Adaptive and local regularization for data fitting by tensor-product spline surfaces

Abstract

We propose to employ a non-constant regularization weight function (RWF) for data fitting via least-squares tensor-product (TP) spline fitting. In the first part of the paper, we formulate the discrete and the continuous version of the problem, and we investigate the influence of the degree of the RWF — which is also chosen as a TP spline function — in the latter situation. The second part presents two methods for automatically generating non-constant RWFs in the discrete situation. These methods are shown to be particularly useful if holes or features are present in the data.

Appendix. A bound on the Sobolev seminorm of the truncated spline projection

The proof of Theorem 1 relies on a bound on the Sobolev seminorm of the truncated spline projection. We use this Appendix to present this result, together with its proof.

Proposition 1

We consider again the assumptions of Theorem 1. The truncated spline projection defined in (5) satisfies

$$\begin{aligned} | u_h|_{W^{2,2}(\varOmega ),\lambda }^2 \le \varepsilon C_3 h^{2p+2}\ , \end{aligned}$$

where the constant $C_3$ depends on f, $\omega$, p and q.

Before presenting the proof, we derive an auxiliary result.

Lemma 3

Given a point ${\textbf {t}}_0\in [0,1]^d$, we consider multi-indices ${\textbf {i}}_h$ satisfying

$$\begin{aligned} {\textbf {t}}_0 \in supp B_{{\textbf {i}}_h,h}^{p} \end{aligned}$$

that are indexed by a zero sequence of mesh sizes h. The values of the associated linear coefficient functionals converge to the value of the function at this point,

$$\begin{aligned} \underset{h\rightarrow 0}{\lim } \mu ^{p}_{{\textbf {i}}_h,h}(\phi ) = \phi ({\textbf {t}}_0) \end{aligned}$$

if $\phi$ is $C^2$ smooth.

Proof

For each mesh size h we denote with $\Delta _h$ the d-dimensional knot span that contains ${\textbf {t}}_0$ (or one of them, if several knot spans with this property exist). As the first step, we exploit the linearity of the coefficient functionals and the equivalence of the norms on the finite-dimensional space of polynomials of degree p to conclude that

$$\begin{aligned}{} & {} \underset{{\textbf {i}}: \Delta _h\subseteq \text {supp} B_{{\textbf {i}},h}^{p}}{\max } |\mu _{{\textbf {i}},h}^{p}(\phi )-\phi ({\textbf {t}}_0)| = \underset{{\textbf {i}}: \Delta _h\subseteq \text {supp} B_{{\textbf {i}},h}^{p}}{\max } |\mu _{{\textbf {i}},h}^{p}\big (\phi -\phi ({\textbf {t}}_0)\big )| \\{} & {} \le \frac{C_9}{h^d} \Big \Vert \sum \limits _{0\le {\textbf {i}} \le {{\textbf {m}}}_h + p} B_{{\textbf {i}},h}^{p}({\textbf {t}}) \mu _{{\textbf {i}},h}^{p}\big (\phi -\phi ({\textbf {t}}_0)\big ) \Big \Vert _{L^2(\Delta _h)} = \frac{C_9}{h^d} \Big \Vert \varPi ^p_h(\phi ) - \phi ({\textbf {t}}_0) \Big \Vert _{L^2(\Delta _h)}, \end{aligned}$$

where the existence of the h-independent constant $C_9$ is guaranteed by the quasi–uniformity of the knot vectors. The factor $1/h^d$ is due to the integration over the knot span that is involved on the definition of the $L^2$ norm on $\Delta _h$. Secondly we apply the triangle inequality, arriving at

$$\begin{aligned} \Big \Vert \varPi ^p_h(\phi ) - \phi ({\textbf {t}}_0) \Big \Vert _{L^2(\Delta _h)}&\le \Big \Vert \varPi ^p_h(\phi ) - \phi \Big \Vert _{L^2(\Delta _h)} + \Big \Vert \phi - \phi ({\textbf {t}}_0) \Big \Vert _{L^2(\Delta _h)}. \end{aligned}$$

