In this work, we extend the so-called mapped bases or fake nodes approach to the barycentric rational interpolation of Floater-Hormann and to AAA approximants. More precisely, we focus on the reconstruction of discontinuous functions by the S-Gibbs algorithm introduced in [12]. Numerical tests show that it yields an accurate approximation of discontinuous functions

Berrut, J. -P.

De Marchi, S.

Elefante, G.

Marchetti, F.

English

J. -P. Berrut

S. De Marchi

G. Elefante

F. Marchetti

Archivio istituzionale della ricerca - Università di Padova

Original Citation:Treating the Gibbs phenomenon in barycentric rational interpolation and approximation via the S-GibbsalgorithmElsevierPublisher:Published version:DOI:Terms of use:Open Access(Article begins on next page)This article is made available under terms and conditions applicable to Open Access Guidelines, as described athttp://www.unipd.it/download/file/fid/55401 (Italian only)Availability:This version is available at: 11577/3323880 since: 2020-01-25T08:37:57Z10.1016/j.aml.2019.106196Università degli Studi di PadovaPadua Research Archive - Institutional RepositoryTreating the Gibbs phenomenon inbarycentric rational interpolation and approximationvia the S-Gibbs algorithmJ.-P. Berrut∗ , S. De Marchi∗∗ , G. Elefante∗ , F. Marchetti∗∗∗∗Département de Mathématiques, Université de Fribourg, Switzerland;∗∗Dipartimento di Matematica “Tullio Levi-Civita”, Università di Padova, Italy;∗∗∗Dipartimento di Salute della Donna e del Bambino, Università di Padova, ItalyAbstractIn this work, we extend the so-called mapped bases or fake nodes approachto the barycentric rational interpolation of Floater-Hormann and to AAA ap-proximants. More precisely, we focus on the reconstruction of discontinuousfunctions by the S-Gibbs algorithm introduced in [? ]. Numerical tests showthat it yields an accurate approximation of discontinuous functions.Keywords: Barycentric rational interpolation, Gibbs phenomenon,Floater-Hormann interpolant, AAA algorithm, fake nodes2010 MSC: : 65D05, 41A05, 65D15.1. Introduction1In the seminal paper [? ], Floater and Hormann (FH) have introduced a2family of linear barycentric rational interpolants, which contains the first Berrut3interpolant [? ? ]. FH interpolants have shown good approximation properties4for smooth functions, in particular using equidistant nodes. Because of their5high accuracy, these interpolants have been applied in several frameworks, such6as for solving Volterra integral equations [? ], or as collocation methods for7nonlinear parabolic partial differential equations [? ]. Among other favourable8properties, the Lebesgue constant grows logarithmically with the number of9nodes [? ]. Other instances in which linear barycentric rational interpolation10is extremely efficient, actually exponentially convergent, is the trigonometric11interpolant presented in [? ] and used with conformally mapped equispaced12points [? ? ], and its special case on the interval, Berrut’s second interpolant [?13] between conformally mapped Chebyshev points [? ]. The Berrut interpolants14Email addresses: jean-paul.berrut@unifr.ch (J.