A reduced basis method for fractional diffusion operators I

Abstract

We propose and analyze new numerical methods to evaluate fractional norms and apply fractional powers of elliptic operators. By means of a reduced basis method, we project to a small dimensional subspace where explicit diagonalization via the eigensystem is feasible. The method relies on several independent evaluations of $({{\,\mathrm{I}\,}}-t_i^2\Delta )^{-1}f$, which can be computed in parallel. We prove exponential convergence rates for the optimal choice of sampling points $t_i$, provided by the so-called Zolotarëv points. Numerical experiments confirm the analysis and demonstrate the efficiency of our algorithm.

1 Introduction

Fractional powers of differential operators are a field of substantial interest in different branches of mathematics. Their augmented appearance in real world problems, such as ecology [11], finance [6], image processing [25], material science [9], and porous media flow [10] has given rise to several approaches in order to understand and analyze problems of this kind.

Typically, direct computations rely on matrix approximations L of the desired operator, whose $s^{th}$-power is computed subsequently. This procedure requires diagonalization of L, which amounts to a large number of time-consuming eigenvalue problems, making this approach inapplicable for general purposes. Adding to this difficulty, many practical scenarios demand numerical methods that allow efficient evaluations in s. In [20], the fractional exponent is determined in a way, such that the observed data matches the mathematical model. In [44], s serves as control parameter to minimize a given cost functional. All these investigations suggest that one is interested in the entire family of solutions for $s\in (0,1)$ rather than one specific value of s. The demand for suitable methods that address these problems has substantially increased throughout the last years.

Fractional powers of the Laplace operator appear to be of particular interest. Widely varying definitions of $(-\Delta )^s$ have emerged, e.g., as pseudo differential operator defined by the Fourier transform, by means of its involved eigensystem, as a singular integral operator, or as inverse of the Riesz potential operator. All these definitions turn out to be equivalent in ${\mathbb {R}}^n$, see [30]. This result no longer holds as bounded domains are incorporated. A detailed excursion of its versatile definitions as well as the comparison of both existing and newly proposed numerical schemes is explained in [32] and [13].

A difficulty that all fractional operators have in common is their nonlocal character. Caffarelli and Silvestre managed to avoid this inconvenience in [22] by relating any fractional power of the Laplacian in ${\mathbb {R}}^n$ to a Dirichlet-to-Neumann map of an involved harmonic extension problem in ${\mathbb {R}}^n\times {\mathbb {R}}^+$, providing a local realization of $(-\Delta )^s$. Adaptions for bounded domains $\Omega$ have been conducted in [19, 21], and [23], yielding a boundary value problem on the semi-infinite cylinder $\mathcal {C} := \Omega \times {\mathbb {R}}^+$. Enhancements for a more general class of operators has been presented in [45].

A large number of methods exploits the structure of harmonic extension techniques to approximate fractional differential operators and their inverses, see [2, 5, 7], and [4]. In [38], the solution of the aforementioned boundary value problem is computed on the truncated cylinder $\mathcal {C}_{\gamma } := \Omega \times [0,\gamma )$, with $\gamma > 0$ of moderate size, by standard finite element techniques, at the cost of one additional space dimension. Truncation can be justified by the fact that the solution decreases exponentially in the extended direction. Other strategies rely on block-wise low-rank approximation of the associated stiffness-matrix [36]. Of particular interest are fractional elliptic operators in context of parabolic equations. Tackling problems of this kind has been a matter of several recent publications, e.g., [3, 15, 37, 43].

In this article, we interpret fractional operators as interpolation operators and make use of the K-method [40] to provide attractive approximations for any arbitrary symmetric, uniformly elliptic operator ${\mathcal {L}}$. This approach requires the knowledge of the map , whose smoothness in t justifies the usage of standard reduced basis technology, see e.g., [8, 33, 34]. Providing an optimal choice of snapshots $t_i$, we pursue a proper orthogonal decomposition (POD) strategy, see e.g., [29], to project a matrix approximation L of the desired operator to a lower dimensional space, where its fractional power can be determined directly.The decoupled structure of the projection trivially admits an efficient implementation in parallel. Multigrid preconditioner can be utilized, whose convergence rates are bounded independently of the shift parameter $t_i$ and uniform mesh size h. The proposed method can be interpreted as model order reduction of the approach devised in [38], without requiring truncation of the domain. Among others, it provides accurate approximations for evaluations of both types $s\mapsto (-\Delta )^su$ and $u\mapsto (-\Delta )^su$ with considerably reduced computational expense. The arising operator incorporates a nonlinear dependency in u. This inconvenience is compensated by its analytically affirmed exponential convergence property, while, at the same time, the computational effort grows only logarithmically in the condition number. Emerging estimates rely on rational approximation of the bivariate function $(1+\lambda ^2t^2)^{-1}$ over a suitable spectral interval of the discrete operator. Realization of the inverse operator and parabolic problems that involve fractional diffusion operators is the matter of a consecutive paper.

We emphasize that this is not the first investigation that recognizes the importance of in context of fractional operators. The approach developed in [17] and later improved in [16] relies on the Dunford-Taylor integral representation of . A sinc quadrature scheme especially tailored for integrals of this type is presented and requires the evaluation of at the quadrature nodes. Exponential decay of the error in the number of nodes is shown. Further modifications are discussed in [14] where computations of the quadrature are accelerated by means of a reduced basis method. While the construction of the reduced basis differs from the one we pursue, the involved proof of convergence also relies on rational approximations and partially follows the outline of our analysis. A similar approach is proposed in [24] in consideration of a different quadrature rule, where exponential convergence rates are observed numerically. A model order reduction that relies on the extension method is proposed in [5].

It remains to be mentioned that we are not the first to relate fractional powers of differential operators to rational approximation. In [27], for instance, the so-called best uniform rational approximation (BURA) of $t^{-s}$, which was originally proposed in [28], is utilized as matrix function to approximate $(-\Delta )^{-s}$.

The paper is organized as follows. In Sect. 2, we introduce three different concepts of Hilbert space interpolation. They serve as abstract template to provide a setting that is suitable for the study of our problem. Equivalence of all three methods appears to be the main result of this section. A reduced basis strategy is applied in Sect. 3. Feasible choices of the reduced space and the efficient implementation of its arising reduced basis interpolation norms are taken into account. Having understood its underlying structure, we proceed, in Sect. 4, to deduce the induced fractional operator, providing the essential definition of this paper. Numerical analysis is performed in Sect. 5. The optimal choice of the reduced space is elaborated, yielding exponential decay in the error for both norm and operator approximation. The core of this paper is summarized in Theorem 12. Thereupon, in Sect. 6, we conduct several numerical examples that illustrate the performance of our method. Eventually, in the appendix, we prove two technical results that are referred to within Sect. 2.

2 Notation and preliminaries

In this section, we establish the notation and terminology we utilize throughout this paper and introduce several function spaces we address in the subsequent.

Throughout what follows, let the induced norm $\Vert \cdot \Vert _{}$ of a Hilbert space $(\mathscr {V},\langle \cdot ,\cdot \rangle _{})$ be defined as

$$\begin{aligned} \Vert \cdot \Vert _{} := \sqrt{\langle \cdot ,\cdot \rangle _{}}. \end{aligned}$$

(2.1)

Conversely, given a Banach space $(\mathscr {V},\Vert \cdot \Vert _{})$, such that $\Vert \cdot \Vert _{}$ satisfies the parallelogram law, we define the induced scalar product of $\Vert \cdot \Vert _{}$ as the unique scalar product $\langle \cdot ,\cdot \rangle _{}$ on $\mathscr {V}$ that satisfies (2.1), obtained by polarization identity. Whenever referring to a Banach space $(\mathscr {V},\Vert \cdot \Vert _{})$ as Hilbert space, we mean that $\Vert \cdot \Vert _{}$ induces a scalar product $\langle \cdot ,\cdot \rangle _{}$, such that $(\mathscr {V},\langle \cdot ,\cdot \rangle _{})$ is a Hilbert space.

