Iterative Schemes for Eigenvalue Problems

We next consider an adaptive method for elliptic eigenvalue problems based on the method intro-duced in [36]. This scheme is applicable to self-adjoint operators; we shall comment on the case of eigenvalue problems for nonsymmetric operators in Remark 3.28 below.

For this subsection, let M be an infinite diagonal matrix defining a bounded operator on `<sub>2</sub>, and let A be symmetric as well as bounded and elliptic on `₂. Note that these assumptions correspond to the observations made in Section 3.5. Furthermore, let

λ(v) := hAv, vi hMv, vi and

λ₀ = inf

v6=0λ(v) ,

where we assume that V₀ := ker(A− λ0M) is one-dimensional and A− λ0Mis elliptic on V₀^⊥. Let u0be such that V0 = span{u0} and ku0k = 1, and let P0 be the orthogonal projector onto V0, that is, P₀ =h·, u0iu0. For v∈ `2, we define

e⊥(v) := (I− P0)(v− u0) ,

which means that if kvk = 1, then ke^⊥(v)k equals the sine of the angle between V0 and v. This provides an adequate measure for the error in the eigenfunction. As part (ii) of Lemma 3.23 below shows, this quantity controls the error in the eigenvalue as well.

For finding approximations to λ₀ and to a normalized element of V₀, we consider a basic Richardson-type method,

ˆ

v_i+1:= v_i− α Avi− λ(vi)Mv_i
, v_i+1:= ˆv_i+1

kˆv_i+1k, (3.39) which amounts to a gradient descent scheme for the Rayleigh quotient, and can also be regarded as a special case of preconditioned inverse iteration. A convergence analysis has been given in [36]. Based on the same arguments, in what follows we obtain a slightly modified analysis with a different choice of iteration parameters.

Lemma 3.23 summarizes some prerequisites for the convergence analysis, following closely the treatment in [36].

Lemma 3.23. Under the above assumptions on A and M, where we set R₀ := A− λ0M, the following hold:

(i) There exists an α > 0 such that for Tα := I− αR0 we have k(I − P0)T_αk =: ρ < 1 . (ii) There exists E > 0 such that for any v∈ `2, we have

λ(v)− λ0 ≤ Eλ(v)ke^⊥(v)k². In particular, if ke^⊥(v)k ≤ E^−1/2, then

λ(v)≤ 1 − Eke^⊥(v)k²
−1

λ₀.

3.6 Adaptive Solvers

(iii) Let C₀ :=kR⁻¹0 kV₀^⊥→V₀^⊥ andC₁ := _2(1+C^C0kMk

0kMk). Provided that ke⊥(v)k < minE^−1/2, (1 + C₁²)^1/2− C1 , with

R(v) := C₀



1−C₀kMkE(1 + ke^⊥(v)k)ke^⊥(v)k 1− Eke⊥(v)k²

−1

(3.40) we have

ke^⊥(v)k ≤ R(v) k(A − λ(v)M)vk .

Proof. See [36], Lemmas 4, 5, and 6; the constants involved in the estimates as given here can be extracted from the corresponding proofs.

The scheme is given in Algorithm 3.2, where we assume the availability of a routine apply as before, and additionally a procedure rayleigh such that for v6= 0, we have

|rayleigh(v; η) − λ(v)| ≤ η .

Note that since M is assumed to be diagonal, Mv can be evaluated exactly for any finitely supported v.

Algorithm 3.2 u_ε= evpsolve(A, v0; ε)

input α, ρ, E, C₁ as in Lemma 3.23, and R as in (3.40); v₀ withkv0k = 1 such that ke⊥(v₀)k ≤ δ with δ as in (3.41); θ, κ∈ (0, 1).

output u_ε withkuεk = 1 and ke^⊥(u_ε)k ≤ ε.

1: i := 0, ε₀ := δ

2: while ε_i > ε

3: w₀ ← vi

4: ξi := ρ + αkMk(1 − Eε²_i)⁻¹Eεi 5: η˜_i := α⁻¹(1 + ε_i)⁻¹ε_i(1− ξi)

6: ε_i+1:= θε_i

7: K_i:= mink : Q^k−1_l=0(1− ξ_i^l+1α˜η_i)
−1

ξ_i^k(ε_i+ kα˜η_i)≤ κεi+1

8: j← 0

9: repeat

10: η_j ← ξ_i^j+1η˜_i

11: r_j ← apply(wj;¹₂η_j)− rayleigh(wj;¹₂kMk⁻¹η_j) Mw_j

12: wˆ_j+1← wj− αrj 13: w_j+1← k ˆw_j+1k⁻¹wˆ_j+1

14: j ← j + 1

15: until(j ≥ Ki ∨ (1− αηj−1)⁻¹(ξ_iR(w_j−1)krj−1k + (α + ξiR(w_j−1))η_j−1)≤ κεi+1)

16: τ_i+1:= (1 + ε²_i+1)⁻¹

κ²+ (1− κ²)(1 + ε²_i+1)
1/2

− κ

17: ˆv_i+1:= coarsen(w_j; τ_i+1ε_i+1)

