Table Of ContentAnalysis of moving least squares approximation revisited
D. Mirzaei
Department of Mathematics, University of Isfahan, 81745-163 Isfahan, Iran.
5
1
0
Abstract
2
n Inthisarticletheerrorestimationofthemovingleastsquaresapproximationisprovided
a
for functions in fractional order Sobolev spaces. The analysis presented in this paper
J
0 extends the previousestimationsandexplainssome unnoticedmathematicaldetails. An
2 application to Galerkin method for partial differential equations is also supplied.
] Keywords: Moving least squares approximation, Error bounds, Sobolev spaces,
A
Meshless methods.
N
.
h
t
a
1. Introduction
m
[
The Moving Least Squares (MLS) approximation was introduced in an early paper
1 by Lancaster and Salkauskas [1] in 1981 with special cases going back to McLain [2, 3]
v
in 1974and1976andto Shepard[4]in1968. Forother earlystudies wecanmentionthe
9
6 work of Farwig [5, 6, 7]. Since, in MLS one writes the value of the unknown function
9
in terms of scattered data, it can be used as an approximation to span the trial space
4
0 in meshless (or meshfree) methods. This approximationhas found many applications in
.
1 curve fitting and numerical solutions of partial differential equations since early nineties
0 [8, 9, 10, 11].
5
1 The error analysis of MLS approximation was provided by some authors, beginning
: with the work of Farwig [7] which is limited to a univariate case. The connection to
v
i Backus-Gilbert optimality was studied by Levin [12] in 1998, and later it was used by
X
Wendland [13, 14, 15] in a more elaborated setting. In Liu et. al. [16] the analysis
r
a is presented for smooth functions in Cm+1(Ω) Hm+1(Ω). Armentano and Dura´n [17]
∩
provederrorestimatesinL forthe functionandits firstderivativesinone dimensional
∞
case. Afterward Armentano [18] generalized this to multi-dimensional cases but it is
still restricted to “convex” domains and Sobolev spaces of order one. One can also find
an estimation in Han and Meng [19] for reproducing kernel particle methods (which is
related to the MLS approximation) for integer order Sobolev spaces. They assumed a
Email address: [email protected] (D.Mirzaei)
Preprint submitted to Elsevier January 22, 2015
constantboundforthenormoftheinversematrix(matrixAintext)andconsidereditfor
special cases in one dimension and first order approximations. Note that the role of this
matrixisverycrucialinanalysis. ThepaperofZuppa[20]isalsolimitedtosomespecific
situations. In Wendland [13, 15] the analysis presented only for the function in classical
function spaces. We can also mention the work of Melenk [21] where the theoretical
and computational aspects of some meshless approximation methods, including MLS,
are considered.
ThecollocationmethodbasedontheMLSapproximationiscalledfinitepointmethod.
Ananalysisforthis methodhasbeenpresentedin[22]. Besides,aninterpolatingMLSis
developed recently. For error analysis and applications to element-free Galerkin method
see [23, 24].
ThepresentworkisbasedonthetheoryofWendlandandextendsalltheaboveresults
to a general case. All mathematical details are provided, special care is taken near the
boundary,andlowerboundfortheminimumeigenvalueoftheMLSlocalmatrixisderived
in general case, independent of the mesh-size. Besides, the analysis is presented for
functions infractionalorderSobolevspaces. Finally anapplicationto Galerkinmethods
for elliptic PDEs is investigated.
2. MLS approximation
Let Ω Rd, for positive integer d, be a nonempty and bounded set. In the next
⊂
section, more conditions on Ω will be imposed. Assume,
X = x ,x ,...,x Ω,
1 2 N
{ }⊂
is a set containing N scattered points, called centers or data site. Distribution of points
should be well enough to pave the way for analysis.
Henceforth, we use Pd , for m N = n Z,n > 0 , as the space of d-variable
m ∈ 0 { ∈ }
polynomials of degree at most m of dimension Q = m+d . A basis for this space is
d
denoted by p ,...,p or p . As usual, B(x,r) stands for the ball of radius
{ 1 Q} { α}06|α|6m (cid:0) (cid:1)
r centered at x.
