Polynomial Approximation of Anisotropic Analytic Functions of Several Variables

Abstract

Motivated by numerical methods for solving parametric partial differential equations, this paper studies the approximation of multivariate analytic functions by algebraic polynomials. We introduce various anisotropic model classes based on Taylor expansions, and study their approximation by finite dimensional polynomial spaces ${{\mathcal {P}}}_\Lambda$ described by lower sets $\Lambda$. Given a budget n for the dimension of ${{\mathcal {P}}}_\Lambda$, we prove that certain lower sets $\Lambda _n$, with cardinality n, provide a certifiable approximation error that is in a certain sense optimal, and that these lower sets have a simple definition in terms of simplices. Our main goal is to obtain approximation results when the number of variables d is large and even infinite, and so we concentrate almost exclusively on the case $d=\infty$. We also emphasize obtaining results which hold for the full range $n\ge 1$, rather than asymptotic results that only hold for n sufficiently large. In applications, one typically wants n small to comply with computational budgets.

Appendix: Proofs of Corollaries 5.6 and 5.7

Proof of Corollary 5.6:

Let $m=m(n)$ be the largest natural number satisfying (5.14). One can check that for $n\ge 2^{16}$, we have $m(n)\ge 6$, and thus it follows from (ii) of Lemma 5.3 that

$$\begin{aligned} \#\Lambda (2^{-m(n)},\rho ^*(1)) \le Cm(n)^{-3/4}2^{m(n)+c\sqrt{m(n)}}\le n, \end{aligned}$$

which gives $\delta _n(\rho ^*(s)) \le 2^{-m(n)s}$, and thus $E_n({{\mathcal {U}}}_{\rho ^*(s),1}) \le 2^{-m(n)s}$. $\square$

Proof of Corollary 5.7:

To show (5.15), we proceed as follows. We consider first the case $n=2^k$, $k\ge 16$. Let m be the largest non-negative natural number satisfying

$$\begin{aligned} m+c\sqrt{m}\le k, \end{aligned}$$

and let $\beta$ be defined by the equation $m=k-\beta \sqrt{k}$. Since $k\ge 16$, the largest m that satisfies the above estimate is greater or equal to 6. Moreover, we can easily show that $\beta \le c$. Therefore, we use the fact that $m=k-\beta \sqrt{k}\ge k-c\sqrt{k}$ and that $k-c\sqrt{k}\ge (1-c/4) k$ for $k\ge 16$, which gives

$$\begin{aligned} \log _2 m\ge \log _2 (1-c/4) +\log _2 k. \end{aligned}$$

Thus if $C_1:=\log _2 C$, we have

$$\begin{aligned} C_1 -\frac{3}{4}\log _2 m+m+c\sqrt{m}\le C_2-\frac{3}{4}\log _2k +k, \quad C_2:=C_1 -\frac{3}{4}\log _2 (1-c/4). \end{aligned}$$

It follows (since $m\ge 6$) that

$$\begin{aligned} \#(\Lambda (2^{-m},\rho ^*(1)))\le Cm^{-3/4}2^{m+c\sqrt{m}}\le {\tilde{C}}k^{-3/4}2^k ={\tilde{C}}\frac{n}{[\log _2 n]^{3/4}}, \quad {\tilde{C}}:=C(1-c/4)^{-3/4}>1. \end{aligned}$$

Therefore, (5.7) and the monotonicity of the sequence $(\delta _n(\rho ^*(s)))_{n\ge 1}$ give

$$\begin{aligned} \delta _{\left\lceil {\tilde{C}}\frac{n}{[\log _2 n]^{3/4}}\right\rceil }(\rho ^*(s))\le 2^{-ms}\le n^{-s}n^{\frac{c s}{\sqrt{\log _2 n}} },\quad n= 2^{k}, \quad k\ge 16, \end{aligned}$$

which, according to Remark 5.2, leads to

$$\begin{aligned} E_{\left\lceil {\tilde{C}}\frac{n}{[\log _2 n]^{3/4}}\right\rceil }({{\mathcal {U}}}_{\rho ^*(s),1})\le n^{-s}n^{\frac{c s}{\sqrt{\log _2 n}} }. \end{aligned}$$

Now, if $k\ge 16$ is such that $2^k\le n<2^{k+1}$, it follows that

$$\begin{aligned} E_{\left\lceil {\tilde{C}}\frac{n}{[\log _2 n]^{3/4}}\right\rceil }({{\mathcal {U}}}_{\rho ^*(s),1})\le E_{\left\lceil {\tilde{C}}\frac{2^k}{k^{3/4}}\right\rceil }({{\mathcal {U}}}_{\rho ^*(s),1})\le 2^{-ks } 2^{cs \sqrt{k} }\le 2^sn^{-s} (2^{\log _2 n})^{\frac{c{ s}}{\sqrt{\log _2 n}}} { =} 2^sn^{-s} n^{\frac{cs}{\sqrt{\log _2 n}}}, \end{aligned}$$

which is (5.15). $\square$