2.1 Hilbert space interpolation

Throughout what follows, let $\mathscr {V}_i,$ $i = 0,1$, denote two real Hilbert spaces with inner products $\langle \cdot ,\cdot \rangle _{i}$, respectively, such that ${\mathscr {V}}_1\subseteq {\mathscr {V}}_0$ is dense with compact embedding. It is well-known that there exists an orthonormal basis $(\varphi _k)_{k=1}^\infty$ of ${\mathscr {V}}_0$ and a sequence of positive real numbers $(\lambda _k)_{k=1}^\infty$ with $\lambda _k\longrightarrow \infty$ as $k\longrightarrow \infty$, satisfying

for all $k\in {\mathbb {N}}$. Along with these premises, we introduce, based on [12, 18, 31], the first of three space interpolation techniques. For each $s\in (0,1)$ we define an interpolation space between ${\mathscr {V}}_0$ and ${\mathscr {V}}_1$ by

equipped with its Hilbert interpolation norm

$\left( [{\mathscr {V}}_0,{\mathscr {V}}_1]_{\mathrm{{H}^s}},\Vert \cdot \Vert _\mathrm{{H}^s({\mathscr {V}}_0,{\mathscr {V}}_1)}\right)$ incorporates a Hilbert space structure and satisfies

Another approach is provided by the real method of interpolation in terms of the K-functional. It was first published by Peetre [40], Lions and Magenes [31] and also works for Banach spaces. Let $\Vert \cdot \Vert _{0}$ and $\Vert \cdot \Vert _{1}$ denote the induced norms on ${\mathscr {V}}_0$ and ${\mathscr {V}}_1$, respectively. We define for all $t>0$ and $u\in {\mathscr {V}}_0$ the K-functional as

to obtain the K-norm

Along with its inner product, obtained by parallelogram law, the norm induces a Hilbert space

which again turns out to be intermediate.

Based on the work of Peetre and Lions, it has been shown that $[{\mathscr {V}}_0,{\mathscr {V}}_1]_{{K}^s}$ can be characterized as space of trace, see [46, 47]. The arising norm, which turns the trace space into a Banach space, is known to be equivalent to the K-norm. We affirm this observation by proving that these norms are not only equivalent but do also coincide up to a multiplicative constant. This result is well-known for some particular choices of ${\mathscr {V}}_0$ and ${\mathscr {V}}_1$, see e.g., [23,  Proposition 2.1], but has not been recorded in its most general setting to the best of our knowledge. To make matters precise, we investigate some technical results.

For each $s\in (0,1)$ let $\alpha := 1-2s$ henceforth. For all $i = 0,1$, we define the space $L_2({\mathbb {R}}^+,\mathscr {V}_i;y^\alpha )$ of all measurable functions $v:{\mathbb {R}}^+\longrightarrow \mathscr {V}_i$, such that

and further

Thereupon, we introduce

and endow it with the norm

This space is amenable to trace evaluations, as the following Theorem shows.

Theorem 1

There exists a linear, surjective trace operator

such that for all $v\in \mathbb {V}{({\mathscr {V}}_0,{\mathscr {V}}_1; y^\alpha )}$ there holds

$$\begin{aligned} \sqrt{d_s}\Vert {{\,\mathrm{tr}\,}}v\Vert _\mathrm{{{H}}^s({\mathscr {V}}_0,{\mathscr {V}}_1)}\le \Vert v\Vert _{\mathbb {V}{({\mathscr {V}}_0,{\mathscr {V}}_1; y^\alpha )}}. \end{aligned}$$

(2.2)

By $d_s$ we refer to a positive constant whose value can be specified by means of the Gamma function,

Proof

See Appendix. $\square$

Theorem 1 justifies the introduction of an interpolation space

$$\begin{aligned}{}[{\mathscr {V}}_0,{\mathscr {V}}_1]_{\mathrm{{E}^s}} := {{\,\mathrm{tr}\,}}(\mathbb {V}{({\mathscr {V}}_0,{\mathscr {V}}_1; y^\alpha )}), \end{aligned}$$

(2.3)

endowed with the extension-norm

$$\begin{aligned} \Vert u\Vert _\mathrm{{E}^s({\mathscr {V}}_0,{\mathscr {V}}_1)} := \inf \limits _{\begin{array}{c} \mathscr {U}\in \mathbb {V}{({\mathscr {V}}_0,{\mathscr {V}}_1; y^\alpha )}\\ {{\,\mathrm{tr}\,}}\mathscr {U} = u \end{array}} \Vert \mathscr {U}\Vert _{\mathbb {V}{({\mathscr {V}}_0,{\mathscr {V}}_1; y^\alpha )}}, \end{aligned}$$

(2.4)

which is is well-defined by standard arguments from calculus of variation. Due to surjectivity, (2.3) coincides with $[{\mathscr {V}}_0,{\mathscr {V}}_1]_{\mathrm{{H}^s}}$.

By means of Euler-Lagrange formalism, we observe that the infimum in (2.4) is the unique solution of an involved variational formulation. Thereupon, we introduce the following definition.

Definition 1

The $\alpha$-harmonic extension $\mathscr {U}$ of $u\in [{\mathscr {V}}_0,{\mathscr {V}}_1]_{\mathrm{{E}^s}}$ is defined as the unique solution of the variational formulation: Find $\mathscr {U}\in \mathbb {V}{({\mathscr {V}}_0,{\mathscr {V}}_1; y^\alpha )}$, such that for all $y\in {\mathbb {R}}^+$ and $\mathscr {W}\in \mathbb {V}{({\mathscr {V}}_0,{\mathscr {V}}_1; y^\alpha )}$ there holds

$$\begin{aligned} \begin{aligned} \langle y^\alpha \mathscr {U}(y),\mathscr {W}(y)\rangle _{1} - \frac{\partial }{\partial y}\left( y^\alpha \langle \mathscr {U}'(y),\mathscr {W}(y)\rangle _{0}\right) = 0,\\ {{\,\mathrm{tr}\,}}\mathscr {U} = u. \end{aligned} \end{aligned}$$

(2.5)

Lemma 1

Let $\mathscr {U}$ denote the $\alpha$-harmonic extension of $u\in [{\mathscr {V}}_0,{\mathscr {V}}_1]_{\mathrm{{{E}}^s}}$. Then there holds

Proof

Follows directly from the fact that (2.5) is the Euler-Lagrange equation of the minimization problem in (2.4). $\square$

As the following Theorem shows, all three interpolation methods coincide.

Theorem 2

Let ${\mathscr {V}}_0, {\mathscr {V}}_1$ denote two Hilbert spaces, such that ${\mathscr {V}}_1\subseteq {\mathscr {V}}_0$ is dense with compact embedding. Then there holds

where $C_s:= \sqrt{\frac{2\sin (\pi s)}{\pi }}$.

Proof

See Appendix. $\square$

Throughout what follows, we denote by $[{\mathscr {V}}_0,{\mathscr {V}}_1]_{s}$ the unique interpolation space emerging from one and hence from all three interpolation methods. The interpolation norms $\Vert \cdot \Vert _\mathrm{{H}^s({\mathscr {V}}_0,{\mathscr {V}}_1)}$ and $\Vert \cdot \Vert _\mathrm{{K}^s({\mathscr {V}}_0,{\mathscr {V}}_1)}$ satisfy the parallelogram law on $[{\mathscr {V}}_0,{\mathscr {V}}_1]_{s}$. By virtue of Theorem 2, this property also applies to $\Vert \cdot \Vert _\mathrm{{E}^s({\mathscr {V}}_0,{\mathscr {V}}_1)}$. We summarize these observations in the following corollary.

Corollary 1

All three interpolation norms, $\Vert \cdot \Vert _\mathrm{{{E}}^s({\mathscr {V}}_0,{\mathscr {V}}_1)}$, $\Vert \cdot \Vert _\mathrm{{{H}}^s({\mathscr {V}}_0,{\mathscr {V}}_1)}$, and $\Vert \cdot \Vert _\mathrm{{{K}}^s({\mathscr {V}}_0,{\mathscr {V}}_1)}$, induce a respective scalar product, $\langle \cdot ,\cdot \rangle _\mathrm{{{E}}^s({\mathscr {V}}_0,{\mathscr {V}}_1)}$, $\langle \cdot ,\cdot \rangle _\mathrm{{{H}}^s({\mathscr {V}}_0,{\mathscr {V}}_1)}$, and $\langle \cdot ,\cdot \rangle _\mathrm{{{K}}^s({\mathscr {V}}_0,{\mathscr {V}}_1)}$, such that for all $v,w\in [{\mathscr {V}}_0,{\mathscr {V}}_1]_{s}$

Definition 2

Let $\Vert \cdot \Vert _{s}\in \{\Vert \cdot \Vert _\mathrm{{E}^s({\mathscr {V}}_0,{\mathscr {V}}_1)}, \Vert \cdot \Vert _\mathrm{{H}^s({\mathscr {V}}_0,{\mathscr {V}}_1)}, \Vert \cdot \Vert _\mathrm{{K}^s({\mathscr {V}}_0,{\mathscr {V}}_1)}\}$. By means of the Riesz-representation Theorem, we define the induced operator of $\Vert \cdot \Vert _s$ as the unique linear function ${\mathcal {L}}_s:[{\mathscr {V}}_0,{\mathscr {V}}_1]_{s}\longrightarrow \mathscr {V}_0$, such that

for each $w\in [{\mathscr {V}}_0,{\mathscr {V}}_1]_{s}$, where $\langle \cdot ,\cdot \rangle _{s}$ refers to the induced scalar product of $\Vert \cdot \Vert _{s}$.