The MLS, as a meshless approximation method, provides an approximation s of
u,X
u in terms of values u(x ) at centers x by
j j
N
u(x) s (x)= a (x)u(x ), x Ω, (2.1)
u,X j j
≈ ∈
j=1
X
where a are MLS shape functions given by
j
Q
a (x)=w(x,x ) λ (x)p (x ), (2.2)
j j k k j
k=1
2X
wheretheinfluenceofthecentersisgovernedbyweightfunctionw (x)=w(x,x ),which
j j
vanishes for arguments x,x Ω with x x greater than a certain threshold, say δ.
j j 2
∈ k − k
Thus we can define w (x) =Φ((x x )/δ) where Φ :Rd R is a nonnegative function
j j
− →
with support in the unit ball B(0,1). Coefficients λ (x) are the unique solution of
k
Q
λ (x) w (x)p (x )p (x )=p (x), 06ℓ6Q, (2.3)
k j k j ℓ j ℓ
kX=1 j∈XJ(x)
where J(x) = j : x x 6δ is the family of indices of points in the support of w.
j 2
{ k − k }
In vector form
a(x)=W(x)PT(PW(x)PT) 1p(x),
−
where W(x) is the diagonal matrix carrying the weights w (x) on its diagonal, P is a
j
Q #J(x)matrixofvaluesp (x ), j J(x),16k 6Q,andp=(p ,...,p )T. InMLS
k j 1 Q
× ∈
one finds the best approximation to u at point x, out of Pd with respect to a discrete
m
ℓ2 norm induced by a moving inner product, where the corresponding weight function
depends not only on points x but also on the evaluation point x in question. Note that
j
A(x) = PW(x)PT is a symmetric positive definite matrix for all x Ω. More details
∈
can be found in Chapter 4 of [15].
In what follows we will assume that Φ is nonnegative and continuous on Rd and
positive on the ball B(0,1/2). In many application we can assume that
Φ(x)=φ( x ), x Rd,
2
k k ∈
meaningthatΦisaradialfunction. Hereφ:[0, ) Rispositiveon[0,1/2],supported
∞ →
in [0,1] and its even extension is nonnegative and continuous on R.
If, further, φ is sufficiently smooth, derivatives of u are usually approximated by
derivatives of s ,
u,X
N
Dαu Dαs (x)= Dαa (x)u(x ), x Ω, (2.4)
u,X j j
≈ ∈
j=1
X
andtheyarecalledstandardderivatives. TheyaredifferentfromGMLSordiffusederiva-
tives [25] which are not the aim of this paper.
3. Error estimation
Since error estimates will be established using a variety of Sobolev spaces, we intro-
duce them now. Let Ω Rd be a domain. For k N , and p [1, ), we define the
0
⊂ ∈ ∈ ∞
Sobolev space Wk(Ω) to consist of all u with distributional derivatives Dαu Lp(Ω),
p ∈
α 6k. The (semi-)norms associated with these spaces are defined as
| |
1/p 1/p
u := Dαu p , u := Dαu p .
| |Wpk(Ω) k kLp(Ω) k kWpk(Ω) k kLp(Ω)
|αX|=k 3 |αX|6k
The case p= is defined in the standard way
∞
u := sup Dαu , u := sup Dαu .
| |W∞k(Ω) α=kk kL∞(Ω) k kW∞k(Ω) α6kk kL∞(Ω)
| | | |
For fractional order Sobolev spaces, we use the norms below. Let p [1, ), k > 0,
∈ ∞
k Z, and let 0<s<1. We define the fractional order Sobolev spaces Wk+s(Ω) to be
∈ p
the space of all u for which the norms below are finite.
1/p
Dαu(x) Dαu(y)p
|u|Wpk+s(Ω) := | x −y d+ps | dxdy ,
|αX|=kZΩZΩ | − |
1/p
u := u + u .
k kWpk+s(Ω) k kWpk(Ω) | |Wpk+s(Ω)
(cid:16) (cid:17)
The first step in deriving error estimates is to consider only local regions that are
D
star-shaped with respect to a ball. A domain Rd is said to be star-shaped with
D ⊂
respect to a ball B = B(y,ρ) = x Rd : x y 6 ρ if for every x , the closed
{ ∈ k − k } ∈ D
convex hull of x B is contained in . Let
{ }∪ D
ρ =sup ρ: is star-shaped with respect to a ball of radius ρ ,
max
{ D }
then the chunkiness parameter of is defined by γ = dD where d is the diameter of
D ρmax D
.
