ADAPTIVE METHODS FOR SOLVING OPERATOR EQUATIONS BY USING FRAMES OF SUBSPACES
H. JAMALI1, KH. SHOKRI TERNONIZ1, §
Abstract. In this paper, using a frame of subspaces we transform an operator equa-tion to an equivalent `2-problem. Then, we propose an adaptive algorithm to solve the
problem and investigate the optimality and complexity properties of the algorithm.
Keywords: Hilbert space, dual space, frame of subspaces, bestN-term approximation, adaptive algorithm.
AMS Subject Classification: 65F10, 65F08
1. Introduction and preliminaries
The aim of the paper is to study the application of frames of subspaces in designing adaptive iterative methods for solving operator equations. Usually these operators are defined on a bounded domain or a closed manifold where a wavelet basis with specific properties is needed to be constructed. Most importantly, during the approach some serious drawbacks such as stability may not be avoided. Therefore, it is suggested to use a slightly weaker concept, namely frame. In [7, 8, 5, 6] some adaptive numerical methods for elliptic operator equations have been developed by using wavelet bases and frames. One of the advantages of frame of subspaces is that they facilitate the construction of frames for special applications and meanwhile it is easier to construct or choose already known frames for smaller spaces.
The main focus of the paper is to findu∈H such that
Lu=f, (1)
whereL:H→H is a bounded, invertible and self adjoint linear operator on a separable Hilbert space H. In general, it is impossible to find the exact solution of the problem (1), because the separable Hilbert space H is infinite dimensional. A natural approach to construct an approximate solution is to solve a finite dimensional counterpart of the problem (1). First, we briefly recall the definitions and basic properties of frames and frames of subspaces.
Assume that H is a separable Hilbert space, Λ is a countable set of indices and Ψ = (ψλ)λ∈Λ⊂H is a frame forH. This means that there exist constants 0< AΨ≤BΨ<∞
such that
AΨkfk2H ≤
X
λ∈Λ
|hf, ψλi|2 ≤BΨkfk2H, ∀f ∈H. (2)
1 Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran.
e-mail: [email protected], [email protected];
§ Manuscript received: May 09, 2016; accepted: September 22, 2016.
TWMS Journal of Applied and Engineering Mathematics, Vol.7, No.1; cI¸sık University, Department of Mathematics 2017; all rights reserved.
For the frame Ψ, the frame operator S : H → H is defined by S(f) = P
λ∈Λhf, ψλiψλ. It was shown thatS is a positive definite and invertible operator satisfyingAΨIH ≤S ≤
BΨIH. Also, the sequence Ψ = (e ψeλ)λ∈Λ = (S−1ψλ)λ∈Λ is a frame (called the canonical
dual frame) forH with boundsBΨ−1, A−1Ψ and everyf ∈H has the expansion
f =X λ∈Λ
hf, ψλiψeλ = X
λ∈Λ
hf,ψeλiψλ. (3)
For an index setΛe ⊂Λ, (ψλ)λ∈
e
Λ is called a frame sequence, if it is a frame for its closed
span. For more details see [1, 3].
For an index set Λ and a family of weights {vλ}λ∈Λ, i.e.,vλ >0 for all λ∈Λ, a family of subspaces{Hλ}λ∈Λ of a Hilbert spaceH is called a frame of subspaces with respect to
{vλ}λ∈Λ forH, if there exist constants 0< A≤B <∞ such that
Akfk2 ≤X λ∈Λ
vλ2kπHλ(f)k
2 ≤Bkfk2 ∀f ∈H, (4)
whereπHλ denotes the orthogonal projection onto the subspace Hλ.
The constants A and B is called the frame bounds of the frame of subspaces. If A =B
then the frame{Hλ}λ∈Λ with respect to{vλ}λ∈Λ, is called a A -tight frame of subspaces.
It is clear, the family {Hλ}λ∈Λ of the frame of subspaces is complete, in the sense that spanλ∈Λ{Hλ}=H.
The following theorem [2], shows how we are able to string together frames for each of the subspacesHλ to get a frame for H.
