H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{H}}$$\end{document}-matrix approximability of inverses of FEM matrices for the time-harmonic Maxwell equations

The inverse of the stiffness matrix of the time-harmonic Maxwell equation with perfectly conducting boundary conditions is approximated in the blockwise low-rank format of H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{H}}$$\end{document}-matrices. Under a technical assumption on the mesh, we prove that root exponential convergence in the block rank can be achieved, if the block structure conforms to a standard admissibility criterion.


Introduction
A backbone of computational electromagnetics is the solution of the time-harmonic Maxwell equations. Since the discovery of Nédélec's edge elements (and their higher order generalizations) finite element methods (FEMs) have become an important discretization technique for these equations with an established convergence theory [35]. While the resulting linear system is sparse, a direct solver cannot Communicated by: Francesca Rapetti up to a small perturbation, be controlled in H 1 so that approximation becomes feasible and one may proceed structurally similarly to the scalar case. The local discrete Helmholtz-type decomposition (Lemma 3.11) may also be of independent interest. This paper is organized as follows. In Section 2, we introduce the time-harmonic Maxwell equations and their discretization with Nédélec's curl-conforming elements. We state the main result of this paper, namely, the existence of H-matrix approximations to the inverse stiffness matrix that converge root exponentially in the block rank. We hasten to add that we do not track the dependence on the frequency ω in our analysis and focus on the case of fixed wave number κ. As in the case of the Helmholtz equation, the high-frequency case of → ∞ would require specialized matrix formats such as directional H 2 -matrices ( DH 2 ) or the butterfly format; we refer to the literature discussions in [4,7,8]. To prove the approximability result of Section 2, we present in Section 3 a local discrete Helmholtz decomposition and prove stability and approximation properties of this decomposition under a certain technical assumption on the mesh. In Section 4, we present a Caccioppoli-type inequality for discrete L-harmonic functions with L being the Maxwell operator. Furthermore, we obtain exponentially convergent approximations to discrete L-harmonic functions. Section 5 is concerned with the proof of the main result of this paper.
Concerning notation: Constants C > 0 may differ in different occurrences but are independent of critical parameters such as the mesh size. a ≲ b indicates the existence of a constant C > 0 such that a ≤ Cb. For a set A ⊂ ℝ 3 , we denote by |A| its Lebesgue measure. For finite sets B, the cardinality of B is also denoted by |B|. We employ standard Sobolev spaces as described in [34]. We also denote c ∶= ℝ 3 ⧵ .

Model problem
Maxwell's equations are a system of first-order partial differential equations that connect the temporal and spatial rates of change of the electric and magnetic fields possibly in the presence of additional source terms. Let ⊂ ℝ 3 be a simply connected polyhedral domain with boundary Γ := ∂Ω that, in physical terms, is filled with a homogeneous isotropic material. Maxwell's equations then connect the electric field E to the magnetic field H by where G is a given function representing the applied current. Homogeneous isotropic materials can be characterized by a positive dielectric constant ε > 0, a positive permeability constant μ > 0, and a non-negative electric conductivity constant σ ≥ 0.
In this paper, we consider perfectly conducting boundary conditions for E , i.e., where n is the unit outward normal vector on Γ.
Multiplying both sides of (2.4) with Ψ ∈ H 0 (curl,Ω) and integrating by parts, we obtain the weak formulation: Find E H 0 (curl,Ω) such that where ⟨⋅, ⋅⟩ 2 ( ) is the L 2 (Ω)-inner product. We assume that κ is not an eigenvalue of the operator ∇ × ∇ ×, see, e.g., [35,Sec. 4]. This implies in particular that κ ≠ 0 since ∇H 1 0 ( ) is contained in the kernel of the operator ∇ × ∇ ×. Then, the Fredholm ∈ alternative provides the existence of a unique solution to the variational problem, and we have the a priori estimate for a constant C stab that depends on Ω and κ, see, e.g., [28,Thm. 5.2].

