Abstract
We analyze rates of approximation by quantized, tensor-structured representations of functions with isolated point singularities in ℝ3. We consider functions in countably normed Sobolev spaces with radial weights and analytic- or Gevrey-type control of weighted semi-norms. Several classes of boundary value and eigenvalue problems from science and engineering are discussed whose solutions belong to the countably normed spaces. It is shown that quantized, tensor-structured approximations of functions in these classes exhibit tensor ranks bounded polylogarithmically with respect to the accuracy ε ∈ (0,1) in the Sobolev space H1. We prove exponential convergence rates of three specific types of quantized tensor decompositions: quantized tensor train (QTT), transposed QTT and Tucker QTT. In addition, the bounds for the patchwise decompositions are uniform with respect to the position of the point singularity. An auxiliary result of independent interest is the proof of exponential convergence of hp-finite element approximations for Gevrey-regular functions with point singularities in the unit cube Q = (0,1)3. Numerical examples of function approximations and of Schrödinger-type eigenvalue problems illustrate the theoretical results.
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Notes
Formally, the strong Kronecker product of two 2 × 2 block matrices is defined as the following 2 × 2 block matrix:
$$ \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix} \Join \begin{bmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{bmatrix} = \begin{bmatrix} A_{11}\otimes B_{11} + A_{12}\otimes B_{21} & A_{11}\otimes B_{12} + A_{12}\otimes B_{22} \\ A_{21}\otimes B_{11} + A_{22}\otimes B_{21} & A_{21}\otimes B_{12} + A_{22}\otimes B_{22} \\ \end{bmatrix}. $$
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Acknowledgements
The authors are grateful to the reviewers for their comments which contributed to the improvement of the paper. Ch. Schwab acknowledges stimulating discussions at WS1936 at the Mathematical Research Institute Oberwolfach 02-06Sep2019.
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Open access funding provided by Swiss Federal Institute of Technology Zurich. M. Rakhuba was supported by ETH Grant ETH-44 17-1.
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Appendices
Appendix A. hp approximation in weighted Gevrey classes
We prove, in this section, the exponential convergence of the hp approximations to functions in the weighted Gevrey class \(\mathcal {J}^{\varpi }_{\gamma }(Q; C, A, {\mathfrak {d}})\) for C, A > 0, γ > 3/2, \({\mathfrak {d}}\geq 1\). Specifically, this corresponds to functions u ∈H1(Q) such that
We recall that the hp space is defined as
The central (novel) result of this section is the existence—for Gevrey-regular functions in \(\mathcal {J}^{\varpi }_{\gamma }(Q)\)—of an exponentially convergent, H1(Q) conforming hp-projector on 1-irregular geometric meshes of hexahedra, as stated in the following proposition.
Proposition 1
Let γ ≥γ0 > 3/2 and \({\mathfrak {d}} \geq 1\). Then, there exists \({\Pi }_{\mathsf {hp}}^{\ell , p} : \mathcal {J}^{\infty }_{\gamma }(Q)\to X_{\mathsf {hp}}^{\ell , p}\) such that for all \(u\in \mathcal {J}^{\varpi }_{\gamma }(Q; C, A, {\mathfrak {d}})\) there exist constants Chp and bhp such that
provided the uniform polynomial degree is \(p\geq c_{0} \ell ^{{\mathfrak {d}}}\) for a sufficiently large constant c0 > 0 (depending on the constant A in Eq. (41) and on \(\mathfrak {d}\)) which is independent of ℓ. The constants Chp, bhp depend on the constants C, A, and \({\mathfrak {d}}\) in \(\mathcal {J}^{\varpi }_{\gamma }(Q)\). In terms of \(N_{\text {dof}}= \dim (X_{\mathsf {hp}}^{\ell , p})\simeq \ell ^{3{\mathfrak {d}}+1}\), (42) reads
The rest of the section will be devoted to an overview of the construction of the conforming projector \({\Pi }_{\mathsf {hp}}^{\ell , p}\). This projector has already been exploited and analyzed in detail, e.g., in [73, 74]; here, we wish to sketch its construction for the sake of self-containedness and to provide the necessary detail of the treatment of non-analytic Gevrey-type estimates (i.e., of the cases where \({\mathfrak {d}}>1\)), which requires some minor modification with respect to the setting of [73, 74]. For positive integers p and s such that 1 ≤ s ≤p, we write Ψp, s = (p −s)!/(p + s)!.
1.1 A.1 Discontinuous projector
We start by introducing a nonconforming projector.
