the linearized Navier-Stokes equations with special boundary conditions in the form of basis functions in the trace space.

Theoretical and numerical results about some weakly singular Volterra-Fredholm equations

In this paper existence, uniqueness results for the solution of some weakly singular linear Volterra and Volterra-Fredholm integral equations are given. For these equations, a numerical model is proposed and its convergence and rate of convergence are analyzed. Numerical results on some polynomial test functions are given.

Thermocapillary long waves in a liquid film flow. Part 1. Low-dimensional formulation

We consider the dynamics of a thin liquid film falling down a uniformly heated wall. The heating sets up surface tension gradients that induce thermocapillary stresses on the free surface, thus affecting the evolution of the film. We model this thermocapillary flow by using a gradient expansion combined with a Galerkin projection with polynomial test functions for both velocity and temperature fields. We obtain equations for the evolution of the velocity and temperature amplitudes at first- and second-order in the expansion parameter. These equations are fully compatible close to criticality with the Benney long-wave expansion. Models of reduced dimensionality for the evolution of the local film thickness, flow rate and interfacial temperature only, are proposed.

Wave dynamics on a thin-liquid film falling down a heated wall

The dynamics of a thin liquid film falling down a uniformly heated wall is studied. The model introduced by Kalliadasis et al. [J. Fluid Mech. 475 (2003) 377] for the same problem is revisited and its deficiencies, namely the prediction of a critical Reynolds number with 20% error, cured. For the energy equation a high-order Galerkin projection in terms of polynomial test functions is developed. It is shown that not only does this more refined formulation correct the critical Reynolds number, but it also gives, with an appropriate expansion close to criticality, the long-wave theory. Bifurcation diagrams for permanent solitary waves are constructed and compared with the solution branches obtained from different models. It is shown that, in all cases, the long-wave theory exhibits limit points and branch multiplicity, while the other models predict the continuing existence of solitary waves. Time-dependent computations show that the free surface and interfacial temperature approach a train of coherent structures that resemble the infinite-domain stationary solitary pulses.

Improving Interval Analysis Bounds by Translations

We explore how a simple linear change of variable affects the inclusion functions obtained with Interval Analysis methods. Univariate and multivariate polynomial test functions are considered, showing that translation-based methods improve considerably the bounds computed by standard inclusion functions. An Interval Branch-and-Bound method for global optimization is then implemented to compare the different procedures, showing that, although with times higher than those given by Taylor forms, the number of clusters and iterations is strongly reduced.

Korovkin tests, approximation, and ergodic theory

We consider sequences of s . k(n) x t . k(n) matrices {A n (f)} with a block structure spectrally distributed as an L 1 p-variate s x t matrix-valued function f, and, for any n, we suppose that A n (.) is a linear and positive operator. For every fixed n we approximate the matrix A n (f) in a suitable linear space M n of s . k(n) x t . k(n) matrices by minimizing the Frobenius norm of A n (f)-X n when X n ranges over M n . The minimizer X n is denoted by P k(n) (A n (f)). We show that only a simple Korovkin test over a finite number of polynomial test functions has to be performed in order to prove the ,following general facts: 1. the sequence {P k(n) (A n (f))} is distributed as f, 2. the sequence {A n (f)-P k(n) (A n (f))} is distributed as the constant function 0 (i.e. is spectrally clustered at zero). The first result is an ergodic one which can be used for solving numerical approximation theory problems. The second has a natural interpretation in the theory of the preconditioning associated to cg-like algorithms.

Piecewise Linear Petrov–Galerkin Error Estimates for the Box Method

The perpendicular bisector box-method (PBM) discretization of the Laplacian on a simplicial element is equivalent to a piecewise linear Petrov–Galerkin method with piecewise constant test functions. This PBM element matrix is expressed in terms of Galerkin’s piecewise linear (PLG) element matrix and the average values assumed by test functions over element faces only. These results are shown to be applicable to the accuracy analysis of PBM discretizations of extremum stable elliptic gradient equations, subject to regularity conditions on variable coefficients in these equations. These gradient equations are shown to satisfy sufficient conditions to validate a discrete extremum principle on N-dimensional Delaunay triangulations. This conclusion need not hold for the PLG equations. The dependence of the discretized Laplacian on only the element-face averages of test functions enables the substitution of piecewise polynomial test functions for the piecewise constant functions. The latter are used to establish piecewise linear accuracy for the PBM by an adaptation of an error estimate for Petrov–Galerkin methods by Strang.