Abstract
The distribution of centered Gaussian random fields (GRFs) indexed by compacta such as smooth, bounded Euclidean domains or smooth, compact and orientable manifolds is determined by their covariance operators. We consider centered GRFs given as variational solutions to coloring operator equations driven by spatial white noise, with an elliptic self-adjoint pseudodifferential coloring operator from the Hörmander class. This includes the Matérn class of GRFs as a special case. Using biorthogonal multiresolution analyses on the manifold, we prove that the precision and covariance operators, respectively, may be identified with bi-infinite matrices and finite sections may be diagonally preconditioned rendering the condition number independent of the dimension p of this section. We prove that a tapering strategy by thresholding applied on finite sections of the bi-infinite precision and covariance matrices results in optimally numerically sparse approximations. That is, asymptotically only linearly many nonzero matrix entries are sufficient to approximate the original section of the bi-infinite covariance or precision matrix using this tapering strategy to arbitrary precision. The locations of these nonzero matrix entries can be determined a priori. The tapered covariance or precision matrices may also be optimally diagonally preconditioned. Analysis of the relative size of the entries of the tapered covariance matrices motivates novel, multilevel Monte Carlo (MLMC) oracles for covariance estimation, in sample complexity that scales log-linearly with respect to the number p of parameters. In addition, we propose and analyze novel compressive algorithms for simulating and kriging of GRFs. The complexity (work and memory vs. accuracy) of these three algorithms scales near-optimally in terms of the number of parameters p of the sample-wise approximation of the GRF in Sobolev scales.
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Acknowledgements
LH, KK and CS acknowledge helpful discussions with Sara van de Geer. This paper was conceived and written in large parts at SAM, D-MATH, ETH Zürich.
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Appendices
A Pseudodifferential operators on manifolds
We consider orientable manifolds satisfying Assumption 1(I). In the case that \(\mathcal {M}\) is not orientable, there exists a covering manifold \(\widetilde{\mathcal {M}}\) of \(\mathcal {M}\) with two sheets such that \(\widetilde{\mathcal {M}}\) is orientable [5, Thm. I.58] and of dimension \(n \ge 1\).
1.1 A.1 Surface differential calculus
A tangent vector at \(x\in \mathcal {M}\) is a mapping \(X:f\rightarrow X(f)\in \mathbb {R}\) which is defined on the set of functions f that are differentiable in a neighborhood of x which satisfies a) for \(\lambda , \mu \in \mathbb {R}\), \(X(\lambda f+ \mu g) = \lambda X(f) + \mu X(g)\), b) \(X(f)=0\) if f is flat, c) \(X(fg) = f(x)X(g) + g(x)X(f)\). The tangent space \(T_x(\mathcal {M})\) to \(\mathcal {M}\) at \(x\in \mathcal {M}\) is the set of tangent vectors at x. In any coordinate system \(\{ x^i \}\) in \(\mathcal {M}\) at x, the vectors \(\partial /\partial x^i\) defined by \((\partial /\partial x^i)_x(f) = [\partial (f\circ \varphi ^{-1})/\partial x^i]_{\varphi (x)}\) belong to \(T_x(\mathcal {M})\), and form a basis of the tangent vector space at \(x\in \mathcal {M}\). Here, \(\varphi \) is any diffeomorphism on a neighborhood of x. Its dual vector space is denoted by \(T_x^*(\mathcal {M})\). The tangent space \(T(\mathcal {M})\) to \(\mathcal {M}\) is \(\bigcup _{x\in \mathcal {M}} T_x(\mathcal {M})\), the dual tangent space \(T^*(\mathcal {M})\) is \(\bigcup _{x\in \mathcal {M}} T^*_x(\mathcal {M})\). The tangent space \(T(\mathcal {M})\) carries a vector fiber bundle structure. More generally, for \(r,s\in \mathbb {N}\), the fiber bundle \(T^r_s(\mathcal {M})\) of (r, s) tensors is \(\bigcup _{x\in \mathcal {M}} T_x(\mathcal {M})^{\otimes ^r} \otimes T_x^*(\mathcal {M})^{\otimes ^s}\).
The manifold \(\mathcal {M}\) gives rise to the compact metric space \((\mathcal {M}, {\text {dist}}(\,\cdot \,, \,\cdot \,) )\), where the distance \({\text {dist}}(\,\cdot \,, \,\cdot \,)\) can be chosen, for example, as the geodesic distance in \(\mathcal {M}\) of two points \(x,x'\in \mathcal {M}\), see [5, Prop. I.35].