Corollary 1 immediately reveals the following result.

Corollary 2

All three interpolation norms, , , and $\Vert \cdot \Vert _\mathrm{{{K}}^s({\mathscr {V}}_0,{\mathscr {V}}_1)}$, induce an operator, , , and , such that for all

Remark 1

Let $\Omega \subseteq {\mathbb {R}}^d$, $d\in {\mathbb {N}}$, be a bounded domain with Lipschitz boundary. Along with the choice ${\mathscr {V}}_0= (L_2(\Omega ),\Vert \cdot \Vert _{L_2})$ and ${\mathscr {V}}_1= (H_0^1(\Omega ),\Vert \nabla \cdot \Vert _{L_2})$, ${\mathcal {L}}_{\textsc {H}^s({\mathscr {V}}_0,{\mathscr {V}}_1)}$ coincides with the spectral fractional Laplacian subject to homogeneous Dirichlet boundary conditions and ${\mathcal {L}}_{\textsc {E}^s({\mathscr {V}}_0,{\mathscr {V}}_1)}$ with the involved Dirichlet-to-Neumann map from Caffarelli and Silvestre.

2.2 The finite element framework

Results from Sect. 2.1 are also valid in a discretized setting. Depending on two fixed Hilbert spaces ${\mathscr {V}}_0$ and ${\mathscr {V}}_1$ which satisfy the premises from Sect. 2.1, we denote by $\textsc {V}_h\subseteq {\mathscr {V}}_1$ a conforming finite element space of dimension N henceforth. Further, let $(b_{k})_{k=1}^N\subseteq \textsc {V}_h$ denote an arbitrary basis of $\textsc {V}_h$. By $M, A\in {\mathbb {R}}^{N\times N}$, we refer to the mass- and stiffness-matrix of $\textsc {V}_h$, arising from finite element discretization in terms of

$$\begin{aligned} M_{ji} = \langle b_i,b_j\rangle _{0},\qquad A_{ji} = \langle b_i,b_j\rangle _{1}. \end{aligned}$$

(2.6)

Due to its finite dimensional nature, the spaces $(\textsc {V}_h,\Vert \cdot \Vert _{0})$ and $(\textsc {V}_h,\Vert \cdot \Vert _{1})$ satisfy the conditions from Sect. 2.1, such that the discrete interpolation norms on $\textsc {V}_h$

$$\begin{aligned} \begin{aligned} \Vert u_h\Vert _{E^s} :=&\Vert u_h\Vert _{{E}^s((\textsc {V}_h,\Vert \cdot \Vert _{0}),(\textsc {V}_h,\Vert \cdot \Vert _{1}))},\quad \Vert u_h\Vert _{H^s} := \Vert u_h\Vert _\mathrm{{H}^s((\textsc {V}_h,\Vert \cdot \Vert _{0}),(\textsc {V}_h,\Vert \cdot \Vert _{1}))}, \\&\Vert u_h\Vert _{K^s}^2 := \int _{0}^{\infty }t^{-2s-1}\underbrace{{{\mathrm {K}}}_{((\textsc {V}_h,\Vert \cdot \Vert _{0}),(\textsc {V}_h,\Vert \cdot \Vert _{1}))}^2(t;u_h)}_{=: K^2(t;u_h)}\,dt, \end{aligned} \end{aligned}$$

(2.7)

are well-defined. The finite element space equipped with each of these norms is a Banach space, inducing both scalar product, $\langle \cdot ,\cdot \rangle _{E^s}$, $\langle \cdot ,\cdot \rangle _{H^s}$, $\langle \cdot ,\cdot \rangle _{K^s}$, and operator, ${\mathcal {L}}_{E^s}$, ${\mathcal {L}}_{H^s}$, ${\mathcal {L}}_{K^s}$, respectively. The aim of this paper is to provide an accurate approximation of these operators with considerably reduced computational expense. By virtue of Corollary 2, it suffices to address this problem in any of those three interpolation settings. Each of them comes with its own benefits and difficulties attached. In the following section, we exploit the advantages of all three strategies to derive a computationally beneficial norm approximation, such that the induced operator satisfies the desired properties.

3 Approximation of the interpolation norms

The goal of this section is to devise an accurate approximation of the discrete interpolation norms, introduced in (2.7), with downsized computational effort. For convenience, we neglect the subscript h for all finite element functions $u_h\in \textsc {V}_h$ and solely write u henceforth. Furthermore, by $(\varphi _k,\lambda _k^2)_{k=1}^N\subseteq \textsc {V}_h\times {\mathbb {R}}^+$ we refer to the 0-orthonormal eigenpairs of $(\textsc {V}_h,\Vert \cdot \Vert _{0})$ and $(\textsc {V}_h,\Vert \cdot \Vert _{1})$ from now on, such that

(3.1)

The eigenvalues are assumed to be in ascending order according to their value, such that

$$\begin{aligned} 0<\lambda _1^2\le \cdots \le \lambda _N^2. \end{aligned}$$

3.1 The reduced basis approach

Utilizing standard reduced basis technology for one dimensional parametric elliptic partial differential equations, we define for each $u\in \textsc {V}_h$ its approximate interpolation norms as follows.

Definition 3

(Reduced basis interpolation norms) For each $t\in {\mathbb {R}}^+$ we define as the unique solution of

(3.2)

for all $w\in \textsc {V}_h$. Given some real parameters $0 = t_0< t_1< \cdots < t_r$, specified in Sect. 5, we introduce the reduced space

(3.3)

The reduced basis interpolation norms on ${\mathcal {V}}_r$ are defined by either of the three equivalent definitions

$$\begin{aligned}&\Vert u\Vert _{E_r^s} := \Vert u\Vert _{{E}^s(({\mathcal {V}}_r,\Vert \cdot \Vert _{0}),({\mathcal {V}}_r,\Vert \cdot \Vert _{1}))}, \end{aligned}$$

(3.4a)

$$\begin{aligned}&\Vert u\Vert _{H_r^s} := \Vert u\Vert _\mathrm{{H}^s(({\mathcal {V}}_r,\Vert \cdot \Vert _{0}),({\mathcal {V}}_r,\Vert \cdot \Vert _{1}))}, \end{aligned}$$

(3.4b)

$$\begin{aligned}&\Vert u\Vert _{K_r^s}^2 := \int _{0}^{\infty }t^{-2s-1}\underbrace{{\mathrm {K}}_{\left( ({\mathcal {V}}_r,\Vert \cdot \Vert _{0}),({\mathcal {V}}_r,\Vert \cdot \Vert _{1})\right) }^2(t;u)}_{=: K_r^2(t;u)}\,dt. \end{aligned}$$

(3.4c)

Remark 2

The choice $t_0 = 0$ yields and hence $u\in {\mathcal {V}}_r$, such that (3.4a)-(3.4c) are well defined. Definition (3.3) is motivated by means of the K-method. The variational problem (3.2), which is uniquely solvable according to Lax-Milgram, appears to be the Euler-Lagrange equation of $K^2(t;u)$, such that coincides with the minimizer of $K^2(t;u)$. Based on a sophisticated selection of $t_1,\ldots ,t_r$, the choice of ${\mathcal {V}}_r$ aims to provide a both accurate and efficient approximation to the family of solutions .

Remark 3

Definition 3 incorporates a nonlinear dependency in u. For simplicity, we neglect this relation in both terminology and notation throughout our discussions. We point out, however, that all ${\mathcal {V}}_r$-connected constructions are subject to this dependency.

In analogy to (3.1), we denote the eigenpairs of $({\mathcal {V}}_r,\Vert \cdot \Vert _{0})$ and $({\mathcal {V}}_r,\Vert \cdot \Vert _{1})$ by $(\phi _j,\mu _j^2)_{j=0}^r\subseteq {\mathcal {V}}_r\times {\mathbb {R}}^+$ from now on, such that

$$\begin{aligned} \langle \phi _j,\phi _i\rangle _{0} = \delta _{ji},\qquad \langle \phi _j,w_r\rangle _{1} = \mu _j^2\langle \phi _j,w_r\rangle _{0}, {\quad w_r\in {\mathcal {V}}_r,} \end{aligned}$$

(3.5)

with

In general, the construction of ${\mathcal {V}}_r$ yields a $r+1$ dimensional space. The proof is carried out in two steps.

