Least squares polynomial chaos expansion: A review of sampling strategies
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Hadigol, Mohammad↗
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Multi-model polynomial chaos surrogate dictionary for Bayesian inference in elasticity problems
Not provided.
Engineering↗
Polynomial Preconditioned GMRES in Trilinos: Practical Considerations for High Performance Computing.
Abstract not provided.
Loe, Jennifer Ann↗
Toward efficient polynomial preconditioning for GMRES
Here, we present a polynomial preconditioner for solving large systems of linear equations. The polynomial is derived from the minimum residual polynomial (the GMRES polynomial) and is more straightforward to compute and implement than many previous polynomial preconditioners. Our current implementation of this polynomial using its roots is naturally more stable than previous methods of computing the same polynomial. We implement further stability control using added roots, and this allows for high degree polynomials. We discuss the effectiveness and challenges of root-adding and give an additional check for stability. In this article, we study the polynomial preconditioner applied to GMRES; however it could be used with any Krylov solver. This polynomial preconditioning algorithm can dramatically improve convergence for some problems, especially for difficult problems, and can reduce dot products by an even greater margin.
97 MATHEMATICS AND COMPUTING↗
Optimal Polynomial Smoothers and One‐Sided V‐Cycles for Poisson Problems
The solution to the Poisson equation arising from the spectral element discretization of the incompressible Navier‐Stokes equations needs robust preconditioning strategies. One such strategy is multigrid. To realize the potential of multigrid methods, effective smoothing strategies are needed. Chebyshev polynomial smoothers, in conjunction with pointwise Jacobi or additive Schwarz methods (ASMs), prove to be an effective smoother. Other polynomial smoothers, however, may provide superior convergence to the multigrid preconditioner. The authors compare the standard Chebyshev polynomial smoothers to both the novel fourth‐kind Chebyshev polynomial smoothers proposed by Lottes as well as smoothers based on the polynomial of best uniform approximation to as proposed by Kraus, Vassilevski, and Zikatanov. At the cost of symmetry, further improvements may be made. For example, a order polynomial smoother on both sides of the V‐cycle may be substituted with an order polynomial smoother on one side at no additional cost. The choice of omitting the postsmoother in favor of higher‐order polynomial presmoothing is advantageous in cases where the multigrid approximation property constant is large. The authors consider a 2D model problem based on finite differences to motivate the choice of polynomial smoother, order, and whether to apply postsmoothing for the target application of high‐order ‐geometric multigrid methods for GPU architectures. Results from both domains demonstrate the substantial improvement of these approaches over the standard Chebyshev polynomial smoother with a symmetric V‐cycle.
97 MATHEMATICS AND COMPUTING↗
Does Bayesian model averaging improve polynomial extrapolations? Two toy problems as tests
We assess the accuracy of Bayesian polynomial extrapolations from small parameter values, x, to large values of x. We consider a set of polynomials of fixed order, intended as a proxy for a fixed-order effective field theory (EFT) description of data. We employ Bayesian model averaging (BMA) to combine results from different order polynomials (EFT orders). Our study considers two 'toy problems' where the underlying function used to generate data sets is known. We use Bayesian parameter estimation to extract the polynomial coefficients that describe these data at low x. A 'naturalness' prior is imposed on the coefficients, so that they are $\mathcal{O}(1)$. We BMA different polynomial degrees by weighting each according to its Bayesian evidence and compare the predictive performance of this BMA with that of the individual polynomials. In conclusion, the credibility intervals on the BMA forecast have the stated coverage properties more consistently than does the highest evidence polynomial, though BMA does not necessarily outperform every polynomial.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Modeling State-Space Aeroelastic Systems Using a Simple Matrix Polynomial Approach for the Unsteady Aerodynamics
A simple matrix polynomial approach is introduced for approximating unsteady aerodynamics in the s-plane and ultimately, after combining matrix polynomial coefficients with matrices defining the structure, a matrix polynomial of the flutter equations of motion (EOM) is formed. A technique of recasting the matrix-polynomial form of the flutter EOM into a first order form is also presented that can be used to determine the eigenvalues near the origin and everywhere on the complex plane. An aeroservoelastic (ASE) EOM have been generalized to include the gust terms on the right-hand side. The reasons for developing the new matrix polynomial approach are also presented, which are the following: first, the "workhorse" methods such as the NASTRAN flutter analysis lack the capability to consistently find roots near the origin, along the real axis or accurately find roots farther away from the imaginary axis of the complex plane; and, second, the existing s-plane methods, such as the Roger s s-plane approximation method as implemented in ISAC, do not always give suitable fits of some tabular data of the unsteady aerodynamics. A method available in MATLAB is introduced that will accurately fit generalized aerodynamic force (GAF) coefficients in a tabular data form into the coefficients of a matrix polynomial form. The root-locus results from the NASTRAN pknl flutter analysis, the ISAC-Roger's s-plane method and the present matrix polynomial method are presented and compared for accuracy and for the number and locations of roots.