1.1.1 A.1.1 Local projector
We denote the reference interval by I = (− 1, 1) and the reference cube by \({\widehat {K}} = (-1, 1)^{3}\). We write also \(H^2_{\text {mix}}({\widehat {K}}) = H^{2}(I)\otimes H^{2}(I)\otimes H^{2}(I)\), where ⊗ denotes the Hilbertian tensor product. Let p ≥ 3: as constructed in [17, Appendix A], there exist univariate projectors \({\widehat {\pi }}_{p} : H^{2}(I) \to \mathbb {P}_{p}(I)\) such that
see [73, Lemma 4.1] (the projector πp is denoted πp,2 there). Then, the Hilbertian tensor product projector given by
has the following property.
Lemma 1
[75, Remark 5.5] For every p ≥ 3 exists a projector \({\widehat {\Pi }}_{p}:H^2_{\text {mix}}({\widehat {K}}) \to \mathbb {Q}_{p}({\widehat {K}})\) such that for all \(v\in H^2_{\text {mix}}({\widehat {K}})\) and all integer s such that 2 ≤s ≤p
with C independent of p, s, and v.
For all \(K\in \mathcal {G}^{\ell }\), we introduce the affine transformation from the reference element to K
it follows that for v defined on K such that \(v\circ {\Phi }_{K} \in H^2_{\text {mix}}({\widehat {K}})\) we can define the local projector on K so that
The projector \({{\Pi }_{p}^{K}}\) is continuous across regular matching faces.
Lemma 2
[73, Lemma 4.2] Let K1, K2 be two axiparallel cubes that share one regular face F (i.e., F is a full face of both K1 and K2). Then, for \(v\in H^{6}(\text {int}(\overline {K}_{1}\cup \overline {K}_{2}))\) the piecewise polynomial
is continuous across F.
Remark 1
By (44) and (45), if a function v on K such that \(v\circ {\Phi }_{K}\in H^2_{\text {mix}}({\widehat {K}})\) vanishes on a face F ⊂∂K, then we also have \(\left ({{\Pi }^{K}_{p}}v\right )_{|_{F}} = 0\).
1.1.2 A.1.2 Globally discontinuous hp projector
Starting from the local, elementwise projector (46), a global, discontinuous projection operator \({\Pi }_{\mathsf {hp}, \text {disc}}^{\ell , p}\) is defined in the usual way: with the nonconforming hp-space
for all \(K\in \mathcal {G}\) and for \(v\in \mathcal {J}^{\infty }_{\gamma }(Q)\), with γ > 3/2,
Note that v(0) is well defined for \(v\in \mathcal {J}^{1}_{\gamma }(Q)\) if γ > 3/2, see [49, Lemma 7.1.3]; hence, a fortiori, \({\Pi }_{\mathsf {hp}, \text {disc}}^{\ell , p} : \mathcal {J}^{\infty }_{\gamma }(Q) \to X_{\mathsf {hp}, \text {disc}}^{\ell , p}\) is well defined if γ > 3/2.
Lemma 3
Let \(u\in \mathcal {J}^{\varpi }_{\gamma }(Q; C_{u}, A_{u}, {\mathfrak {d}})\). Then, if \(p\simeq \ell ^{{\mathfrak {d}}}\), there exist constants C, b > 0 such that
with \(\dim \left (X_{\mathsf {hp}, \text {disc}}^{\ell , p} \right )\simeq \ell ^{3{\mathfrak {d}}+1}\).
Proof
The proof follows along the lines of the proof of [75, Proposition 5.13]. Denote \(\eta = u - {\Pi }_{\mathsf {hp}, \text {disc}}^{\ell , p} u\) and \(N_{K}[v]^{2} = \| v \|^{2}_{L^{2}(K)} / {h_{K}^{2}} + \| \nabla v \|^{2}_{L^{2}(K)}\).