Theorem 1.1. Let Λ be an index set, vλ >0 for each λ∈Λ, and {ψλi}i∈IΛ be a frame sequence in H with frame bounds Aλ and Bλ. Define Hλ =spani∈IΛ{ψλi} for all λ∈Λ, and suppose that 0 < A = infλ∈ΛAλ ≤ B = supλ∈ΛBλ < ∞. Then {vλψλi}λ∈Λ,i∈IΛ is a frame forHif and only if{Hλ}λ∈Λis a frame of subspaces with respect to{vλ}λ∈ΛforH.
For a frame of subspaces {Hλ}λ∈Λ with respect to{vλ}λ∈Λ define
(X
λ∈Λ
⊕Hλ)`2 ={{ψλ}λ∈Λ|ψλ∈Hλ, X
λ∈Λ
kψλk2 <∞}
with inner product given by h{ψλ}λ∈Λ,{ϕλ}λ∈Λi = Pλ∈Λhψλ, ϕλi. Now the synthesis operatorTH,v : (Pλ∈Λ⊕Hλ)`2 → H for the frame of subspace {Hλ}λ∈Λ with respect to
{vλ}λ∈Λ is defined by
TH,v(f) =
X
λ∈Λ
vλfλ ∀f ={fλ}λ∈Λ∈(
X
λ∈Λ
⊕Hλ)`2.
Also, the adjoint TH,v∗ of the synthesis operator is called the analysis operator. In fact
TH,v∗ : H → (P
λ∈Λ⊕Hλ)`2 is given by T
∗
H,v(f) = {vλπHλ(f)}λ∈Λ. It is proved that the
synthesis operatorTH,vis bounded, linear and onto. [2]. Also, the analysis operatorTH,v∗ is an (possibly into) isomorphism. As in the well known frame situation, the frame operator
SH,v for{Hλ}λ∈Λ and{vλ}λ∈Λ is defined by
SH,v(f) =TH,vTH,v∗ (f) =TH,v({vλπHλ(f)}λ∈Λ) =
X
λ∈Λ
v2λπHλ(f).
the following reconstruction formula satisfies:
f =X λ∈Λ
v2λSH,v−1πHλ(f) ∀f ∈H.
It is proved that{SH,v−1Hλ}λ∈Λ is a frame of subspaces with respect to {vλ}λ∈Λ. [2].
Proposition 1.1. Let {Hλ}λ∈Λ be a frame of subspaces with respect to {vλ}λ∈Λ, and let L : H → H be a bounded invertible operator on H. Then {L(Hλ)}λ∈Λ is a frame of
subspaces with respect to{vλ}λ∈Λ.
Proof. See [2].
2. Preconditioning by using frames of subspaces
The most straight forward approach to an iterative solution of a linear system is to rewrite the equation (1) as a linear fixed-point iteration. One way to do this is the Richardson iteration. The abstract method reads as follows:
writeLu=f asu= (I−L)u+f. For givenu0 ∈H, define for n≥0,
un+1= (I−L)un+f. (5) SinceLu−f = 0,
un+1−u= (I−L)un+f −u−(f−Lu) = (I −L)un−u+Lu = (I−L)(un−u).
Hencekun+1−ukH ≤ kI−LkH→Hkun−ukH,so that (5) converges ifkI−LkH→H <1. It is sometimes possible to precondition (1) by multiplying both sides by a matrixB,
BLu=Bf,
so that convergence of iterative methods is improved. This is a very effective technique for solving differential equations, integral equations, and related problems [2, 3]. We shall apply this technique by using frames of subspaces.
Let{Hλ}λ∈Λbe a frame of subspaces with respect to{vλ}λ∈Λfor a separable Hilbert space H with the frame operator SH,v. By Proposition 1.1, {L(Hλ)}λ∈Λ also is a frame with
respect to{vλ}λ∈Λ. We denote the frame operator for {L(Hλ)}λ∈Λ and{vλ}λ∈Λ, by SH,v0
and we note thatSH,v0 f =P
λ∈Λv2λLπHλL
−1f =LP
λ∈Λv2λπHλL
−1f =LS
H,vL−1f, that means,SH,v0 =LSH,vL−1.
Also sinceL is bounded invertible then there exist two positive constants c1 and c2 such that
c1kukH ≤ kLukH ≤c2kukH, ∀u∈H. (6) Now we design an algorithm in order to approximate the solutionu of the equation (1). The convergence rate of the algorithm depends on the values of the bounds of the frames.