D
Approximatinga functionu Wm+1( ) byaveragedTaylorpolynomialsQ u Pd
∈ q D m ∈ m
is discussed in [26, Chapter 4]. The averaged Taylor polynomials are defined as follows.
Let B be a ball with respect to which is star-shaped having radius ρ> 1ρ . Then
D 2 max
1
Q u(x):= Dαu(y)(x y)αϕ(y)dy,
m
α! −
|αX|6m ZB
where ϕ(y)>0 is a C “bump” function supported in B satisfying both ϕ(y)dy =1
∞ B
and maxϕ6Cρ d.
− R
In [26] the Wℓ bounds on u Q u are given for integer ℓ when u Wm+1( ) and
p − m ∈ p D
ℓ6m+1. AversionoftheseresultsthatapplieswhenubelongstoWm+s( ),06s<1,
p D
was proved in [27]. An improvement of conditions (range of s) was discussed in [28] by
the same authors.
Lemma 3.1. Let B be a ball in such that is star-shaped with respect to B and
D D
such that its radius ρ > (1/2)ρ . Let Q u be the Taylor polynomial of order m of u
max m
averaged over B where u Wm+s( ) for 0 6 s < 1 and p [1, ). Let m > d/p for
∈ p D ∈ ∞
p>1 and m>d for p=1. Then there exists constant C =C(m,d,p,γ) such that
ku−QmukL∞(D) 6CdDm+s−d/p|u|Wpm+s(D), (3.1)
where d is the diameter of .
D D
4
We should note that the identity
DαQmu=Qm−|α|Dαu, for all u∈W1|α|(D). (3.2)
which is found in [26, Section 4], holds for α 6 m. Applying Lemma 3.1 on Dαu
| |
instead of u, using the identity (3.2) and the inequality Dαu 6 u ,
| |Wpk+s−|α|(D) | |Wpk+s(D)
we obtain
Corollary 3.2. Let 06s<1. For u Wm+s( ),
∈ p D
kDαu−DαQmukL∞(D) 6CdDm+s−|α|−d/p|u|Wpm+s(D), (3.3)
provided that m> α +d/p for p>1 and m> α +d for p=1.
| | | |
Analogousto the errorbound(3.1), onecaneasilyderive the Wℓ bound foru Q u
q − m
when u Wm+s( ). We give the results in the following Lemma.
∈ p D
Lemma 3.3. Let q [1, ], p [1, ) and α be a multi-index satisfying m> α +d/p
∈ ∞ ∈ ∞ | |
for p>1 and m> α +d for p=1. With the notation and assumptions of Lemma 3.1,
| |
we have
ku−QmukWq|α|(D) 6CdDm+s−|α|+d(1/q−1/p)|u|Wpm+s(D), (3.4)
where C =C(m,d,p,q,α,γ).
Proof. Although the proof can be implicitly extracted from [27], but we present it here
for the reader’s conveniences. Let q [1, ). Using the definition of Sobolev norms, we
∈ ∞
have
u Q u q = Dβ(u Q u)qdx
k − m kWq|α|(D) |βX|6|α|ZD| − m |
q
6#{β ∈Nd0 :|β|6|α|}×vol(D) βm6axα kDβ(u−Qmu)kL∞(D)
(cid:16)| | | | (cid:17)
6C(d,α)dd max Dβ(u Q u) q
D β6α k − m kL∞(D)
| | | |
6C(m,d,p,α,γ)dDq(m+s−|α|+d(1/q−1/p))|u|qWpm+s(D).
At the third line above, we use the facts that vol( )6C dd and
d
D D
α
| | i+d 1 α +d
# β Nd : β 6 α = − = | | = (αd).
{ ∈ 0 | | | |} d 1 d O | |
i=0(cid:18) − (cid:19) (cid:18) (cid:19)
X
In the last line, Corollary 3.2 has been applied. Finally taking the q-th root of the both
sides completes the proofwith the new constantC =C(m,d,p,q,α,γ). The case q =
∞
can be proved by a similar argument (see also [15, Proposition 11.29]).