Pototzky, Anthony S.↗
A new surrogate modeling technique combining Kriging and polynomial chaos expansions – Application to uncertainty analysis in computational dosimetry
In numerical dosimetry, the recent advances in high performance computing led to a strong reduction of the required computational time to assess the specific absorption rate (SAR) characterizing the human exposure to electromagnetic waves. However, this procedure remains time-consuming and a single simulation can request several hours. As a consequence, the influence of uncertain input parameters on the SAR cannot be analyzed using crude Monte Carlo simulation. The solution presented here to perform such an analysis is surrogate modeling. This paper proposes a novel approach to build such a surrogate model from a design of experiments. Considering a sparse representation of the polynomial chaos expansions using least-angle regression as a selection algorithm to retain the most influential polynomials, this paper proposes to use the selected polynomials as regression functions for the universal Kriging model. The leave-one-out cross validation is used to select the optimal number of polynomials in the deterministic part of the Kriging model. The proposed approach, called LARS-Kriging-PC modeling, is applied to three benchmark examples and then to a full-scale metamodeling problem involving the exposure of a numerical fetus model to a femtocell device. The performances of the LARS-Kriging-PC are compared to an ordinary Kriging model and to a classical sparse polynomial chaos expansion. The LARS-Kriging-PC appears to have better performances than the two other approaches. A significant accuracy improvement is observed compared to the ordinary Kriging or to the sparse polynomial chaos depending on the studied case. This approach seems to be an optimal solution between the two other classical approaches. A global sensitivity analysis is finally performed on the LARS-Kriging-PC model of the fetus exposure problem.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Extending Romanovski polynomials in quantum mechanics
Some extensions of the (third-class) Romanovski polynomials (also called Romanovski/pseudo-Jacobi polynomials), which appear in bound-state wavefunctions of rationally extended Scarf II and Rosen-Morse I potentials, are considered. For the former potentials, the generalized polynomials satisfy a finite orthogonality relation, while for the latter an infinite set of relations among polynomials with degree-dependent parameters is obtained. Both types of relations are counterparts of those known for conventional polynomials. In the absence of any direct information on the zeros of the Romanovski polynomials present in denominators, the regularity of the constructed potentials is checked by taking advantage of the disconjugacy properties of second-order differential equations of Schrödinger type. It is also shown that on going from Scarf I to Scarf II or from Rosen-Morse II to Rosen-Morse I potentials, the variety of rational extensions is narrowed down from types I, II, and III to type III only.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Polynomial solutions of the Monge-Ampère equation
The question of the existence of polynomial solutions to the Monge-Ampère equation z{sub xx}z{sub yy}−z{sub xy}{sup 2}=f(x,y) is considered in the case when f(x,y) is a polynomial. It is proved that if f is a polynomial of the second degree, which is positive for all values of its arguments and has a positive squared part, then no polynomial solution exists. On the other hand, a solution which is not polynomial but is analytic in the whole of the x, y-plane is produced. Necessary and sufficient conditions for the existence of polynomial solutions of degree up to 4 are found and methods for the construction of such solutions are indicated. An approximation theorem is proved. Bibliography: 10 titles.
97 MATHEMATICS AND COMPUTING↗
Multiple zeros of polynomials
For polynomials of higher degree, iterative numerical methods must be used. Four iterative methods are presented for approximating the zeros of a polynomial using a digital computer. Newton's method and Muller's method are two well known iterative methods which are presented. They extract the zeros of a polynomial by generating a sequence of approximations converging to each zero. However, both of these methods are very unstable when used on a polynomial which has multiple zeros. That is, either they fail to converge to some or all of the zeros, or they converge to very bad approximations of the polynomial's zeros. This material introduces two new methods, the greatest common divisor (G.C.D.) method and the repeated greatest common divisor (repeated G.C.D.) method, which are superior methods for numerically approximating the zeros of a polynomial having multiple zeros. These methods were programmed in FORTRAN 4 and comparisons in time and accuracy are given.
