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Sparse operator compression of higherorder elliptic operators with rough coefficients
 Thomas Y. Hou^{1} and
 Pengchuan Zhang^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s4068701701131
© The Author(s) 2017
 Received: 9 December 2016
 Accepted: 26 June 2017
 Published: 4 December 2017
Abstract
We introduce the sparse operator compression to compress a selfadjoint higherorder elliptic operator with rough coefficients and various boundary conditions. The operator compression is achieved by using localized basis functions, which are energy minimizing functions on local patches. On a regular mesh with mesh size h, the localized basis functions have supports of diameter \(O(h\log (1/h))\) and give optimal compression rate of the solution operator. We show that by using localized basis functions with supports of diameter \(O(h\log (1/h))\), our method achieves the optimal compression rate of the solution operator. From the perspective of the generalized finite element method to solve elliptic equations, the localized basis functions have the optimal convergence rate \(O(h^k)\) for a (2k)thorder elliptic problem in the energy norm. From the perspective of the sparse PCA, our results show that a large set of Matérn covariance functions can be approximated by a rankn operator with a localized basis and with the optimal accuracy.
1 Background
1.1 Main objectives and the problem setting
Without imposing the sparsity constraints on the basis \({\varPsi }\), the compression error \(E_{{\mathrm {oc}}}({\varPsi }; {\mathcal {L}}^{1})\) achieves its minimum \(\lambda _{n+1}({\mathcal {L}}^{1})\) if we use the first n eigenfunctions of \({\mathcal {L}}^{1}\) to form \({\varPsi }\) (\(\lambda _n\) is the nth eigenvalue arranged in a descending order). However, the eigenfunctions are expensive to compute and do not have localized support [20, 40, 49]. In many cases, localized/sparse basis functions are preferred. For example, in the multiscale finite element method [12], localized basis functions lead to sparse linear systems and thus result in more efficient algorithms, see, e.g., [1, 2, 5, 10, 11, 22, 23, 28, 36, 39, 44]. In quantum chemistry, localized basis functions like the Wannier functions have better interpretability of the local interactions between particles (see, e.g., [26, 29, 30, 40, 47]), and also lead to more efficient algorithms [15]. In statistics, the sparse principal component analysis (SPCA) looks for sparse vectors to span the eigenspace of the covariance matrix, which leads to better interpretability compared with the PCA, see, e.g., [8, 25, 45, 46, 49].
1.2 Summary of our main results
 1.They are optimally localized up to a logarithmic factor, i.e.,Here, \(\text {supp}(\psi _i^{{\mathrm {loc}}})\) denotes the area/volume of the support of the localized function \(\psi _i^{{\mathrm {loc}}}\) in \(\mathbb {R}^d\), and the constant \(C_l\) is independent of n.$$\begin{aligned} \left \text {supp}(\psi _i^{{\mathrm {loc}}})\right \le \frac{C_l \log (n)}{n} \quad \forall 1 \le i \le n. \end{aligned}$$(1.4)
 2.If we use a generalized finite element method [1, 10, 22, 44] to solve the elliptic equations, we achieve the optimal convergence rate in the energy norm, i.e.,where \(L_n\) is the stiffness matrix under the basis \({\varPsi }^{{\mathrm {loc}}}\), \(\Vert \cdot \Vert _H\) is the associated energy norm, and \(C_e\) is independent of n.$$\begin{aligned} \Vert {\mathcal {L}}^{1} f  {\varPsi }^{{\mathrm {loc}}} L_n^{1} ({\varPsi }^{{\mathrm {loc}}})^T f \Vert _H \le C_e \sqrt{\lambda _{n}({\mathcal {L}}^{1})} \Vert f\Vert _2 \quad \forall f \in L^2(D), \end{aligned}$$(1.5)
 3.For the sparse operator compression problem, we achieve the optimal approximation error up to a constant, i.e.,where \(E_{{\mathrm {oc}}}({\varPsi }^{{\mathrm {loc}}}; {\mathcal {L}}^{1})\) is the operator compression error defined in Eq. (1.3).$$\begin{aligned} E_{{\mathrm {oc}}}({\varPsi }^{{\mathrm {loc}}}; {\mathcal {L}}^{1}) \le C_e^2 \lambda _n({\mathcal {L}}^{1}), \end{aligned}$$(1.6)
1.3 Our construction
 1.they achieve the optimal convergence rate to solve the elliptic equation, i.e.,where the constant \(C_e\) is independent of n.$$\begin{aligned} \Vert {\mathcal {L}}^{1} f  {\varPsi }^{{\mathrm {loc}}} L_n^{1} ({\varPsi }^{{\mathrm {loc}}})^T f \Vert _H \le C_e h^k \Vert f\Vert _2 \quad \forall f \in L^2(D), \end{aligned}$$(1.8)
 2.they achieve the optimal approximation error to approximate the elliptic operator, i.e.,$$\begin{aligned} E_{{\mathrm {oc}}}({\varPsi }^{{\mathrm {loc}}}; {\mathcal {L}}^{1}) \le C_e^2 h^{2k}. \end{aligned}$$(1.9)
1.4 Comparison with other existing methods
Our approach for operator compression originates at the MsFEM and numerical homogenization, where localized multiscale basis functions are constructed to approximate the solution space of some elliptic PDEs with multiscale coefficients; see [1, 2, 5, 10, 12, 22, 28, 35, 36, 39, 44]. Specifically, our work is inspired by the work presented in [28, 36], in which multiscale basis functions with support size \(O(h\log (1/h))\) are constructed for secondorder elliptic equations with rough coefficients and homogeneous Dirichlet boundary conditions. In this paper, we generalize the construction [36] and propose a general framework to compress higherorder elliptic operators with optimal compression accuracy and optimal localization.
We remark that although we use the framework presented in [36] as the direct template for our method, to the best of our knowledge, the local orthogonal decomposition (LOD) [28], in the context of multidimensional numerical homogenization, contains the first rigorous proof of optimal exponential decay rates with a priori estimates (leading to localization to subdomains of size \(h \log (1/h)\), with basis functions derived from the Clement interpolation operator). The idea of using the preimage of some continuous or discontinuous finite element space under the partial differential operator to construct localized basis functions in Galerkintype methods was even used earlier, e.g., in [16], although it did not provide a constructive local basis. In addition to establishing the exponential decay of the basis (for general nonconforming measurements of the solution, we will generalize the proof of this result to higherorder PDEs and measurements formed by local polynomials), a major contribution of [36] was to introduce a multiresolution operator decomposition for secondorder elliptic PDEs with rough coefficients.
