Terms Of Orthogonal Polynomials Research Articles

Application of the Polynomial Chaos Expansion to the Uncertainty Propagation in Fault Transients in Nuclear Fusion Reactors: DTT TF Fast Current Discharge

Nuclear fusion reactors are composed of several complex components whose behavior may be not certain a priori. This uncertainty may have a significant impact on the evolution of fault transients in the machine, causing unexpected damage to its components. For this reason, a suitable method for the uncertainty propagation during those transients is required. The Monte Carlo method would be the reference option, but it is, in most of the cases, not applicable due to the large number of required, repeated simulations. In this context, the Polynomial Chaos Expansion has been considered as a valuable alternative. It allows us to create a surrogate model of the original one in terms of orthogonal polynomials. Then, the uncertainty quantification is performed repeatedly, relying on this much simpler and faster model. Using the fast current discharge in the Divertor Tokamak Test Toroidal Field (DTT TF) coils as a reference scenario, the following method has been applied: the uncertainty on the parameters of the Fast Discharge Unit (FDU) varistor disks is propagated to the simulated electrical and electromagnetic relevant effects. Eventually, two worst-case scenarios are analyzed from a thermal–hydraulic point of view with the 4C code, simulating a fast current discharge as a consequence of a coil quench. It has been demonstrated that the uncertainty on the inputs (varistor parameters) strongly propagates, leading to a wide range of possible scenarios in the case of accidental transients. This result underlines the necessity of taking into account and propagating all possible uncertainties in the design of a fusion reactor according to the Best Estimate Plus Uncertainty approach. The uncertainty propagation from input data to electrical, electromagnetic, and thermal hydraulic results, using surrogate models, is the first of its kind in the field of the modeling of superconducting magnets for nuclear fusion applications.

Solvable random-matrix ensemble with a logarithmic weakly confining potential.

This work identifies a solvable (in the sense that spectral correlation functions can be expressed in terms of orthogonal polynomials), rotationally invariant random matrix ensemble with a logarithmic weakly confining potential. The ensemble, which can be interpreted as a transformed Jacobi ensemble, is in the thermodynamic limit characterized by a Lorentzian eigenvalue density. It is shown that spectral correlation functions can be expressed in terms of the nonclassical Gegenbauer polynomials C_{n}^{(-1/2)}(x) with n≥2, which have been proven to form a complete orthogonal set with respect to the proper weight function. A procedure to sample matrices from the ensemble is outlined and used to provide a numerical verification for some of the analytical results. This ensemble is pointed out to potentially have applications in quantum many-body physics.

Krylov complexity and orthogonal polynomials

Krylov complexity measures operator growth with respect to a basis, which is adapted to the Heisenberg time evolution. The construction of that basis relies on the Lanczos algorithm, also known as the recursion method. The mathematics of Krylov complexity can be described in terms of orthogonal polynomials. We provide a pedagogical introduction to the subject and work out analytically a number of examples involving the classical orthogonal polynomials, polynomials of the Hahn class, and the Tricomi-Carlitz polynomials.

Non-classical light state transfer in su(2) resonator networks

We use a normal mode approach to show full and partial state transfer in a class of coupled resonator networks with underlying su(2) symmetry that includes the so-called J_{x} photonic lattice. Our approach defines an auxiliary Hermitian coupling matrix describing the network that yields the normal modes of the system and its time evolution in terms of orthogonal polynomials. Our results provide insight on the full quantum state reconstruction time in a general su(2) network of any size and the full quantum transfer time in the J_{x} network of size 4 n + 1 with n=1,2,3,ldots For any other network sizes, the Fock state probability distribution of the initial state is conserved but the amplitudes suffer a phase shift proportional to pi /2 that results in partial quantum state transfer.

Fourier Transforms of Some Special Functions in Terms of Orthogonal Polynomials on the Simplex and Continuous Hahn Polynomials

In this paper, Fourier transform of multivariate orthogonal polynomials on the simplex is presented. A new family of multivariate orthogonal functions is obtained using the Parseval’s identity and several recurrence relations are derived.