We start by considering \(K\in {\mathscr{L}}^{\ell }_{j}\) for j ≥ 1 and write dK = dist(K, (0, 0, 0)). By Lemma 6, scaling inequalities (see [75, Equations (5.26)–(5.31)]), and the regularity of u (see (41)),
Then, using the fact that for sufficiently large s and c = 2Au + 1,
see [20, Equation (42)], choosing \(s = (p/c)^{1/{\mathfrak {d}}}\simeq \ell \), with c > 1 sufficiently large, and summing over all mesh layers not touching the origin (“interior mesh layers”), we obtain that there exist C1, b1 > 0 such that for every ℓ ≥ 1 holds
We now consider the element \(K \in {\mathscr{L}}^{\ell }_{0}\), i.e., K = (0, 2−ℓ)3. By Hardy’s inequality and choosing γ > 1,
Finally, the dimension of the \(\mathbb {Q}_{p}(K)\) space in each element \(K\in \mathcal {G}^{\ell }\) is given by (1 + p)3; since each non-terminal mesh layer \({\mathscr{L}}^{\ell }_{j}\), j > 0, contains 7 elements, we have that \(\dim (X_{\mathsf {hp}, \text {disc}}^{\ell , p}) = (1+p)^{3}(1+7\ell )\). The observation that \(p\simeq \ell ^{{\mathfrak {d}}}\) concludes the proof. □
1.2 A.2 Conforming hp approximation
A conforming hp approximation is obtained by locally lifting the polynomial face jumps of the discontinuous, piecewise polynomial approximation. Their construction is detailed in [73, Section 5.3].
1.2.1 A.2.1 Edge and face liftings
Since our discontinuous interpolant is the same as in [73], apart from the nonzero constant in \({\mathscr{L}}^{\ell }_{0}\) (see (47) and [73, Equation (4.10)]), the construction of the polynomial face jump liftings can be replicated verbatim as in [73]. We recall it here briefly, referring the reader to the aforementioned [73, Section 5.3] for the details.
We start by considering the interface between two mesh levels \({\mathscr{L}}^{\ell }_{k}\) and \({\mathscr{L}}^{\ell }_{k+1}\), \(k\in \mathbb {N}\). We introduce a local coordinate system \(\hat {x}, \hat {y}, \hat {z}\) and label the faces and edges belonging to the interface as Fi, i = 1, 2, 3 and Ei, \(i=1, \dots , 9\), respectively, see Fig. 10. Furthermore, we denote by hE the maximum length of all edges Ei. We refer to Fig. 10 for the precise numbering of edges and faces and for the location of the local system of coordinates. Given two neighboring elements Ka and Kb with interface \(f_{ab} = \overline {K}_{a}\cap \overline {K}_{b}\), the jump of a function
on fab is given by
where \(n_{K_{a}}\) (resp. \(n_{K_{b}}\)) is the normal pointing outwards from element Ka (resp. Kb).
Consider face F1 of Fig. 10: we define the jump of the discontinuous interpolant on this face as
where f1j are the four parts of the face F1, see Fig. 10. The jumps on the other faces are defined similarly. The edge jump, e.g., on E1, is then defined as . Let n denote the normal on face F1 pointing outwards from \(K^{\ell }_{010, k+1}\); the lifting of the jump on edge E1 is given by
After having defined the other edge liftings \({\mathfrak {L}}^{E_{i}}\), \(i=2,\dots , 9\), in the same way, we can introduce the full edge lifting operator
We now introduce the face lifting operator for the face F1, the other liftings being derived in the same way. We have
where n is again the normal on face F1 pointing outwards from \(K^{\ell }_{010, k+1}\). Then, the global lifting \({\mathfrak {L}}^{k}\) between mesh levels \({\mathscr{L}}^{\ell }_{k}\) and \({\mathscr{L}}^{\ell }_{k+1}\) is the sum of the local liftings on the three interfaces:
Note that the lifting thus defined has support only in the elements belonging to mesh level \({\mathscr{L}}^{\ell }_{k}\).
We now turn to the terminal layer, i.e., to the jumps of \({\Pi }_{\mathsf {hp}, \text {disc}}^{\ell , p} u\) between the element \(K^{\ell }_{000, 0} = (0, 2^{-\ell })^{3}\) and the elements of \({\mathscr{L}}^{\ell }_{1}\). The (three) faces belonging to the interface are all regular, but \({\Pi }_{\mathsf {hp}, \text {disc}}^{\ell , p} u\) is defined as a constant in \(K^{\ell }_{000,0}\), see (47). One has to lift the nodal jumps at all the nodes of \(K^{\ell }_{000,0}\) except the origin. Then, the same procedure as for the other mesh layers (applied to the nodally lifted polynomial) gives a lifting operator \({\mathfrak {L}}^{0}\).
The full lifting operator is thus given by the sum of the local operators, as
with all \({\mathfrak {L}}^{k}\) constructed as in (51). Such a lifting permits to obtain a conforming projector into \(X_{\mathsf {hp}}^{\ell , p}\), with approximation error bounded by a multiple of the approximation error of the discontinuous operator \({\Pi }_{\mathsf {hp}, \text {disc}}^{\ell , p}\), as stated in the next proposition, that is proven in [73].