Discretization by edge elements
where the elements T j ∈ T h are open tetrahedra. The mesh T h is assumed to be regular in the sense of Ciarlet, i.e., there are no hanging nodes.
The assumption of quasi-uniformity includes the assumption of γ-shape regularity, i.e., there is γ > 0 such that diam(T j ) ≤ γ|T j | 1/3 for all T j ∈ T h . For the Galerkin discretization of (2.5), we use lowest order Nédélec's H(curl,Ω)-conforming elements of the first kind, see, e.g., [35,Sec. 5]. That is, on T ∈ T h , we introduce the lowest order local Nédélec space and set The standard degrees of freedom of h (T h , ) are the line integrals of the tangential component of U h on the edges of T h , see, e.g., [35,Sec. 5.5.1], [3,Sec. 2.3.2]. Hence, the dimension of h (T h , ) is the number of edges of T h . The standard basis X h ∶= { e } of h (T h , ) consists of the so-called (lowest order) edge elements, where the function e ∈ h (T h , ) is associated with the edge e of T h and is supported by the union of the tetrahedra sharing the edge e. More specifically, for an edge e with endpoints V 1 , V 2 and a tetrahedron T with edge e, one has is obtained by taking the e ∈ X h , whose edge e satisfies e ⊂ Ω; that is, X h,0 is obtained from X h by removing the shape functions associated with edges lying on Γ.
Using h,0 (T h , ) ⊆ 0 (curl, ) as ansatz and test space in (2.5), we arrive at the Galerkin discretization of finding h ∈ h,0 (T h , ) such that Using the basis X h,0 , the Galerkin discretization (2.7) can be formulated as a linear system of equations, where the system matrix ∈ ℂ N×N is given by .
For unique solvability of the discrete problem (2.7) or, equivalently, the invertibility of A, we recall the following Lemma 2.2. In that result and throughout the paper, we denote by the L 2 (Ω)-orthogonal projection onto h (T h , ). [28,Thm. 5.7] Assume (2.6). There exists h 0 > 0 depending on the parameters of the continuous problem and the γ-shape regularity of T h , such that, for h < h 0 , the discrete problem (2.7) has a unique solution, and there holds the stability estimate

Lemma 2.2
Here, C > 0 is a constant depending solely on the γ-shape regularity of T h and the parameters of the continuous problem.

Hierarchical matrices
The goal of this paper is to obtain an H-matrix approximation of the inverse matrix B := A − 1 . An H-matrix is a blockwise low-rank matrix, where suitable blocks for low-rank approximation are chosen by the concept of admissibility, which is defined in the following.

Definition 2.3 (bounding boxes and η-admissibility)
A cluster τ is a subset of the index set I = {1, 2, … , N} . For a cluster ⊂ I , an axis-parallel box B R ⊆ ℝ 3 is called a bounding box, if B R is a cube with side length R τ and ∪ i∈ supp i ⊆ B R . Let η > 0. Then, a pair of clusters is called η-admissible, if there exist bounding boxes B R and B R of τ and σ such that Definition 2.4 (Concentric boxes) Axis-parallel boxes B R of side length R are called boxes. Two boxes B R and B R ′ of side length R and R ′ are said to be concentric, if they have the same barycenter and B R can be obtained by a stretching of B R ′ by the factor R∕R � taking their common barycenter as the origin.

Definition 2.5 (cluster tree)
A cluster tree with leaf size n leaf ∈ ℕ is a binary tree I with root I such that each cluster ∈ I is either a leaf of the tree and satisfies | | ≤ n leaf , or there exist disjoint subsets ′ , �� ∈ I of τ, called sons, with = �∪ �� . The level function level ∶ I → ℕ 0 is inductively defined by level(I) = 0 and level( � ) ∶= level( ) + 1 for ′ a son of τ. Furthermore, depth( I ) ∶= max ∈ I level( ) is called the depth of a cluster tree.
Definition 2.6 (block cluster tree, sparsity constant and partition) Let I be a cluster tree with root I and η > 0 be a fixed admissibility parameter. The block cluster tree I×I is a tree constructed recursively from the root I × I such that for each block × ∈ I×I with , ∈ I , the set of sons of τ × σ is defined as The sparsity constant C sp of a block cluster tree, see, e.g., [23,30], is given as The leaves of the block cluster tree induce a partition P of the set I × I , which we call a partition based on I . For such a partition P and a fixed admissibility parameter η > 0, we define the far field and the near field as For clusters τ, ⊂ I , we adopt the notation For ∈ ℂ N and ∈ ℂ N×N , the restrictions x| τ and A| τ×σ are understood as (x| τ ) i = χ τ (i)x i and (A| τ×σ ) ij = χ τ (i)χ σ (j)A ij , where χ τ and χ σ are the characteristic functions of the sets τ, σ. For integers r ∈ ℕ , matrices ℂ ×r are understood as matrices in ℂ N×r such that each column is in ℂ .