1.1.1 A.1.1 Coordinate charts and triangulations
Provided that the manifold \(\mathcal {M}\) of dimension n satisfies Assumption 1(I), it can be locally represented as parametric surface consisting of smooth coordinate patches. Specifically, denote by \(\Box = [0,1]^n\) the unit cube. Then we assume that \(\mathcal {M}\) is partitioned into a finite number M of closed patches \(\mathcal {M}_i\) such that
Here, each \(\gamma _i :\Box \rightarrow \mathcal {M}_i\) is assumed to be a smooth diffeomorphism. We also assume that there exist smooth extensions \(\widetilde{\mathcal {M}}_i\supset \mathcal {M}_i\) and \(\widetilde{\gamma }_i :\widetilde{\Box } \rightarrow \widetilde{\mathcal {M}}_i\) such that \(\widetilde{\gamma }_i\vert _\Box = \gamma _i\), where \(\widetilde{\Box } = (-1,2)^n\). Note that, in the notation of Section 2, \(G = \widetilde{\Box }\).
The intersections \(\mathcal {M}_i \cap \mathcal {M}_j\) for \(i\ne j\) are either assumed to be empty or to be diffeomorphic to \([0,1]^k\) for some \(0\le k <n\). We assume the charts \(\gamma _i\) to be \(C^0\)-compatible in the sense that for every \(\hat{x} \in \mathcal {M}_i \cap \mathcal {M}_{i'}\) exists a bijective mapping \(\Theta :\Box \rightarrow \Box \) such that \((\gamma _{i'}\circ \Theta )(x) = \hat{x}\) for \(x=(x_1,\ldots ,x_n)\in \Box \) with \(\gamma _i(x) = \hat{x}\). Note that \(C^0\)-compatibility admits \(\mathcal {M}= \partial G\) for certain polytopal domains G. In the case that \(\mathcal {M}= \partial G\) is smooth, we shall assume that the extensions satisfy \(\widetilde{\mathcal {M}}_i \subset \mathcal {M}\) and that the charts \(\gamma _i\) are smoothly compatible.
In the construction of MRAs on \(\mathcal {M}\), we shall require triangulations of \(\mathcal {M}\). We shall introduce these in the Euclidean parameter domain \(\Box \) and lift them to the coordinate patches \(\mathcal {M}_i\) on \(\mathcal {M}\) via the charts \(\gamma _i\).
A mesh of refinement level j on \(\mathcal {M}\) is obtained by dyadic subdivisions of depth j of \(\Box \) into \(2^{jn}\) subcubes \(C_{j,k} \subseteq \Box \), where the multi-index \(k = (k_1,\ldots ,k_n) \in \mathbb {N}_0^n\) tags the location of \(C_{j,k}\) with \(0\le k_m < 2^j\). With this construction in each co-ordinate patch, and taking into account the inter-patch compatibility of the charts \(\gamma _i\), this results in a regular quadrilateral triangulation of \(\mathcal {M}\) consisting of \(2^{jn}M\) cells \(\Gamma _{i,j,k} := \gamma _i(C_{j,k}) \subset \mathcal {M}_i \subset \mathcal {M}\).
1.1.2 A.1.2 Sobolev spaces
Let \(\mathcal {M}\) denote a compact manifold as in Section A.1.1. Sobolev spaces on \(\mathcal {M}\) are invariantly defined in the usual fashion, i.e., in local coordinates of a smooth atlas \(\{\widetilde{\gamma }_i\}_{i=1}^{M}\) of coordinate charts on \(\mathcal {M}\).
As in Assumption 1(I), we assume that \(\mathcal {M}\) has dimension \(n\in \mathbb {N}\), \(\partial \mathcal {M}= \emptyset \), and is equipped with a (surface) measure \(\mu \). It is given in terms of the first fundamental form on \(\mathcal {M}\) which, on \(\mathcal {M}_i\), is given by
The matrix \(K_i\) in (A.1) is symmetric and positive definite uniformly in \(x\in \widetilde{\mathcal {M}}_{i}\). The \(L^2(\mathcal {M})\) inner product on \(\mathcal {M}\) can then be expressed in the local chart coordinates via
where \(\{\chi _i\}_{i=1}^M\) denotes a smooth partition of unity which is subordinate to the atlas \(\{\widetilde{\gamma }_i\}_{i=1}^M\). For \(1\le p \le \infty \), \(L^p(\mathcal {M})\) shall denote the usual space of real-valued, strongly measurable maps \(v:\mathcal {M}\rightarrow \mathbb {R}\) which are p-integrable with respect to \(\mu \).