There are several new ingredients in our analysis that are essential for us to obtain our results for higherorder elliptic operators with rough coefficients. First of all, we prove an inverse energy estimate for functions in \(\Psi \), which is crucial in proving the exponential decay. In particular, Lemma 4.1 is an essential step to obtaining the inverse energy estimate for higherorder PDEs that is not found in [28] nor [36]. We remark that Lemma 3.12 in [36] provides such an estimate for secondorder elliptic operators, by utilizing a relation between the Laplacian operator \(\Delta \) and the ddimensional Brownian motion. It is not straightforward to extend this probabilistic argument to higherorder cases. In contrast, our inverse energy estimate is valid for any 2kthorder elliptic operators and is tighter than the estimation in [36] for the secondorder case. Secondly, we prove a projectiontype polynomial approximation property in \(H^k(D)\). This polynomial approximation property plays an essential role in both estimating the compression accuracy and in localizing the basis functions. Thirdly, we propose the notion of the strong ellipticity to analyze the higherorder elliptic operators and show that strong ellipticity is only slightly stronger than the standard uniform ellipticity. Very recently, the authors of [37] introduce the Gaussian cylinder measure and successfully generalize the probabilistic framework in [35, 36] to a much broader class of operators, including higherorder elliptic operators without requiring the strong ellipticity.
As in [28, 36], the error bound in our convergence analysis blows up for fixed oversampling ratio r / h. To achieve the desired \(O(h^k)\) accuracy in the energy norm, we require \(r/h=O(\log (1/h))\). There has been some previous attempt to study the convergence of MsFEM using oversampling techniques with r / h being fixed, see, e.g., [18, 41]. In particular, the authors of [18, 41] showed that if the oversampling ratio r / h is fixed, the accuracy of the numerical solution will depend on the regularity of the solution and cannot be guaranteed for problems with rough coefficients. By imposing \(r/h=O(\log (1/h))\), the authors of [18, 41] proved that the MsFEM with constrained oversampling converges with the desired accuracy O(h).
There has been some previous work for secondorder elliptic PDEs by using basis functions of support size O(h), see, e.g., [2, 21]. However, they need to use \(O(\log (1/h))\) basis functions associated with each coarse finite element to recover the O(h) accuracy. The computational complexity of this approach is comparable to the one that we present in this paper. It is worth mentioning that the authors of [21] use a local oversampling operator to construct the optimal local boundary conditions for the nodal multiscale basis and enrich the nodal multiscale basis with optimal edge multiscale basis. Moreover, the method in [21] allows an explicit control of the approximation accuracy in the offline stage by truncating the SVD of the oversampling operator. In [21], the authors demonstrated numerically that this method is robust to highcontrast problems and the number of basis functions per coarse element is typically small. We remark that the recently developed generalized multiscale finite element method (GMsFEM) [5, 10] has provided another promising approach in constructing multiscale basis functions with support size O(h).
In comparison with the \(l^1\)based optimization method or the SPCA, our approach has the advantage that this construction will guarantee that \(\psi _{i,q}\) decays exponentially fast away from \(\tau _i\). This exponential decay justifies the local construction of the basis functions in Eq. (1.7). Moroever, our construction (1.7) is a quadratic optimization with linear constraints, which can be solved as efficiently as solving an elliptic problem on the local domain \(S_r\). The computational complexity to obtain all n localized basis functions \(\{\psi _i^{{\mathrm {loc}}}\}_{i=1}^n\) is only of order \(N\log ^{3d}(N)\) if a multilevel construction is employed, where N is the degree of freedom in the discretization of \({\mathcal {L}}\); see [36]. In contrast, the orthogonality constraint in Eq. (1.10) is not convex, which introduces additional difficulties in solving the problem. Finally, our construction of \(\{\psi _i^{{\mathrm {loc}}}\}_{i=1}^n\) is completely decoupled, while all the basis functions in Eq. (1.10) are coupled together. This decoupling leads to a simple parallel execution and thus makes the computation of \(\{\psi _i^{{\mathrm {loc}}}\}_{i=1}^n\) even more efficient.
The rest of the paper is organized as follows. In Sect. 2, we introduce the abstract framework of the sparse operator compression. In Sect. 3, we prove a projectiontype polynomial approximation property for the Sobolev spaces, which can be seen as a generalization of the Poincare inequality for functions with higher regularity. This polynomial approximation property is critical in our analysis of the higherorder case. It plays a role similar to that of the Poincare inequality in the analysis of the secondorder elliptic operator. In Sect. 4, we prove the inverse energy estimate by scaling. In Sect. 5, we use the secondorder elliptic PDE to illustrate the main idea of our analysis. In Sect. 6, we first introduce the notion of strong ellipticity and then prove the exponential decay of the constructed basis function for strongly elliptic operators. In Sect. 7, we localize the basis functions and provide the convergence rate for the corresponding MsFEM and the compression rate for the corresponding operator compression. Finally, we present several numerical results to support the theoretical findings in Sect. 8. Some concluding remarks are made in Sect. 9 and a few technical proofs are deferred to the “Appendix.”
2 Operator compression
In this section, we provide an abstract and general framework to compress a bounded selfadjoint positive semidefinite operator \({\mathcal {K}}: X \rightarrow X\), where X can be any separable Hilbert space with inner product \((\cdot , \cdot )\). In the case of operator compression of an elliptic operator \({\mathcal {L}}\), \({\mathcal {K}}\) plays the role of the solution operator \({\mathcal {L}}^{1}\) and \(X = L^2(D)\). In the case of the SPCA, \({\mathcal {K}}\) plays the role of the covariance operator. In Sect. 2.1, we introduce the Cameron–Martin space, which plays the role of the solution space of \({\mathcal {L}}\). In Sect. 2.2, we provide our main theorem to estimate the compression error. We will use this abstract framework to compress elliptic operators in the rest of the paper.
2.1 The Cameron–Martin space
Suppose \(\{(\lambda _n, e_n)\}_{n=1}^{\infty }\) are the eigen pairs of the operator \({\mathcal {K}}\) with the eigenvalues \(\{\lambda _n\}_{n=1}^{\infty }\) in a descending order. We have \(\lambda _n\ge 0\) for all n since \({\mathcal {K}}\) is selfadjoint and positive semidefinite. From the spectral theorem of a selfadjoint operator, we know that \(\{e_n)\}_{n=1}^{\infty }\) forms an orthonormal basis of X.