Multivariate Orthogonal Laurent Polynomials and Integrable Systems

An ordering for Laurent polynomials in the algebraic torus $(\mathbb{C}^\*)^D$, inspired by the Cantero–Moral–Vel´azquez approach to orthogonal Laurent polynomials in the unit circle, leads to the construction of a moment matrix for a given Borel measure in the unit torus $\mathbb{T}^D$. The Gauss–Borel factorization of this moment matrix allows for the construction of multivariate biorthogonal Laurent polynomials in the unit torus, which can be expressed as last quasi-determinants of bordered truncations of the moment matrix. The associated second-kind functions are expressed in terms of the Fourier series of the given measure. Persymmetries and partial persymmetries of the moment matrix are studied and Cauchy integral representations of the second-kind functions are found, as well as Plemelj-type formulae. Spectral matrices give string equations for the moment matrix, which model the three-term relations as well as the Christoffel–Darboux formulae. Christoffel-type perturbations of the measure given by the multiplication by Laurent polynomials are studied. Sample matrices on poised sets of nodes, which belong to the algebraic hypersurface of the perturbing Laurent polynomial, are used to find a Christoffel formula that expresses the perturbed orthogonal Laurent polynomials in terms of a last quasi-determinant of a bordered sample matrix constructed in terms of the original orthogonal Laurent polynomials. Poised sets exist only for prepared Laurent polynomials, which are analyzed from the perspective of Newton polytopes and tropical geometry. Then, an algebraic geometrical characterization of prepared Laurent polynomial perturbation and poised sets is given; full-column-rankness of the corresponding multivariate Laurent–Vandermonde matrices and a product of different prime prepared Laurent polynomials leads to such sets. Some examples are constructed in terms of perturbations of the Lebesgue–Haar measure. Discrete and continuous deformations of the measure lead to a Toda-type integrable hierarchy, being the corresponding flows described through Lax and Zakharov–Shabat equations; bilinear equations and vertex operators are found. Varying size matrix nonlinear partial difference and differential equations of two-dimensional Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows are connected with a Gauss–Borel factorization of the Jacobi-type matrices and its quasi-determinants allow for expressions for the multivariate orthogonal polynomials in terms of shifted quasi-tau matrices, which generalize those that relate the Baker functions with ratios of Miwa shifted $\tau$-functions in the one-dimensional scenario. It is shown that the discrete and continuous flows are deeply connected and determine nonlinear partial difference-differential equations that involve only one site in the integrable lattice behaving as a Kadomtsev–Petviashvili-type system.

Poncelet–Darboux, Kippenhahn, and Szegő: Interactions between projective geometry, matrices and orthogonal polynomials

We study algebraic curves that are envelopes of families of polygons supported on the unit circle T. We address, in particular, a characterization of such curves of minimal class and show that all realizations of these curves are essentially equivalent and can be described in terms of orthogonal polynomials on the unit circle (OPUC), also known as Szegő polynomials. Our results have connections to classical results from algebraic and projective geometry, such as theorems of Poncelet, Darboux, and Kippenhahn; numerical ranges of a class of matrices; and Blaschke products and disk functions. This paper contains new results, some old results presented from a different perspective or with a different proof, and a formal foundation for our analysis. We give a rigorous definition of the Poncelet property, of curves tangent to a family of polygons, and of polygons associated with Poncelet curves. As a result, we are able to clarify some misconceptions that appear in the literature and present counterexamples to some existing assertions along with necessary modifications to their hypotheses to validate them. For instance, we show that curves inscribed in some families of polygons supported on T are not necessarily convex, can have cusps, and can even intersect the unit circle. Two ideas play a unifying role in this work. The first is the utility of OPUC and the second is the advantage of working with tangent coordinates. This latter idea has been previously exploited in the works of B. Mirman, whose contribution we have tried to put in perspective.

Hankel determinants of linear combinations of moments of orthogonal polynomials, II

We present a formula that expresses the Hankel determinants of a linear combination of length d+1 of moments of orthogonal polynomials in terms of a dtimes d determinant of the orthogonal polynomials. This formula exists somehow hidden in the folklore of the theory of orthogonal polynomials but deserves to be better known, and be presented correctly and with full proof. We present four fundamentally different proofs, one that uses classical formulae from the theory of orthogonal polynomials, one that uses a vanishing argument and is due to Elouafi (J Math Anal Appl 431:1253–1274, 2015) (but given in an incomplete form there), one that is inspired by random matrix theory and is due to Brézin and Hikami (Commun Math Phys 214:111–135, 2000), and one that uses (Dodgson) condensation. We give two applications of the formula. In the first application, we explain how to compute such Hankel determinants in a singular case. The second application concerns the linear recurrence of such Hankel determinants for a certain class of moments that covers numerous classical combinatorial sequences, including Catalan numbers, Motzkin numbers, central binomial coefficients, central trinomial coefficients, central Delannoy numbers, Schröder numbers, Riordan numbers, and Fine numbers.