Theorem 2.1. Let {Hλ}λ∈Λ be a frame of subspaces with respect to{vλ}λ∈Λ for H with
frame operatorSH,v and letL be as in (1). Let u0 = 0 and for k≥1,
uk=uk−1+
2
c2
1A+c22B
LSH,v0 (f −Luk−1),
where SH,v0 is the frame operator for the frame of subspaces {L(Hλ)}λ∈Λ with respect to
{vλ}λ∈Λ with bounds A, B, and c1, c2 as in (6). Then
ku−ukkH ≤(
c22B−c21A c2
1A+c22B
In particular the vectorsuk converges tou as k→ ∞.
Proof. By definition ofuk we obtain
u−uk=u−uk−1+
2
c21A+c22BLS 0
H,v(f−Luk−1)
= (I− 2
c2
1A+c22B
LSH,v0 L)(u−uk−1)
= (I− 2
c2
1A+c22B
LSH,v0 L)2(u−uk−2)
=...= (I− 2
c21A+c22BLS 0
H,vL)k(u−u0),
therefore
ku−ukkH ≤ kI− 2
c21A+c22BLS 0
H,vLkkkukH. (7)
But for everyv∈H we have
h(I− 2
c2
1A+c22B
LSH,v0 L)v, vi=kvk2H − 2
c2
1A+c22B
hSH,v0 Lv, Lvi
=kvk2
H−
2
c21A+c22Bh
X
λ∈Λ
vλ2πLHλ(Lv), Lvi
=kvk2
H −
2
c21A+c22B
X
λ∈Λ
v2λkπLHλ(Lv)k 2
H
≤ kvk2H− 2A
c21A+c22BkLvk 2
H
≤ kvk2H − 2A
c2
1A+c22B
c21kvk2H = (c
2
2B−c21A c21A+c22B)kvk
2
H,
where in the first inequality we used the property of the lower bound of the frame of subspaces and in the second inequality we used the property of c1 in (6). Similarly we have
−(c
2
2B−c21A c21A+c22B)kvk
2
H ≤ h(I− 2
c21A+c22BLS 0
H,vL)v, vi,
and so we conclude that
kI− 2
c2
1A+c22B
LSH,v0 Lk ≤ c
2
2B−c21A c2
1A+c22B
. (8)
Combining this inequality with (7) gives the result.
Now, let ube the solution of the equation(1) and TH,v be the synthesis operator of the frame of subspaces{Hλ}λ∈Λ forH. Since TH,v is onto then there exists U ∈Pλ∈Λ⊕Hλ such thatu =TH,vU, so the equation (1) is equivalent to LTH,vU =f, orTH,v∗ LTH,vU =
TH,v∗ f, where TH,v∗ is the analysis operator of the frame of subspaces. Therefore finding the solutionuof the equation (1) is equivalent to finding the solution U of the equation
M U =F, (9)
whereM :=TH,v∗ LTH,v and F :=TH,v∗ f.
Note that we can consider the equation (9) as a matrix equation from (P
itself , where the entries ofM are the operators of the formmλ,λ0 =vλvλ0πH λ0L.
We note that (P
λ∈Λ⊕Hλ)`2 =RanT
∗
H,v⊕KerTH,v,and the following lemma holds.
Lemma 2.1. The orthogonal projector onto RanTH,v∗ is Q=TH,v∗ SH,v−1TH,v.
Proof. Forx∈TH,v∗ f,
Qx = Q({vλπHλf}λ∈Λ)
= TH,v∗ SH,v−1TH,v({vλπHλf}λ∈Λ)
= TH,v∗ SH,v−1(X λ∈Λ
v2λπHλf)
= TH,v∗ (X λ∈Λ
SH,v−1v2λπHλf)
= {vλπHλ(
X
λ∈Λ
SH,v−1v2λπHλf)}λ∈Λ
= {vλπHλ(f)}λ∈Λ.
That isQ=idon RanTH,v∗ andQ= 0 on KerTH,v.
Therefore,M|RanT∗ :RanT∗
H,v −→RanTH,v∗ is boundedly invertible and we havekMk ≤
BkLk and kM|−1RanT∗ H,v
k ≤A−1kL−1k.
3. An adaptive algorithm based on a frame of subspaces
In this section, we construct an adaptive algorithm in order to give an approximate solution to the exact solutionU of the equation (9). In order to analyze adaptive methods, we compare them with the bestN-term approximation. The aim is to balance between the accuracy and computational complexity at the same time.