Definition 2.7 ( H-matrices)
Let P be a partition of I × I based on a cluster tree I and admissibility parameter η > 0. A matrix ∈ ℂ N×N is an H-matrix, if, for every admissible pair (τ,σ) ∈ P far , we have a rank r factorization where ∈ ℂ ×r and ∈ ℂ ×r .

Main result
The following theorem is the main result of this paper. It states that the inverse of the Galerkin matrix A from (2.8) can be approximated at an exponential rate in the block rank by an H-matrix.
Theorem 2.8 Let η > 0 be a fixed admissibility parameter and P be a partition of I × I based on the cluster tree I and η. Let the mesh T h be such that Assumption 3.4 holds true for any box. Let h < h 0 with h 0 given by Proposition 2.2, and let A be the stiffness matrix given by (2.8). Then, there exists an H-matrix H with blockwise rank r such that The constants C apx , b > 0 depend only on κ, Ω, η, and the γ-shape regularity of the quasi-uniform triangulation T h . The constant C sp (defined in (2.11)) depends only on the partition , , d, and Ω.

Remark 2.9
The low-rank structure of the far-field blocks allow for efficient storage of H-matrices as the memory requirement to store an H-matrix is O(C sp depth( I )rN) . Standard clustering methods such as the geometric clustering for quasi-uniform meshes (see, e.g., [26,Sec. 5.4.2]) lead to balanced cluster trees, i.e., depth( I ) ∼ log(N) and a uniformly (in the mesh size h) bounded sparsity constant. In total this gives a storage complexity of O(rN log(N)) for the matrix H instead of the O(N 2 ) for the fully populated inverse A − 1 .

Helmholtz decompositions: continuous and localized discrete
Helmholtz decompositions, i.e., writing a vector field as a sum of a divergence-free field and a gradient field, play a key role in our analysis. In fact, we use two different decompositions, the regular decomposition (see, e.g., [28,Lem. 2.4] and [29,Thm. 11]) and a localized discrete version (Definition 3.6).

Lemma 3.1 (Regular decomposition) Let ⊂ ℝ 3 be a bounded Lipschitz domain.
Then, there is a constant C > 0 depending only on Ω such that any E ∈ H 0 (curl,Ω) can be written as E = z + ∇p with ∈ 1 0 ( ) and p ∈ H 1 0 ( ) and Proof Regular decompositions are available in the literature, see, e.g., [28,Lem. 2.4] and [29,Thm. 11]. The statement that ‖ ‖ 2 ( ) and ‖∇p‖ 2 ( ) are controlled by ‖ ‖ 2 ( ) is a variation of these estimates. For a proof, see [32] or the appendix. ◻ The function z of the regular decomposition provided by Lemma 3.1 is not necessarily divergence-free. This can be corrected by subtracting a gradient. To that end, we introduce, for a given open set The mapping 2 ( ) ∋ ↦ ∈ H 1 0 ( ) has the following properties: The constant C in statement (i) reflects the Poincaré constant of the simply connected domain Ω. The property (ii) follows by construction. ◻

Remark 3.3 (classical Helmholtz decomposition) Selecting
Regular decompositions as in Lemma 3.1 can also be done locally for discrete functions. Let P 1 (T) denote the space of polynomials of degree at most 1 on T ∈ T h . We introduce spaces of globally continuous, piecewise linear polynomials by We will require the following assumption on the meshes T h :

