Constructive tractability of the Helmholtz problem Arthur G. - - PowerPoint PPT Presentation

Constructive tractability

• f the Helmholtz problem

Arthur G. Werschulz Henryk Wo´ zniakowski

Fordham University Department of Computer and Information Sciences Columbia University Department of Computer Science University of Warsaw Department of Mathematics

ICERM HDA-IBC Providence, RI 15 September 2014

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Helmholtz problem

For f ∈ Fd, q ∈ Qd, find an approximation of the solution u = uf ,q to Lqu := −∆u + qu = f in I d := (0, 1)d, with either Dirichlet u = g

• n ∂I d
• r Neumann

∂νu = g

• n ∂I d

boundary conditions. Actually consider variational form of this problem (H1(I d)-error). Here, d can be huge!!!

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Our previous work

◮ “Tractability of quasilinear problems I: general results”,

• J. Approx. Theory, 2007.

◮ “Tractability of quasilinear problems II: second-order elliptic

problems”, Math. Comp., 2007.

◮ “Tractability of multivariate approximation over a weighted

unanchored Sobolev space”, Constr. Approx., 2009.

◮ “Tractability of the Helmholtz equation with

non-homogeneous Neumann boundary conditions: relation to L2-approximation”, J. Comp., 2009 (only W).

◮ “Tight tractability results for a model second-order Neumann

problem”, J. FoCM., 2014.

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What we really want

◮ General q. ◮ General Dirichlet or Neumann boundary conditions. ◮ Wide range of weighted spaces. ◮ Necessary and sufficient weight conditions for various flavors

• f tractability.

◮ Explicit optimal tractability algorithms.

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What you’re getting today

◮ General q, but sometimes specializing to q ≡ 1. ◮ Homogeneous Neumann boundary conditions. ◮ Variety of Fd and Qd. ◮ Tractability results, but sometimes optimal tractability

algorithms.

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Problem definition

◮ Let

Fd = unit ball of Hd,γ = H(Kd,γ), Qd = { q ∈ Fd : q(·) ≥ q0 } where q0 ∈ (0, 1).

◮ Let

Bd(v, w; q) =

• I d[∇v·∇w+qvw]

∀ v, w ∈ H1(I d), q ∈ Qd.

◮ Seek u = Sd(f , q) ∈ H1(I d):

Bd(u, w; q) = f , wL2(I d) ∀ w ∈ H1(I d), for (f , q) ∈ Fd × Qd, i.e., the variational solution of −∆u + qu = f in I d, ∂νu = 0

• n ∂I d.

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Quasilinearity

◮ Sd is quasilinear:

◮ Linear in first argument. ◮ Lipschitz in both arguments.

◮ See WW, 2007a.

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“Usual IBC stuff”

◮ Continuous linear information. ◮ Error of algorithm An using at most n info evals:

e(An, Sd) = sup

[f ,q]∈Fd×Qd

Sd(f , q) − An(f , q)H1(I d)

◮ nth minimal error:

e(n, Sd) = inf

An e(An, Sd) ◮ Info complexity:

n(ε, Sd) = inf{ n ∈ N0 : e(n, Sd) ≤ CRId ε } ABS : CRId ≡ 1 NOR : CRId = e(0, Sd)

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Reduction to approximation problem

◮ Easy lower bound: Sd APPHd,γ →[H1(I d)]∗ ◮ Known upper bound:

◮ Sd(·, q) is [H1(I d)]∗-Lipschitz ◮ Sd(f , ·) is L2(I d)-Lipschitz ◮ Sd APPHd,γ→L2(I d)

◮ New upper bound: Sd is [H1(I d)]∗-Lipschitz . . . vile constants. ◮ Interpolatory algorithm An(f , q) = Sd(˜

f , ˜ q)

◮ yields sharp info complexity bounds ◮ no joy implementation-wise when q ≡ const

◮ How to find easily-implementable “good” algorithms?

We are currently investigating Galerkin algorithms, but we are not yet done . . .

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Tractability results for arbitrary q ∈ Qd [WW 2007]

◮ Hd,γ = H(Kd,γ), where

Kd,γ(x, y) =

• u⊆{1,2,...,d}

|u|≤ω

γd,u

• j∈u

K(xj, yj), with

• [0,1]2 K(x, y) dx dy ∈ (0, ∞)

◮ Finite-order weights

γd,u = 0 for all |u| > ω

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Tractability results for arbitrary q ∈ Qd [WW 2007]

◮ ABS + FOW =

⇒ PT: n(ε, Sd) ≤ Cε−2d 2ω

◮ ABS + FOW + supd

• |u|≤ω γd,u < ∞ =

⇒ SPT: n(ε, Sd) ≤ Cε−2

◮ NOR + FOW =

⇒ PT: n(ε, Sd) ≤ Cε−2d ω

◮ NOR + FOW + supd

• |u|≤ω γd,u < ∞ =

⇒ SPT: n(ε, Sd) ≤ Cε−2

◮ Also have results for Λstd, as well as for Dirichlet problems. ◮ Based on Wasilkowski+W, 2004.