ForN ∈N, define
X
N
:={V ∈(X λ∈Λ
⊕Hλ)`2 : #supp V ≤N},
and the corresponding error
ρN(V) := inf VN∈PN
kV −VNk(P
λ∈Λ⊕Hλ)`2, V ∈( X
λ∈Λ
⊕Hλ)`2,
where #supp V is the number of nonzero entries ofV. A best approximation to V from P
N (called the best N-term approximation to V) is obtained by taking a set ΛN ⊂ Λ with #ΛN ≤ N on which kvλk takes its N largest values. Note that the set ΛN is not unique.
Given a sequenceV = (vλ)λ ∈(
P
λ∈Λ⊕Hλ)`2, for each n≥1 letv
∗
n be then-th largest of the valueskvλk and define the decreasing rearrangement V∗ of V by V∗ := (vn∗)∞n=1. For
each 0< τ < 2 we let `ωτ(Λ) denote the collection of all vectors V ∈ (P
λ∈Λ⊕Hλ)`2 for
which |V|`ω
τ(Λ) := supn≥1n
1
τv∗
n is finite. This expression defines a quasi norm for `ωτ(Λ). A corresponding norm is defined by kVk`ω
τ(Λ) := |V|`ωτ(Λ)+kVk( P
λ∈Λ⊕Hλ)`2. Also there
exists a constantCτ such that
|V +W|`ω
whereab means that, there is a constantc such that a≤cb. Now let VN be the best
N-term approximation of V such that kV −VNk(P
λ∈Λ⊕Hλ)`2 ≤. If for somes >0
kV −VNk(P
λ∈Λ⊕Hλ)`2 N
−s, (11)
thenN −s1.Forτ = (1 2 +s)
−1, (11) means thatV ∈`ω
τ(Λ) and for 0< τ <2,
supN∈N{N
skV −V Nk(P
λ∈Λ⊕Hλ)`2} ' |V|`ωτ(Λ). (12)
One can see [4, 9] for further details on the quasi-Banach spaces`ωτ(Λ).
Proposition 3.1. Let s >0 and τ = (s+21)−1. IfV ∈`ωτ(Λ), then
ρN(V)N−skVk`ω
τ(Λ), (13)
with a constant only depending on τ for τ &0.
Proof. See [4].
Assumption. We assume that the matrixM is s∗-compressible, in the sense that for 0 < s < s∗ there exists a sequence α = (αj)j ∈ `1(Λ), and a matrix Mj having at most
αj2j nonzero entries per row and column, and a positive constantCM such that
kM −Mjk ≤CMαj2−js, (14)
for all j∈N. (kM−Mjk is the spectral norm of (M−Mj).) Such a matrix maps`ωτ(Λ) boundedly into itself forτ = (12 +s)−1. [4].
Remark 3.1. If M is s∗ compressible then M maps`ωτ(Λ) boundedly into itself for every
τ = (12+s)−1. [4].
For a finite support vector V, N := (#supp(V)) < ∞, we denote the best 2j-term approximation toV in (P
λ∈Λ⊕Hλ)`2 byV[j], forj= 1, 2, ..., blogNc, and let V[j]=V,
forj >logN. For a givenK ∈Ndefine
WK :=MKV[0]+
K−1
X
j=0
Mj(V[K−j]−V[K−j−1]),
whereMj is as (14). In this case
M V −WK=M V −MKV[0]−
K−1
X
j=0
Mj(V[K−j]−V[K−j−1])
=M V −MKV[0]−MK−1(V[1]−V[0])−...−M0(V[K]−V[K−1])
=M(V −V[K]) + (M −M0)(V[K]−V[K−1]) +...+ (M−MK)V[0],
and sinceM iss∗-compressible then
kM V −WKk(P
λ∈Λ⊕Hλ)`2 ≤
kMkkV −V[K]k(P
λ∈Λ⊕Hλ)`2 +kM−M0kkV[K]−V[K−1]k(
P
λ∈Λ⊕Hλ)`2
+...+kM −MKkkV[0]k(P
λ∈Λ⊕Hλ)`2 ≤ kMkkV −V[K]k(
P
λ∈Λ⊕Hλ)`2+
CM(α0kV[K]−V[K−1]k(P
λ∈Λ⊕Hλ)`2 +...+αK2
−KskV
[0]k(P
λ∈Λ⊕Hλ)`2).