Sobolev spaces on \(\mathcal {M}\) are invariantly defined by lifting their Euclidean versions on \(\widetilde{\Box }\) to \(\widetilde{\mathcal {M}}_{i}\) via \(\widetilde{\gamma }_i\). For \(s\ge 0\), the respective norm on \(H^s(\mathcal {M})\) may be defined by
This definition is equivalent to the definition of \(H^s(\mathcal {M})\) and \( \Vert \,\cdot \, \Vert _{ H^s(\mathcal {M}) }\) in (). For further details the reader is referred to [42, pp. 30–31 of Appendix B] and the references therein. (Note that the proof of this equivalence on the sphere \(\mathcal {M}=\mathbb {S}^2\) as elaborated in [42] exploits only compactness and smoothness of \(\mathbb {S}^2\). Thus, it can be generalized to any manifold as considered in this work.)
For \(s<0\), the spaces \(H^s(\mathcal {M})\) are defined by duality, here and throughout identifying \(L^2(\mathcal {M})\) with its dual space.
1.2 A.2 (Pseudo)differential operators
We review basic definitions and notation from the Hörmander–Kohn–Nirenberg calculus of pseudodifferential operators, to the extent that they are needed in our analysis of covariance kernels and operators.
1.2.1 A.2.1 Basic definitions
Let G be an open, bounded subset of \(\mathbb {R}^n\), r, \(\rho \), \(\delta \in \mathbb {R}\) with \(0\le \rho \le \delta \le 1\). The Hörmander symbol class \(S^r_{\rho ,\delta }(G)\) consists of all \(b\in C^\infty (G\times \mathbb {R}^n)\) such that, for all \(K\subset \subset G\) and for any \(\alpha ,\beta \in \mathbb {N}_0^n\), there is a constant \(C_{K,\alpha ,\beta }>0\) with
When the set G is clear from the context, we write \(S^r_{\rho ,\delta }\). In what follows, we shall restrict ourselves to the particular case \(\rho = 1\), \(\delta = 0\), and consider \(S^r_{1,0}\). In addition, we write \(S^{-\infty }_{1,0} := \bigcap _{r\in \mathbb {R}} S^r_{1,0}\). A symbol \(b\in S^r_{1,0}\) gives rise to a pseudodifferential operator B via the relation ().
When \(b\in S^r_{1,0}\), the operator B is said to belong to \(OPS^r_{1,0}(G)\) and it is (in a suitable topology) a continuous operator \(B:C_0^\infty (G) \rightarrow C^\infty (G)\), cf. [69, Thm. II.1.5]. We write \(OPS^{-\infty }(G) = \bigcap _{r\in \mathbb {R}} OPS^{r}_{1,0}(G)\). We say that the operator \(B \in OPS^r_{1,0}(G)\) is elliptic of order \(r \in \mathbb {R}\) if, for each compact \(K\subset \subset G\), there exist constants \(C_K>0\) and \(R>0\) such that
1.2.2 A.2.2 (Pseudo)differential operators on manifolds
We suppose Assumption 1. Having introduced the class \(OPS^r_{1,0}(G)\) for an Euclidean domain G, the operator class \(OPS^r_{1,0}(\mathcal {M})\) is defined by the usual “lifting to \(\mathcal {M}\) in local coordinates” as described, e.g., in [69, Sec. II.5]. The definition is based on the behavior of \(OPS^r_{1,0}(G)\) under smooth diffeomorphic changes of coordinates which we consider first.
Let \(G, \mathcal {O}\subset \mathbb {R}^n\) be open and let \(\gamma :G \rightarrow \mathcal {O}\) be a diffeomorphism. Consider \(B \in OPS^r_{1,0}(G)\), so that \(B:C_0^\infty (G) \rightarrow C^\infty (G)\). We define the transported operator \(\widetilde{B}\) by
For \(r\in \mathbb {R}\), we then consider \(OPS^r_{1,0}(\mathcal {M})\), the Hörmander class of pseudodifferential operators on \(\mathcal {M}\) (investigated earlier by Kohn and Nirenberg [50]). We alert the reader to the use of the notation \(OPS^r(\mathcal {M})\) for the so-called classical pseudodifferential operators which afford (pseudohomogeneous) symbol expansions and comprise a strict subset of \(OPS^r_{1,0}(\mathcal {M})\), see, e.g., [66].
Pseudodifferential operators in \(OPS^{r}_{1,0}(\mathcal {M})\) on manifolds \(\mathcal {M}\) are defined in local coordinates. A linear operator \(\mathcal {B}:C^\infty (\mathcal {M})\rightarrow C^\infty (\mathcal {M})\) is a pseudodifferential operator of order \(r\in \mathbb {R}\) on \(\mathcal {M}\), \(\mathcal {B}\in OPS^r_{1,0}(\mathcal {M})\), if for any finite, smooth partition of unity \(\bigl \{ \chi _i \in C_0^\infty (\widetilde{\mathcal {M}}_i): i=1,\ldots ,\bar{m} \bigr \}\) with respect to any atlas \(\bigl \{ \bigl (\widetilde{\mathcal {M}}_i, \widetilde{\gamma }_i \bigr ) \bigr \}_{i=1}^{\bar{m}}\) of \(\mathcal {M}\) all transported operators satisfy
The class of all such operators is denoted \(OPS^r_{1,0}(\mathcal {M})\). Importantly, \(OPS^r_{1,0}(\mathcal {M})\) defined in this way does not depend on the choice of the atlas of \(\mathcal {M}\) and is invariantly defined [69, Sec. II.5], [46, Def. 18.1.20].
