[Paper Review] Approximation of mixed order Sobolev functions on the $d$-torus -- Asymptotics, preasymptotics and $d$-dependence

Thomas Kuehn, Winfried Sickel+1 more|arXiv (Cornell University)|Dec 22, 2013

Mathematical Approximation and Integration19 references4 citations

TL;DR

This paper investigates the approximation of $d$-variate periodic functions in Sobolev spaces of dominating mixed smoothness on the $d$-torus, focusing on the asymptotic and preasymptotic behavior of approximation numbers and their dependence on dimension $d$. It establishes super-exponential decay of approximation constants in $d$ for fixed smoothness $s>0$, and proves quasi-polynomial tractability of the approximation problem, resolving key issues in high-dimensional numerical analysis.

ABSTRACT

We investigate the approximation of $d$-variate periodic functions in Sobolev spaces of dominating mixed (fractional) smoothness $s>0$ on the $d$-dimensional torus, where the approximation error is measured in the $L_2-$norm. In other words, we study the approximation numbers of the Sobolev embeddings $H^s_{ m mix}(\mathbb{T}^d)\hookrightarrow L_2(\mathbb{T}^d)$, with particular emphasis on the dependence on the dimension $d$. For any fixed smoothness $s>0$, we find the exact asymptotic behavior of the constants as $d o\infty$. We observe super-exponential decay of the constants in $d$, if $n$, the number of linear samples of $f$, is large. In addition, motivated by numerical implementation issues, we also focus on the error decay that can be achieved by low rank approximations. We present some surprising results for the so-called ``preasymptotic'' decay and point out connections to the recently introduced notion of quasi-polynomial tractability of approximation problems.

Motivation & Objective

• To analyze the dependence of approximation numbers on dimension $d$ for Sobolev spaces of mixed smoothness on the $d$-torus.
• To clarify the asymptotic and preasymptotic behavior of approximation rates in high dimensions.
• To establish tractability properties, particularly quasi-polynomial tractability, for the approximation problem.
• To provide explicit, sharp constants in the approximation rate estimates that depend on $d$ and $s$.

Proposed method

• Derives the exact asymptotic behavior of approximation numbers $a_n(I_d)$ as $n \to \infty$ for fixed $s > 0$, showing convergence to $\left[\frac{2^d}{(d-1)!}\right]^s (\ln n)^{(d-1)s} n^{-s}$.
• Uses spectral analysis and singular value decomposition for compact operators between Hilbert spaces to characterize approximation numbers as singular values.
• Employs combinatorial estimates and bounds on sums over multi-indices to control the growth of the number of terms in the approximation error decomposition.
• Applies the notion of quasi-polynomial tractability by deriving bounds on the minimal number of samples $n(\varepsilon, d)$ needed to achieve error $\varepsilon$, showing $\ln n(\varepsilon, d) \leq C \ln(\varepsilon^{-1})(1 + \ln d)$.
• Compares different equivalent norms on $H^{s}_{\rm mix}(\mathbb{T}^d)$, showing that the constant $C_s(d)$ in the upper bound decays super-exponentially as $\left[\frac{\lambda^d}{(d-1)!}\right]^s$ with $2 \leq \lambda \leq 6$, depending on the norm.
• Establishes connections to tensor product structures and uses known results on tensor product operators to deduce tractability from univariate behavior.

Experimental results

Research questions

• RQ1 What is the precise asymptotic behavior of the approximation numbers $a_n(I_d)$ as $n \to \infty$ for fixed $s > 0$ and growing $d$?
• RQ2 How does the approximation error decay in the preasymptotic regime, particularly for $n \leq 4^d$, and what does this imply for practical algorithms?
• RQ3 What is the dependence of the approximation constants on the dimension $d$, and does it decay super-exponentially?
• RQ4 Is the approximation problem quasi-polynomially tractable, and how does this relate to the curse of dimensionality?
• RQ5 How do different equivalent norms on $H^s_{\rm mix}(\mathbb{T}^d)$ affect the constants in the approximation rate estimates?

Key findings

• The approximation numbers satisfy $\lim_{n \to \infty} \frac{n^s a_n(I_d)}{(\ln n)^{(d-1)s}} = \left[\frac{2^d}{(d-1)!}\right]^s$, showing exact asymptotic behavior with super-exponential decay in $d$.
• The upper bound constant $C_s(d)$ in the approximation rate is $\left[\frac{\lambda^d}{(d-1)!}\right]^s$ with $2 \leq \lambda \leq 6$, indicating super-exponential decay in $d$ for fixed $s > 0$.
• For $n \leq 4^d$, the preasymptotic decay is slow, with the rate $n^{-s}(\ln n)^{(d-1)s}$ peaking at $\left(\frac{d-1}{e}\right)^{s(d-1)}$, which grows super-exponentially in $d$, delaying the onset of decay.
• The approximation problem is quasi-polynomially tractable: $n(\varepsilon, d) \leq C_t \exp\left(t \ln(\varepsilon^{-1})(1 + \ln d)\right)$ for some $t, C_t > 0$, implying that the number of samples grows slowly with $d$ and $\varepsilon^{-1}$.
• The problem is not polynomially tractable but is weakly tractable, and quasi-polynomial tractability holds for all considered equivalent norms on $H^s_{\rm mix}(\mathbb{T}^d)$, including $H^{s,\#}_{\rm mix}$, $H^{s,+}_{\rm mix}$, $H^{s,*}_{\rm mix}$, and $H^m_{\rm mix}$.
• The results are derived via combinatorial estimates on multi-index sets and spectral analysis, with connections to tensor product operators and known tractability theory.