In the other words there exists a constantC3 such that
kM V −WKk(P
λ∈Λ⊕Hλ)`2 ≤ (15)
C3(kV −V[K]k(P
α0kV[K]−V[K−1]k(P
λ∈Λ⊕Hλ)`2 +...+αK2
−KskV
[0]k(P
λ∈Λ⊕Hλ)`2).
Now by proposition 3.1 and the definition ofV[j] there exists a constant C4 such that
kV −V[j]k(P
λ∈Λ⊕Hλ)`2 =ρ2j(V)≤C42
−jskVk `ω
τ(Λ),
hence
kV[j]−V[j−1]k(P
λ∈Λ⊕Hλ)`2 ≤ kV −V[j]k(
P
λ∈Λ⊕Hλ)`2 +kV −V[j−1]k(
P
λ∈Λ⊕Hλ)`2
=ρ2j(V) +ρ2j−1(V)
≤C4(2−s+ 1)2−(j−1)skVk`ω τ(Λ).
Applying the above inequalities and the fact that kV[0]k(P
λ∈Λ⊕Hλ)`2 ≤ kV[0]k`ωτ(Λ), the
inequality (15) induces a constantC such that
kM V −WKk(P
λ∈Λ⊕Hλ)`2 ≤C2
−KskVk `ω
τ(Λ). (16)
Now we are ready to design our algorithm. First, following [4], we introduce the following routine.
APPLY [M, V, ]:
i) Compute V[0], V[j]−V[j−1], j= 1, ...,blogNc and defineV[j]:=V forj > logN.
ii) Compute K as the smallest integer such that 2K ≥C1s−
1
skVk
1
s
`ω τ(Λ).
iii) Fork= 1 to K compute
Rk:=kMkkV−V[k]k(P
λ∈Λ⊕Hλ)`2+CA(α0kV[k]−V[k−1]k(
P
λ∈Λ⊕Hλ)`2+...+αk2
−kskV
[0]k(P
λ∈Λ⊕Hλ)`2). iv) IfRk ≤then exit.
v) WK :=MKV[0]+
PK−1
j=0 Mj(V[k−j]−V[k−j−1]).
Remark 3.2. Since K is the smallest integer such that 2K ≥ C1s−
1
skVk
1
s
`ω
τ(Λ), then
2K−1 < C1s−
1
skVk
1
s
`ω
τ(Λ). Thus2
K <2C1s−1
skVk
1
s
`ω
τ(Λ),which 2
K −1
skVk
1
s
`ω τ(Λ).
Lemma 3.1. Let V ∈`ωτ(Λ) with τ = (s+ 12)−1. For a given accuracy >0, the output
WK of APPLY [M, V, ]satisfies
kM V −WKk(P
λ∈Λ⊕Hλ)`2 ≤, (17) and
#supp(WK)−1skVk
1
s
`ω
τ(Λ). (18)
Also the number of arithmetic operations to computeWKis at most a multiple of−
1
skVk
1
s
`ω τ(Λ).
Proof. The inequality (17) comes from the inequality (16) and the definition of K in
APPLY.
Recalling the number of nonzero entries inV[j], and definition ofMj we conclude #supp(WK)≤#rows(MK) + #rows(MK−1) +...+ #rows(M0)
≤αK2K+αK−12K−1+...+α0 ≤(|αK|+|αK−1|+...+|α0|)2K 2K,
where the last inequality is induced by (αj)j ∈`1(Λ). Now by using remark 3.2, we obtain
the inequality (18). Also if NK denotes the number of arithmetic operation needed to computeWK we have
NK ≤#rows(MK)#supp(V[0]) + #rows(MK−1)#supp(V[1]−V[0])
≤αK2K+αK−12K−1+...+α02K−12K −
1
skVk
1
s
`ω τ(Λ).
Also, as in [4], for an accuracy > 0 and a finitely supported vector W with N = #(spport(W)), we introduce the following basic numerical ingredient that we will use in our algorithm.
COARSE [W, ]→(Λ,W¯)
(i) Sort the nonzero entries of W into decreasing order in modulus and obtain the vector λ∗ := (λ1, ..., λN) of indices which gives the decreasing rearrangement W∗ = (kWλ1kH, ...,kWλNkH) of nonzero entries ofW; then computekWk
2 (P
λ∈Λ⊕Hλ)`2 = PN
i=1kWλik 2
H.