Proposition 2
[73, Proposition 5.3] The discontinuous projection operator \({\varPi }_{\mathsf {hp}}^{\ell , p}\) defined in (47) and the lifting operator \(\mathfrak {L}\) defined in (52) are such that
is conforming in H1(Q) and there exists C > 0 independent of p such that
Here, \(\eta = u - {\varPi }_{\mathsf {hp}, \text {disc}}^{\ell , p} u\).
The exponential convergence of the conforming approximation, stated in Proposition 3, is a direct consequence of the last results.
Proof of Proposition 3
Inequality (42) follows from Proposition 4 and Lemma 8, once the algebraic term in p of inequality (48) has been absorbed in the exponential by a change of constants. □
Remark 2
Recall that \({\varGamma } = \left \{ (x_{1},x_{2},x_{3})\in \partial Q: x_{1} x_{2} x_{3} \not = 0\right \}\) contains the faces of the boundary of Q not abutting at the singularity. All liftings obtained by the operator (52) admit traces which vanish on Γ. I.e., for all \(v\in \mathcal {J}^{\infty }_{\gamma }(Q)\),
Therefore, by Remark 7, if \(v_{|_{{\varGamma }}} = 0\), then also \(\left ({\Pi }_{\mathsf {hp}}^{\ell , p} v \right )_{|_{{\varGamma }}} = 0\).
1.3 A.3 Combination of patches
We conclude this section by considering the approximation in a domain which contains the singular point in its interior. Let then R = (− 1, 1)3. The definition of the weighted space follows directly from the definition of the spaces in Q, by keeping the weight r = |x| to be the distance from the origin.
The construction of the graded mesh is done by decomposing R into eight sub-cubes of unitary edge and by collecting the elements of the sub-meshes (called here “patches”) obtained by symmetry from \(\mathcal {G}^{\ell }\). The projector \({\Pi }_{\mathsf {hp}}^{\ell , p}\) in R can also be straightforwardly constructed by combining local projectors obtained by symmetry; we show that it is continuous on interpatch faces, hence conforming on the whole cube R.
We detail the construction for two patches; the rest follows by iterating this argument. Specifically, we consider the two cubes
and introduce the reflection operator
Note that (ψ±)− 1 = ψ±. Then, the mesh on the domain \(Q^{\pm } = \overline {Q^{+}\cup Q^{-}}\) is given by
see Fig. 11.
The projection operator for functions \(v\in \mathcal {J}^{\infty }_{\gamma }(Q^{\pm })\) can be easily constructed by reflection
The operator thus obtained is continuous hence conforming, as discussed in the next lemma.
Lemma 4
The operator \({\Pi }_{\mathsf {hp}}^{\ell , p, \pm }\) is conforming in H1(Q±). Furthermore, if γ ≥γ0 > 3/2 and \({\mathfrak {d}} \geq 1\), then for all \(u\in \mathcal {J}^{\varpi }_{\gamma }(Q^{\pm }; C, A, {\mathfrak {d}})\) there exist \(C_{\mathsf {hp}}^{\pm }\), bhp such that, for all \(\ell \in \mathbb {N}\), with \(p \geq c_{0}^{\pm } \ell ^{{\mathfrak {d}}}\) for a sufficiently large \(c_{0}^{\pm }>0\) independent of ℓ, there holds
Furthermore, there holds \(\dim (X_{\mathsf {hp}}^{\ell , p})\simeq \ell ^{3{\mathfrak {d}}+1}\).
Proof
\({\Pi }_{\mathsf {hp}}^{\ell , p, \pm }\) is continuous in the sub-patches Q+ and Q−. It remains to check the continuity across the interpatch interface F± = {0}× (0, 1)2. By construction, all elemental faces belonging to the interface are regular, hence, by Lemma 7, the discontinuous projector \({\Pi }_{\mathsf {hp}, \text {disc}}^{\ell , p}\) is continuous across these faces.
