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.
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Acknowledgements
The authors would like to thank David Großmann, Thomas Takacs and the reviewers for their helpful suggestions on how to improve this paper.
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This work is supported by European Research Council through the CHANGE project (GA No. 694515).
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Appendix. A bound on the Sobolev seminorm of the truncated spline projection
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
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
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,
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
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
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
since \(\phi\) is required to be \(C^2\) smooth. Finally we complete the proof by combining these observations to obtain
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
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.
We split the domain into several subdomains and obtain
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
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
Moreover, for all \(\nu\), \(\eta\), \({\textbf {i}}\) and \({\textbf {j}}\) there exists a constant \(C_5\) such that
thus
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
We now use the assumption (4), which implies
and the fact that
to get
Finally we arrive at
The claimed result then follows since \(q\ge 2p+4+d\) is assumed in the theorem. \(\square\)
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Merchel, S., Jüttler, B. & Mokriš, D. Adaptive and local regularization for data fitting by tensor-product spline surfaces. Adv Comput Math 49, 58 (2023). https://doi.org/10.1007/s10444-023-10035-1
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DOI: https://doi.org/10.1007/s10444-023-10035-1