18: vi+1:=kˆvi+1k⁻¹ˆvi+1 19: i← i + 1

20: end while

21: u_ε:= vi

Proposition 3.24. Let α, ρ, E, C₁ be chosen as in Lemma 3.23, and let v₀ ∈ `2 with kv0k = 1 such thatke⊥(v0)k ≤ δ, where with C2:= ¹₂(1− ρ)⁻¹kMk we have

0 < δ < min(1 + C₁²)^1/2− C1, (E⁻¹+ (C₂α)²)^1/2− (C2α) . (3.41)

3 Higher-Dimensional Approximation and Adaptive Wavelet Methods

Then the iterates v_i in Algorithm 3.2 satisfy ke^⊥(v_i)k ≤ θⁱδ with kvik = 1, and in particular, we have ke⊥(u_ε)k ≤ ε with kuεk = 1.

Proof. Note first that the assumption on δ implies that δ < E^−1/2, and hence the hypothesis of part (iii) of Lemma 3.23 is satisfied.

We consider the first outer iteration, i.e., i = 0. With T_α as in Lemma 3.23, for any j we have ˆ

Using the estimate (3.42) recursively, on the one hand we obtain

ke^⊥(w_j)k ≤ ξ^j₀ε₀

and thus ke^⊥(w_j)k ≤ κε1 holds as well if the inner loop terminates with the second criterion in line 15.

It remains to show that there exists a uniform bound for K_i, which ensures that the inner loop finishes after a finite, uniformly bounded number of steps. To this end, note that

ln

3.6 Adaptive Solvers

the infinite sum on the right hand side converges, which implies

j−1

Y

l=0

(1− αηl)⁻¹ <

∞

Y

l=0

(1− ξ0^lα˜η₀)⁻¹≤ C < ∞ ,

and hence by monotonicity of ξ_iand ˜η_iwe obtain that for given problem parameters, K_iis uniformly bounded with respect to i.

By definition of ˆv₁, we havekwj−ˆv₁k ≤ τ1ε₁as well askˆv₁k² ≥ 1−(τ1ε₁)², and as a consequence ke⊥(ˆv₁)k ≤ ke⊥(w_j)k + k(I − P0)(w_j− ˆv₁)k = κε1+ τ₁ε₁.

For the normalized iterate v1, we thus obtain ke^⊥(v₁)k = ke^⊥(ˆv₁)k

kˆv₁k ≤ κ + τ₁

p1 − (τ1ε1)² ε₁, and with our choice of τ₁, it follows that ke⊥(v₁)k ≤ ε1.

As the above steps can be repeated for general i, the statement follows by induction.

In [36], a complexity estimate similar to Theorem 3.21 is shown for the eigenvalue solver. The modifications we have made to the constants in the iterative scheme do not affect this result, which is summarized in the following theorem.

Theorem 3.25. Let s > 0, u₀∈ A^s∞, let w_η := apply(v; η) satisfy kwηk^As∞ .kvk^As∞, # supp wη . η^−1skvk

1 s

A^s_∞,

where the order of arithmetic operations required for the evaluation of w_η and of rayleigh(v; η) are both of order O(η^−1skvk

1 s

A^s_∞ + # supp v), and let coarsen be as in Theorem 3.21. Then Algorithm 3.2 requiresO(ε^−1sku0kA^1ss_∞) operations, and the result u_ε satisfies

# supp u_ε. ε^−1sku0kA^1ss_∞, kuεkA^s_∞ .ku0kA^s_∞

with constants independent ofε and u0.

Proof. The result follows with the same arguments as in the proof of [36, Theorem 3].

Remark 3.26. On the basis of a suitable routine apply, one can obtain a procedure rayleigh with the required properties by setting

rayleigh(v; η) := happly(v; hMv, viη), vi

hMv, vi .

In this manner, the evaluation of apply required in each step of the iteration can also be used for the approximation of the Rayleigh quotient. However, different constructions of a procedure rayleigh that lead to better complexity are possible, see [36, Section 4.3.1].

Remark 3.27. Further results along these lines for preconditioned inverse iteration with more general preconditioners have been obtained in [127]. In [156] it has been shown that inexact inverse iteration, which in the present setting can be realized by an iteration of the form

ˆ

v_i+1:= solve(A, v_i, ε_i) , v_i+1:= ˆv_i+1 kˆv_i+1k,

3 Higher-Dimensional Approximation and Adaptive Wavelet Methods

shares the optimality properties as in Theorem 3.25 of the scheme considered above, provided that the tolerances ε_i are chosen appropriately.

Remark 3.28. The convergence analysis for the eigenvalue solvers mentioned thus far has been carried out in the case of self-adjoint operators. For nonsymmetric problems, as in the case of the explicitly correlated formulation (2.11) of the electronic Schr¨odinger equation, preconditioned inverse iteration can generally not be expected to converge, but inexact inverse iteration as outlined in Remark 3.27 may serve as the basis of an adaptive solver.