5
Remark 3.4. In cases=1, ifwe assumem+1> α +d/p forp>1andm+1> α +d
| | | |
for p =1 then estimations (3.1) and (3.4) are still valid, due to [26]. The reader should
becautiousthattheseerrorboundscannotbeobtainedbyinsertings=0andreplacing
m by m+1 in fractional cases, because the later produces Q u.
m+1
Up to this point, we reviewed some Sobolev error bounds for a function which is
approximatedbytheaveragedTaylorpolynomialonastar-shapeddomain. Thesebounds
areusuallyusedforanalyzingthefiniteelementmethod(FEM).NowweturntotheMLS,
as a meshless approximation method, and employ the above bounds to analyze it. The
finalboundwillbepresentedforfunctionsinfractionalSobolevspaces. Althoughonecan
usethe interpolationarguments(for examplethe “real”methodbasedonK-functionals)
to extend the integer order Sobolev spaces to fractional ones, here we follow the direct
approach because all materials are provided via (3.1) and (3.4).
First we introduce some other notations. For a set of points X = x ,x ,...,x in
1 2 N
{ }
a bounded domain Ω Rd, the fill distance is defined to be
⊂
h = sup min x x ,
X,Ω j 2
x Ω16j6Nk − k
∈
and the separation distance is defined by
1
q = min x x .
X i j 2
2 i=j k − k
6
A set X of data sites is said to be quasi-uniform with respect to a constant c >0 if
qu
q 6h 6c q . (3.5)
X X,Ω qu X
AsetX = x ,...,x Rd withN >QiscalledPd -unisolventifthe zeropolynomial
{ 1 N}⊂ m
is the only polynomial from Pd that vanishes on X.
m
A set Ω Rd is said to satisfy an interior cone condition if there exist an angle
⊂
θ (0,π/2) and a radius r >0 such that for every x Ω a unit vector ξ(x) exists such
∈ ∈
that the cone
C(x,ξ,θ,r):= x+ty :y Rd, y =1,yTξ >cosθ,t [0,r]
2
∈ k k ∈
(cid:8) (cid:9)
is contained in Ω.
Assuming the compact set Ω satisfies an interior cone condition with radius r and
angle θ, and data site X Ω satisfies the quasi-uniform condition (3.5), Wendland
⊂
[15, Chapter 4] proved that shape functions a (x) from MLS approximation (2.1)
j
{ }
provideastable local polynomial reproductionofdegreemonΩ,i.e. thereexistconstants
h ,C ,C >0 independent of X such that for every x Ω
0 1 2
∈
1. N a (x)p(x )=p(x), for all p Pd ,
j=1 j j ∈ m
6
P
2. N a (x) 6C
j=1| j | 1
3. a (x)=0 if x x >δ =2C h ,
Pj j 2 2 X,Ω
k − k
for all X with h 6h . ConstantC depends onthe weightfunction φ, andconstants
X,Ω 0 1
C and h are
2 0
16(1+sinθ)2m2 r
C = , h = . (3.6)
2 3sin2θ 0 C2
He also proved that, if the weight function possesses k continuous derivatives then the
approximant s is also in Ck. Using the above properties, he proved the error bound
u,X
u s 6Chm+1 u ,
k − u,XkL∞(Ω) X,Ω| |Cm+1(Ω∗)
where u := max Dβu in which Ω = B(x,C h ) can be
Cm+1(Ω∗) β=m+1 L∞(Ω∗) ∗ x Ω 2 0
| | | | k k ∪ ∈
obviously larger than the exact domain Ω. Here Dβ is the classical derivative operator
on space Cm+1. The results of the present paper (in a special case) extend this bound
for functions u in fractional order Sobolev space Wm+s(Ω).
p
Takingderivativesoforderα, α 6m,undersomemildconditions,wecanshowthat
| |
functions Dαa (x) inapproximation(2.4)formanotherlocalpolynomialreproduction
j
{ }
in the followingsense: there existconstants h ,C ,C >0 independent ofX suchthat
0 1,α 2
for every x Ω
∈
1. N Dαa (x)p(x )=Dαp(x), for all p Pd ,
j=1 j j ∈ m
2. PNj=1|Dαaj(x)|6C1,αhX−|,αΩ|,
3. Dαa (x)=0 if x x >δ =2C h ,
P j j 2 2 X,Ω
k − k
for all X with h 6 h . The first and the last items are immediately followed from
X,Ω 0
the previous local polynomial reproduction system. But proving item 2 invites more
challenges. First we prove the following straightforwardresult.