While we may use the inequality (2) with $\ell =p$ and domain $\Delta _h$ to bound the first term on the right–hand side, the second term can be estimated via the first derivatives, which gives

$$\begin{aligned} | \phi ({\textbf {t}}) - \phi ({\textbf {t}}_0) | \le | h \underset{{\textbf {t}}'\in \Delta _h}{\max }\Vert \nabla \phi ({\textbf {t}}')\Vert \, | \quad \text {if} \quad {\textbf {t}}\in \Delta _h \end{aligned}$$

since $\phi$ is required to be $C^2$ smooth. Finally we complete the proof by combining these observations to obtain

$$\begin{aligned} |\mu ^{p}_{{\textbf {i}}_h,h}(\phi )-\phi ({\textbf {t}}_0) |\le & {} \underset{{\textbf {i}}: \Delta _h\subseteq \text {supp} B_{{\textbf {i}},h}^{p}}{\max } |\mu _{{\textbf {i}},h}^{p}(\phi )-\phi ({\textbf {t}}_0)| \\\le & {} \frac{C_{9}}{h^d} \big (C_1 h \underbrace{| \phi |_{ W^{1,2}(\Delta _h)}}_{\le C_{11} h^d | \phi |_{ W^{2,\infty }(\varOmega )}} + C_{10} h^{d+1} |\nabla \phi |_{W^{1,\infty }(\varOmega )}\big ) \end{aligned}$$

and noting that the right-hand side converges to zero as $h\rightarrow 0$. $\square$

Now we are ready to prove the proposition.

Proof (Proposition 1)

Let

$$\begin{aligned} H^{s} = \varOmega \setminus \bigcup _{\textbf{i}:\, \omega B_{\textbf{i},h}^{\textbf{p}}\ne 0} \text {supp} B_{\textbf{i},h}^{\textbf{p}} \subset H \end{aligned}$$

be the subdomain of $\varOmega$ with the property that no basis function possesses a support which simultaneously overlaps $\varOmega _0$ and $H^s$, see Fig. 12.

Fig. 12

A domain $\varOmega$, a subdomain H (hole, light and dark blue) and the resulting subdomain $H^s$ (dark blue), for $p=3$

We split the domain into several subdomains and obtain

$$\begin{aligned}{} & {} | u_h|_{W^{2,2}(\varOmega ),\lambda }^2 = \int \limits _{\varOmega } \lambda ({\textbf {t}}) {\sum _{\nu ,\eta =1}^d (\partial _{\nu \eta } u_h({\textbf {t}}))^2} \textrm{d}{} {\textbf {t}} \\{} & {} = \underbrace{\int \limits _{\varOmega _0} \lambda {\sum \limits _{\nu ,\eta =1}^d (\partial _{\nu \eta } u_h)^2} \textrm{d}{} {\textbf {t}}}_{(a)} + \underbrace{\int \limits _{H \setminus H^{s}} \lambda {\sum \limits _{\nu ,\eta =1}^d (\partial _{\nu \eta } u_h)^2} \textrm{d}{} {\textbf {t}}}_{(b)} + \underbrace{\int \limits _{H^{s}} \lambda {\sum \limits _{\nu ,\eta =1}^d (\partial _{\nu \eta } u_h)^2} \textrm{d}{} {\textbf {t}}}_{(c)}. \end{aligned}$$

We note that $\lambda |_{\varOmega _0} = 0$, thus the integral (a) does not contribute to the squared $L^2$ norm. Similarly, the integral (c) also vanishes since $T_h^p (\varPi _h^p f)|_{H^{s}}=0$. The integral (b) can be estimated by

$$\begin{aligned}{} & {} \int \limits _{H \setminus H^{s}} \lambda {\sum \limits _{\nu ,\eta =1}^d (\partial _{\nu \eta } u_h)^2} \textrm{d}{} {\textbf {t}} = \int \limits _{H \setminus H^{s}} \lambda {\sum \limits _{\nu ,\eta =1}^d (\partial _{\nu \eta }T^p_h \varPi ^p_h f)^2} \textrm{d}{} {\textbf {t}} \\\le & {} \int \limits _{H \setminus H^{s}} \lambda {\sum \limits _{\nu ,\eta =1}^d} {\underset{B_{{\textbf {i}},h}^{p}\cdot B_{{\textbf {j}},h}^{p}|_{H\setminus H^{s}}\ne 0}{\underset{{\textbf {j}}:\, 0\le {\textbf {j}} \le {\textbf {m}}_{h}+p}{\underset{{\textbf {i}}:\, 0\le {\textbf {i}} \le {\textbf {m}}_{h}+p}{\sum }}}} |\varvec{\mu }^{p}_{{\textbf {i}},h}(f) \varvec{\mu }^{p}_{{\textbf {j}},h}(f)| \; |\partial _{\nu \eta } B_{{\textbf {i}},h}^{p} {\partial _{\nu \eta }} B_{{\textbf {j}},h}^{p}|\, \textrm{d}{} {\textbf {t}}. \end{aligned}$$