Lemma 2.1
 1.\({\mathcal {K}}(X)\) is an inner product space with inner product defined by$$\begin{aligned} ( {\mathcal {K}}\varphi _1, {\mathcal {K}}\varphi _2 )_H = ({\mathcal {K}}\varphi _1, \varphi _2) \qquad \forall \varphi _1, \varphi _2 \in X. \end{aligned}$$(2.1)
 2.
\({\mathcal {K}}(X)\) is continuously imbedded in X.
 3.
\({\mathcal {K}}(X)\) is dense in X if the null space of \({\mathcal {K}}\) only contains the origin, i.e., \(\mathrm {null}({\mathcal {K}}) = \{\varvec{0}\}\).
Proof
 1.
Since \({\mathcal {K}}\) is selfadjoint, we have \(( {\mathcal {K}}\varphi _1, {\mathcal {K}}\varphi _2 )_H = ( {\mathcal {K}}\varphi _2, {\mathcal {K}}\varphi _1 )_H\). The linearity and nonnegativity are obvious. Finally, if \(( {\mathcal {K}}\varphi , {\mathcal {K}}\varphi )_H = 0\) for some \(\varphi \in X\), then \(({\mathcal {K}}\varphi , \varphi ) = 0\). Suppose that \(\varphi = \sum _n \alpha _n e_n\) by expanding \(\varphi \) with eigenvectors of \({\mathcal {K}}\). Then, we have \(({\mathcal {K}}\varphi , \varphi ) = \sum _n \lambda _n \alpha _n^2 = 0\). Therefore, \(\alpha _n = 0\) for all \(\lambda _n > 0\). Equivalently, we obtain \(\varphi \in \mathrm {null}({\mathcal {K}})\), i.e., \({\mathcal {K}}\varphi = 0\).
 2.Since \(\lambda _n^2 \le \lambda _1 \lambda _n\) for all \(n\in \mathbb {N}\), we have \({\mathcal {K}}^2 \preceq \lambda _1 {\mathcal {K}}\). Then, we obtainwhere we have used the definition of \(( \cdot , \cdot )_H\) in Eq. (2.1) in the last step.$$\begin{aligned} \sqrt{({\mathcal {K}}\varphi , {\mathcal {K}}\varphi )} \le \sqrt{ \lambda _1 ({\mathcal {K}}\varphi , \varphi )} = \sqrt{ \lambda _1} \sqrt{ ( {\mathcal {K}}\varphi , {\mathcal {K}}\varphi )_H}, \end{aligned}$$(2.2)
 3.
If \(\mathrm {null}({\mathcal {K}}) = \{\varvec{0}\}\), we have \(\text {span}\{e_n, n\ge 1\} \subset {\mathcal {K}}(X)\). Then, \({\mathcal {K}}(X)\) is dense in X. \(\square \)
We define the Cameron–Martin space H as the completion of \({\mathcal {K}}(X)\) with respect to the norm \(\sqrt{( \cdot , \cdot )_H}\). Then, H is a separable Hilbert space and we have the following lemma.
Lemma 2.2
 1.
H can be continuously embedded into X.
 2.
H is dense in X if \(\mathrm {null}({\mathcal {K}}) = \{\varvec{0}\}\).
 3.For all \(\psi \in X\) and all \(f \in H\), we have$$\begin{aligned} (f, {\mathcal {K}}\psi )_H = (f, \psi ). \end{aligned}$$(2.3)
Proof
 1.
By the continuous imbedding from \({\mathcal {K}}(X)\) to X, we know that a Cauchy sequence in \({\mathcal {K}}(X)\) is also a Cauchy sequence in X. Therefore, we have \(H \subset X\). By Eq. (2.2) and the the continuity of norms, we have \((\psi , \psi ) \le \lambda _1 (\psi , \psi )_H\) for any \(\psi \in H\).
 2.
It is obvious from item 3 in Lemma 2.1.
 3.
If \(f \in {\mathcal {K}}(X)\), Eq. (2.3) is exactly the definition of \((\cdot , \cdot )_H\) in Eq. (2.1). By the continuity of the inner product, Eq. (2.3) is true for any \(f \in H\). \(\square \)
2.2 Operator compression
Suppose H is an arbitrary separable Hilbert space and \(\Phi \subset H\) is ndimensional subspace in H with basis \(\{\varphi _i\}_{i=1}^n\). In the rest of the paper, \({\mathcal {P}}_{\Phi }^{(H)}\) denotes the orthogonal projection from a Hilbert space H to its subspace \(\Phi \). With this notation, we present our theorem for error estimates below.
Theorem 2.1
 1.For any \(u \in {\mathcal {K}}(X)\) and \(u = {\mathcal {K}}f\), we have$$\begin{aligned} \Vert u  {\mathcal {P}}_{\Psi }^{(H)} u \Vert _{H} \le k_n \Vert f\Vert _X. \end{aligned}$$(2.5)
 2.For any \(u \in {\mathcal {K}}(X)\) and \(u = {\mathcal {K}}f\), we have$$\begin{aligned} \Vert u  {\mathcal {P}}_{\Psi }^{(H)} u \Vert _{X} \le k_n^2 \Vert f\Vert _X. \end{aligned}$$(2.6)
 3.We havewhere \(\Vert \cdot \Vert \) is the induced operator norm on \(\mathcal {B}(X,X)\). Moreover, the rankn operator \({\mathcal {P}}_{\Psi }^{(H)} {\mathcal {K}}: X \rightarrow X \) is selfadjoint.$$\begin{aligned} \Vert {\mathcal {K}} {\mathcal {P}}_{\Psi }^{(H)} {\mathcal {K}}\Vert \le k_n^2, \end{aligned}$$(2.7)
In Theorem 2.1, by using a projectiontype approximation property of \(\Phi \) in H, i.e., Eq. (2.4), we obtain the error estimates of the multiscale finite element method with finite element basis \(\{{\mathcal {K}}\varphi _i\}_{i=1}^n\) in the energy norm, i.e., Eq. (2.5). We will take \(\Phi \) as the discontinuous piecewise polynomial space later, which is a poor finite element space for elliptic equations with rough coefficients. However, after smoothing \(\Phi \) with the solution operator \({\mathcal {K}}\), the smoothed basis functions \(\{{\mathcal {K}}\varphi _i\}_{i=1}^n\) have the optimal convergence rate. This datadependent methodology to construct finite element spaces was pioneered by the generalized finite element (GFEM) [1, 44], the multiscale finite element method (MsFEM) [12, 22, 24], and numerical homogenization [28, 36].