Lemma 2

For all $t\in {\mathbb {R}}^+$ there holds

Proof

Both and u provide expansions in the eigenbasis

Plugging into (3.2) with $w = \varphi _i$ yields

Resolving the equation for concludes the proof. $\square$

Lemma 3

Let $|\{\lambda _k^2\in \{\lambda _1^2,\ldots ,\lambda _N^2\}:u_k\ne 0\}| =: m\in {\mathbb {N}}$ denote the number of pairwise distinct eigenvalues, whose corresponding eigenspaces contribute nontrivially to the linear combination of u. Then there holds

Proof

W.l.o.g. we assume that $\{k\in \{1,\ldots ,N\}:u_k\ne 0\} = \{1,\ldots ,l\}$, where $l\ge m$. We show linear independency straight forward with the result from the previous Lemma. Assume that

for some coefficients $\alpha _0,\ldots ,\alpha _r\in {\mathbb {R}}$. The orthonormal system $(\varphi _k)_{k= 1}^{l}$ is linearly independent, yielding

(3.6)

Let now $\mathbb {I}\subseteq \{1,\ldots ,l\}$, such that $\{\lambda _k^2\in \{\lambda _1^2,\ldots ,\lambda _N^2\}:k\in \mathbb {I}\} = \{\lambda _1^2,\ldots ,\lambda _l^2\}$ and $|\mathbb {I}| = m$. Then, condition (3.6) is equivalent to

Exploiting $r+1\le m$ and the pairwise distinctness of both $(t_j)_{j=0}^r$ and $(\lambda _k^2)_{k\in \mathbb {I}}$, one deduces that $\alpha _0 = \cdots = \alpha _r = 0$. $\square$

The proof of Lemma 3 immediately reveals that the set becomes linearly dependent as $r+1>m$. In practice, we observe two possible constellations. In the common case, where u provides contributions from multiple basis vectors $\varphi _k$, solutions of (3.2) for different shift parameters $t_i$ indeed lead to an enrichment of ${\mathcal {V}}_r$, as long as r is small enough. If the amount of non-zero Fourier-components of u is rather small, we might be confronted with the case, where augmenting r does not affect the dimension of ${\mathcal {V}}_r$ any further. In this case, however, enlargement of ${\mathcal {V}}_r$ is no longer necessary, as the following Theorem shows.

Theorem 3

Let $|\{\lambda _k^2\in \{\lambda _1^2,\ldots ,\lambda _N^2\}:u_k\ne 0\}| =: m\in {\mathbb {N}}$. If $r+1\ge m$, then the reduced basis interpolation norms (3.4a)-(3.4c) coincide with the exact finite element interpolation norms from (2.7), respectively.

Proof

It suffices to validate the claim with respect to the Hilbert space interpolation norm for the case $r + 1 = m$. Let therefore $\mathbb {I}=\{i_0,\ldots ,i_{r}\}\subseteq \{1,\ldots ,N\}$, such that $\{\lambda _{i_0}^2,\ldots ,\lambda _{i_{r}}^2\} = \{\lambda _k^2\in \{\lambda _1^2,\ldots ,\lambda _N^2\}:u_k\ne 0\}$. Moreover, let

refer to the orthogonal decomposition of u according to the corresponding eigenspaces. Due to

and the regularity of the matrix $B\in \mathbb {R}^{(r+1)\times (r+1)}$ with $B_{kl} := (1+t_k^2\lambda _l^2)^{-1}$, we have that $(u^i)_{i\in \mathbb {I}}$ is a basis of ${\mathcal {V}}_r$, which is orthogonal by construction. We deduce $\lambda _{i_j}^2 = \mu _j^2$ for all $j = 0,\ldots ,r$. Direct computations reveal

$\square$

With exception of Theorem 6, we only consider the case where $r+1\le m$ for the rest of this paper, such that the dimension of ${\mathcal {V}}_r$ coincides with $r+1$.

3.2 Computational aspects

The remainder of this section reviews the major ingredients to supply the reduced basis interpolation norms with a computationally applicable form. Before addressing this issue explicitly, we specify some further notation. Throughout what follows, for each $v\in \textsc {V}_h$ we denote by $\underline{v}\in {\mathbb {R}}^N$ its uniquely assigned coefficient vector, such that

where $(b_k)_{k=1}^N$ denotes the finite element basis from (2.6). Moreover, we introduce the $s^{th}$-power of any symmetric matrix $Q\in {\mathbb {R}}^{l\times l}$, $l\in {\mathbb {N}}$, by diagonalization, i.e.,

where $\Phi \in {\mathbb {R}}^{l\times l}$ denotes the matrix of eigenvectors of Q and $\Lambda ^s$ the involved diagonal matrix, containing the $s^{th}$-power of all corresponding eigenvalues. If Q is also positive definite, we set

Based on these definitions, we follow the idea of POD [29] to propose an accurate procedure that computes (3.4a) - (3.4c) for one and hence for all norms efficiently. The structure of $\Vert u\Vert _{E_r^s}$ and $\Vert u\Vert _{K_r^s}$ is less amenable to direct computations, since this would require quadrature rules on the unbounded domain ${\mathbb {R}}^+$. A more convenient setting can be provided by the eigensystem. Targeting at the computation of ${\mathcal {V}}_r$, we introduce the matrix

whose $j^{th}$ column consists of the coefficient vector of , i.e.,

(3.7)

Thereupon, we introduce an orthonormal basis of ${\mathcal {V}}_r$ that is suitable for the study of our problem.

Definition 4

The reduced basis matrix $V_r\in {\mathbb {R}}^{N\times (r+1)}$ is defined as the unique matrix that arises from Gram-Schmidt orthonormalization chronologically applied to the columns of $\widehat{V}_r$ with respect to the scalar product $\langle \cdot ,\cdot \rangle _{M}$.

Remark 4

The chronological performance of Gram-Schmidt orthonormalization in Definition 4 yields that the first column of $V_r$ coincides with $\beta ^{-1}\underline{u}$, where $\beta := \Vert u\Vert _{0}$.

The reduced basis matrix suggests a canonical basis on ${\mathcal {V}}_r$ by referring to the unique functions $b_{1}^r,\ldots ,b_{r}^r\in {\mathcal {V}}_r\subseteq \textsc {V}_h$, whose assigned coefficient vectors coincide with the columns of $V_r$, i.e.,

Thereupon, we introduce for all $v_r\in {\mathcal {V}}_r$ its uniquely assigned coefficient vector $\underline{\underline{v_r}}\in {\mathbb {R}}^{r+1}$, such that

There holds

$$\begin{aligned}&\langle v_r,w_r\rangle _{1} = \langle \underline{v_r},\underline{w_r}\rangle _{A} = \underline{v_r}^TA\underline{w_r} = \underline{\underline{v_r}}^TV_r^TAV_r\underline{\underline{w_r}}, \end{aligned}$$

(3.8a)

$$\begin{aligned}&\langle v_r,w_r\rangle _{0} = \underline{\underline{v_r}}^TV_r^TMV_r\underline{\underline{w_r}} = \underline{\underline{v_r}}^T\underline{\underline{w_r}} \end{aligned}$$

(3.8b)

for all $v_r, w_r\in {\mathcal {V}}_r$, where the last equality follows by the orthonormal property of $V_r$. Thereupon, we introduce the following definition.

Definition 5

The projected stiffness matrix $A_r\in {\mathbb {R}}^{(r+1)\times (r+1)}$ is defined by

Theorem 4

Let $\underline{\underline{e_1}}\in {\mathbb {R}}^{r+1}$ denote the first unit vector and $\beta = \Vert u\Vert _{0}$. Then there holds

The proof is postponed to Sect. 4. Theorem 4 highlights the beneficial structure of the proposed reduced basis algorithm. The arising problem size is of much smaller magnitude $r\ll N$, making direct computations of the eigensystem affordable.

4 Approximation of the operators

All three reduced basis interpolation norms (3.4a)–(3.4c) induce an operator, ${\mathcal {L}}_{E_r^s}$, ${\mathcal {L}}_{H_r^s}$, and ${\mathcal {L}}_{K_r^s}$ on ${\mathcal {V}}_r$, which do all coincide up to s-dependent constants. In the same way as ${\mathcal {L}}_{H_r^s}$ serves as reduced basis surrogate for the spectral finite element operator

${\mathcal {L}}_{E_r^s}$ can be interpreted as model order reduction of the generalized Dirichlet-to-Neumann map from [45]. The latter coincides with ${\mathcal {L}}_{\textsc {E}^s({\mathscr {V}}_0,{\mathscr {V}}_1)}$, if ${\mathscr {V}}_0$ and ${\mathscr {V}}_1$ are chosen appropriately. The purpose of ${\mathcal {L}}_{K_r^s}$ is more of a technical than theoretical kind. We make use of the K-method as vital tool to proof convergence for one and hence for all three reduced basis operators. However, due to its computationally beneficial form, we stick to the equivalent spectral setting at first and refer to ${\mathcal {L}}_{H_r^s}$ as our truth reduced basis approximation. In dependency of $u\in \textsc {V}_h$, we state the essential definition of this paper.