Assumption 3.4 For a simply connected domain D ⊂ ℝ 3 , define the sets of elements touching D as
For any box D ⊂ ℝ 3 , there is a set D which is a union of elements in T h such that We call D a mesh-conforming region for D. If a box D has more than one meshconforming region D , one is selected as "the" mesh-conforming one. Remark 3. 5 The reason behind Assumption 3.4 is that the region D may not be simply connected, but by adding elements of the mesh holes may be filled to obtain a simply connected set D .
The spaces localized to a mesh-conforming region D are given by Definition 3.6 (Local discrete regular decomposition) Let D ⊂ ℝ 3 be a box and D be the corresponding mesh-conforming region from Assumption 3.4. We denote by as ηE h = z + ∇p, where ∈ 1 0 ( ) and p ∈ H 1 0 ( ) are given by Lemma 3.1. Then, the local discrete regular decomposition is given by For future reference, we note that , is not unique. However, its gradient ∇p h is unique. 2. Due to the cut-off function η, the decomposition depends on E h on supp η only, which is quantified in the stability assertions of Lemma 3.11. 3. The local regular decomposition provides, for a function E h that is a discrete function on D , two representations in view of η ≡ 1 on D , namely, which is a discrete Helmholtz decomposition as described in, e.g., [ The following lemma formulates a local exact sequence property.
Proof We recall from, e.g., [35,Thm. 3.37] the following commuting diagram property: for a simply connected Lipschitz domain ω the condition ∇ × = 0 implies = ∇ for some ψ ∈ H 1 (ω); furthermore, ψ is unique up to a constant. The discrete commuting diagram property for a tetrahedron T is: . The function φ h is unique up to a constant, which we fix, for example, by the condition In order to prove the following lemmas, we need to introduce some projections and their properties. Let D ⊂ ℝ 3 be a box and D be defined according to Assumption 3.4. We define the space here, τ is a unit vector parallel to the edge e. A key property of the operators D and D is that they commute, i.e., (see, e.g., [35, (5.59 Moreover, the lowest order elemental Nédélec interpolants have first-order approximation properties. In the following, we show local stability and approximation properties for the local discrete regular decomposition of Definition 3.6. This will be based on Lemma 3.8 with D = B R , where B R is a box with side length R. It is an important geometric observation that, due to the assumption that Ω is a Lipschitz polyhedron, the intersection B R ∩Ω is a Lipschitz domain and the intersection B R ∩ c is connected provided R is sufficiently small. Then, the additional assumptions on D ∩ Ω = B R ∩ Ω in Lemma 3.8 can be satisfied. We formulate this as an assumption on R in terms of a number R max > 0 that depends on Ω: where the constant C > 0 depends only on Ω, the γ-shape regularity of the quasiuniform triangulation T h , and C η . Proof The proof is done in two steps. We note that the condition on the parameter ε and the assumption on the mesh-conforming region (Assumption 3.4) ensures that � B R ⊆ B (1+ )R .
Step 1: In this step we provide a proof of the stability estimate. Recalling the stability estimate Lemma 3.1 and using the product rule for the curl operator, it follows that Since ∇p h satisfies (3.6), we get with (3.7) and the aid of (3.11) The definition of z h gives The combination of the above inequalities provides the desired local stability result.
Step 2: To prove the approximation property, we first need to ascertain the exist- To that end, we note that h ∈ h (T h ,B R ) , use the commuting diagram property (3.10) of B R and B R , and the fact that B R is a projection operator to compute on B R : Lemma 3.8 then provides the existence of Since p h satisfies (3.6), we get from z + ∇p = E h = z h + ∇p h on B R and the approximation property of B R given in Lemma 3.9 The combination of the above inequality and (3.11) implies which finishes the proof. ◻ (3.11)

Low-dimensional approximation of discrete L-harmonic functions
such a space will be formally introduced as H c,h (D) below. In this section, we show that discrete L-harmonic functions can be approximated from low-dimensional spaces on compact subsets of D . Discrete interior regularity estimates, introduced in the following, play a key role.

The Caccioppoli-type inequalities
Caccioppoli inequalities usually estimate higher order derivatives by lower order derivatives on (slightly) enlarged regions. The following discrete Caccioppoli-type inequalities are formulated with an h-weighted H(curl)-norm and an h-weighted H 1 -norm. For a box B R of side length R > 0, we define the norms |||⋅||| c,h,R and |||⋅||| g,h,R (the subscripts c and g abbreviate "curl" and "gradient") as follows: For any bounded open set B ⊂ ℝ 3 , we define and The following lemma provides a discrete Caccioppoli-type estimate for functions in H c,h (B (1+ )R ∩ ).
Step 1: Using the vector identity we get