1.2.3 A.2.3 Principal symbols
For a bounded, open set \(\mathcal {O}\subset \mathbb {R}^n\), the principal symbol \(b_0(x,\xi )\) of \(B\in OPS^r_{1,0}(\mathcal {O})\) is the equivalence class in \(S^r_{1,0}(\mathcal {O})/S^{r-1}_{1,0}(\mathcal {O})\) (see, e.g., [69, p. 49]). Any member of the equivalence class will be called a principal symbol of B. For \(\mathcal {B}\in OPS^r_{1,0}(\mathcal {M})\), its principal symbol \(b_0(x,\xi )\) is invariantly (with respect to the choice of atlas on \(\mathcal {M}\)) defined on \(T^*(\mathcal {M})\) (see [69, Eq. (5.6)]).
1.2.4 A 2.4 Pseudodifferential calculus
The symbol class \(S^r_{1,0}\) admits a symbolic calculus (e.g., [69, Prop. II.1.3]). In the sequel, we assume all pseudodifferential operators to be properly supported, see [69, Def. II.3.6]. This is not restrictive, as every \(\mathcal {B}\in OPS^r_{1,0}(\mathcal {M})\) can be written as \(\mathcal {B}= \mathcal {B}_1 + \mathcal {R}\), where \(\mathcal {B}_1\in OPS^r_{1,0}(\mathcal {M})\) is properly supported and where \(\mathcal {R}\in OPS^{-\infty }(\mathcal {M})\) [46, Prop. 18.1.22].
Proposition 1
Let \(r,t\in \mathbb {R}\) and \(\mathcal {A}\in OPS^r_{1,0}(\mathcal {M})\), \(\mathcal {B}\in OPS^t_{1,0}(\mathcal {M})\) be properly supported. Then, it holds
-
(i)
\(\mathcal {A}+ \mathcal {B}\in OPS^{\max \{ r , t \}}_{1,0}(\mathcal {M})\),
-
(ii)
\(\mathcal {A}\mathcal {B}\in OPS^{ r+t }_{1,0}(\mathcal {M})\),
-
(iii)
\(\forall s\in \mathbb {R}: \; \mathcal {A}:H^{s}(\mathcal {M}) \rightarrow H^{s-r}(\mathcal {M})\) is continuous.
Proof
The symbol class \(S^{r}_{1,0}\) is constructed such that by (A.2), \(S^{r}_{1,0} \subseteq S^{\max \{r, t\}}_{1,0}\) and also \(S^{t}_{1,0} \subseteq S^{\max \{r,t\}}_{1,0}\). Assertion (i) follows from the construction of the class \(OPS^{r}_{1,0}(\mathcal {M})\) via an atlas, see Section A.2.2.
Recall the atlas \(\{ \widetilde{\gamma }_i \}_{i=1}^{M}\) of \({\mathcal {M}}\) with subordinate smooth partition of unity \(\{ \chi _i \}_{i=1}^{M}\). The transported operators \(A_{i,i'}\) and \(B_{i,i'}\) defined according to (A.3) belong to \(OPS^r_{1,0}(\widetilde{\Box })\) and to \(OPS^t_{1,0}(\widetilde{\Box })\). By [69, Thm. II.4.4], \(A_{j,j'} B_{j,j'} \in OPS^{r +t}_{1,0}(\widetilde{\Box })\). Thus, claim (ii) holds by the construction of the class \(OPS^{r+t}_{1,0}(\mathcal {M})\) in Subsection A.2.2.
Finally, the third assertion (iii) follows from [69, Thm. II.6.5], which is elucidated on [69, p. 53 of Sec. II.7].\(\square \)
Proposition 2
Let \(r\in \mathbb {R}\) and \(\mathcal {A}\in OPS^r_{1,0}(\mathcal {M})\) be self-adjoint, positive definite, and elliptic, i.e., there exists a constant \(a_- > 0\) such that
Then, for every \(\beta \in \mathbb {R}\), \(\mathcal {A}^\beta \in OPS^{\beta r}_{1,0}(\mathcal {M})\).