Lemma 3.5. Let X = x ,...,x Ω has fill distance h . Suppose that function
1 N X,Ω
{ } ⊂
φ:[0, ) R,issupportedin[0,1]anditsevenextensionbelongstoCm(R)form N .
0
∞ → ∈
Then for w (x)=φ( x x /δ), x Rd and α 6m we have
j j 2
k − k ∈ | |
|Dαwj(x)|6Cαh−X|,αΩ|, ∀x∈Ω, j =1,...,N, (3.7)
provided that δ =2C h .
2 X,Ω
Proof. Since φ is a compactly supported and Cm function, derivatives of φ up to order
m are continuous and bounded. The absolute value of Dαw , has a bound with a factor
j
δ α times derivatives of φ. This immediately gives the desired bound for sufficiently
−| |
small h .
X,Ω
7
The MLS approximation can be implemented in a more stable fashion, if a shifted
and scaled polynomial basis function is used as a basis for Pd . In this case, we use the
m
basis
(x z)α
− , (3.8)
( h|α| )
X,Ω 06α6m
| |
where z is fixed and depends on the evaluation point to be considered. If x is the
evaluation point, the best result will be obtained if we finally set z = x. In fact, MLS
usesdifferentbasesforeachevaluationpoint. Wecandothis,becausetheformulbationof
MLS approximationand equations (2.2) and(2.3) are independent of thbe choice of basis
functions. Thus the MLS shape functions can be written as
(x z)α
j
a (x)=w (x) λ (x) − , j =1,2,...,N, (3.9)
j j |αX|6m α h|Xα,|Ω
where λ (x) is obtained by solving the positive definite system
α
T
(x z)α
A(x)λ(x)=p(x), p (x)= − , (3.10)
α ( h|α| )
X,Ω 06α6m
| |
where
N
x z x z
j j
A (x)= w (x)p − p − , i,k=1,...,Q. (3.11)
i,k j i k
h h
j=1 (cid:18) X,Ω (cid:19) (cid:18) X,Ω (cid:19)
X
N
Since φ is supported in the unit ball, we used the summation index instead of
Xj=1 j∈XJ(x)
in the above formulation. Since the set point X satisfies the quasi uniform condition
(3.5), the number #J(x) of points in J(x) can be bounded independent of h [15]. In
X,Ω
fact, for x ,x B(x,δ) and x =x the balls B(x ,q ) and B(x ,q ) are disjoint. All
i k i k i X k X
∈ 6
of these balls with x J(x) are contained in the ball B(x,q +δ). It is clear that
j X
∈
vol B(x ,q ) 6vol B(x,q +δ) ,
j X X
(cid:16)j∈[J(x) (cid:17) (cid:0) (cid:1)
which simply gives #J(x)qd 6 (δ +q )d. Using the quasi-uniform condition and δ =
X X
2C h , we have
2 X,Ω
#J(x)6(1+2C c )d =:C .
2 qu #
Lemma 3.6. If the weight function φ satisfies the assumptions of Lemma 3.5, then for
a fixed but arbitrary evaluation point x Ω we have
∈
|DαA(x)|6Cαbh−X|,αΩ|, ∀α with |α|6m, (3.12)
where C is a constant matrix independent of h .
α b X,Ω
8
Proof. Equation (3.11) gives
N
x x x x
DαA (x)= Dαw (x)p j − p j − .
i,k j i k
h h
j=1 (cid:18) X,Ω (cid:19) (cid:18) X,Ω (cid:19)
X b b
Evaluatingatx,takingabsolutevaluefrombothsidesandusing x x 6δ =2C h
j 2 2 X,Ω
k − k
we obtain
b b
N
|DαAi,k(x)|6C |Dαwj(x)|6CC#Cαh−X|,αΩ|.
j=1
X
b b
This completes the proof.
Since A(x) is positive definite for all x Ω, all eigenvalues are real and positive. If
∈
thebasis(3.8)isemployed,wecanprovethatthesmallesteigenvalueofA(x)hasalower
bound away from zero and independent of h . Proving this assertion helps us to find
X,Ω
a bound for DαA 1(x). First, recall
−
| |
vTA(x)v
λ (A(x))= min , (3.13)
min v RQ 0 vTv
∈ \{ }
for symmetric matrix A(x). Since A(x) is also positive definite, we necessarily have
λ (A(x))>0. To bound λ we followsome parts ofMelenk’s argument[21]and the
min min
concept of norming sets presented in Appendix .1.