Young's inequality then gives
Kicking back the term 1 to the left-hand side, we arrive at Since � � + ‖∇ ‖ 2 L ∞ ≲ ( R) −2 with implied constant depending on κ, we are left with estimating Re a(ηE h ,ηE h ).
Step 2: Using the orthogonality relation in the definition of the space H c,h (B (1+ )R ∩ ) , we get For each element T ∈ T h , Lemma 3.9 yields To proceed further, we observe that h | T ∈ N 0 (T) has the form E h = a + b × x so that curl E h | T = 2b and hence ∑ 3 j=1 � x j h � ≲ �∇ × h � pointwise on T so that we get with an implied constant independent of the function η

Using (4.7) we obtain
Computing ∇ × 2 h = ∇ 2 × h + 2 ∇ × h , using the product rule and the fact that x j (∇ × h ) = 0 since ∇ × E h is constant gives again in view of (4.7) and R ≲ 1 Summing the squares of (4.8), (4.9) over all elements T with T ∩ supp η ≠ Ø, which is ensured if we sum over all T with T ⊂ B (1+ε)R ∩Ω, and inserting the result in (4.6) yields Using Young's inequality, h ≲ 1 and 0 ≤ η ≤ 1 as well as the definition of the norm |||⋅||| c,h,R , we obtain Inserting this in (4.4) produces Using again Young's inequality to kick the term ‖ ‖ ∇ × h ‖ ‖ 2 (B (1+ )R ∩ ) of the righthand side back to the left-hand side produces the desired estimate. ◻ For functions in H g,h (B (1+ )R ∩ ) , a discrete Caccioppoli-type estimate has already been established in [16,Lem. 2], which we state in the following for the sake of completeness.

Low-dimensional approximation in H c,h (B R ∩˝).
In this subsection, we apply the Caccioppoli-type estimates from Lemmas 4.1 and 4.2 to find approximations of the Galerkin solutions from low-dimensional spaces. We will need a Poincaré inequality as given in [24, (7.45)]: for open sets D ⊂ ω with |D| > 0 and u ∈ H 1 (ω), we have In the following, we consider low-dimensional approximations of discrete harmonic functions in Lemma 4.3 that generalizes [16,Lem. 4].
The fact that Ω c is Lipschitz (see [31,Thm. 2] for details) implies the existence of a constant c > 0 depending only on Ω such that for all x ∈ Ω c and all r ∈ (0,1) we have |B r (x) ∩ c | ≥ cr 3 , where B r (x) denotes the ball of radius r centered at x. Selecting an x ∈ B (1+ )R ∩ c and noting that B εR/2 (x) ⊂ B (1 + 2ε)R , we conclude .
Due to (4.11), [16,Lem. 4] provides a subspace W m of H g,h (B R ∩ ) such that where C dim depends only on Ω and the γ-shape regularity of the quasi-uniform triangulation T h . We denote by û h the extension by zero of u h to Ω c . It follows from the Poincaré inequality (4.10) and Combining (4.14) and (4.12) leads to

Remark 4.4
The factor ε − 2 instead of ε − 0 for boxes B R near the boundary is a consequence of not assuming a relation between the orientation of the boxes and the boundary. Aligning boxes with the boundary allows one to better exploit boundary conditions and improve the factor ε − 2 .
In the following, we will need a simplified version of Lemma 4.3:

Corollary 4.5
Let R ∈ (0,2diam(Ω)), ε ∈ (0,1), q ∈ (0,1). There are constants C ′′ dim and C ′′ app depending only on Ω and the γ-shape regularity of the quasi-uniform triangulation T h such that, for any concentric boxes B R , B (1 + 2ε)R and any m ∈ ℕ , there exists a subspace W m ⊂ H g,h (B R ∩ ) of dimension such that for any u h ∈ H g,h (B (1+2 )R ∩ ) there holds Proof The case that the parameters satisfy (4.11) is covered by Lemma 4.3. For the converse case h∕R > q ∕(8m max{1, C app }) , we take W m ∶= H g,h (B R ∩ ) so that the minimum in (4.18) is zero and observe in view of the quasi-uniformity of T h which finishes the proof. ◻ If E h is locally discrete divergence-free, then the function ∇(p + φ z ) in the decomposition E h = z −∇φ z + ∇(p + φ z ) given by Definition 3.6 is also locally discrete divergence-free since z −∇φ z is divergence-free. The following lemma shows that ∇ B (1+2 )R ∇(p + ) is discrete divergence-free as well:  Step 4: Let T h (B (1+2 )R ∩ ) and B (1+2 )R be given according to Assumption 3.4.
Note that ηE h ∈ H 0 (curl,Ω). Decompose ηE h ∈ H 0 (curl,Ω) as ηE h = z + ∇p with ∈ 1 0 ( ) and p ∈ H 1 0 ( ) according to Lemma 3.1. Let φ z be given by (3.1) . We apply Corollary 4.5 with the pair (R,ε) replaced with (R,̃ ) = (R(1 + ), 2(1+ ) ) to get a subspace W m ⊂ H g,h (B (1+ )R ∩ ) for the box B (1+ε)R ∩Ω and an w m ∈ W m such that Of these 6 terms, the first three terms are shown to be small, the next two terms are from a low-dimensional space, and the last term is exponentially (in m) close to ∇w m by (4.23), which is also from a low-dimensional space, namely, ∇W m . As the approximation of E h , we thus take with the |||⋅||| c,h,R -orthogonal projection B R of (4.20). Property (i) is then satisfied by construction. In order to prove (ii), we compute