We consider the error contribution from interior mesh layers, i.e., from all elements in \({\mathscr{L}}^{\ell }_{j}\), j > 0. For all faces F in interior mesh layers which are situated perpendicular to F±, we have
We now consider any edge E belonging to F± and separating the mesh levels \({\mathscr{L}}^{\ell }_{k}\) and \({\mathscr{L}}^{\ell }_{k+1}\), see Fig. 11 for an example. By the continuity of the discontinuous projector across regular faces
where \({\Pi }_{\mathsf {hp}, \text {disc}}^{\ell , p, -}\) is the discontinuous projector in patch \(\mathcal {G}^{\ell , -}\). Therefore, from definitions (49), (50), and (51), we conclude that the projection operator is continuous across interior mesh layers \({\mathscr{L}}^{\ell }_{k}\), k > 0.
When dealing with the terminal layer \({\mathscr{L}}^{\ell }_{0}\), we note that the discontinuous projector is constant hence continuous. The nodal liftings are continuous by the symmetry of their construction; the edge liftings are then continuous by the same argument as in interior mesh layers, and this gives the continuity between patches.
Finally, (53) follows from the application of the corresponding approximation results in each patch. □
We can directly extend the construction in the proof of the above lemma to the remaining patches Rm = (0, a1) × (0, a2) × (0, a3) with \((a_{1}, a_{2}, a_{3})\in \{-1, 1\}^{3}\), \(m=0, \dots , 7\). Recall that \({\Pi }_{\mathsf {hp}}^{\ell , p, m}\) is the conforming hp projector in patch Rm, obtained by reflection from the one defined in (0, 1)3, see (55). Recall also that the functions ψm are the reflections from (0, 1)3 to Rm. For γ > 3/2, given the finite element space on \(R = \bigcup _{m} R^{m}\),
we define the global projector
Then, by the same arguments as in Lemma 9 applied to all interpatch interface, there holds the following result.
Corollary 1
The operator \({\Pi }_{\mathsf {hp}}^{\ell , p, R}\) defined in (54) is conforming in H1(R). Furthermore, if γ ≥γ0 > 3/2 and \({\mathfrak {d}} \geq 1\), then for all \(u\in \mathcal {J}^{\varpi }_{\gamma }(R; C, A, {\mathfrak {d}})\) exist constants \(C^{R}_{\mathsf {hp}}\), \(b^{R}_{\mathsf {hp}}\) (that depend on C, A, and \({\mathfrak {d}}\)) such that, for every \(\ell \in \mathbb {N}\) there holds, with \(p \geq {c_{0}^{R}} \ell ^{{\mathfrak {d}}}\) for some \({c_{0}^{R}}>0\) independent of ℓ, the error bound
Furthermore, \(\dim (X_{\mathsf {hp}}^{\ell , p})\simeq \ell ^{3{\mathfrak {d}}+1}\).
Appendix B. Extension of rank bounds to domains with internal singularity
As a corollary to Theorem 1, we show here how the result can be generalized to functions that have the singularity in an internal point of the domain. As an example, we will consider the case of the axiparallel cube R = (− 1, 1)3 and of functions in the weighted analytic class \(\mathcal {J}^{\varpi }_{\gamma }(R;C,A,{\mathfrak {d}})\) with singularity at the origin. The cube R can be decomposed into eight congruent cubes, all with the singularity situated at one corner, that we will denote by Rm, \(m=0, \dots , 7\). For each m, there exist \((a_{1}, a_{2}, a_{3})\in \{-1, 1\}^{3}\) such that Rm = (0, a1),×(0, a2) × (0, a3). We do not need to specify any particular ordering, but choose, without loss of generality R0 = Q. We will denote ψm : Q →Rm the linear transformation from Q = R0 to Rm such that for all (x1, x2, x3) ∈Q
i.e., ψm only operates reflections with respect to interpatch interfaces. Note that ψ0 is the identity.
Furthermore, we define by \({\mathscr{A}}^{\ell .m}\) the analysis operator (see Section 3.1.4) of patch Rm, such that
1.1 B.1 Quasi interpolation on R
We can then define the local hp projection and interpolation operators in the patches Rm, \(m=0, \dots , 7\), in the same way, i.e., as
in each Rm. The definition of the local quasi interpolation operators also follows directly, by setting \({\mathfrak {P}}^{\ell , m} = \mathcal {I}^{\ell , m}{\Pi }_{\mathsf {hp}}^{\ell , p, m}\), for \(m = 0, \dots , 7\). Then, the global (on R) quasi interpolation operator is the operator \({\mathfrak {P}}^{\ell , R}\) such that \({\mathfrak {P}}^{\ell , R}_{|_{R^{m}}} = {\mathfrak {P}}^{\ell , m}\) for all \(m =0, \dots , 7\).