(ii)Find the smallest K ∈Nsuch that PKi=1kWλik
2 exceeds kWk2 (P
λ∈Λ⊕Hλ)`2
−2. For thisK define Λ :={λi:i= 1, ..., K}and ¯W by ¯Wλ =Wλ forλ∈Λ and ¯Wλ = 0 forλ /∈Λ.
Now, let 0< <kVk(P
λ∈Λ⊕Hλ)`2andW be a finitely supported approximation toV such
that kV −Wk(P
λ∈Λ⊕Hλ)`2 ≤d for some d < 1, then it is obvious that the COARSE
[W,(1−d)] produces ¯W supported on Λ whichkV −W¯k(P
λ∈Λ⊕Hλ)`2 ≤.(Note that the
output ¯W ofCOARSE, by construction, satisfieskW −W¯k(P
λ∈Λ⊕Hλ)`2 ≤). Moreover,
we have the following lemma [4].
Lemma 3.2. If V ∈ `ωτ(Λ), τ = (s+ 12)−1, for some s > 0 then the outputs W ,¯ Λ of COARSE[W,(1−d)]requires at most2N arithmetic operations andN logN sorts,where
N = #supp(W). Moreover,
|W¯|`ω
τ(Λ) |V|`ωτ(Λ), (19)
and#Λ (the cardinality of supp( ¯W) ) satisfies
#(Λ) |V|
1
s
`ω τ(Λ)
1
s. (20)
Also for F =TH,v∗ f we assume that the routine:
RHS[ε, F]→F
determines a finitely supported vectorF∈(Pλ∈Λ⊕Hλ)`2 satisfying
kF −Fk(P
λ∈Λ⊕Hλ)`2 ≤.
Assuming Q is bounded on `ω
τ0(Λ) for τ0 = (12 +s0)−1, 0 < s0 < s (hence Q is bounded
on `ωτ, [9])we construct our algorithm for the target accuracy > 0. At first, for some 0< d < 13 and ρ:=kI−αMk<1 (since M is a positive definite matrix this real number
αexists) setK :=min{k∈N: 3ρk< d min{1,[C1C2|I−Q|`ω τ→`ωτ]
s
s0−s}},whereC1, C2
are two constants induced from (10) and (19) forτ0.
SOLVE[, M, F]→(U,Λ)
(i) Seti= 0, U(0)= 0,Λ0 =∅, 0 :=kM|−1Ran(T∗)k kFk(P
λ∈Λ⊕Hλ)`2.
(ii) Ifi ≤stop and set U :=Ui, otherwise
(ii.1)i:=i+ 1, i := 3ρK i−d1.
(ii.2)Fi:=RHS [F, di 6αK].
(ii.4)Forj= 1, ..., K compute
(1) Wj−1 :=APPLY [M, V(i,j−1), di 6αK]. (2) V(i,j) :=V(i,j−1)+α(Fi−Wj−1).
(iii) Ui :=COARSE [V(i,K),(1−d)i] and go to (ii).
Remark 3.3. By definition ofV(i,j),Fi inSOLVEand lemma (3.1) and since M QU =
F,
kQU−V(i,1)−(I−αM)(QU −Ui−1)k(P
λ∈Λ⊕Hλ)`2
=kQU −(V(i,0)−α(APPLY [M, V(i,0),di
6α]−F
i))
−(I−αM)(QU −Ui−1)k(P
λ∈Λ⊕Hλ)`2
=kQU −Ui−1+α(APPLY [M, V(i,0),di
6α]−αF
i)
−QU+Ui−1+αM QU−αM Ui−1k(P
λ∈Λ⊕Hλ)`2
=kαAPPLY [M, V(i,0),di
6α]−αF
i+αF −αM Ui−1k (P
λ∈Λ⊕Hλ)`2
=kα(APPLY [M, V(i,0),di
6α]−M U
i−1) +α(F−Fi)k
(P
λ∈Λ⊕Hλ)`2
≤αkAPPLY [M, V(i,0),di
6α]−M U
i−1k (P
λ∈Λ⊕Hλ)`2 +αkF −F
ik
(P
λ∈Λ⊕Hλ)`2
≤αdi
6α +α di 6α =
di 3 .