Wood, C. A.↗
Necklaces, symmetries and self-reciprocal polynomials
The connection between a certain class of necklaces and self-reciprocal polynomials over finite fields is shown. For equal to or greater than 2, self-reciprocal polynomials of degree 2n arising from monic irreducible polynomials of degree n are shown to be either irreducible or the product of two irreducible factors which are necessarily reciprocal polynomials. DeBruijn's (1959) method is used to count the number of necklaces in this class and hence obtain a formula for the number of irreducible self-reciprocal polynomials, showing that they exist for every even degree. Thus every extension of a finite field of even degree can be obtained by adjoining a root of an irreducible self-reciprocal polynomial.
Miller, R. L.↗
On polynomial preconditioning for indefinite Hermitian matrices
The minimal residual method is studied combined with polynomial preconditioning for solving large linear systems (Ax = b) with indefinite Hermitian coefficient matrices (A). The standard approach for choosing the polynomial preconditioners leads to preconditioned systems which are positive definite. Here, a different strategy is studied which leaves the preconditioned coefficient matrix indefinite. More precisely, the polynomial preconditioner is designed to cluster the positive, resp. negative eigenvalues of A around 1, resp. around some negative constant. In particular, it is shown that such indefinite polynomial preconditioners can be obtained as the optimal solutions of a certain two parameter family of Chebyshev approximation problems. Some basic results are established for these approximation problems and a Remez type algorithm is sketched for their numerical solution. The problem of selecting the parameters such that the resulting indefinite polynomial preconditioners speeds up the convergence of minimal residual method optimally is also addressed. An approach is proposed based on the concept of asymptotic convergence factors. Finally, some numerical examples of indefinite polynomial preconditioners are given.
Freund, Roland W.↗
To the Theory of C{sub 0}-Operator Orthogonal Polynomials
Operator orthogonal polynomials are considered whose arguments are generators of strongly continuous semigroups of transformations of class C{sub 0} acting in a Banach space. Earlier such polynomials were considered by the authors in the case of the Chebyshev polynomials of the first and second kind. In this paper, more general classes of operator orthogonal polynomials are considered, which include the Jacobi and Aptekarev polynomials. Integral representations of operator fractional-rational functions and also of Bessel operator-valued functions of an imaginary argument are presented.
97 MATHEMATICS AND COMPUTING↗
Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations
The principal aim of the current paper is to present and analyze two new spectral algorithms for solving some types of linear and nonlinear fractional-order differential equations. The proposed algorithms are obtained by utilizing a certain kind of shifted Chebyshev polynomials called the shifted fifth-kind Chebyshev polynomials as basis functions along with the application of a modified spectral tau method. The class of fifth-kind Chebyshev polynomials is a special class of a basic class of symmetric orthogonal polynomials which are constructed with the aid of the extended Sturm–Liouville theorem for symmetric functions. An investigation for the convergence and error analysis of the proposed Chebyshev expansion is performed. For this purpose, a new connection formulae between Chebyshev polynomials of the first and fifth kinds are derived. The obtained numerical results ascertain that our two proposed algorithms are applicable, efficient and accurate.
97 MATHEMATICS AND COMPUTING↗
Long-time uncertainty propagation using generalized polynomial chaos and flow map composition
We present an efficient and accurate method for long-time uncertainty propagation in dynamical systems. Uncertain initial conditions and parameters are both addressed. The method approximates the intermediate short-time flow maps by spectral polynomial bases, as in the generalized polynomial chaos (gPC) method, and uses flow map composition to construct the long-time flow map. In contrast to the gPC method, this approach has spectral error convergence for both short and long integration times. The short-time flow map is characterized by small stretching and folding of the associated trajectories and hence can be well represented by a relatively low-degree basis. The composition of these low-degree polynomial bases then accurately describes the uncertainty behavior for long integration times. The key to the method is that the degree of the resulting polynomial approximation increases exponentially in the number of time intervals, while the number of polynomial coefficients either remains constant (for an autonomous system) or increases linearly in the number of time intervals (for a non-autonomous system). The findings are illustrated on several numerical examples including a nonlinear ordinary differential equation (ODE) with an uncertain initial condition, a linear ODE with an uncertain model parameter, and a two-dimensional, non-autonomous double gyre flow.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