1.2 B.2 Patchwise QTT formats
It is now straightforward to consider the “patchwise QTT” formats which are constructed by adding a patch index to the formats considered so far. For a function \(u\in \mathcal {J}^{\infty }_{\gamma }(R)\), we consider the tensor \(A \in \mathbb {R}^{8\times 2^{\ell }\times 2^{\ell }\times 2^{\ell }}\) such that for \(m=0, \dots , 7\)
Then, writing with the usual notation \(i=\overline {i_{1}{\dots } i_{\ell }}\), \(j=\overline {j_{1}{\dots } j_{\ell }}\) and \(k=\overline {k_{1}{\dots } k_{\ell }}\),
-
A admits a patchwise, classic QTT decomposition if
$$ A_{m,i,j,k} = U^{1}_{m, :}(i_{1}) {\cdots} U^{\ell}(i_{\ell})V^{1}(j_{1}) {\cdots} V^{\ell}(j_{\ell})W^{1}(k_{1}) {\cdots} W^{\ell}(k_{\ell}) $$for all \(m=0, \dots , 7\), \((i,j,k)\in \{0, \dots , 2^{\ell }-1\}^{3}\) and where \(U^{1}_{m, :}(i_{1})\) indicates the m th row of U1(i1) with cores defined as in (14) and the following convention on ranks
$$ r_{0} := 8, \qquad r_{3\ell} := 1. $$Note that the only modification with respect to Definition 3 is the convention r0 = 8.
-
A admits a patchwise, transposed order QTT decomposition if
$$ A_{m,i,j,k} = U^{1}_{m, :}(\overline{i_{1}j_{1}k_{1}}) {\cdots} U^{\ell}(\overline{i_{\ell} j_{\ell} k_{\ell} }) $$(56)with cores as in Definition 4 and with the restriction on the ranks r0 = 8, rℓ = 1.
-
A admits a patchwise, Tucker QTT decomposition if
$$ \begin{aligned} A_{m,i,j,k} = \sum\limits_{\beta_{1}, \beta_{2}, \beta_{3} = 1}^{R_{1}, R_{2}, R_{3}} &G_{\beta_{1}, \beta_{2}, \beta_{3}}^{m} U^{1}_{{\beta_{1}}}(i_{1}) U^{2}(i_{2}) \ldots U^{\ell}(i_{\ell}) \\ V^{1}_{{\beta_{2}}}(j_{1}) &V^{2}(j_{2}){\ldots} V^{\ell}(j_{\ell}) W^{1}_{{\beta_{3}}}(k_{1}) W^{2}(k_{2}){\ldots} W^{\ell}(k_{\ell}). \end{aligned} $$(57)where, clearly, the Tucker core is now a four-dimensional array of size 8 ×R1 ×R2 ×R3.
Let \(\mathcal {T}^{\ell , R}\) be the tensor product mesh on R given by
We define the finite element space in R as
The following proposition is then a direct consequence of Theorem 1 and of Corollary 1.
Proposition 3
Assume γ > 3/2, Cu > 0, Au > 0, \({\mathfrak {d}}\geq 1\), and 0 < ε0 ≪ 1. Furthermore, assume the function u belongs to the weighted Gevrey class \(u\in \mathcal {J}^{\varpi }_{\gamma }(R; C_{u}, A_{u}, \gamma , {\mathfrak {d}})\cap {H^{1}_{0}}(R)\). Then, for all 0 < ε ≤ε0, there exists \(\ell \in \mathbb {N}\) and \({v_{\mathsf {qtd}}^{\ell }} \in X^{\ell ,R}\) such that
and the multi-dimensional array \(V_{\mathsf {qtd}}^{\ell }\in \mathbb {R}^{8\times 2^{\ell }\times 2^{\ell }\times 2^{\ell }}\) such that \((V_{{\mathsf {qtd}}}^{\ell })_{m, :,:,:} = {\mathscr{A}}^{\ell ,m}v_{\mathsf {qtt}}^{\ell }\), \(m=0,\dots , 7\) admits a patchwise representation with
degrees of freedom, with a positive constant C independent of ε and
Proof
Here, we retrace the steps of the proofs of Lemmas 2, 3, and 5, generalizing them to the multipatch case.