Proof of Theorem 2.1
 1.For an arbitrary \(v \in \Psi \), due to the definition of \(\Psi \), we can write \(v = {\mathcal {K}}( \sum _{i=1}^n c_i \varphi _i )\), and thus we get \(u  v = {\mathcal {K}}( f  \sum _{i=1}^n c_i \varphi _i )\). By Lemma 2.2, we haveBy choosing \(c_i\) such that \(\sum _{i=1}^n c_i \varphi _i = {\mathcal {P}}_{\Phi }^{(X)} (f)\), the second term vanishes. Then, we obtain$$\begin{aligned} \begin{aligned} \Vert u  v\Vert _{H}^2&= \left( u  v, f  \sum _{i=1}^n c_i \varphi _i\right) \\&= \left( u  v  {\mathcal {P}}_{\Phi }^{(X)} (uv), f  \sum _{i=1}^n c_i \varphi _i\right) + \left( {\mathcal {P}}_{\Phi }^{(X)} (uv), f  \sum _{i=1}^n c_i \varphi _i\right) . \end{aligned} \end{aligned}$$Therefore, we conclude \(\Vert u  v\Vert _{H} \le k_n \Vert f\Vert _X\).$$\begin{aligned} \begin{aligned} \Vert u  v\Vert _{H}^2&= \left( u  v  {\mathcal {P}}_{\Phi }^{(X)} (uv), f  \sum _{i=1}^n c_i \varphi _i\right) \\&\le \Vert u  v  {\mathcal {P}}_{\Phi }^{(X)} (uv)\Vert _X \Vert f  {\mathcal {P}}_{\Phi }^{(X)} (f)\Vert _X \le k_n \Vert u  v\Vert _{H} \Vert f\Vert _X \end{aligned} \end{aligned}$$
 2.We use the Aubin–Nistche duality argument to get the estimation in item 2. Let \(v = {\mathcal {K}}( u  {\mathcal {P}}_{\Psi }^{(H)} u )\). On one hand, we getOn the other hand, we obtain$$\begin{aligned} (u  {\mathcal {P}}_{\Psi }^{(H)} u, v  {\mathcal {P}}_{\Psi }^{(H)} v)_{H} = (u  {\mathcal {P}}_{\Psi }^{(H)} u, v)_{H} = (u  {\mathcal {P}}_{\Psi }^{(H)} u, u  {\mathcal {P}}_{\Psi }^{(H)} u)_{X} = \Vert u  {\mathcal {P}}_{\Psi }^{(H)} u\Vert _{X}^2. \end{aligned}$$We have used the result of item 1 in the last step. Combining these two estimates, the result follows.$$\begin{aligned} (u  {\mathcal {P}}_{\Psi }^{(H)} u, v  {\mathcal {P}}_{\Psi }^{(H)} v)_{H} \le \Vert u  {\mathcal {P}}_{\Psi }^{(H)} u\Vert _{H} \Vert v  {\mathcal {P}}_{\Psi }^{(H)} v\Vert _{H} \le k_n \Vert f\Vert _X \, k_n \Vert u  {\mathcal {P}}_{\Psi }^{(H)} u\Vert _X. \end{aligned}$$
 3.From the last item, we obtain that \(\Vert {\mathcal {K}}f  {\mathcal {P}}_{\Psi }^{(H)} {\mathcal {K}}f \Vert _X \le k_n^2 \Vert f\Vert _X\) for any \(f \in X\). Therefore, we conclude \(\Vert {\mathcal {K}} {\mathcal {P}}_{\Psi }^{(H)} {\mathcal {K}}\Vert \le k_n^2\). Now, we prove that \({\mathcal {P}}_{\Psi }^{(H)} {\mathcal {K}}\) is selfadjoint. For any \(x_1, x_2 \in X\), by definition of Hnorm we haveSince \({\mathcal {P}}_{\Psi }^{(H)}\) is selfadjoint in H, we have$$\begin{aligned} (x_1, {\mathcal {P}}_{\Psi }^{(H)} {\mathcal {K}}x_2) = ({\mathcal {K}}x_1, {\mathcal {P}}_{\Psi }^{(H)} {\mathcal {K}}x_2)_H. \end{aligned}$$where we have used the definition of Hnorm again in the last step. \(\square \)$$\begin{aligned} ({\mathcal {K}}x_1, {\mathcal {P}}_{\Psi }^{(H)} {\mathcal {K}}x_2)_H = ({\mathcal {P}}_{\Psi }^{(H)} {\mathcal {K}}x_1, {\mathcal {K}}x_2)_H = ({\mathcal {P}}_{\Psi }^{(H)} {\mathcal {K}}x_1, x_2), \end{aligned}$$
Theorem 2.2
 1.The optimization problem (2.9) admits a unique minimizer \(\psi _i\), which can be written as$$\begin{aligned} \psi _i = \sum _{j=1}^n \Theta _{i,j}^{1} {\mathcal {K}}\varphi _j. \end{aligned}$$(2.11)
 2.For \(w \in \mathbb {R}^n\), \(\sum _{i=1}^n w_i\psi _i\) is the minimizer of \(\Vert \psi \Vert _{H}\) subject to \((\varphi _j, \psi ) = w_j\) for \(j = 1,2, \ldots , n\). Moreover, for any \(\psi \) which satisfies \((\varphi _j, \psi ) = w_j\) for \(j = 1,2, \ldots , n\), we have$$\begin{aligned} \Vert \psi \Vert _H^2 = \left\ \sum _{i=1}^n w_i\psi _i\right\ _H^2 + \left\ \psi  \sum _{i=1}^n w_i\psi _i\right\ _H^2. \end{aligned}$$(2.12)
 3.
\((\psi _i, \psi _j)_H = \Theta ^{1}_{i,j}\).
 1.
The basis function \(\psi _i\) decays exponentially fast away from its associated patch; see Theorems 6.3 and 6.4.
 2.
The localized basis function \(\psi _i^{{\mathrm {loc}}}\) approximates \(\psi _i\) accurately; see Theorem 7.1. Meanwhile, the compression rate \(E_{{\mathrm {oc}}}({\varPsi }^{{\mathrm {loc}}}; {\mathcal {L}}^{1})\) is the same as \(E_{{\mathrm {oc}}}({\varPsi }; {\mathcal {L}}^{1})\); see Theorem 7.2 and Corollary 7.3.
3 A projectiontype polynomial approximation property
The following projectiontype polynomial approximation property in the Sobolev space \(H^k(D)\) plays an essential role in both obtaining the optimal approximation error and proving the exponential decay of the energyminimizing basis functions. It can be viewed as a generalized Poincare inequality.