Lemma 3.7. Suppose that the bounded set Ω Rd satisfies an interior cone condition
⊂
with radius r and angle θ (0,π/2). Let X = x ,...,x Ω has fill distance
1 N
∈ { } ⊆
h = h and satisfies h 6 r/C =: h where C is defined in (3.6). Suppose that
X,Ω 2 0 2
δ = 2C h is the size of supports of the weight functions, and for a fixed but arbitrary
2
x in Ω the set x X : j J(x) is Pd -unisolvent. Then there exists constant C
{ j ∈ ∈ } m λ
independent of h such that
b λb (A(x))>C >0,
min λ
provided that the shifted scaled basis functions (3.8) are employed.
b
Proof. Let v RQ, v = 0, be a vector at which the minimum in (3.13) occurs for
∗ ∗
∈ 6
x=x. Define
(x x)α
π(x):= vα∗ h−α . (3.14)
b |αX|6m | b|
Now using (3.11) we simply have
N
v TA(x)v = w (x)π2(x ).
∗ ∗ j j
j=1
X
b b
9
Since x : j J(x) is Pd -unisolvent, the functionals Z = δ : j J(x) form a
{ j ∈ } m { xj ∈ }
norming set for Pd , i.e. there exists an injective mapping T : Pd T(Pd ) R#J(xb),
m m → m ⊆
where T(p) = p(xb) (See Appendix .1). Set Z allows us to equip Pbd with an
j j J(xb) m
equivalent norm(cid:0) via t(cid:1)he∈operator T. We define the norm on R#J(xb) by
p 2 := w (x)p2(x ),
k k2,w j j
j J(xb)
∈X
b
and the norm on Pd by the infinity norm. Using the properties of norming sets and
m
setting p=π, we have
1
kπk2,w > kT−1kkπk∞,B(xb,δ).
Obviously,the set ∂B(x,h)= x: x x =h is a subset of B(x,δ). Definition of π in
{ k − k }
(3.14) ensures that the values of π on ∂B(x,h) are independent of h, because h will be
canceled from the numberators and the bdenominators. In fact πb ,∂B(xb,h) is bounded
k k∞
from below by a positive factor times v b:= v , and thus we have
k ∗k1 |α|6m| α∗|
P
π ,B(xb,δ) > π ,∂B(xb,h) >Cπ v∗ 1 >Cπ v∗ 2,
k k∞ k k∞ k k k k
where C is the mentioned factor which is independent of h. The last inequality follows
π
from the standard relations between one and two norms in RQ.
It remains to bound T 1 . By assumptions, Ω satisfies a cone condition with angle
−
k k
θ and radius r, and h 6 r/C . The later gives δ/2 6 r. Of course the cone condition
2
will be obeyedif we useany radiuslessthanr. Thus for everyx Ω, there existsa cone
∈
C(x) = C(x,ξ,θ,δ/2) Ω B(x,δ/2), and using Lemma Appendix .2 there exists a
⊂ ∩
closed ball b
1 sinθ
b b B =B(xb,ρδ) C(x), ρ= .
⊂ 21+sinθ
We are going to prove
e e b
kpk∞,B(xb,δ) 6C1kpk∞,Be 6C|p(xk)|, xk ∈B ⊂B(x,δ/2), (3.15)
forallp∈Pdm. UsingLemmaAppendix .4,thefirstinequaleitysatisbfieswithC1 = ρ2 m.
To prove the second, let kpk ,Be = |p(xM)|, for xM ∈ B. Since ball B itself sa(cid:0)tis(cid:1)fies
∞
an interior cone condition with radius ρδ and angle π/3, Theorem Appendix .3 can be
appliedprovidedthath6 ρδ√3/2 . Onecaneasilycheeckthatthisconeditionisalways
4(1+√3/2)m2
satisfiedbecauseδ =2C h= 32h(1+sinθ)2m2. TheproofofthementionedTheoremshows
2 3sin2θ
that there exists a point xk ∈X ∩B such that |p(xk)|> 12|p(xM)|= 12kpk ,Be. Thus in
(3.15) the constant C can be chosen as C :=2(4(1+sinθ))m. Letting ∞
θ,m sinθ
e
w := min φ(s),
min
s [0,1/2]
∈
10