Definition 6

(Reduced basis operator) For all $r\in {\mathbb {N}}$ we define the reduced basis operator ${\mathcal {L}}_{H_r^s}$ of ${\mathcal {L}}_{H^s}$ as the induced operator of $\Vert \cdot \Vert _{H_r^s}$.

Remark 5

In order to indicate its nonlinear nature, we write arguments of the reduced basis operator in brackets, i.e., ${\mathcal {L}}_{H_r^s}(u)$ instead of ${\mathcal {L}}_{H_r^s} u$.

As one can show, the matrix representation of ${\mathcal {L}}_{H^s}$, i.e., the unique matrix $L_{H^s}\in {\mathbb {R}}^{N\times N}$, such that

for all $u\in \textsc {V}_h$, is given by $(M^{-1}A)^s$. It serves as matrix approximation of the original fractional operator ${\mathcal {L}}_{\textsc {H}^s({\mathscr {V}}_0,{\mathscr {V}}_1)}$ in a finite element setting. Direct computations, however, are not feasible due to the typically large problem size. The proposed algorithm provides a remedy for this difficulty. In the following, we derive the u-dependent matrix representation $L_{H_r^s}$ of the reduced basis operator, providing the necessary information to carry out the actual computations.

Theorem 5

For all $u\in \textsc {V}_h$ there holds

$$\begin{aligned} \Vert u\Vert _{H_r^s}^2 = \underline{u}^TMV_rA_r^sV_r^TM\underline{u}. \end{aligned}$$

(4.1)

Moreover, the induced scalar product $\langle \cdot ,\cdot \rangle _{H_r^s}$ on $({\mathcal {V}}_r,\Vert \cdot \Vert _{H_r^s})$ satisfies

$$\begin{aligned} \langle v_r,w_r\rangle _{H_r^s} = \underline{v_r}^TMV_rA_r^sV_r^TM\underline{w_r} \end{aligned}$$

(4.2)

for all $v_r,w_r\in {\mathcal {V}}_r$. The matrix representation of ${\mathcal {L}}_{H_r^s}$ is given by

Proof

It suffices to prove (4.1). Recalling (3.5), we define

Equation (3.5) combined with (3.8a) and (3.8b) yields

Thus,

$\square$

We catch up on the postponed proof from the previous section.

Proof of Theorem 4

Follows directly from Theorem 5 and Remark 4 by utilizing the substitution $\underline{u} = \beta V_r\underline{\underline{e_1}}$. $\square$

$L_{H_r^s}$ serves as efficient approximation of $(M^{-1}A)^s$. Ahead of investigating its accuracy, we examine the arising computational costs, requiring knowledge of the map $r\mapsto (t_1,\ldots ,t_r)$ and its involved complexity. We address this problem adequately in Sect. 5, indicating for now that its essential computational expense amounts to finding a lower and upper bound for the spectrum of $M^{-1}A$. The overall complexity has to be regarded from two different perspectives, the so-called offline and online phase, and depends on the particular problem.

At first, we consider evaluations of type $u\mapsto {\mathcal {L}}_{H_r^s}(u)$, which are of nonlinear character. The underlying offline phase encompasses computations of the spectral bounds as a one-time investment. The online phase has to be performed for each argument separately. It incorporates computations of r finite element solutions of the original, expensive problem size, followed by orthonormalization of $\widehat{V}_r\in {\mathbb {R}}^{N\times (r+1)}$ to obtain the reduced basis matrix $V_r$. Furthermore, the projected matrix $A_r = V_r^TAV_r$ has to be established in order to determine its eigensystem. The assembly of first $A_r^s$ and second of $V_rA_r^sV_r^TM\underline{u}$ completes the computations. Despite its nonlinear nature, the savings gained substantially outweigh the arising inaccuracy, if r is of moderate size.

The offline-online decomposition is of particular interest, if we target at approximations of type $s\mapsto {\mathcal {L}}_{H_r^s}(u)$ for fixed u and several values of $s\in (0,1)$. In this case, the online phase breaks down to the assembly of $A_r^s$ and $V_rA_r^sV_r^TM\underline{u}$, while all remaining computations are the matter of a one-time investment within the offline phase.

Several properties of the reduced basis interpolation norms also apply to Definition 6. Exemplary, the operator counterpart of Theorem 3 is procured in the following.

Theorem 6

Let $|\{\lambda _k^2\in \{\lambda _1^2,\ldots ,\lambda _N^2\}:u_k\ne 0\}| =: m\in {\mathbb {N}}$. If $r+1\ge m$, then there holds

Proof

Follows the very same arguments as the proof of Theorem 3. $\square$

5 Convergence analysis

The goal of this section is to specify the choice of snapshots in Definition 3 to gain optimal convergence properties. We affirm that there exists a tuple of positive numbers $t_1,\ldots ,t_r$, naturally arising from the analysis, such that exponential decay in the error for both norm and operator action is obtained. While computations are carried out in the spectral setting, the approach involving the K-functional turns out to provide a more beneficial environment for the analysis. We adopt Theorem 5 in this context and take advantage of the, up to a multiplicative constant, interchangeable role of the reduced basis scalar products. Before going further into detail, we investigate two fundamental definitions that are based on [26, 48], see also [39], involving the theory of elliptic integrals and Jacobi elliptic functions, see [1, Section 16 & 17].

Definition 7

Let $\delta \in (0,1)$. For each $r\in {\mathbb {N}}$ we define the Zolotarëv points $\mathcal {Z}_1,\ldots ,\mathcal {Z}_r$ on the interval $[\delta ,1]$ by

where ${{\,\mathrm{dn}\,}}(\theta ,k)$ denotes the Jacobi elliptic function, $\mathcal {K}(k)$ the corresponding complete elliptic integral of first kind with elliptic modulus k, and $\delta ':= \sqrt{1-\delta ^2}$.

Definition 8

Let $0<a<b\in {\mathbb {R}}^+$. For each $r\in {\mathbb {N}}$ we define the transformed Zolotarëv points $\widehat{\mathcal {Z}}_1,\ldots ,\widehat{\mathcal {Z}}_r$ on [a, b] by

where $\mathcal {Z}_1,\ldots ,\mathcal {Z}_r$ refer to the Zolotarëv points on $\left[ \frac{a}{b},1\right]$.

As shown in the further course of action, the transformed Zolotarëv points on $[\lambda _N^{-2},\lambda _1^{-2}]$ turn out to be perfectly tailored for our reduced basis strategy. We recap that $\lambda _1^2$ and $\lambda _N^2$ refer to the minimal and maximal eigenvalue of the discrete operator and agree on the following nomenclature.

Definition 9

A reduced space is called Zolotarëv space, if and only if there exist two constants $\lambda _L^{2},\lambda _U^2\in {\mathbb {R}}^+$ with

such that the squared snapshots $t_1^2,\ldots ,t_r^2$ coincide with the transformed Zolotarëv points on $\sigma ^{inv} := [\lambda _U^{-2}, \lambda _L^{-2}]$. We call $\sigma := [\lambda _L^2,\lambda _U^2]$ the spectral interval of ${\mathcal {V}}_r$.

5.1 Error of the reduced basis interpolation norms

We specify some further notation. By $a\preceq b$ we mean that there exists a constant $C\in {\mathbb {R}}^+$, independent of a, b, and r, such that $a \le Cb$. Along with this premiss, we prove that Definition 9 provides an optimal choice for the reduced space.

Theorem 7

(Exponential convergence of the reduced basis interpolation norms) Let $u\in \textsc {V}_h$ and ${\mathcal {V}}_r\subseteq \textsc {V}_h$ a Zolotarëv space with $\sigma = [\lambda _L^2,\lambda _U^2]$ and $\delta = \nicefrac {\lambda _L^2}{\lambda _U^2}$. Then there holds

(5.1)

The constant C only depends on $\delta$ and satisfies

Its precise value coincides with $2C^*$, where $C^*$ refers to the constant from Remark 7.

Verifying the claim of Theorem 7 is challenging, which is why we conduct the proof in several steps. Under the prescribed assumptions, we start with the first inequality of (5.1).

Lemma 4

There holds

Proof

The relation ${\mathcal {V}}_r\subseteq \textsc {V}_h$ immediately reveals

for all $t\in {\mathbb {R}}^+$. Hence,

$\square$

Due to

(5.2)

the error of the norms can be traced back to the error of the K-functionals. To make matters precise, we conduct some technical preparations.

Definition 10

For all $t\in {\mathbb {R}}^+$ we denote by the unique minimizer of .

Remark 6

Similarly to , utilizing Euler-Lagrange formalism, the minimizer is the unique solution of the variational problem

or equivalently, solves the linear system of equations

where $I_r\in {\mathbb {R}}^{(r+1)\times (r+1)}$ represents the identity matrix.