Theorem 3.1
To prove Theorem 3.1, we use a basic result about the Sobolev spaces, due to J. Deny and J.L. Lions, which pervades the mathematical analysis of the finite element method: over the quotient space \(H^{k+1}(D)/{\mathcal {P}}_k(D)\), the seminorm \(\cdot _{k+1, D}\) is a norm equivalent to the quotient norm. We will use the following theorem (Theorem 3.1.4 in [6]), to prove Theorem 3.1.
Theorem 3.2
By specializing the operator \(\widehat{\Pi }\) to be the projection of \(H^{k+1}(\widehat{\Omega })\) to the polynomial space \({\mathcal {P}}_k(\widehat{\Omega })\) in \(L^2(\widehat{\Omega })\), we can prove Theorem 3.1.
Proof of Theorem 3.1
Finally, applying Theorem 3.2 with \(\widehat{\Pi }\) and \(\Pi _{\Omega }\) above, we prove Theorem 3.1 with the constant \(C(k, \widehat{\Omega }) := C(\widehat{\Pi }, \widehat{\Omega })\) in Eq. (3.3). \(\square \)
We also give the following theorem, which is a direct result of the Friedrichs’ inequality; see, e.g., [34].
Theorem 3.3
4 An inverse energy estimation by scaling
In the sparse operator compression, we will show that for a large set of compact operators, the basis functions \(\{\psi _i\}_{i=1}^n\) constructed in (2.9) have exponentially decaying tails, which makes localization of these basis functions possible. The following lemma plays a key role in proving such exponential decay property.
Lemma 4.1
Proof
5 Exponential decay of basis functions: the secondorder case
The analysis for a general higherorder elliptic PDE is quite technical. In this section, we will prove that the basis function \(\psi _i\) for a secondorder elliptic PDE has exponential decay away from \(\tau _i\). When \(c \equiv 0\), this problem has been studied in [36]. When \(c \ne 0\), it has been recently studied in [38] independently of our work. The results presented in this secondorder case are not new [36]. We would like to use the simpler secondorder elliptic PDE example to illustrate the main ingredients in the proof of exponential decay for a higherorder elliptic PDE, namely the recursive argument, the projectiontype approximation property and the inverse energy estimate.
Lemma 5.1
Now, we are ready to prove the exponential decay of the basis function \(\psi _i\).
Theorem 5.1
Proof
Let \(b_k := \Vert \psi _i\Vert _{H(S_0^c)}^2\), and from definition we have \(b_{0} = \Vert \psi _i\Vert _{H(D)}^2\), \(b_{k+1} = \Vert \psi _i\Vert _{H(S_1)}^2\) and \(b_{k}b_{k+1} = \Vert \psi _i\Vert _{H(S^*)}^2\). The strategy is to prove that for any \(k \ge 1\), there exists constant C such that \(b_{k+1} \le C(b_{k}  b_{k+1})\). Then, we have \(b_{k+1} \le \frac{C}{C+1} b_k\) for any \(k \ge 1\) and thus we get the exponential decay \(b_k \le (\frac{C}{C+1})^{k1} b_1 \le (\frac{C}{C+1})^{k1} b_0\). We will choose l such that \(C \le \frac{1}{\mathrm {e} 1}\) and thus get \(b_k \le e^{1k} b_0\), which gives the result (5.4). We start from \(k=1\) because we want to make sure \(\tau _i \in S_0\); otherwise, \(S_0 = \emptyset \) and \(\tau _i \in S^*\).
Now, we prove that for any \(k \ge 1\), there exists constant C such that \(b_{k+1} \le C(b_{k}  b_{k+1})\), i.e., \( \Vert \psi _i\Vert _{H(S_1)}^2 \le C \Vert \psi _i\Vert _{H(S^*)}^2\). Let \(\eta \) be the function on D defined by \(\eta (x) = {\mathrm {dist}}(x, S_0)/\left( {\mathrm {dist}}(x, S_0) + {\mathrm {dist}}(x, S_1)\right) \). Observe that (1) \(0 \le \eta \le 1\) (2) \(\eta \) is equal to zero on \(S_0\) (3) \(\eta \) is equal to one on \(S_1\) (4) \(\Vert \nabla \eta \Vert _{L^{\infty }(D)} \le \frac{1}{l h}\).^{1}
Remark 5.1
We point out that boundary conditions may be important in several applications. For example, the Robin boundary condition is useful in the application of the SPCA. The periodic boundary condition is useful in compressing a Hamiltonian with a periodic boundary condition in quantum physics.
6 Exponential decay of basis functions: the higherorder case
6.1 Construction of basis functions and the approximation rate
 1.For any \(u \in H\) and \({\mathcal {L}}u = f\), we haveHere, \(C_p\) plays the role of the Poincare constant \(1/\pi \).$$\begin{aligned} \Vert u  {\mathcal {P}}_{\Psi }^{(H)} u \Vert _{H} \le \frac{C_p h^k}{\sqrt{a_{\min }}} \Vert f\Vert _{L^2(D)}. \end{aligned}$$(6.11)
 2.For any \(u \in H\) and \({\mathcal {L}}u = f\), we have$$\begin{aligned} \Vert u  {\mathcal {P}}_{\Psi }^{(H)} u \Vert _{L^2(D)} \le \frac{C_p^2 h^{2 k}}{a_{\min }} \Vert f\Vert _{L^2(D)}. \end{aligned}$$(6.12)
 3.We have$$\begin{aligned} \Vert {\mathcal {K}} {\mathcal {P}}_{\Psi }^{(H)} {\mathcal {K}}\Vert \le \frac{C_p^2 h^{2 k}}{a_{\min }}. \end{aligned}$$(6.13)
6.2 The strong ellipticity condition
In our proof, we need the following strong ellipticity condition of the operator \({\mathcal {L}}\) to obtain the exponential decay.
Definition 6.1
For a 2kthorder partial differential operator \({\mathcal {L}}u = (1)^k \sum \nolimits _{\alpha \le 2k} a_{\alpha } D^{\alpha } u\), \({\mathcal {L}}\) is strongly elliptic if there exists a strongly elliptic operator in the divergence form \(\widetilde{{\mathcal {L}}}\) such that \({\mathcal {L}}u = \widetilde{{\mathcal {L}}} u\) for all \(u \in C^{2k}(D)\).
Remark 6.1
The strong ellipticity condition guarantees that for any local subdomain \(S \subset D\), the seminorm \(\cdot _{k,2,S}\) can be controlled by the local energy norm \(\Vert \cdot \Vert _{H(S)}\).