Proof
Since \(a_- >0\), \(\mathcal {A}\) is invertible. The assertion follows from [66, Thm. 3].\(\square \)
B Multiresolution bases on manifolds
In this section we briefly explain how the single-scale basis \(\mathbf {\Phi }_j\), the dual single-scale basis \(\widetilde{\mathbf {\Phi }}_j\) as well as the biorthogonal complement bases \(\mathbf {\Psi }_j\) and \(\widetilde{\mathbf {\Psi }}_j\) in (), () and () can be constructed on a manifold \(\mathcal {M}\) which satisfies Assumption 1(I). We collect some of their basic properties.
We recall from () that, for \(j>j_0\), the subspaces \(V_j \subset V_{j+1} \subset \ldots \subset L^2(\mathcal {M})\) are spanned by single-scale bases \(\mathbf {\Phi }_j := \{ \phi _{j,k}: k\in \Delta _j \}\), where \(\Delta _j\) denote suitable index sets describing spatial localization of the \(\phi _{j,k}\). Furthermore, the subspaces are of cardinality \(\dim (V_j) = \mathcal {O}( 2^{nj} )\). We assume elements \(\phi _{j,k}\in V_j\) to be normalized in \(L^2(\mathcal {M})\), and their supports to scale according to \({\text {diam}} ({\text {supp}} \phi _{j,k}) \simeq 2^{-j}\). We associate with these bases so-called dual single-scale bases \(\widetilde{\mathbf {\Phi }}_j := \{ \widetilde{\phi }_{j,k}: k\in \Delta _j \}\), for which one has \(\langle \phi _{j,k}, \widetilde{\phi }_{j,k'}\rangle = \delta _{k,k'}\) for \(k,k'\in \Delta _j\). Such dual systems of one-scale bases on \(\mathcal {M}\) can be lifted in charts \(\mathcal {M}_j\) via parametrizations \(\gamma _j\) from tensor products of univariate systems in the parameter domains \(\Box \subset \mathbb {R}^n\). For example, for primal bases \(\mathbf {\Phi }_j\) obtained from tensorized, univariate B-splines of order d in \(\Box \) with dual bases of order \(\widetilde{d}\) such that \(d + \widetilde{d}\) is even, the \(\mathbf {\Phi }_j\) and \(\widetilde{\mathbf {\Phi }}_j\) have approximation orders d and \(\widetilde{d}\), respectively, see (). The respective regularity indices \(\gamma \) and \(\widetilde{\gamma }\), see (), satisfy \(\gamma = d - 1/2\), whereas \(\widetilde{\gamma } \sim \widetilde{d}\). We refer to [22, 24, 55, 56, 59] for detailed constructions.
The biorthogonality of the systems \(\mathbf {\Phi }_j\), \(\widetilde{\mathbf {\Phi }}_j\) allows to introduce canonical projectors \(Q_j\) and \(Q^*_j\) for \(j\in \mathbb {N}\) with \(j > j_0\):
associated with corresponding multiresolution sequences \(\{ V_j \}_{j>j_0}\) and \(\{ \widetilde{V}_j \}_{j>j_0}\).
The \(L^2(\mathcal {M})\)-boundedness of \(Q_j\) implies the Jackson and Bernstein inequalities,
for all \(-\widetilde{d} \le s\le t \le d\), \(s<\gamma \), \(-\widetilde{\gamma } < t\), and
for all \(t\le s \le \gamma \), with constants implied in \(\lesssim \) which are uniform with respect to j.
To define MRAs, we start by introducing index sets \(\nabla _j := \Delta _{j+1}\backslash \Delta _j\), \(j>j_0\). Given single-scale bases \(\mathbf {\Phi }_j\), \(\widetilde{\mathbf {\Phi }}_j\), the biorthogonal complement bases \(\mathbf {\Psi }_j\) and \(\widetilde{\mathbf {\Psi }}_j\) in () satisfying the biorthogonality relation () can be constructed such that () holds. We refer to [55, 56, 59] for particular constructions.
With the convention \(Q_{j_0} = Q^*_{j_0} = 0\), one has for \(v_J \in V_J\) and for \(\widetilde{v}_J \in \widetilde{V}_J\) that
From this observation, a second wavelet basis \(\widetilde{\mathbf {\Psi }}\) such that \(\mathbf {\Psi }\) and \(\widetilde{\mathbf {\Psi }}\) are mutually biorthogonal in \(L^2(\mathcal {M})\) is now obtained from the union of the coarse single-scale basis and complement bases, i.e.,
where we use the convention \(\mathbf {\Psi }_{j_0} := \mathbf {\Phi }_{j_0+1}\), \(\widetilde{\mathbf {\Psi }}_{j_0} := \widetilde{\mathbf {\Phi }}_{j_0+1}\) and assume that all basis functions are normalized in \(L^2(\mathcal {M})\). The bases \(\mathbf {\Psi }\) and \(\widetilde{\mathbf {\Psi }}\) are called the primal and dual MRAs, respectively.