Lemma 5

For all $t\in {\mathbb {R}}^+$ and $w_r\in {\mathcal {V}}_r$ there holds

Proof

For convenience, we omit the dependency in t in consecutive elaborations. One observes

We define the bilinear form $a(u,w) := \langle u,w\rangle _{0} + t^2 \langle u,w\rangle _{1}$ on $\textsc {V}_h\times \textsc {V}_h$. Due to (3.2), there holds

Thus,

$\square$

The accuracy of $K_r^2(t;u)$ rests upon the approximation quality of the minimizer , as the following result shows.

Corollary 3

For all $t\in {\mathbb {R}}^+$ there holds

(5.3)

Proof

Follows directly from Lemma 5 with . $\square$

Dealing with (5.3) is challenging. We derive an upper bound for the error which turns out to be more amenable to analytical considerations.

Corollary 4

For all $t\in {\mathbb {R}}^+$ and $w_r\in {\mathcal {V}}_r$ there holds

(5.4)

Proof

Due to the minimization property of , there holds

for all $t\in {\mathbb {R}}^+$ and $w_r\in {\mathcal {V}}_r$. Utilizing Lemma 5 concludes the proof. $\square$

In the subsequent, we aim to choose

(5.5)

from Corollary 4 in a clever way, such that the upper bound in (5.4) becomes small. The idea of how to choose its coefficients $\alpha _j$ emerges from the following investigation.

Theorem 8

For all $\alpha _0,\ldots ,\alpha _r\in {\mathbb {R}}$ there holds

Proof

Due to Corollary 4, there holds for any function of type (5.5)

Spectral decomposition combined with Lemma 2 yields

Based on the orthogonal property of $(\varphi _k)_{k=1}^N$, we observe

Computations in the 0-norm can be concluded analogously, proving the claim. $\square$

Theorem 8 reveals that (5.5) has to be chosen in a way, such that for all $\lambda _1,\ldots ,\lambda _N$, or more generally, for all values of $\lambda \in [\lambda _1,\lambda _N]$, the difference

(5.6)

becomes small. Typically, neither $\lambda _1$ nor $\lambda _N$ are known a-priori, which is why we consider (5.6) with respect to the spectral interval from Definition 9, admitting $\lambda \in [\lambda _L,\lambda _U]$ instead. Any possible bound of (5.6) then trivially also holds on $[\lambda _1,\lambda _N]$.

In the further course of action, we derive two different candidates for the coefficients $(\alpha _j)_{j=0}^r$ in dependency of t. The first one ensures that (5.6) becomes small for $t\ge 1$, while the second achieves the same as $t<1$. To this extent, we make a first ansatz and set $\alpha _0 = 0$. The latter coefficients are determined by means of a rational interpolation problem, which we inquire in the subsequent Lemma.

Lemma 6

Assume that $\kappa \in {\mathbb {R}}^+, \kappa _1,\ldots ,\kappa _r\in \sigma ^{inv}$, and $\kappa _i\ne \kappa _j$ for $i\ne j$. Consider the space $\mathcal {R}$ of all rational functions R, which admit a representation

for coefficients $\alpha _1,\ldots ,\alpha _r\in {\mathbb {R}}$. Further define

Then the solution of the rational interpolation problem: Find $q\in \mathcal {R}$, such that

(5.7)

satisfies

(5.8)

Proof

Let

denote the unique solution of (5.7). Then there holds

$$\begin{aligned} g_{\kappa }(x) - q(x) = \frac{1}{1+\kappa x} - \sum \limits _{j = 1}^{r}{\hat{\alpha }}_j\frac{1}{1+\kappa _j x} = \frac{p(x)}{(1+\kappa x)\prod \nolimits _{j = 1}^{r}(1 + \kappa _j x)} \end{aligned}$$

(5.9)

for a suitable polynomial p of degree r. The interpolation property yields

The fundamental Theorem of algebra affirms the existence of a constant $c \in {\mathbb {R}}$, such that

The constant c can be further specified. Multiplying (5.9) by $(1+\kappa x)$ and setting $x = -\frac{1}{\kappa }$ yields

All together, we obtain for all $x\in \sigma$

$\square$

Minimizing the maximal deviation of the upper bound in (5.8) leads to a min-max problem of the following kind: Find $\kappa _1,\ldots ,\kappa _r\in \sigma ^{inv}$, such that

$$\begin{aligned} \min \limits _{\theta _1,\ldots ,\theta _r\in \sigma ^{inv}}\max \limits _{x\in \sigma } \prod \limits _{j=1}^{r}\left| \frac{1 - \theta _j x}{1 + \theta _j x}\right| = \max \limits _{x\in \sigma }\prod \limits _{j=1}^{r}\left| \frac{1 - \kappa _j x}{1 + \kappa _j x}\right| . \end{aligned}$$

(5.10)

Closely related problems have been investigated in [26, 48]. We summarize the essential results. Consider the slightly modified problem: Find $\kappa _1,\ldots ,\kappa _r\in [\delta ,1]$, $\delta = \nicefrac {\lambda _L^2}{\lambda _U^2}$, such that

$$\begin{aligned} \min \limits _{\theta _1,\ldots ,\theta _r\in [\delta ,1]}\max \limits _{x\in [\delta ,1]} \prod \limits _{j=1}^{r}\left| \frac{x-\theta _j}{x + \theta _j}\right| = \max \limits _{x\in [\delta ,1]}\prod \limits _{j=1}^{r}\left| \frac{x - \kappa _j}{x+\kappa _j}\right| . \end{aligned}$$

(5.11)

Zolotarëv and Gonchar showed that its unique solution is given by the Zolotarëv points $\mathcal {Z}_1,\ldots ,\mathcal {Z}_r$ on $[\delta ,1]$. They further approved that there exists a positive constant C, depending on $\delta$ only, such that

$$\begin{aligned} \max \limits _{x\in [\delta ,1]}\prod \limits _{j=1}^{r}\left| \frac{x - \mathcal {Z}_j}{x+\mathcal {Z}_j}\right| \preceq e^{-Cr}. \end{aligned}$$

(5.12)

The product in (5.12), considered as function in x, features $r+1$ points of alternance and has the least deviation from zero on $[\delta ,1]$ among all functions of this form. We set results from problem (5.11) in correspondence with (5.10).

Theorem 9

The unique solution $\kappa _1,\ldots ,\kappa _r$ of problem (5.10) satisfies $\kappa _j = \widehat{\mathcal {Z}}_j$ for all $j=1,\ldots ,r$, where $\widehat{\mathcal {Z}}_j$ denotes the $j^{th}$ transformed Zolotarëv point on $\sigma ^{inv}$.

Proof

Consider the linear transformation $\Psi :[\delta ,1]\longrightarrow \sigma ^{inv}$ with $\Psi (x) := \lambda _L^{-2}x$. Direct computations, based on the results of [48], reveal

$\square$

Within the proof of Theorem 9, we have additionally shown that

which, due to (5.12), immediately reveals the following result.

Corollary 5

Let $\widehat{\mathcal {Z}}_1,\ldots ,\widehat{\mathcal {Z}}_r$ denote the transformed Zolotarëv points on $\sigma ^{inv}$. Then there holds

(5.13)

Remark 7

As shown in [35], constants from Corollary 5 can be further specified. More precisely, the maximal deviation in (5.13) can be bounded by means of

with

and $\mathcal {K}$ the elliptic integral from Definition 7. The following asymptotic formulas are known to hold,

see e.g., [1, Section 17]. This yields the asymptotic behaviour of $C^*$ in dependency of $\delta$,

Along with the choice $\lambda _L := \lambda _1$ and $\lambda _U := \lambda _N$, this reveals that the constant $C^*$ only deteriorates at logarithmical rate as the condition number $\nicefrac {\lambda _N^2}{\lambda _1^2} = \delta ^{-1}$ increases. The number of solves required to achieve a prescribed precision $\varepsilon > 0$ behaves like

We eventually return to the original problem of interest in (5.6).

Lemma 7

Denote by $\hat{\alpha }_1,\ldots ,\hat{\alpha }_r\in {\mathbb {R}}$ the coefficients of the unique solution $q\in \mathcal {R}$ from the rational interpolation problem (5.7) with $\kappa := t^2$ for some $t\in {\mathbb {R}}^+$ and $\kappa _j := t_j^2$. Let $C^*$ denote the constant from Remark 7. Then there holds

(5.14)

Proof

According to Lemma 6, Corollary 5, and Remark 7, there holds

for all $\lambda \in [\lambda _L,\lambda _U]$. By construction, there holds $\lambda _k\in [\lambda _L,\lambda _N]$ for all $k = 1,\ldots ,N$, which is why (5.14) is valid. $\square$

Assembling results from above finally enables us to derive an upper bound for the error in the reduced basis K-functional.