- Patchwise classic QTT :
-
The tensor \(V_{{\mathsf {qtt}}}^{\ell }\) can be written as the product
$$ (V_{\mathsf{qtd}}^{\ell})_{m, i, j, k} = U^{0}(m)U^{1}(i_{1}){\cdots} U^{\ell}(i_{\ell})V^{1}(j_{1}){\cdots} V^{\ell}(j_{\ell})W^{1}(k_{1}){\cdots} W^{\ell}(k_{\ell}), $$where the bounds on the ranks of the cores \(U^{1}, \dots , U^{\ell }\) and the first rank of the core V1 are multiplied by 8, while the other bounds are left unchanged with respect to the single patch case. The multiplication of the cores U0 and U1 gives the multipatch formulation.
- Patchwise transposed order QTT :
-
The row space of the unfolding matrices
$$ V^{(q)}_{\overline{m \xi_{1}\eta_{1}\zeta_{1}}, \overline{\xi_{2}\eta_{2}\zeta_{2}}} = (V_{\mathsf{qtt}}^{\ell})_{m, \overline{\xi_{1}\xi_{2}}, \overline{\eta_{1}\eta_{2}}, \overline{\zeta_{1}\zeta_{2}}} $$defined for \(m\in {0, \dots , 7}\), \(\xi _{1}, \eta _{1}, \zeta _{1} \in \{0, \dots , 2^{q}-1\}\), and \(\xi _{2},\eta _{2}, \zeta _{2}\in \{0, \dots , 2^{\ell -q}-1\}\) is bounded asymptotically by the same quantity as the one of the unfolding matrix in (30), by symmetry. Thus, \(V_{\mathsf {qtd}}^{\ell }\) admits a decomposition such that
$$ (V_{\mathsf{qtd}}^{\ell})_{m, i, j, k} = U^{0} (m) U^{1}(\overline{i_{1}j_{1}k_{1}}){\cdots} U^{\ell}(\overline{i_{\ell} j_{\ell} k_{\ell}}). $$By multiplying U0(m) and \( U_{1}(\overline {i_{1}j_{1}k_{1}})\) for all \(m=0, \dots , 7\) and i1, j1, k1 ∈{0, 1}, we obtain a representation of the form (56).
- Patchwise Tucker QTT :
-
We Tucker-decompose the tensor \(V_{{\mathsf {qtd}}}^{\ell }\), thus obtaining, by the same arguments that we used for (32),
$$ V_{\mathsf{qtd}}^{\ell} = \sum\limits_{\beta_{1} ,\beta_{2}, \beta_{3} = 1}^{R_{T}} \sum\limits_{\beta_{0}=1}^{R_{P}} G_{\beta_{0}, \beta_{1}, \beta_{2}, \beta_{3}} Z_{\beta_{0}} \otimes U_{\beta_{1}} \otimes V_{\beta_{2}}\otimes W_{\beta_{3}}, $$where \(R_{T}\leq C \ell ^{{\mathfrak {d}}+1}\). Then, by contracting the core G and the factor Z over the index β0 and by deriving the existence of the block QTT decomposition of the Tucker factors U, V, W as in (33), we obtain the existence of a representation of Vqtdℓ of the form (57).
□
Remark 3
In Proposition 5, we consider the approximation of functions in the cube R = (− 1, 1) for ease of notation. Nonetheless, the argument and the result extend, without major modification, to \(\widetilde {R} = (-a_{1}, b_{1}) \times (-a_{2}, b_{2}) \times (-a_{3}, b_{3})\), with ai, bi > 0, i = 1, 2, 3, and with a point singularity at the origin. This implies, by translation, that given a cube of fixed size, we can obtain bounds on patchwise quantized tensor representations that are uniform in the location of the singularity.