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Tractability results for q ≡ 1 [WW 2014]

◮ Now Sd is linear! ◮ Here, Fd is unit ball of Hd,γ.

◮ Choose general weights (not necessarily FOW)

γ = { γd,u ≥ 0 : u ⊆ [d] := {1, 2, . . . , d}, d ∈ N } with γd,∅ = 1.

◮ Then

Hd,γ = { w ∈ [H1(I)]⊗d : ∂uw ≡ 0 whenever γd,u = 0 }, with ∂u =

j∈u ∂j, where

v, wHd,γ =

• u⊆[d]

γ−1

d,u ∂uv, ∂uwL2(I d)

∀ v, w ∈ Hd,γ.

◮ ABS=NOR, since e(0, Sd) = 1.

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Tractability results for q ≡ 1 [WW 2014]

Everything depends on eigenpairs (βk,γ, ek) of S∗

dSd for k ∈ Nd: ◮ Eigenvalues:

βk,γ = 1 1 + π2 d

j=1(kj − 1)2 ·

1 1 +

∅=u⊆[d] γ−1 d,u

• j∈u[π2(kj − 1)2]

◮ Eigenvectors:

ek(x) =

d

• j=1

cos[π(kj − 1)xi]

◮ Optimal algorithm:

Let n = |M(ε, d, γ)|, where M(ε, d, γ) = { k ∈ Nd : βk,γ > ε2 }. Then An(f , 1) =

• k∈M(ε,d,γ)

f , ekHd,γ ek2

Hd,γ

Sdek

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Tractability results for q ≡ 1 [WW 2014]

◮ Relation to L2-approximation: Let APPd,γ = APPHd,γ →L2(I d)

and Md = max

j=1,2,...,d γd,{j} < ∞

and cd = min{1, M−1

d }.

Then n(√εc−1/4

d

, APPd,γ) ≤ n(ε, Sd) ≤ n(ε, APPd,γ).

◮ APPd,γ was studied in WW, 2009.

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Tractability for q ≡ 1 + product weights [WW 2014]

◮ Let 1 ≥ γ1 ≥ γ2 ≥ · · · > 0. Then γd,u = j∈u γj. ◮ Quasi-polynomial tractability

n(ε, Sd) ≤ C exp

• t(1 + ln ε−1)(1 + ln d)
• ,

holds for all product weights, with t = 2 ln(1 + π2) . = 0.838233

◮ Polynomial tractability holds iff ∃ τ > 1 2:

Bτ = ζ(2τ) π2τ lim sup

d→∞

1 ln d

d

• j=1

γτ

j < ∞,

in which case n(ε, Sd) ≤ CτdBτ ε−2τ.

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Tractability for q ≡ 1 + product weights [WW 2014]

◮ Strong polynomial tractability iff ∃ τ > 1 2:

Aτ := sup

d∈N d

• j=1

γτ

d,j < ∞,

When this holds, let τ ∗ = inf{ τ > 1

2 : Aτ < ∞ }.

Then for all τ > τ ∗, we have n(ε, Sd) ≤ ε−2τ exp

• ζ(2τ) π−2τ Aτ
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Future work

◮ Study general q. ◮ We believe that results for q ∈ Qd will be analogous to those

for q ≡ 1, but work is still in progress.

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A new approximation problem

◮ For s ∈ N0, let

v2

Hs(Hd,γ ) =

• u⊆[d]

γ−1

d,u∂uv2 Hs(I d). ◮ APPr,s: Approximate Hs(Hd,γ)-functions in the Hr(I d)-norm,

where r, s ∈ N0 and r ≤ s.

◮ Then Galerkin error is bounded by e(APP1,2), modulo a

constant, with still-unknown dependence on d.

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Spectral results for APPr,s

◮ Let Wr,s = APP∗ r,s APPr,s. Then e(n, APPr,s) =

• βn+1,r,s,

where β1,r,s ≥ β2,r,s ≥ · · · > 0 are eigenvalues of Wr,s.

◮ For k ∈ Nd, let ek(x) = d j=1 cos[π(kj − 1)xj]. Then

Wr,sek = βr,s,γ,kek ∀ k ∈ Nd, where βr,s,γ,k = ζr,k ζs,k αk,γ, ζs,k =

• 0≤|m|≤s

d

• j=1

[π(kj − 1)]2mj, αk,γ =

u⊆[d]

γ−1

d,u

• j∈u

[π(kj − 1)]2 −1 .

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