Skew-Orthogonal Polynomials in the Complex Plane and Their Bergman-Like Kernels

Non-Hermitian random matrices with symplectic symmetry provide examples for Pfaffian point processes in the complex plane. These point processes are characterised by a matrix valued kernel of skew-orthogonal polynomials. We develop their theory in providing an explicit construction of skew-orthogonal polynomials in terms of orthogonal polynomials that satisfy a three-term recurrence relation, for general weight functions in the complex plane. New examples for symplectic ensembles are provided, based on recent developments in orthogonal polynomials on planar domains or curves in the complex plane. Furthermore, Bergman-like kernels of skew-orthogonal Hermite and Laguerre polynomials are derived, from which the conjectured universality of the elliptic symplectic Ginibre ensemble and its chiral partner follow in the limit of strong non-Hermiticity at the origin. A Christoffel perturbation of skew-orthogonal polynomials as it appears in applications to quantum field theory is provided.

Orthogonal functions related to Lax pairs in Lie algebras

We study a Lax pair in a 2-parameter Lie algebra in various representations. The overlap coefficients of the eigenfunctions of L and the standard basis are given in terms of orthogonal polynomials and orthogonal functions. Eigenfunctions for the operator L for a Lax pair for mathfrak {sl}(d+1,mathbb {C}) is studied in certain representations.

Non-local Solvable Birth\u2013Death Processes

In this paper, we study strong solutions of some non-local difference–differential equations linked to a class of birth–death processes arising as discrete approximations of Pearson diffusions by means of a spectral decomposition in terms of orthogonal polynomials and eigenfunctions of some non-local derivatives. Moreover, we give a stochastic representation of such solutions in terms of time-changed birth–death processes and study their invariant and their limit distribution. Finally, we describe the correlation structure of the aforementioned time-changed birth–death processes.

IMPLEMENTATION OF QUADRATIC AXIAL TRIAL FUNCTIONS IN THE HIGH-FIDELITY TRANSPORT CODE PROTEUS-MOC

PROTEUS-MOC is a pin-resolved high-fidelity transport code, in which the axial variation of angular flux is represented in terms of orthogonal polynomials. Currently, PROTEUS-MOC employs linear functions and requires relatively fine axial meshes to achieve high accuracy, which increases the number of axial meshes and hence the memory requirement. In this study, aiming to reduce the memory requirement and potentially the computational time by allowing larger axial meshes, we have extended the PROTEUS-MOC transport solution method to quadratic trial functions. Preliminary tests for the performance of quadratic trial functions have been performed using the 3-D C5G7 benchmark problem. Test results showed that for the same axial mesh configuration with relatively large sizes, the quadratic approximation yields about 2 to 5 times more accurate pin powers than the linear approximation, depending on the degree of axial variation of angular fluxes. The quadratic approximation also allows the use of about 3 times coarser axial meshes than the linear approximation for comparable pin power accuracy, which consequently reduces the memory requirement by about 2 times. The memory reduction is not proportional because of the increased number of coefficients in each element from 2 to 3. However, the quadratic approximation did not reduce the computational time as expected because of the deteriorated performance of the pCMFD acceleration scheme due to large axial mesh sizes.

Bound states and the potential parameter spectrum

In this article, we answer the following question: If the wave equation possesses bound states, but it is exactly solvable for only a single non-zero energy, can we find all bound state solutions (energy spectrum and associated wavefunctions)? To answer this question, we use the “tridiagonal representation approach” to solve the wave equation at the given energy by expanding the wavefunction in a series of energy-dependent square integrable basis functions in configuration space. The expansion coefficients satisfy a three-term recursion relation, which is solved in terms of orthogonal polynomials. Depending on the selected energy, we show that one of the potential parameters must assume a value from within a discrete set called the “potential parameter spectrum” (PPS). This discrete set is obtained from the spectrum of the above polynomials and can be either a finite or an infinite set. Inverting the relation between the energy and the PPS gives the bound state energy spectrum. Therefore, the answer to the above question is affirmative.