The key to the preconditioning results for the covariance and precision matrices in Subsection 3.2 is the effect of diagonal preconditioning for pseudodifferential operators in MRAs.
To address this, we let \(\mathcal {B}\in OPS^r_{1,0}(\mathcal {M})\) be a pseudodifferential operator which satisfies Assumption 1(II), so that \(\mathcal {B}:H^{r/2}(\mathcal {M}) \rightarrow H^{-r/2}(\mathcal {M})\) is an isomorphism. Assume that \(\gamma > 0\). By (), \(\mathbf {\Psi }\) is a Riesz basis for \(L^2(\mathcal {M})\), so that the corresponding finite section matrices
are ill-conditioned, \({\text {cond}}_{2}({\textbf{B}}_J) \simeq 2^{|r|J}\). Stability of the Galerkin projection in \(H^{r/2}(\mathcal {M})\) and the Riesz-basis property () in \(H^{r/2}(\mathcal {M})\) imply the following result on diagonal preconditioning of \({\textbf{B}}_J\).
Proposition 3
For \(r\in \mathbb {R}\), define the diagonal matrix \(\textbf{D}^r_J \in \mathbb {R}^{p(J) \times p(J)}\) by
where \(|\lambda |=j\) for \(\lambda =(j,k)\) and, as in (),
Suppose that the manifold \(\mathcal {M}\) and the operator \(\mathcal {B}\in OPS^r_{1,0}(\mathcal {M})\) satisfy Assumptions 1(I) and (II), respectively. Furthermore, assume that () holds with
Then, for every \(J\in \mathbb {N}\), the diagonal matrices \(\textbf{D}^r_J\) define uniformly spectrally equivalent preconditioners for \({\textbf{B}}_J\), i.e.,
with constants implied in \(\simeq \) independent of J.
Proof
Under Assumptions 1(I)–(II) the operator \(\mathcal {B}\in OPS^{r}_{1,0}(\mathcal {M})\) defines an isomorphism between \(H^{r/2}(\mathcal {M})\) and \(H^{-r/2}(\mathcal {M})\), see Proposition 11(iii) and Proposition 12, and the norm equivalence
holds. Here, \(\langle \,\cdot \,, \,\cdot \,\rangle \) denotes the \((H^{-r/2}(\mathcal {M}),H^{r/2}(\mathcal {M}))\) duality pairing. The assertion then follows from the Riesz basis property ().\(\square \)
C Coloring of Whittle–Matérn type
Three essential characteristics of the covariance structure of a random field are given by its smoothness, the correlation length, and the marginal variance. A convenient approach to define models, for which these important properties can be parametrized, i.e., controlled in terms of certain numerical parameters, is to generalize the Matérn covariance family. Such a parametrization in turn facilitates for instance likelihood-based inference in spatial statistics.
Specifically, let us consider the white noise equation () for an elliptic, self-adjoint coloring pseudodifferential operator \(\mathcal {A}\) which is a fractional power \(\beta >0\) of an elliptic “base (pseudo)differential coloring operator” \(\mathcal {L}\in OPS^{\bar{r}}_{1,0}(\mathcal {M})\) of order \(\bar{r}>0\), shifted by the multiplication operator with respect to a nonnegative length-scale function \(\kappa :\mathcal {M}\rightarrow \mathbb {R}\), i.e.,
Here, \(\beta >0\) and \(\mathcal {L}\) are such that the resulting coloring operator \(\mathcal {A}\) fulfills Assumption 1(II). In particular, \(\kappa \in C^\infty (\mathcal {M})\).
For a linear, second-order (so that \(\bar{r}=2\)) elliptic (surface) differential operator \(\mathcal {L}\) on \(\mathcal {M}\) in divergence form, models of this type have been developed, e.g., in [12, 52]. Moreover, computationally efficient methods to sample from such random fields or to employ the models in statistical applications, involving for instance inference or spatial predictions, have been discussed recently, e.g., in [10, 11, 19, 41]. The following proposition extends and unifies these approaches, admitting rather general operators \(\mathcal {L}\) (which, in the classic Matérn case, see [53, 72], is the Laplace–Beltrami operator \(\mathcal {L}= -\Delta _\mathcal {M}\in OPS^{2}_{1,0}(\mathcal {M})\), with \(\bar{r}=2\) and constant correlation length parameter \(\kappa > 0\)).