Similarly we can prove
kQU−V(i,K)−(I−αM)K(QU−Ui−1)k(P
λ∈Λ⊕Hλ)`2 ≤
di
3 . (21)
Theorem 3.1. If U is a solution for (9) then the following inequalities hold for the algo-rithm SOLVE:
kQ(U−U)k(P
λ∈Λ⊕Hλ)`2 ≤
kQU + (I−Q)Ui−1−V(i,K)k(P
λ∈Λ⊕Hλ)`2 ≤
2
3di, (i≥1) (22)
Proof. In order to prove the first inequality, it is enough to provekQ(U−Ui)k
(P
λ∈Λ⊕Hλ)`2 ≤
i for each i≥0. Fori= 0, sinceQU =M|−1RanT∗
H,vF then
kQ(U −U0)k(P
λ∈Λ⊕Hλ)`2 =kQUk(
P
λ∈Λ⊕Hλ)`2 =kM|
−1
RanT∗Fk(P
λ∈Λ⊕Hλ)`2
≤ kM|−1RanT∗kkFk(P
λ∈Λ⊕Hλ)`2 =0.
Now for ani≥1, let kQ(U −Ui−1)k(P
λ∈Λ⊕Hλ)`2 ≤i−1. SinceM QU =F and
(I−αM)K(QU−Ui−1) = (I−αM)KQ(U−Ui−1)−(I−Q)Ui−1,
by using the inequality (21)
kQU+ (I−Q)Ui−1−V(i,K)k(P
λ∈Λ⊕Hλ)`2
=kQU + (I−Q)KQ(U −Ui−1)−(I−Q)K(QU −Ui−1)−V(i,K)k(P
≤ k(I−αM)KQ(U−Ui−1)k(P
λ∈Λ⊕Hλ)`2+
kQU−V(i,K)−(I−αM)K(QU−Ui−1)k(P
λ∈Λ⊕Hλ)`2
≤ρKkQ(U−Ui−1)k(P
λ∈Λ⊕Hλ)`2 +
di 3
≤ρKi−1+ di
3 =
2 3di, as we desired in (22).
Now by using (22) and the definition ofUi inSOLVE we obtain, kQU + (I−Q)Ui−1−Uik(P
λ∈Λ⊕Hλ)`2
=kQU+ (I−Q)Ui−1−V(i,K)+V(i,K)−Uik(P
λ∈Λ⊕Hλ)`2
≤ kQU+ (I−Q)Ui−1−V(i,K)k(P
λ∈Λ⊕Hλ)`2 +kV
(i,K)−Uik
(P
λ∈Λ⊕Hλ)`2
≤(2d
3 + (1−d))i= (1−
d
3)i. Therefore
kQ(U −Ui)k2(P
λ∈Λ⊕Hλ)`2
≤ kQ(U −Ui)k2(P
λ∈Λ⊕Hλ)`2+
k(I−Q)(Ui−1−Ui)k2(P
λ∈Λ⊕Hλ)`2
=kQ(U−Ui) + (I−Q)(Ui−1−Ui)k2(P
λ∈Λ⊕Hλ)`2
=kQU + (I−Q)Ui−1−Uik2(P
λ∈Λ⊕Hλ)`2
≤(1−d 3)
22
i ≤2i, and so
kQ(U −Ui)k(P
λ∈Λ⊕Hλ)`2 ≤.
In the following theorems, we investigate the optimal computational complexity of the algorithm SOLVE as it recovers an approximate solution with desired accuracy at a computational expense that stays proportional to the number of terms in a corresponding wavelet-bestN-term approximation.
Theorem 3.2. Assume that the solution U of (9) belongs to`ωτ. Then
#(supp(U))
1
s|U|−
1
s
`ω τ .