Proof of main results
The results of the preceding Section 4 allow us to show that the Galerkin approximation E h of (2.7) can be approximated from low-dimensional spaces in regions B R away from the support of the right-hand side F.

Theorem 5.1
Let h 0 > 0 be given by Lemma 2.2, and let T h be a quasi-uniform mesh with mesh size h ≤ h 0 . Fix q ∈ (0,1) and η > 0. Set ζ = 1/(1 + η). For every cluster pair (τ,σ) with bounding boxes B R and B R with dist(B R , B R ) ≥ diam(B R ) and each k ∈ ℕ , there exists a space k ⊂ 2 (B R ∩ ) with such that for an arbitrary right-hand side F ∈ L 2 (Ω) with supp ⊂ B R ∩ , the corresponding Galerkin solution E h of (2.7) can be approximated from V k such that Here, L 2 h is the L 2 -orthogonal projection onto h (T h , ) and C box , C dim are constants depending only on κ, Ω, and the shape regularity of T h . In the following, we employ Lemma 4.8. In order to do so, boxes have to have side length smaller than R max ∕2 , which may not hold for general bounding boxes B R . However, as bounding boxes can always be chosen to satisfy R τ < 2 diam(Ω), there exists a constant L ∈ ℕ independent of R τ such that R ∕L ≤ R max ∕2 with R max given in Definition 3.10. Consequently, we can decompose a box B R = int Such a dual basis of {Ψ i : i = 1,…,N} can be constructed as (discontinuous) piecewise polynomials of degree 1 as described in, e.g., [11,Sec. 4.8] for classical Lagrange elements. In fact, suppλ i can be taken to be a single tetrahedron in suppΨ i . The constant C depends solely on the γ-shape regularity of T h . We emphasize that our choice of scaling of the functions Ψ i is responsible for the factor h − 1/2 . For clusters ′ , define the mappings where ′ is the characteristic function of ′ . For v ∈ L 2 (Ω) and a cluster ′ with bounding box B R ′ , we observe for the ℓ 2 -norm ∥⋅∥ 2 on ℂ ′ that We observe that, for h ∈ h,0 (T h , ) expanded as h = ∑ i∈I i i , we have i = ( I ( h )) i . In particular, we have for the coefficients μ i with i ∈ � Step 2: Let V k be the space given by Theorem 5.1 for the boxes B R , B R . For arbitrary ∈ ℂ , define the function ∶= ∑ i∈ i i and observe: Let h ∈ h,0 (T h , ) be the Galerkin solution corresponding to the right-hand side f b and ̃ h ∈ k be the approximation to E h asserted in Theorem 5.1. Then, We define the low-rank factor X τσ as an orthogonal basis of the space V ∶= { (̃ k )∶̃ k ∈ k } and set Then, the rank of X τσ is bounded by dim k ≤C dim (1 + ) 3 k 4 q −3 + ln 4 (k(1 + )) . Since H is the orthogonal projection from ℂ N onto V , we conclude that ∶= H ( h ) is the ∥⋅∥ 2 -best approximation of the Galerkin solution in V , which results in By (5.5) and ∈ ℂ , we have Since = H , we conclude As b was arbitrary, we obtain the stated norm bound. ◻

Conflict of interest The authors declare no competing interests.
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