Proposition 4
Suppose that the manifold \(\mathcal {M}\) satisfies Assumption 1(I) and that \(\mathcal {L}\in OPS^{\bar{r}}_{1,0}(\mathcal {M})\) for some \(\bar{r}>0\) is self-adjoint and positive. Let \(\beta > 0\) be such that \(\bar{r}\beta > n/2\) and let \(\mathcal {Z}_\beta \) denote the GRF solving the white noise equation () with coloring operator \(\mathcal {A}= (\mathcal {L}+\kappa ^2 )^\beta \) on \(\mathcal {M}\), where \(\kappa :\mathcal {M}\rightarrow \mathbb {R}\) is smooth. Then, the covariance operator \(\mathcal {C}_\beta \) of the GRF \(\mathcal {Z}_\beta \) is a self-adjoint operator, (strictly) positive definite, compact operator on \(L^2(\mathcal {M})\), with finite trace. Furthermore, the covariance operator of \(\mathcal {Z}_\beta \) is given by
It defines an isomorphism between \(H^{s}(\mathcal {M})\) and \(H^{s+2\bar{r}\beta }(\mathcal {M})\) for all \(s\in \mathbb {R}\).
The associated precision operator \(\mathcal {P}_\beta \) satisfies, for all \(s\in \mathbb {R}\),
and, for any \(s\in \mathbb {R}\), it defines an isomorphism between \(H^s(\mathcal {M})\) and \(H^{s-2\bar{r}\beta }(\mathcal {M})\).
A GRF \(\mathcal {Z}_\beta \) defined as in () with coloring operator \(\mathcal {A}= (\mathcal {L}+\kappa ^2 )^{\beta }\) admits the regularity
Proof
We first note that by Proposition 11(i) \(\mathcal {L}+\kappa ^2 \in OPS^{\bar{r}}_{1,0}(\mathcal {M})\), since the multiplication operator with the function \(\kappa ^2\in C^\infty (\mathcal {M})\) is an element of \(OPS^0_{1,0}(\mathcal {M})\). By Proposition 12 \(\mathcal {A}= (\mathcal {L}+\kappa ^2)^\beta \in OPS^{\bar{r}\beta }_{1,0}(\mathcal {M})\). Therefore all results follow by the same arguments as used in the proof of Proposition 1.\(\square \)
Proposition 14 shows that the covariance and precision operator of the GRF \(\mathcal {Z}_\beta \) defined by the white noise driven \(\mathrm {S\Psi DE}\) () with coloring operator of Whittle–Matérn type, \(\mathcal {A}=(\mathcal {L}+\kappa ^2)^\beta \), satisfy \(\mathcal {C}_\beta \in OPS^{-2\bar{r}\beta }_{1,0}(\mathcal {M})\) and \(\mathcal {P}_\beta \in OPS^{2\bar{r}\beta }_{1,0}(\mathcal {M})\). For this reason, all results of Subsections 3.2 and 3.3 on optimal preconditioning and matrix compression are applicable for covariance operators \(\mathcal {B}=\mathcal {C}_\beta \) and precision operators \(\mathcal {B}= \mathcal {P}_\beta \) of Whittle–Matérn type, where the order \(r\in \mathbb {R}\) is given by \(-2\bar{r}\beta \) and \(2\bar{r}\beta \), respectively.
Remark 1
The coefficient \(\beta >0\) in the Whittle–Matérn like coloring operator \(\mathcal {A}= (\mathcal {L}+ \kappa ^2 )^\beta \) and the order \(\bar{r}>0\) of the base operator \(\mathcal {L}\in OPS^{\bar{r}}_{1,0}(\mathcal {M})\) govern the spatial regularity of the GRF \(\mathcal {Z}_\beta \) (in \(L^p(\Omega )\)-sense and \(\mathbb {P}\)-a.s.). The shift \(\kappa ^2\) does not influence the smoothness, but controls the spatial correlation length of \(\mathcal {Z}_\beta \). Allowing for a function-valued shift \(\kappa ^2\in C^\infty (\mathcal {M})\) thus corresponds to models with a spatially varying correlation length which form an important extension of the classical Matérn model.
As noted in Proposition 14 above, the corresponding Whittle–Matérn like covariance operator \(\mathcal {C}_\beta \) is a self-adjoint, positive definite, compact operator on the Hilbert space \(L^2(\mathcal {M})\). By the spectral theorem and by the (assumed) nondegeneracy of \(\mathcal {C}\), there exists a countable system \(\{ e_j \}_{j\in \mathbb {N}}\) of eigenvectors for \(\mathcal {C}_\beta \) which forms an orthonormal basis for \(L^2(\mathcal {M})\). The corresponding positive eigenvalues \(\{\lambda _j(\mathcal {C}_\beta ) \}_{j\in \mathbb {N}}\) accumulate only at zero and we may assume that they are in non-increasing order. This gives rise to a Karhunen–Loève expansion of the centered GRF \(\mathcal {Z}_\beta \),
with equality in \(L^2(\Omega ; L^2(\mathcal {M}))\). Here, \(\{ \xi _j \}_{j\in \mathbb {N}}\) are i.i.d. \(\textsf{N}(0,1)\)-distributed random variables.