Theorem 10

Let $C^*$ denote the constant from Remark 7. Then there holds for all $t\in {\mathbb {R}}^+$

Proof

Theorem 8 combined with Lemma 7 yields

$\square$

As it turns out in the further course of action, the upper bound derived in Theorem 10 is only sharp enough in case of $t\ge 1$, but not as $t<1$. We overcome this inconvenience by subtle adjustments of the interpolation problem (5.7), such that its arising solution leads to the desired properties.

Theorem 11

Let $C^*$ denote the constant from Remark 7. Then there holds for all $t\in {\mathbb {R}}^+$

Proof

In analogy to Lemma 6, we consider the following rational interpolation problem: For $\kappa \in {\mathbb {R}}^+$ and $\kappa _1,\ldots ,\kappa _r\in \sigma ^{inv}$ pairwise distinct, find $q\in \hat{\mathcal {R}}$, such that

where $\hat{\mathcal {R}}$ denotes the linear span of $\mathcal {R}$ enriched by constant functions. We remark that $\alpha _0 = 0$ is no longer constrained. Similarly to the proof of Lemma 6, one affirms that

The transformed Zolotarëv points on $\sigma ^{inv}$ ensure exponential convergence of the product. Proceeding in the same manner as before, one concludes the proof. $\square$

We have now all the required tools to prove exponential convergence of the reduced basis interpolation norms in the K-setting.

Proof

(Proof of Theorem 7) Theorem 10 and 11 together with (5.2) yield

$\square$

5.2 Error of the reduced basis operator

The reduced basis interpolation norms provide an exponential decay in the error, granting good chances that similar results are valid with respect to the induced operators. Indeed, convergence of the operators is based on the results from Sect. 5.1. The core of this paper is summarized in the following Theorem, relying on the notation

Theorem 12

(Exponential convergence of the reduced basis operator) Let $u\in \textsc {V}_h$ and ${\mathcal {V}}_r\subseteq \textsc {V}_h$ a Zolotarëv space with $\sigma = [\lambda _L^2,\lambda _U^2]$ and $\delta = \nicefrac {\lambda _L^2}{\lambda _U^2}$. Then there exists a constant $C\in {\mathbb {R}}^+$, such that

$$\begin{aligned} \Vert {\mathcal {L}}_{H_r^s}(u) - {\mathcal {L}}_{H^s}u\Vert _{0} \preceq e^{-Cr}\Vert u\Vert _{2}. \end{aligned}$$

(5.15)

The constant C only depends on $\delta$ and satisfies

Its precise value coincides with the constant $C^*$ from Remark 7. Moreover, if $s\in \left( 0,\frac{1}{2}\right)$, the 2-norm of u in (5.15) can be replaced by $\Vert u\Vert _{1}$.

The rest of this section is dedicated to the proof of Theorem 12 and therefore subject to the prescribed assumptions. All arising matrix-valued integrals are understood component-by-component.

Lemma 8

There holds

Moreover, the induced scalar product $\langle \cdot ,\cdot \rangle _{K^s}$ on $(\textsc {V}_h,\Vert \cdot \Vert _{K^s})$ satisfies

for all $v,w\in \textsc {V}_h$.

Proof

We show the first equality. Due to (3.2), there holds

Utilizing the identity $t^2A = (M+t^2A) - M$ yields

This reveals

Hence,

$\square$

Throughout the rest of this paper, the identity matrix on ${\mathbb {R}}^{(r+1)\times (r+1)}$ is denoted by $I_r$, giving rise to the following claim.

Lemma 9

There holds

Moreover, the induced scalar product $\langle \cdot ,\cdot \rangle _{K_r^s}$ on $({\mathcal {V}}_r,\Vert \cdot \Vert _{K_r^s})$ satisfies

(5.16)

for all $v,w\in \textsc {V}_h$. The matrix representation of ${\mathcal {L}}_{K_r^s}$ is given by

Proof

In analogy to the proof of Lemma 8, one shows that

Plugging into the integral representation of $\Vert u\Vert _{K_r^s}^2$ eventually proves the claim. $\square$

Definition 11

For all $v\in \textsc {V}_h$ and $w_r\in {\mathcal {V}}_r$ we define

Lemma 8 and 9 are fundamental to show pointwise convergence of the induced operator in the K-setting. We proceed in two steps.

Theorem 13

For all $w\in \textsc {V}_h$ there holds

Proof

Due to Lemma 8 and 9, $\langle w,u\rangle _{K_r^s} - \langle w,u\rangle _{K^s}$ can be expressed as

One ascertains that the first term cancels out the third, i.e.,

Computations of the remaining terms together with (3.7) and Remark 6 reveal

Cauchy-Schwarz inequality leads to

There holds for all $t\in {\mathbb {R}}^+$

which validates the conjecture by virtue of Corollary 3. $\square$

Theorem 14

Let $C^*$ denote the constant from Remark 7. Then there holds

(5.17)

Moreover, if $s\in \left( 0,\frac{1}{2}\right)$, the 2-norm of u in (5.17) can be replaced by $\Vert u\Vert _{1}$.

Proof

We prove that for any $w\in \textsc {V}_h$ and sufficiently small $\varepsilon >0$, satisfying $4s+2\varepsilon < 4$, there holds

which directly implies (5.17). Moreover, if $s<\frac{1}{2}$, we can choose $\varepsilon <1-2s$ to verify the latter claim and conclude the proof. To this extent, let $\varepsilon \in (0,2-2s)$, if $s\ge \frac{1}{2}$, and $\varepsilon \in (0,1-2s)$ otherwise. Applying Theorem 11 followed by Cauchy-Schwarz inequality yields

Define $i := \min \{k\in \{1,\ldots ,N\}: \lambda _k \ge 1\}$ to observe

Similarly, we obtain for the rest of the sum

such that

On the interval $[1,\infty )$, we make use of Theorem 10 to conclude for all $s\in (0,1)$

Adding up the integrals in combination with Theorem 13 proves the claim. $\square$

As shown in the subsequent, equality of the scalar products is also valid with respect to Definition 11, providing the crucial link to transfer the error analysis form the K-setting to the spectral setting.

Lemma 10

For all $v\in \textsc {V}_h$ and $w_r\in {\mathcal {V}}_r$ there holds

(5.18)

Proof

If $v\in {\mathcal {V}}_r$, then (5.18) holds due to Corollary 1. Let now $\Pi _r:\textsc {V}_h\rightarrow {\mathcal {V}}_r$ denote the 0-orthogonal projection on ${\mathcal {V}}_r$, such that

for each $v\in \textsc {V}_h$. By virtue of the identity $\underline{\underline{(\Pi _rv)}}^T = \underline{v}^TMV_r$, equation (4.2) and (5.16), there holds for any arbitrary $v\in \textsc {V}_h$ and $w_r\in {\mathcal {V}}_r$

$\square$

We are eventually able to conduct the proof of Theorem 12.

Proof

(Proof of Theorem 12) Due to Corollary 1 and Lemma 10, there holds

Hence,

where the last inequality relies on (5.17). The fact that $\Vert u\Vert _{2}$ reduces to $\Vert u\Vert _{1}$ as $s<\frac{1}{2}$ follows directly from the latter claim of Theorem 14. $\square$

Remark 8

Results from Theorem 7 and 12 can be ameliorated in a sense that the lower and upper bound, $\lambda _L^2$ and $\lambda _U^2$, solely have to be chosen with respect to that minimal and maximal eigenvalue, whose corresponding eigenfunction nontrivially contributes to the linear combination of the argument u. This leads to improvements of the constant $C^*$ and hence to faster convergence.

6 Numerical examples

In the subsequent, we underpin our analytical results by several numerical examples, which can be found on GitHub¹. To make matters precise, let $\Omega \subseteq {\mathbb {R}}^2$ be an open, bounded Lipschitz domain with Lipschitz boundary $\partial \Omega$. We define ${\mathscr {V}}_0:= (L_2(\Omega ),\Vert \cdot \Vert _{L_2})$ and ${\mathscr {V}}_1:= (H_0^1(\Omega ),\Vert \nabla \cdot \Vert _{L_2})$ to consider its arising interpolation space $[{\mathscr {V}}_0,{\mathscr {V}}_1]_{s}$. The induced operator ${\mathcal {L}}_{\textsc {H}^s({\mathscr {V}}_0,{\mathscr {V}}_1)}$ of $\Vert \cdot \Vert _\mathrm{{H}^s({\mathscr {V}}_0,{\mathscr {V}}_1)}$ coincides with the spectral fractional Laplacian subject to homogeneous Dirichlet boundary conditions, i.e.,

where $(\varphi _k,\lambda _k^2)_{k=1}^{\infty }\subseteq H_0^1(\Omega )\times {\mathbb {R}}^+$ denote the $L_2$-orthonormal eigenfunctions and eigenvalues of ${\mathscr {V}}_0$ and ${\mathscr {V}}_1$. This setting is utilized to adequately study and analyze the performance of our method. All tests were implemented within the NGS-Py interface of the open source finite element packages NETGEN and NGSolve², see [41, 42]. Computations of the Zolotarëv points are performed by means of the special function library from Scipy³.