Lemma 6.1

\({\mathcal {L}}\) is nonnegative, i.e.,$$\begin{aligned} \sum _{ 0 \le \sigma , \gamma  \le k} a_{\sigma \gamma }(x) \varvec{\zeta }_{\sigma } \varvec{\zeta }_{\gamma } \ge 0 \qquad \forall \varvec{\zeta } \in \mathbb {R}^{\left( {\begin{array}{c}k+d\\ k\end{array}}\right) }, \end{aligned}$$(6.19)

\({\mathcal {L}}\) is bounded, i.e., there exist \(\theta _{0,\max }\ge 0\) and \(\theta _{k,\max } > 0\) such that$$\begin{aligned} \sum _{ 0 \le \sigma , \gamma  \le k} a_{\sigma \gamma }(x) \varvec{\zeta }_{\sigma } \varvec{\zeta }_{\gamma } \le \theta _{k,\max } \sum _{\sigma  = k} \varvec{\zeta }_{\sigma }^2 + \theta _{0,\max } \sum _{\sigma  < k} \varvec{\zeta }_{\sigma }^2 \qquad \forall x \in D \quad \forall \varvec{\zeta } \in \mathbb {R}^{\left( {\begin{array}{c}k+d\\ k\end{array}}\right) }, \end{aligned}$$(6.20)

and \({\mathcal {L}}\) is strongly elliptic, i.e., there exists \(\theta _{k,\min } > 0\) such that$$\begin{aligned} \sum _{ \sigma  = \gamma  = k} a_{\sigma \gamma }(x) \varvec{\zeta }_{\sigma } \varvec{\zeta }_{\gamma } \ge \theta _{k,\min } \sum _{\sigma  = k} \varvec{\zeta }_{\sigma }^2 \qquad \forall x \in D \quad \forall \varvec{\zeta } \in \mathbb {R}^{\left( {\begin{array}{c}k+d1\\ k\end{array}}\right) }. \end{aligned}$$(6.21)

If \({\mathcal {L}}\) contains only highest order terms, i.e., \({\mathcal {L}}u = \sum \nolimits _{\sigma  = \gamma  = k} (1)^{\sigma } D^{\sigma } (a_{\sigma \gamma }(x) D^{\gamma } u)\), then we have$$\begin{aligned} \psi _{k,2,S} \le \theta _{k,\min }^{1/2} \Vert \psi \Vert _{H(S)} \qquad \forall \psi \in H^{k}(D). \end{aligned}$$(6.23)

If \({\mathcal {L}}\) contains loworder terms, for any regular domain partition \(D = \cup _{i=1}^m \tau _i\) with diameter \(h>0\) satisfying \(\frac{h^2(1h^{2k})}{1h^2} \le \frac{\theta _{k,\min }^2}{16 \theta _{0,\max }\theta _{k,\max } C_p^2}\), and any subdomain \(S = \cup _{_{j\in \Lambda }} \tau _j\), we haveHere, \(\Lambda \) is any subset of \(\{1, 2, \ldots , m\}\), and \(\psi _{i,q}\) is defined by Eq. (6.9).$$\begin{aligned} \psi _{i,q}_{k,2,S} \le \left( 2/\theta _{k,\min }\right) ^{1/2} \Vert \psi _{i,q}\Vert _{H(S)} \qquad \forall \tau _i \not \in \mathcal {S},\quad 1\le q\le Q. \end{aligned}$$(6.24)
Proof
Remark 6.2
When \({\mathcal {L}}\) contains loworder terms but there is no crossing term between \(D^{\sigma } u\) (\(\sigma  = k\)) and \(D^{\sigma } u\) (\(\sigma  < k\)), i.e., \(J_3 = 0\), we can directly get the same bound in Eq. (6.23) for all \(h >0\).
Theorem 6.2

\(d=1\) or 2 : one or twodimensional physical domain,

\(k=1\) : secondorder partial differential operators.
For the case \((d, k)= (3, 2)\), we are not able to prove that strong ellipticity is equivalent to uniform ellipticity for elliptic operators with smooth and multiscale coefficients, but we suspect that it is true. For all other cases, there are uniformly but not strongly elliptic operators. Fortunately, for small physical dimensions d and differential orders k, strongly elliptic operators approximate uniformly elliptic operators well and counter examples are difficult to construct.
6.3 Exponential decay of basis functions I
In this subsection, we prove the exponential decay of basis functions constructed in Eq. (6.9) for higherorder elliptic operators that contain only the highest order terms. We will leave the proof for the general operators to the next subsection. The proof follows exactly the same structure as that in the secondorder elliptic case.
Theorem 6.3

\({\mathcal {L}}\) is bounded, i.e., there exist nonnegative \(\theta _{k,\max }\) such that$$\begin{aligned} \sum _{ \sigma = \gamma = k} a_{\sigma \gamma }(x) \varvec{\zeta }_{\sigma } \varvec{\zeta }_{\gamma } \le \theta _{k,\max } \sum _{\sigma  = k} \varvec{\zeta }_{\sigma }^2 \qquad \forall \varvec{\zeta } \in \mathbb {R}^{\left( {\begin{array}{c}k+d1\\ k\end{array}}\right) }, \end{aligned}$$(6.32)

and \({\mathcal {L}}\) is strongly elliptic, i.e., there exists \(\theta _{k,\min } > 0\) such that$$\begin{aligned} \sum _{ \sigma  = \gamma  = k} a_{\sigma \gamma }(x) \varvec{\zeta }_{\sigma } \varvec{\zeta }_{\gamma } \ge \theta _{k,\min } \sum _{\sigma  = k} \varvec{\zeta }_{\sigma }^2 \qquad \forall \varvec{\zeta } \in \mathbb {R}^{\left( {\begin{array}{c}k+d1\\ k\end{array}}\right) }. \end{aligned}$$(6.33)
Proof
The proof follows the same structure as that of Theorem 5.1 and [36] (Thm. 3.9). Let \(k\in \mathbb {N}\), \(l > 0\) and \(i \in \{1, 2, \ldots , m\}\). Let \(S_0\) be the union of all the domains \(\tau _j\) that are contained in the closure of \(B(x_i, k l h) \cap D\), let \(S_1\) be the union of all the domains \(\tau _j\) that are not contained in the closure of \(B(x_i, (k+1) l h) \cap D\) and let \(S^*=S_0^c \cap S_1^c \cap D\) (be the union of all the remaining elements \(\tau _j\) not contained in \(S_0\) or \(S_1\)). In the following, we will prove that for any \(k \ge 1\), there exists constant C such that \(\Vert \psi _{i,q}\Vert ^2_{H(S_1)} \le C \Vert \psi _{i,q}\Vert ^2_{H(S^*)}\). Then, the same recursive argument in the proof of Theorem 5.1 can be used to prove the exponential decay.