Partial sums of the Karhunen–Loève expansion (C.2) are of great importance for deterministic numerical approximations of PDE models in UQ which take \(\mathcal {Z}_\beta \) as a model for a distributed uncertain input data, see, e.g., [15, 31, 41] and the references there.
The error in a J-term truncation of the expansion (C.2) is governed by the eigenvalue decay \(\lambda _j(\mathcal {C}_\beta )\rightarrow 0\) as \(j\rightarrow \infty \). Assuming that \(\kappa >0\) is constant on \(\mathcal {M}\), we find by using the spectral asymptotics \( \lambda _j(\mathcal {L}) = c' j^{\bar{r}/n} + o( j^{\bar{r}/n} )\) for \(\mathcal {L}\) [69, Thm. XII.2.1] as well as the spectral mapping theorem that
This shows that the asymptotic behavior \(\lambda _j(\mathcal {C}_\beta ) \simeq j^{-2\beta \bar{r}/n}\), which is expected from [69, Thm. XII.2.1] applied for the operator \(\mathcal {C}_\beta \in OPS^{-2\beta \bar{r}}_{1,0}(\mathcal {M})\), is only visible for \(j > J^* = J^*(\kappa ,\mathcal {L}) = \mathcal {O}\bigl ( \kappa ^{2n/\bar{r}} \bigr )\), where the constant implied in \(\mathcal {O}( \,\cdot \, )\) is independent of the value of \(\beta >0\). For \(1\le j \le J^*\), one expects an eigenvalue “plateau”
Since in models of Whittle–Matérn type with \(\mathcal {L}\in OPS^{\bar{r}}_{1,0}(\mathcal {M})\) the (nondimensional) spatial correlation length \(\bar{\lambda }\) is \(\kappa ^{-2/\bar{r}}\), (C.3) indicates that for small values of \(\bar{\lambda }\), the plateau in the spectrum of \(\mathcal {C}_\beta \) scales as \(J^* = \mathcal {O}( \kappa ^{2n/\bar{r}} ) = \bar{\lambda }^{-n}\). Due to \(\lambda _j(\mathcal {P}_\beta ) = \lambda _j(\mathcal {C}_\beta ^{-1}) = 1/\lambda _j(\mathcal {C}_\beta )\), analogous statements hold for the precision operator \(\mathcal {P}_\beta \).
D Proof of Theorem 2
The idea of this proof is similar to techniques in [43] and the references therein.
Proof
We recall the asymptotic estimates of the computational work and error of the MLMC covariance estimation from () and ()
and
We seek to find sample numbers \(\widetilde{M}_{j}\), \(j=0,\ldots ,J\) that optimize the computational work to achieve a certain accuracy. We consider \(\widetilde{M}_j\) as a continuous variable and seek to find stationary points of the Lagrange multiplier function
Hence, we seek \(\widetilde{M}_j\), \(j=j_0,\ldots ,J\) such that \(\partial g(\xi )/\partial \widetilde{M}_j =0\), \(j=j_0,\ldots ,J\). This results in the conditions \(\widetilde{M}_j = 2^{-j(n+\alpha )2/3}\), \(j=j_0 + 1,\ldots ,J\), and we thus choose
where \(\widetilde{M}_{j_0}\) is still to be determined. This yields
and
where \(E_j = 2^{-j\alpha 2/3 + jn/3}\), \(j=j_0,\ldots ,J\). It holds that
We choose \(\widetilde{M}_{j_0}\) to equilibrate the error contributions in \(2^{-J\alpha _0} + \widetilde{M}_{j_0}^{-1/2} \sum _{j=0}^J E_j\), which leads us to
By inserting the corresponding value of \(\widetilde{M}_{j_0}\) and of \(\sum _{j=j_0}^J E_j\) into (D.1), we obtain that
The assertion now follows by expressing the computational work as a function of \(\varepsilon \) with the choice \(\varepsilon = 2^{2J\alpha _0}\).\(\square \)
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Harbrecht, H., Herrmann, L., Kirchner, K. et al. Multilevel approximation of Gaussian random fields: Covariance compression, estimation, and spatial prediction. Adv Comput Math 50, 101 (2024). https://doi.org/10.1007/s10444-024-10187-8
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DOI: https://doi.org/10.1007/s10444-024-10187-8