Example 1

Consider the unit square $\Omega = (0,1)^2$ and a finite element space $\textsc {V}_h\subseteq H_0^1(\Omega )$ of polynomial order $p = 3$ on a quasi-uniform, triangular mesh $\mathcal {T}_h$ with mesh size $h = 0.08$, together with its arising eigenbasis $(\varphi _k)_{k=1}^N$ and $N = 1762$. We set

for some randomly chosen coefficients $c_i\in (-1,1)$ with $n = 300$, such that the reduced basis norm is exact for $r\ge 299$. On $\Omega$, the exact eigenvalues $(\nu _k^2)_{k=1}^\infty$ of $-\Delta$ are given in closed form in terms of

Consistent with Remark 8, we set $\lambda _L^2 := \nu _{1}^2\le \lambda _1^2$ and utilize the power method to obtain an upper bound $\lambda _U^2 := 4200 \ge \lambda _{n}^2$. In accordance with Theorem 7, Figure 1 affirms the exponential decay of the error $E_{u}^{Norm}(s,r) := \Vert u\Vert _{H_r^s}^2 - \Vert u\Vert _{H^s}^2$ in r. Furthermore, we observe, as indicated in Sect. 5, that the error increases as s is augmented.

Fig. 1

Error $E_{u}^{Norm}(s,r)$ of the reduced basis interpolation norm of a randomly chosen $u\in {{\,\mathrm{span}\,}}\{\varphi _1,\ldots ,\varphi _n\}$ for three different values of s, $\sigma = [2\pi ^2,4200]$, and $2C^*\approx 1.65$

Fig. 2

$L_2$-error $E_{u_h}^{Op}(s,r)$ of the reduced basis operator for three different choices of s

Fig. 3

Impact of the mesh parameters $h = 2^{-i}$ on the error $E_{u_h^{rand}}^{Op}(0.1,r)$

Example 2

Consider the unit circle $\Omega = \{\mathbf {x}\in {\mathbb {R}}^2: \Vert \mathbf {x}\Vert <1\}$, where $\Vert \cdot \Vert$ denotes the Euclidean norm. For the sake of simplicity, we approximate $\Omega$ using a quasi-uniform triangular mesh $\mathcal {T}_h$ of mesh-size $h = 0.06$. In particular, the boundary $\partial \Omega$ is approximated using (non-curved) line segments. We highlight, however, that our theory equally holds true when curved elements are used. Provided the mesh, we use polynomials of order $p = 2$ to construct the finite element space $V_h$. Let now $u_h := \Pi _h u\in \textsc {V}_h$ denote the $L_2$-orthogonal projection of

onto $\textsc {V}_h$. Having no further information which eigenfunctions contribute to the linear combination of $u_h$, we set $\lambda _L^2 := 1\le \lambda _1^2$ and $\lambda _U^2 := \tilde{\lambda }_N^2+1\approx 49723.4$, where $\tilde{\lambda }_N^2$ denotes a numerical approximation of $\lambda _N^2$ obtained by power iteration. The error of the reduced basis operator, $E_{u_h}^{Op}(s,r) := \Vert {\mathcal {L}}_{H_r^s}(u_h) - {\mathcal {L}}_{H^s}u_h\Vert _{L_2}$, from Theorem 12 is examined in Figure 2. Here, the exact operator action ${\mathcal {L}}_{H^s}u_h$ has been replaced by ${\mathcal {L}}_{H_{r^*}^s}(u_h)$, where $r^*\in {\mathbb {N}}$ is taken large enough to neglect the arising inaccuracy. The observed convergence rate is is in accordance with $\mathcal {O}\left( e^{-C^*r}\right)$, where $C^*\approx 0.41$.

The dominant computational effort for evaluating ${\mathcal {L}}_{H_r^s}(u_h)$ comes from the offline phase, in which a basis for the reduced basis space is computed. The construction of the latter requires the solution of r parametric reaction-diffusion equations, which can be obtained in $\mathcal {O}(rN)$ operations using an optimal solver such as a multigrid method. For each new exponent queried during the online phase, the surrogate is found in the coordinate space with complexity $\mathcal {O}(k^3)$ by diagonalizing the compressed matrix $A_r\in \mathbb {R}^{(r+1)\times (r+1)}$. Since typically $k\ll N$, this allows us to efficiently query the surrogate for multiple values of s.

It is evident that the performance of our method relies on the condition of the problem and thus on the mesh parameter h. The exponential convergence property of ${\mathcal {L}}_{H_r^s}$ for $h = 2^{-i}$, $i = 3,5,7,9$, on $\Omega$ with $p = 1$ and $s = 0.1$ is shown in Figure 3 with respect to a randomly chosen $u_h^{rand}\in \textsc {V}_h$. The example confirms that the rate of convergence deteriorates as $h\rightarrow 0$. The error approximately behaves like $\mathcal {O}\left( e^{-0.6r}\right)$ for $h = 2^{-3}$ and $\mathcal {O}\left( e^{-0.28r}\right)$ for $h = 2^{-9}$.

Appendix

Proof of Theorem 1

The proof follows the outline of [23, Proposition 2.1] and [19,  Lemma 2.2]. Let $v\in \mathbb {V}{({\mathscr {V}}_0,{\mathscr {V}}_1; y^\alpha )}$ with

and $v_k(y) = \langle v(y),\varphi _k\rangle _{0}$. Then there holds

$$\begin{aligned} \begin{aligned} \Vert v\Vert _{\mathbb {V}({\mathscr {V}}_0,{\mathscr {V}}_1; y^\alpha )}^2&= \sum \limits _{k=1}^\infty \int _{{\mathbb {R}}^+}y^\alpha \left( \lambda _k^2 v_k(y)^2 + v_k'(y)^2\right) \,dy \\&\ge \sum \limits _{k=1}^\infty v_k(0)^2\min \limits _{\begin{array}{c} \psi _k\in H^1({\mathbb {R}}^+) \\ \psi _k(0) = 1 \end{array}}\int _{{\mathbb {R}}^+}y^\alpha \left( \lambda _k^2 \psi _k(y)^2 + \psi _k'(y)^2\right) \,dy. \end{aligned} \end{aligned}$$

(7.1)

It is well-known that the minimizer $\psi _k$ coincides with the solution of a Bessel-type differential equation

which admits the following representation

Here, $c_k$ denotes a constant and ${\textbf {K}}_s$ the modified Bessel function of second kind, see [1]. The constant is chosen in a way, such that $\psi _k(0) = 1$, which allows us to write

$$\begin{aligned} \psi _k(y) = \psi (\lambda _ky) \end{aligned}$$

(7.2)

for a suitable function $\psi \in H^1({\mathbb {R}}^+)$ with $\psi (0) = 1$. For all $k\in {\mathbb {N}}$ the value of the minimum in (7.1) is given by

Integration by parts and incorporating the asymptotic behaviour of ${\textbf {K}}_\textsc {s}$ yields

see e.g., [19,  Remark 2.3]. Thus,

We conclude that ${{\,\mathrm{tr}\,}}$ defines a linear, bounded operator that satisfies (2.2), such that its range is contained in $[{\mathscr {V}}_0,{\mathscr {V}}_1]_{\mathrm{{H}^s}}$. To evidence surjectivity, we observe that for each $u\in [{\mathscr {V}}_0,{\mathscr {V}}_1]_{\mathrm{{H}^s}}$ the function

is contained in $\mathbb {V}{({\mathscr {V}}_0,{\mathscr {V}}_1; y^\alpha )}$ and satisfies ${{\,\mathrm{tr}\,}}\mathscr {U} = u$. $\square$

Proof of Theorem 2

Let $u\in [{\mathscr {V}}_0,{\mathscr {V}}_1]_{\mathrm{{E}^s}}$ and consider the ansatz function

with $\psi$ as in (7.2). The proof of Theorem 1 validates that $\mathscr {U}\in \mathbb {V}{({\mathscr {V}}_0,{\mathscr {V}}_1; y^\alpha )}$ and

Moreover, $\mathscr {U}$ satisfies (2.5), by construction of $\psi$. What follows is that $\mathscr {U}$ is the $\alpha$-harmonic extension of u. Due to its minimization property from Lemma 1, we further have

To evidence the second equality, we refer to [18, Theorem A.2]. $\square$