-P. Berrut∗), demarchi@math.unipd.it(S. De Marchi∗∗), giacomo.elefante@unifr.ch (G. Elefante∗),francesco.marchetti.1@phd.unipd.it (F. Marchetti∗∗∗)Preprint submitted to Elsevier January 22, 2020enjoy a Lebesgue constant that grows logarithmically for a wide class of nodes15as well [? ? ].16In [? ], the Adaptive Antoulas-Anderson (AAA) greedy algorithm for com-17puting a barycentric rational approximant has been presented. This recent18method leads to impressively well-conditioned bases, which can be used in var-19ious fields, such as in computing conformal maps, or in rational minimax ap-20proximations (see [? ? ]). Note that a similar approach has been considered in21[? ] for kernel-based interpolation.22The FH interpolants and the approximants obtained by the AAA algorithm23suffer from the well-known Gibbs phenomenon, when the underlying function24presents jump discontinuities. For a general overview of that phenomenon,25interested readers may refer to [? ? ].26In [? ], a new interpolation procedure has been suggested in the frame-27work of univariate polynomial interpolation to reduce the Gibbs phenomenon.28The method essentially maps the polynomial basis, in which the interpolant is29expressed, by a suitable map S, or equivalently uses the so-called fake nodes,30without resampling the underlying function. This led to the S-Gibbs algorithm,31which basically constructs the map S that is then used for eliminating the Gibbs32effect (see S-Gibbs [? , Algorithm 2]).33In this work, we propose an extension of the fake nodes approach (cf. [? ]) to34the framework of barycentric rational interpolation, focusing on FH interpolants35and the AAA algorithm.362. Barycentric polynomial interpolation37Let Xn := {xi : i = 0, . . . , n} be a set of n+1 distinct nodes in I = [a, b] ⊂ R,38increasingly ordered from x0 = a to xn = b. We consider the problem of39interpolating a function f : I −→ R given the set of samples Fn := {fi = f(xi) :40i = 0, . . . , n}.41It is well-known (see e.g. [? ]) that it is possible to write the unique42interpolating polynomial Pn[f ] of degree at most n of f at Xn for any x ∈ I in43the second barycentric form44Pn[f ](x) =∑ni=0λix−xi fi∑ni=0λix−xi, (1)where λi =∏j 6=i1xi − xjare the so-called weights. This expression is one of45the most stable formulas for evaluating Pn[f ] (see [? ]). If the weights λi are46changed to other nonzero weights, say wi, then the corresponding barycentric47rational function48rn[f ](x) =∑ni=0wix−xi fi∑ni=0wix−xi(2)still satisfies the interpolation conditions rn[f ](xi) = fi, i = 0, . . . , n. For more49details about barycentric rational interpolation, we refer to [? ].5022.1. The Floater-Hormann family51Let n ∈ N, d ∈ {0, . . . , n}. Let pi, i = 0, . . . , n−d denote the unique polyno-mial interpolant of degree at most d interpolating the d+ 1 points (xk, fk), k =i, . . . , i+ d. One can write the FH rational interpolant asRn,d[f ](x) =∑n−di=0 λi(x)pi(x)∑n−di=0 λi(x), where λi(x) =(−1)i(x− xi) · · · (x− xi+d),which interpolates f at the set of nodes Xn. It has been proved in [? ] that52Rn,d[f ] has no real poles and that it reduces to the unique interpolating poly-53nomial of degree at most n when d = n.54One can derive the barycentric form of this family of interpolants as well.Indeed, with considering the sets Ji = {k ∈ {0, 1, . . . , n − d} : i − d ≤ k ≤ i},one hasRn,d[f ](x) =∑ni=0wix−xi fi∑ni=0wix−xi, where wi = (−1)i−d∑k∈Jij+d∏j=kj 6=i1|xi − xj |.2.2. The AAA algorithm55Let us consider a set of points XN with a large value of N and a function f .The AAA algorithm [? ] is a greedy technique that in the step m ≥ 0 considersthe set X (m) = XN \ {x0, . . . , xm} and constructs the interpolantrm[f ](x) =∑mi=0wix−xi fi∑mj=0wjx−xj=n(x)d(x),by solving the discrete least squares problemmin‖fd− n‖X (m) ‖w‖2 = 1,for the unknown vector w = (w0, . . . , wm), where‖·‖X (m) is the discrete 2-norm56over X (m). The subsequent data