Proof. Let (QU)Ni be the best Ni-term approximation for QU such that
kQU −(QU)Nik(Pλ∈Λ⊕Hλ)`2 ≤ di
3 , (23)
where 0< d < 13.Since U ∈`ωτ, by (11) we have
Ni
−1
s
i |QU|
1
s
`ω τ
−1
s
i |U|
1
s
`ω
τ. (24)
Sinces0 < s, then for a vectorV with #supp(V) =N
|V|`ω τ0 ≤N
s−s0|V| `ω
τ, (25)
combination (25) and (24) gives
1− s0
s
i |(QU)Ni|`ωτ0 1−s0
s
i N
(s−s0)
1− s0 s i ( −1 s i )
(s−s0)
(|U|
1
s
`ω τ)
(s−s0)
|(QU)Ni|`ωτ |U| 1−s0
s
`ω
τ |QU|` ω τ |U|
−s0 s `ω τ , therefore 1− s0 s
i |(QU)Ni|`ωτ0 |U| s0
s
`ω
τ. (26)
Using (22) and (23),
k(QU)Ni+ (I −Q)U
i−1−VKk
(P
λ∈Λ⊕Hλ)`2 ≤
k(QU)Ni−QUk(Pλ∈Λ⊕Hλ)`2 +kQU + (I −Q)U
i−1−
VKk(P
λ∈Λ⊕Hλ)`2 ≤
di
3 +
2di 3 =di, then by lemma 3.2 and 10 we have
|Ui|`ω
τ0 ≤C2|(QU)Ni+ (I−Q)U
i−1|
`ω τ0
≤C1C2|(QU)Ni|`ωτ0 +C1C2|I−Q|`ωτ0→`ωτ0|U
i−1|
`ω τ0,
by (26) and sincei = 3ρ
k i−1 d 1− s0 s
i |Ui|`ω
τ0 ≤C|U| −s0
s
`ω
τ +C1C2|I −Q|` ω τ0→`
ω τ0(
3ρK d )
1−s0 s(1−
s0 s
i−1 |Ui −1|
`ω
τ0), (27)
for some constantC. Now by the properties of K we obtain
1− s0
s
i |Ui|`ω τ0 |U|
−s0 s
`ω
τ . (28)
Finally by (28), (26) and lemma 3.2 we conclude
#(supp(Ui))−
1
s0
i |(QU)Ni+ (I−Q)U
i−1|s10
`ω τ0 − 1 s i (
1−ss0
i [|(QU)Ni|`ωτ0 +|I−Q|`ωτ0→`ωτ0|U
i−1|
`ω τ0])
1
s0
1
s
i |U|
−1
s
`ω τ ,
and it is obvious that this proves our request.
Theorem 3.3. Assume that the solution U of (9) belongs to `ωτ. Then the number of arithmetic operations needed to compute U is bounded by a multiple of
1
s|U| −1
s
`ω τ .
Proof. Since M U = F and M is bounded on `ωτ then |F|`ω
τ |U|`ωτ, and therefore by
lemma 3.2
#supp(Fi)−
1
s
i |U|
1
s
`ω
τ, (29)
and
|Fi|`ω
τ |U|`ωτ. (30)
Now by (25), (29) and (30) we obtain
|Fi|`ω
τ0 (#supp(F
i))s−s0
|Fi|`ω τ
(−
1
s
i |U|
1
s
`ω τ)
s−s0
|U|`ω τ
s0 s−1
i |U|
−s0 s
`ω τ .
Thus
(i)1−s
0 s|Fi|`ω
τ0 |U| −s0
s
`ω
Lemma 3.1 together with (28) and (31) give (i)1−
s0
s|V(i,j)|`ω
τ0 |U| −s0
s
`ω
τ , 0 ≤ j ≤ K,
therefore by lemma 3.1 (for s), #supp(Wj−1) −s10
i |V(i,j−1)|
1
s0
`ω τ0
1
s
i |U|
−1s
`ω
τ .Also by the
previous theorem #supp(Ui−1) = #supp(Vi,0)
1
s
i−1|U| −1
s
`ω
τ ,while by step (2) inSOLVE,
#supp(V(i,j))≤#supp(V(i,j−1)) + #supp(Fi) + #supp(Wj−1).
Therefore we conclude
#supp(Vi,j)
1
s
i |U|
−1
s
`ω
τ . (32)
Now by lemmas 3.2 and 3.1 together with (32), the number of arithmetic operations needed
from i to i+1 is at most a multiple of1s|U|−
1
s
`ω
τ ,which is the desired result.
References
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Hassan Jamaliwas born in 1977 in Mashhad, Iran. He completed his Ph.D. degree in mathematics from Vali-e-Asr University of Rafsanjan, Iran in 2008. He has been a member of Faculty of Mathematics at Vali-e-Asr University of Rafsanjan since 2005. His research interests comprise: wavelet analysis, frame theory, and numerical analysis.