Let \(\eta (x)\) be a smooth function which satisfies (1) \(0 \le \eta \le 1\), (2) \(\eta _{B(x_i, k l h)} = 0\), (3) \(\eta _{B^c(x_i, (k+1) l h)} = 1\) and (4) \(\Vert D^{\sigma } \eta \Vert _{L^{\infty }(D)} \le \frac{C_{\eta }}{(l h)^{\sigma }}\) for all \(\sigma \).
6.4 Exponential decay of basis functions II
The following theorem gives the exponential decay property of \(\psi _{i,q}\) for an operator \({\mathcal {L}}\) with lowerorder terms. Similar to the proof of Theorem 6.4, we need the polynomial approximation property (6.5) and the Friedrichs’ inequality (3.4) to bound the lowerorder terms, and we get an extra factor of 2 in our error bound.
Theorem 6.4

\({\mathcal {L}}\) is nonnegative, i.e.,$$\begin{aligned} \sum _{ 0 \le \sigma , \gamma  \le k} a_{\sigma \gamma }(x) \varvec{\zeta }_{\sigma } \varvec{\zeta }_{\gamma } \ge 0, \qquad \forall x \in D \quad \forall \varvec{\zeta } \in \mathbb {R}^{\left( {\begin{array}{c}k+d\\ k\end{array}}\right) }, \end{aligned}$$(6.44)

\({\mathcal {L}}\) is bounded, i.e., there exist \(\theta _{0,\max }\ge 0\) and \(\theta _{k,\max } > 0\) such that$$\begin{aligned} \sum _{ 0 \le \sigma , \gamma  \le k} a_{\sigma \gamma }(x) \varvec{\zeta }_{\sigma } \varvec{\zeta }_{\gamma } \le \theta _{k,\max } \sum _{\sigma  = k} \varvec{\zeta }_{\sigma }^2 + \theta _{0,\max } \sum _{\sigma  < k} \varvec{\zeta }_{\sigma }^2 \qquad \forall x \in D , \quad \forall \varvec{\zeta } \in \mathbb {R}^{\left( {\begin{array}{c}k+d\\ k\end{array}}\right) }, \end{aligned}$$(6.45)

and \({\mathcal {L}}\) is strongly elliptic, i.e., there exists \(\theta _{k,\min } > 0\) such that$$\begin{aligned} \sum _{ \sigma  = \gamma  = k} a_{\sigma \gamma }(x) \varvec{\zeta }_{\sigma } \varvec{\zeta }_{\gamma } \ge \theta _{k,\min } \sum _{\sigma  = k} \varvec{\zeta }_{\sigma }^2, \qquad \forall \varvec{\zeta } \in \mathbb {R}^{\left( {\begin{array}{c}k+d1\\ k\end{array}}\right) }. \end{aligned}$$(6.46)
Proof
Remark 6.3
6.5 Lemmas
In this subsection, we will prove the following lemma, which is used in the proof of Theorem 6.3 and Theorem 6.4.
Lemma 6.2
If the operator \({\mathcal {L}}\) contains only the highest order terms, i.e., \({\mathcal {L}}u = (1)^k \sum \nolimits _{\sigma =\gamma =k} D^{\sigma }(a_{\sigma \gamma } D^{\gamma } u)\), we have \(\Vert {\mathcal {L}}v\Vert _{L^2(\tau _j)} \le \sqrt{\theta _{k,\max }} C(k,d,\delta ) h^{k} \Vert v\Vert _{H(\tau _j)}\) for all \(h > 0\).
We will use Lemma 4.1 to prove this result, but we need to deal with the variable coefficients \(a_{\sigma \gamma }\) and the loworder terms \(a_{\sigma \gamma }\) with \(\sigma +\gamma  < 2 k\) before we can apply Lemma 4.1. Our strategy is to transfer the variable coefficients to constant ones by the variational formulation (see Lemma 6.3) and to use the polynomial approximation property to deal with the loworder terms; see Lemma 6.4. For this purpose, we first introduce the following two lemmas.
Lemma 6.3
Proof
Lemma 6.4
Proof
Now, we are ready to prove Lemma 6.2.
Proof of Lemma 6.2
7 Localization of the basis functions
From now on, to simplify the expression of constants, we will assume without loss of generality that the domain is rescaled so that \(\text {diam}(D) \le 1\).
Lemma 7.1
Proof
Theorem 7.1
Proof
When the operator \({\mathcal {L}}\) contains only the highest order terms, i.e., \({\mathcal {L}}u = (1)^k \sum \nolimits _{\sigma =\gamma =k} D^{\sigma }(a_{\sigma \gamma } D^{\gamma } u)\), Eqs. (7.14) and (7.18) hold true for all \(h > 0\). In this case, we can get rid of the factor “2” in both Eqs. (7.14) and (7.18). Therefore, we obtain the estimate on \(C_3\) stated in the theorem. \(\square \)
Theorem 7.2
Proof
By applying the Aubin–Nistche duality argument, we can get the following corollary.
Corollary 7.3
Corollary 7.3 shows that we can compress the symmetric positive semidefinite operator \({\mathcal {K}}\) with the optimal rate \(h^{2 k}\) and with the nearly optimal localized basis (with support size of order \(h \log (1/h)\)).
Remark 7.1
All the results and proofs presented above can be carried over to other homogeneous boundary conditions. Given a specific homogeneous boundary condition, one only needs to modify the proof of Lemma 7.1. Specifically, when the patch \(\tau _i\) intersects with the boundary of D, the constructed function \(\zeta _{i,q}\) should honor the same boundary condition on \(\partial D\). The scaling argument in the proof of Lemma 7.1 still works for other homogeneous boundary conditions.
8 Numerical examples
In this section, we present several numerical results to support the theoretical findings and to show how the sparse operator compression is utilized in higherorder elliptic operators. In Sect. 8.1, we apply our method to compress the Matérn covariance function (8.1) with \(\nu = 1/2\). We show that our method is able to achieve the optimal compression error with nearly optimally localized basis functions, which means that we are able to get optimality on both ends of the accuracy–sparsity tradeoff in the sparse PCA. In Sect. 8.2, we apply our method to a 1D fourthorder elliptic equation with the homogeneous Dirichlet boundary condition and show that our basis functions, when used as multiscale finite element basis, can achieve the optimal \(h^2\) convergence rate in the energy norm. In Sect. 8.3, we apply our method to a 2D fourthorder elliptic equation and show that the energyminimizing basis functions decays exponentially fast away from its associated patch.