site xm+1 ∈ X (m) is chosen by maximizing the57residual |f(x)− n(x)/d(x)| with respect to x ∈ X (m).583. Mapped bases and fake nodes in barycentric rational interpolation59Here we investigate the extension of the interpolation method presented in60[? ] to the Floater-Hormann interpolants and to the approximants produced61via the AAA algorithm.62Let S : I −→ R be a mapping that we assume injective. We construct the“new” interpolant rSn : I −→ R at the nodes Xn and function values Fn asrSn [f ](x) :=∑ni=0wiS(x)−S(xi)fi∑ni=0wiS(x)−S(xi).3As discussed in [? , p. 3] in the polynomial interpolation setting, we can see63rSn [f ] from two different perspectives.64First, since the interpolant rn[f ] defined in (2) admits a cardinal basis form65rn[f ](x) =∑nj=0 fjbj(x), where bj(x) =wjx−xj∑ni=0wix−xiis the j-th basis function, in66the same spirit, we can write rSn [f ] in the mapped cardinal basis form rSn [f ](x) =67 ∑ni=0 fibSi (x), where bSj (x) =wjS(x)−S(xj)∑ni=0wiS(x)−S(xi)is the j-th mapped basis function.68Using the S mapping approach, a more stable interpolant may arise. We69present an upper bound on the S-Lebesgue constant which involves the classical70Lebesgue constant.71Theorem 1. Let Λn(Xn) = maxx∈In∑j=0|bj(x)| and ΛSn(Xn) := maxx∈In∑j=0|bSj (x)| bethe classical and the S-Lebesgue constants, respectively. We then haveΛSn(Xn) ≤ CΛn(Xn),where C = maxk Akmink AkwithAk = maxx∈In∏l=0l 6=k∣∣∣∣S(x)− S(xl)x− xl∣∣∣∣ , Ak = minx∈In∏l=0l 6=k∣∣∣∣S(x)− S(xl)x− xl∣∣∣∣ .Proof. We bound each basis function bSj in terms of bj for all x ∈ I. We compute|bSj (x)| =∣∣∣∣∣wjS(x)−S(xj)∑ni=0wiS(x)−S(xi)∣∣∣∣∣ =∣∣∣∣∣wjS(x)−S(xj)∏nl=0(S(x)− S(xl))∑ni=0wiS(x)−S(xi)∏nl=0(S(x)− S(xl))∣∣∣∣∣=∣∣∣∣∣∣∣wj∏nl=0l 6=j(S(x)− S(xl))∑ni=0 wi∏nl=0l 6=i(S(x)− S(xl))∣∣∣∣∣∣∣=∣∣∣∣∣∣∣∣∣∣∣∣∣wj∏nl=0l 6=j(S(x)− S(xl))∏nm=0m 6=j x−xm∏nm=0m 6=jx−xm∑ni=0 wi∏nl=0l 6=i(S(x)− S(xl))∏nm=0m6=i x−xm∏nm=0m6=ix−xm∣∣∣∣∣∣∣∣∣∣∣∣∣=∣∣∣∣∣∣∣wjx−xj∏nl=0l 6=jS(x)−S(xl)x−xl∑ni=0wix−xi∏nl=0l 6=iS(x)−S(xl)x−xl∣∣∣∣∣∣∣≤ maxk Akmink Ak∣∣∣∣∣wjx−xj∑ni=0wix−xi∣∣∣∣∣ = maxk Akmink Ak |bj(x)| = C|bj(x)|.724Second, equivalently to the above mapped basis perspective, we can discussthe construction of the interpolant rSn [f ] via the so-called fake nodes approach.Let r̃n[g] be the barycentric interpolant as in (2) that interpolates, at the setof fake nodes S(Xn), the function g : S(I) −→ R, making use of the samefunctional values Fn, that isg|S(Xn) = f|Xn .Observe that rSn [f ](x) = r̃n[g](S(x)) for every x ∈ I. Hence, we may also build73rSn [f ] upon a standard barycentric interpolation process, thereby providing a74more intuitive interpretation of the method.75The choice of the mapping S is crucial for the accuracy of the proposed76interpolant rSn [f ]. Here, we assume that f presents jump discontinuities and77we adopt the so-called S-Gibbs Algorithm (SGA) [? ] to construct an effective78mapping S.794. Numerical Examples80In this section, we test the fake nodes approach with SGA in the framework81of FH interpolants and the AAA algorithm for approximation. We fix k = 1082in the SGA. As observed in [? ], also in this setting the choice of the shifting83parameter is non-critical as long as it is taken "sufficiently large".84In I = [−5, 5] we consider the discontinuous functionsf1(x) =e1x+5.5 , −5 ≤ x < −3cos(3x), −3 ≤ x < 2−x330 + 2, 2 ≤ x ≤ 5.f2(x) =log(− sin(x/2)), −5 ≤ x < −2.5tan(x/2), −2.5 ≤ x < 2arctan(e−1x−5.1 ), 2 ≤ x ≤ 5.We evaluate the constructed interpolants on a set of 5000 equispaced evaluation85points Ξ = {x̄i = −5 + i1000 : i = 0, . . . , 