According to Lemma 3 there exists a constant $C_4$ — which depends on f — such that the coefficient functionals satisfy $|\mu ^{p}_{{\textbf {i}},h}(f)|\le C_4$ if the value of h is sufficiently small, hence

$$\begin{aligned} \int \limits _{H \setminus H^s} \lambda \sum \limits _{\nu ,\eta =1}^d {(\partial _{\nu \eta } u_h)^2} \textrm{d}{} {\textbf {t}} \le&\ C_4^2 \int \limits _{H \setminus H^s} \lambda {\sum \limits _{\nu ,\eta =1}^d} {\underset{B_{{\textbf {i}},h}^{p}\cdot B_{{\textbf {j}},h}^{p}|_{H\setminus H^{s}}\ne 0}{\underset{{\textbf {j}}:\, 0\le {\textbf {j}} \le {\textbf {m}}_{h}+p}{\underset{{\textbf {i}}:\, 0\le {\textbf {i}} \le {\textbf {m}}_{h}+p}{\sum }}}} (|\partial _{\nu \eta } B_{{\textbf {i}},h}^{p} {\partial _{\nu \eta }} B_{{\textbf {j}},h}^{p}|) \textrm{d}{} {\textbf {t}}. \end{aligned}$$

Moreover, for all $\nu$, $\eta$, ${\textbf {i}}$ and ${\textbf {j}}$ there exists a constant $C_5$ such that

$$\begin{aligned} |\partial _{\nu \eta } B_{{\textbf {i}},h}^{p} {\partial _{\nu \eta }} B_{{\textbf {j}},h}^{p}| \le C_5 h^{-4}, \end{aligned}$$

thus

$$\begin{aligned} \int \limits _{H \setminus H^s} \lambda \sum \limits _{\nu ,\eta =1}^d {(\partial _{\nu \eta } u_h)^2} \textrm{d}{} {\textbf {t}} \le&\ C_4^2 C_5 h^{-4} \int \limits _{H \setminus H^s} \lambda {\sum \limits _{\nu ,\eta =1}^d} {\underset{B_{{\textbf {i}},h}^{p}\cdot B_{{\textbf {j}},h}^{p}|_{H\setminus H^{s}}\ne 0}{\underset{{\textbf {j}}:\, 0\le {\textbf {j}} \le {\textbf {m}}_{h}+p}{\underset{{\textbf {i}}:\, 0\le {\textbf {i}} \le {\textbf {m}}_{h}+p}{\sum }}}} 1 \; \textrm{d}{} {\textbf {t}}. \end{aligned}$$

The number of terms in the summation can be bounded by a constant multiplied with the number of elements that intersect $H\setminus H^s$, i.e., by $\frac{C_6}{h^{d-1}}$, where the constant $C_6$ depends on p, d and $\omega$. We thus arrive at

$$\begin{aligned} \int \limits _{H \setminus H^s} \lambda \sum \limits _{\nu ,\eta =1}^d {(\partial _{\nu \eta } u_h)^2} \textrm{d}{} {\textbf {t}} \le C_4^2 C_5 C_6 h^{-d-3} \int \limits _{H \setminus H^s} \lambda \; \textrm{d}{} {\textbf {t}}. \end{aligned}$$

We now use the assumption (4), which implies

$$\begin{aligned} \lambda ({\textbf {t}}) \le \varepsilon {\text {dist}}({\textbf {t}}, \partial H)^{q} \le \varepsilon C_7 h^q \text { on } H\setminus H^{s}\ \end{aligned}$$

and the fact that

$$\begin{aligned} \textrm{vol}_d(H\setminus H^s) \le C_8 h \end{aligned}$$

to get

$$\begin{aligned} \int \limits _{H \setminus H^s} \lambda \; \textrm{d}{} {\textbf {t}} \le \varepsilon C_7C_8 h^{q+1}. \end{aligned}$$

Finally we arrive at

$$\begin{aligned} \int \limits _{H \setminus H^s} \lambda \sum \limits _{\nu ,\eta =1}^d {(\partial _{\nu \eta } u_h)^2} \textrm{d}{} {\textbf {t}} \le \varepsilon C_4^2 C_5 C_6 C_7 C_8 h^{q-d-2}. \end{aligned}$$

The claimed result then follows since $q\ge 2p+4+d$ is assumed in the theorem. $\square$