8.1 The compression of a Matérn covariance kernel
8.2 The 1D fourthorder elliptic operator
We partition the physical space [0, 1] uniformly into \(m = 2^6\) patches, where the ith patch \(I_i = [(i1) h, i h]\) with \(h = 1/m\). In this fourthorder case, our theory requires the piecewise polynomial space \(\Phi \) be the space of (discontinuous) piecewise linear functions, which has dimension \(n = 2 m\). We have two \(\varphi \)’s, denoted as \(\varphi _{i,1}\) and \(\varphi _{i,2}\), associated with the patch \(I_i\). Solving the quadratic optimization problem (6.9), we obtain the exponentially decaying basis functions. We also have two \(\psi \)’s, denoted as \(\psi _{i,1}\) and \(\psi _{i,2}\), associated with the patch \(I_i\). We plot \(\varphi _{i,1}\) and \(\varphi _{i,2}\) associated with the patch \(I_{32} = [1/2h, 1/2]\) in Fig. 7 A. In Fig. 7b, c, we plot the basis functions \(\psi _{32,1}\) and \(\psi _{32,2}\), which clearly show exponential decay.
To demonstrate the necessity for \(\Psi \) to contain all piecewise linear functions, in the third column of Fig. 7, we also plot the basis functions associated the patch \(I_{32}\) when \(\Phi \) is the space of piecewise constant functions. In this case, we have only one \(\varphi \), denoted as \(\varphi _{i}\), associated with the patch \(I_i\). In the third column of Fig. 7a, b, we plot \(\varphi _{32}\) and \(\psi _{32}\). Solving the quadratic optimization problem (6.9), we obtain only one basis function \(\psi \), denoted as \(\psi _{i}\), associated with the patch \(I_i\). In Fig. 7c, we plot the basis function \(\psi _{32}\) in the third column. Note that \(\psi _{32}\) also shows an exponential decay, but its decay rate is much smaller than that of \(\psi _{32,1}\) and \(\psi _{32,2}\).
8.3 The 2D fourthorder elliptic operator
Based on the uniform partition with grid size \(h_x = h_y = \frac{1}{8}\), we construct the piecewise linear function space \(\Phi \), which has dimension \(n = 3 m = 192\). We solve the quadratic optimization problem (6.9) with the weighted extended Bsplines (Websplines [19]) of degree 3 on the uniform refined grid with grid size \(h_{x,f} = h_{y,f} = \frac{1}{32}\). The 2D Gaussian quadrature with 5 points on each axis is utilized to compute the integral on each fine grid cell. The three basis functions associated with the patch \([1/2h_x, 1/2]\times [1/2h_y, 1/2]\) are shown in Fig. 9. We also show them in the logscale in Fig. 10. We can clearly see that the basis functions decay exponentially fast away from its associated patch, which validates our Theorem 6.3.
We point out that the stiffness matrix for the fourthorder elliptic operator (8.9) becomes illconditioned very quickly when we refine the grid size. A carefully designed numerical strategy is required to validate the optimal convergence rate. We will leave this to our future work.
9 Concluding remarks
In this paper, we have developed a general strategy to compress a class of selfadjoint higherorder elliptic operators by minimizing the energy norm of the localized basis functions. These energyminimizing localized basis functions are obtained by solving decoupled local quadratic optimization problems with linear constraints, and they give optimal approximation property of the solution operator. For a selfadjoint, bounded and strongly elliptic operator of order 2k (\(k \ge 1\)), we have proved that with support size \(O(h \log (1/h))\), our localized basis functions can be used to compress higherorder elliptic operators with the optimal compression rate \(O(h^{2k})\). We have applied our new operator compression strategy in different applications. For elliptic equations with rough coefficients, our localized basis functions can be used as multiscale basis functions, which gives the optimal convergence rate \(O(h^k)\) in the energy norm. In the application of the sparse PCA, our localized basis functions achieve nearly optimal sparsity and the optimal approximation rate simultaneously when the covariance operator to be compressed is the solution operator of an elliptic operator. We remark that a number of Matérn covariance kernels are related to the Green’s functions of some elliptic operators.
There are several directions we can explore in the future work. First of all, the constants in both the compression error and the localization depend on the contrast of the coefficients, which makes the existing methods inefficient for coefficients with high contrast. Other methods (e.g., [16, 28, 35, 36]) also suffer from the same limitation. Our sparse operator compression framework can be used to deal with this high contrast case, and we will report our findings in our upcoming paper. Secondly, in the application of the sparse PCA, our current construction requires the knowledge of the underlying elliptic operator \({\mathcal {L}}\). We believe that it is possible to construct these localized basis functions using only the covariance function. Moreover, given any covariance operator, which may not be the solution operator of an elliptic operator, we can still define the Cameron–Martin space and the corresponding energyminimizing basis functions. We are interested in the localization and compression properties of these energyminimizing basis functions in this general setting. Our preliminary results show that the energyminimizing basis functions still enjoy fast decay rate away from its associated patch, although the exponential decay may not hold true any more. Thirdly, it is interesting to apply our framework to the graph Laplacians, which can be viewed as discretized elliptic operators. Along this direction, we would like to develop an algorithm with nearly linear complexity to solve linear systems with graph Laplacians. Finally, we are also interested in applying our method to construct localized Wannier functions and to compress the Hamiltonian in quantum chemistry. Unlike the secondorder elliptic operators with multiscale diffusion coefficients, all multiscale features of the Hamiltonian \({\mathcal {H}}= \Delta + V(x)\) lie in its potential V(x). Some adaptive domain partition strategy may prove to be useful in this application.
\(\Vert \nabla \eta \Vert _{L^{\infty }(D)} := \mathop {{{\mathrm{ess\,sup}}}}\limits _{x\in D} \nabla \eta (x)\).
Declarations
Acknowledgements
The research was in part supported by NSF Grants DMS 1318377 and DMS 1613861. We would like to thank Professor Lei Zhang and Venkat Chandrasekaran for several stimulating discussions, and Professor Houman Owhadi for valuable comments.
Dedication
Honor of Bjorn Engquist on the occasion of his 70th birthday.
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