5000} and compute the Relative86Maximum Absolute Error RMAE = maxi|rn(x̄i)−f(x̄i)||f(x̄i)| and the same for rSn .87The FH interpolants88Here, we take various sets of equispaced nodes Xn = {−5 + 5in : i = 0, . . . , n},89varying the size of n. The results are displayed in Figure 1. We observe that90the proposed reconstruction via the fake nodes approach by far outperforms the91standard technique.92Figure 2 displays a comparison between the direct application of the FH93interpolant and the one modified by the SGA.94The AAA algorithm95As the starting set for the AAA algorithm, we consider 10000 nodes randomly96uniformly distributed in I, which we denote by Xrand.975(a) Interpolation of f1 with d = 1. (b) Interpolation of f1 with d = 8.(c) Interpolation of f2 with d = 1. (d) Interpolation of f2 with d = 8.Figure 1: The RMAE for f1 and f2 when one doubles the number of nodes from 40 to 2560.In blue, the standard interpolant Rn,d. In red, the proposed interpolant Rsn,d.Looking at Table 1, we observe that using the AAA algorithm with starting98set S(Xrand) (indicated in the Table as AAAS), that is, constructing the ap-99proximants via the fake nodes approach, does not suffer from the effects of the100Gibbs phenomenon. For both approximants we fix the maximum degree to 20101and to 40 (by default 100 in the algorithm).102f mmax AAA AAAsf1 20 1.5674e+02 8.5189e-0540 1.4308e+00 2.9550e-09f2 20 3.6034e+02 2.2066e-0740 1.4656e+00 6.3485e-11Table 1: RMAE for AAA and AAAs approximants5. Conclusions103This work introduces an extension of the fake nodes approach to barycentric104rational approximation, in particular to the family of FH interpolants and to1056(a) Direct approximation. (b) Using the proposed approach.Figure 2: Interpolation of the function f1 using 80 nodes and d = 8 in the FH interpolant. Inred the function and in blue the interpolant.the AAA algorithm for approximation, focusing on the treatment of the Gibbs106phenomenon via the S-Gibbs algorithm. The results show that the proposed107reconstructions outperform their classical versions, as they are not affected by108distortions and oscillations.109Acknowledgement110This work has been performed within the Italian Network on Approxi-111mation (RITA), and with the support of the GNCS-INdAM funds 2019 and112NATIRESCO BIRD181249 project.113References114[1] A. Abdi, J.-P. Berrut, and S. A. Hosseini, The linear barycentric115rational method for a class of delay Volterra integro-differential equations,116J. Sci. Comput., 75 (2018), pp. 1757–1775.117[2] R. Baltensperger, Some results on linear rational trigonometric inter-118polation, Comput. Math. Appl., 43 (2002), pp. 737–746.119[3] R. Baltensperger and J.-P. Berrut, The linear rational collocation120method, J. Comput. Appl. Math., 134 (2001), pp. 243–258.121[4] J.-P. 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Treating the Gibbs phenomenon in barycentric rational interpolation and approximation via the S-Gibbs algorithm

In this work, we extend the so-called mapped bases or fake nodes approach to the  barycentric rational interpolation of Floater–Hormann and to AAA approximants. More  precisely, we focus on the reconstruction of discontinuous functions by the S-Gibbs  algorithm introduced in De Marchi et al. (2020). Numerical tests show that it yields an  accurate approximation of discontinuous functions

Treating the Gibbs phenomenon in barycentric rational interpolation and approximation via the S-Gibbs algorithm

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