Appendix C. QTT representation of prolongation matrices
In order to evaluate the error εℓ, we need a tensor of \({\nabla }u_{\mathsf {qtd}}^{\ell ,\delta }\) evaluated on the background mesh with L levels of refinement and have it represented using the respective tensor decomposition without accessing all its elements. This can be implemented as a multiplication by the prolongation matrices in the respective tensor format. To introduce the prolongation matrices, we start by considering the one-dimensional piecewise linear space on the background mesh with ℓ levels (recall that \(I^{\ell }_{j} = (2^{-\ell }j, 2^{-\ell }(j + 1))\))
Furthermore, we introduce the one-dimensional analysis operator \({\mathscr{A}}^{\ell }_{\mathrm {1d}} : H^{1}((0,1))\to \mathbb {R}^{2^{\ell }}\) as
Then, for every L > ℓ, the one-dimensional prolongation operator \(P^{(\ell \to L)}_{\mathrm {p.l.}} \in \mathbb {R}^{2^{L}\times 2^{\ell }}\) is realized by the matrix such that
In the same vein, the one-dimensional prolongation operator for piecewise constant function is such that
for all
Recall that we consider functions u such that \(u_{|_{{\varGamma }}}=0\), where Γ = ∂Q∖{x = (x1, x2, x3) ∈∂Q : x1x2x3 = 0}. In this case, the three-dimensional prolongation matrices from mesh level ℓ to L > ℓ, can be written as a tensor product of the one-dimensional piecewise linear and piecewise constant prolongation matrices, which read:
and
respectively, where we used the notation
The matrix \(P_{\mathrm {p.c.}}^{(\ell \to L)}\) can be represented with QTT ranks \(1,1,\dots , 1\), as it has Kronecker product structure, since \(I^{(\ell )} = \left (I^{(1)} \right )^{\otimes \ell }\) and
We now show, in Proposition 6 below, that \(P_{\mathrm {p.l.}}^{(\ell \to L)}\) also has low-rank QTT structure. For convenience, we introduce the matricization operator \({\mathscr{M}}: \mathbb {R}^{r_{1}\times m \times n \times r_{2}} \to \mathbb {R}^{m r_{1}\times n r_{2}}\) such that:
that allows to recast tensor cores as matrices. The following proposition holds.
Proposition 4
The matrix \(P_{\mathrm {p.l.}}^{(\ell \to L)}\), L > ℓ, defined in (59) has explicit QTT representation with ranks \(2,2,\dots ,2\). In particular, for each \(i_{1},\dots ,i_{L} \in \{0,1\}\), \(j_{1},\dots ,j_{\ell } \in \{0,1\}\) and \(j_{\ell +1},\dots ,j_{L} \in \{0\}\):
where the matricizations read
with blocks given by
Proof
First, we introduce the notation
so that
Since I(ℓ) = I ⊗I(ℓ− 1) and \(J^{(\ell )} = I \otimes J^{(\ell -1)} + J \otimes \left (J^{\top } \right )^{\otimes (\ell -1)}\) and using the operation ⋊⋉ that denotes the strong Kronecker product between block matrices, in which matrix-matrix multiplication of blocks is replaced by a Kronecker productFootnote 1 , we obtain
We complete the proof by the observations that
□
Corollary 2
Let \(v_{\mathsf {qtt}}^{\ell } \in X_{\mathrm {1d}, 0}^{\ell }\) and let \({\mathscr{A}}^{\ell }_{\mathrm {1d}} v_{\mathsf {qtt}}^{\ell }\) have QTT ranks \(r_{1},r_{2},\dots ,r_{\ell -1}\). Then, for any L > ℓ, the vector \({\mathscr{A}}^{L}_{\mathrm {1d}}v_{\mathsf {qtt}}^{\ell } = \{v_{\mathsf {qtt}}^{\ell }(x_{i})\}_{i=0}^{2^{L} -1}\), xi = 2−Li can be represented with QTT ranks equal to \(2r_{1},2r_{2},\dots ,2r_{\ell -1}\).
Proof
According to Proposition 6, the matrix \(P_{\mathrm {p.l.}}^{(\ell \to L)}\) has ranks \(2,2,\dots ,2\). The statement then follows from the fact that the multiplication in (58) of a TT-matrix with ranks \(R_{1},\dots ,R_{\ell -1}\) by a TT-vector with the ranks \(r_{1},\dots ,r_{\ell -1}\), leads to the TT representation with ranks \(R_{1}r_{1},\dots ,R_{\ell -1}r_{\ell -1}\), see [65]. □
The multidimensional prolongation matrices are assembled as Kronecker products of the one-dimensional matrices \(P_{\mathrm {p.l.}}^{(\ell \to L)}\) and/or \(P_{\mathrm {p.c.}}^{(\ell \to L)}\). For example, to find the values of \(v_{\mathsf {qtt}}^{\ell }\in X^{\ell }\) on a mesh with L levels, the matrix
represented in the respective format is applied to the coefficient vector \({\mathscr{A}}^{\ell } v_{\mathsf {qtt}}^{\ell }\).
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Marcati, C., Rakhuba, M. & Schwab, C. Tensor rank bounds for point singularities in ℝ3. Adv Comput Math 48, 18 (2022). https://doi.org/10.1007/s10444-022-09925-7
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DOI: https://doi.org/10.1007/s10444-022-09925-7
Keywords
- Quantized tensor train
- Tensor networks
- Low-rank approximation
- Exponential convergence
- Schrödinger equation