A nonlinear moment model for radiative transfer equation in slab geometry

This paper is concerned with the approximation of the radiative transfer equation for a grey medium in the slab geometry by the moment method. We develop a novel moment model inspired by the classical PN model and MN model. The new model takes the ansatz of the M1 model as the weight function and follows the primary idea of the PN model to approximate the specific intensity by expanding it around the weight function in terms of orthogonal polynomials. The weight function uses the information of the first two moments, which brings the new model the capability to approximate an anisotropic distribution. Mathematical properties of the moment model are investigated, and particularly the hyperbolicity and the characteristic structure of the Riemann problem of the model with three moments are studied in detail. Some numerical simulations demonstrate its numerical efficiency and show its superior in comparison to the PN model.

Erratum: “Series solutions of Heun-type equation in terms of orthogonal polynomials” [J. Math. Phys. 59, 113507 (2018)

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Representing by Orthogonal Polynomials for Sums of Finite Products of Fubini Polynomials

In the classical connection problem, it is dealt with determining the coefficients in the expansion of the product of two polynomials with regard to any given sequence of polynomials. As a generalization of this problem, we will consider sums of finite products of Fubini polynomials and represent these in terms of orthogonal polynomials. Here, the involved orthogonal polynomials are Chebyshev polynomials of the first, second, third and fourth kinds, and Hermite, extended Laguerre, Legendre, Gegenbauer, and Jabcobi polynomials. These representations are obtained by explicit computations.

Analysis of Electromagnetic Wave Propagation in Variable Magnetized Plasma via Polynomial Chaos Expansion

A 3-D stochastic finite-difference time-domain algorithm is developed and applied to electromagnetic (EM) wave propagation in collisional magnetized plasma characterized by a variable electron density, collision frequency, and background magnetic field. The proposed stochastic model is based on the expansion of the random/variable time-domain electric and magnetic fields in terms of orthogonal polynomials in independent random variables representative of the variable ionospheric content. EM wave propagation in magnetized plasma having low variability (small deviations) and also high variability (large deviations) of the electron density, collision frequency, and background magnetic field is studied. The stochastic algorithm is validated against brute-force Monte Carlo results. The algorithm is considerably more computationally efficient than Monte Carlo. When applied to EM wave propagation in the ionosphere, the variability of the Earth’s magnetic field and ionospheric parameters can be accounted for due to naturally varying space weather conditions and day-to-day variations, measurement errors, and so on. Although only electrons are considered here, positive and negative ions may be accommodated in a straightforward manner.

A New Hybrid form of Krawtchouk and Tchebichef Polynomials: Design and Application

In the past decades, orthogonal moments (OMs) have received a significant attention and have widely been applied in various applications. OMs are considered beneficial and effective tools in different digital processing fields. In this paper, a new hybrid set of orthogonal polynomials (OPs) is presented. The new set of OPs is termed as squared Krawtchouk–Tchebichef polynomial (SKTP). SKTP is formed based on two existing hybrid OPs which are originated from Krawtchouk and Tchebichef polynomials. The mathematical design of the proposed OP is presented. The performance of the SKTP is evaluated and compared with the existing hybrid OPs in terms of signal representation, energy compaction (EC) property, and localization property. The achieved results show that SKTP outperforms the existing hybrid OPs. In addition, face recognition system is employed using a well-known database under clean and different noisy environments to evaluate SKTP capabilities. Particularly, SKTP is utilized to transform face images into moment (transform) domain to extract features. The performance of SKTP is compared with existing hybrid OPs. The comparison results confirm that SKTP displays remarkable and stable results for face recognition system.

Series solutions of Heun-type equation in terms of orthogonal polynomials

We introduce a nine-parameter Heun-type differential equation and obtain three classes of its solution as series of square integrable functions written in terms of the Jacobi polynomial. The expansion coefficients of the series satisfy three-term recursion relations, which are solved in terms of orthogonal polynomials with continuous and/or discrete spectra. Some of these are well-known polynomials while the others are either new or modified versions of the known ones.

Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane

We study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated to a certain normal matrix model. The model depends on a parameter and the asymptotic distribution of the eigenvalues undergoes a transition for a special value of the parameter, where it develops a corner-type singularity. In the double scaling limit near the transition we determine the asymptotic behaviour of the orthogonal polynomials in terms of a solution of the Painlev\'e IV equation. We determine the Fredholm determinant associated to such solution and we compute it numerically on the real line, showing also that the corresponding Painlev\'e transcendent is pole-free on a semiaxis.