B-Spline-Based Monotone Multigrid Methods Markus Holtz, Angela Kunoth no. 252

25 Pages·2006·0.37 MB·English

B-Spline-Based Monotone Multigrid Methods

Markus Holtz, Angela Kunoth

no. 252

Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungs-
gemeinschaft getragenen Sonderforschungsbereiches 611 an der Univer-
sität Bonn entstanden und als Manuskript vervielfältigt worden.
Bonn, Dezember 2005
B{SPLINE{BASED MONOTONE MULTIGRID METHODS(cid:3)
MARKUS HOLTZ AND ANGELA KUNOTHy
Abstract. Forthee(cid:14)cientnumericalsolutionofellipticvariationalinequalitiesonclosedconvex
sets,multigridmethodsbasedonpiecewiselinear(cid:12)niteelementshavebeeninvestigatedoverthepast
decades. Essentialfortheirsuccessistheappropriateapproximationoftheconstraintsetoncoarser
gridswhichisbasedonfunctionvaluesforpiecewiselinear(cid:12)niteelements. Ontheotherhand,there
are a number of problems which pro(cid:12)t from higher order approximations. Among these are the
problem of prizing American options, formulated as a parabolic boundary value problem involving
Black{Scholes’ equation with a free boundary. In addition to computing the free boundary, the
optimal exercise prize of the option, of particular importance are accurate pointwise derivatives of
thevalueofthestockoptionuptoordertwo,theso{calledGreekletters.
Inthispaper,weproposeamonotonemultigridmethodfordiscretizationsintermsofB{splines
ofarbitraryordertosolveellipticvariationalinequalitiesonaclosedconvexset. Inordertomaintain
monotonicity (upper bound) and quasi{optimality (lower bound) ofthe coarse gridcorrections, we
proposeanoptimizedcoarsegridcorrection(OCGC)algorithmwhichisbasedonB{splineexpansion
coe(cid:14)cients. WeprovethattheOCGCalgorithmisofoptimalcomplexityofthedegreesoffreedomof
thecoarsegridand,therefore,theresultingmonotonemultigridmethodisofasymptoticallyoptimal
multigridcomplexity.
Finally, the method is applied to a standard model for the valuation of American options. In
particular,itisshownthat adiscretization based onB{splinesoforderfourenables ustocompute
thesecondderivativeofthevalueofthestockoptiontohighprecision.
Keywords. Variationalinequality,linearcomplementaryproblem,monotonemultigridmethod,
cardinalhigherorderB{spline,systemoflinearinequalities,optimizedcoarsegridcorrection(OCGC)
algorithm,optimalcomplexity,convergence rates,Americanoption,Greekletters,highprecision.
AMS subject classi(cid:12)cations. 65M55,35J85,65N30,65D07.
1. Introduction. The motivation for this paper stems from an application in
Mathematical Finance, the fair prizing of American options. In a standard model,
this problem can be formulated as a parabolic boundary value problem involving
Black{Scholes’equation[BS]with afreeboundary. Inadditiontocomputingthe free
boundary(theoptimalexerciseprizeoftheoption),pointwisehigherorderderivatives
of the solution (the value of the stock option) are particularly important. These so{
calledGreeklettersareneededwithhighprecisionastheyplayacrucialroleashedge
parameters in the analysis of market risks. Thus, a discretization in terms of higher
order basis functions is preferable.
Ontheotherhand,forthefastnumericalsolutionoftheresulting(semi{discrete)
ellipticvariationalinequality,themethodofchoiceisthemonotonemultigridmethod
developed in [Ko1, Ko2]. Multigrid methods have been proposed previously for such
problems using second order discretizations (i.e., standard (cid:12)nite di(cid:11)erence stencils
or piecewise linear (cid:12)nite elements) in di(cid:11)erent variants [BC, HM, Ho, Ma] where,
however,notallof themhaveassuredconsequentlythat the obstaclecriterionismet.
Usingpiecewiselinear(cid:12)niteelementansatzfunctions,geometricconsiderationsbased
on point values are used in [Ko1] to represent the problem{inherent obstacles on
coarsergridsin suchawaythat aviolationof the obstacleisexcluded. The di(cid:14)culty
tocorrectlyidentifyingcoarsegridapproximationshasalsobeen the motivationfora
(cid:3) Thisworkhasbeen supportedinpartbytheDeutsche Forschungsgemeinschaft SFB611,Uni-
versita(cid:127)tBonn.
yInstitut fu(cid:127)r Numerische Simulation, Universita(cid:127)t Bonn, Wegelerstr. 6, 53115 Bonn, Germany,
fholtz,[email protected], www.ins.uni-bonn.de/(cid:24)kunoth.
1
2 MarkusHoltzandAngelaKunoth
cascadic multigrid algorithm for variationalinequalities in [BBS] for which, however,
no convergencetheory is yet available.
In this paper, we generalize the monotone multigrid (MMG) method from [Ko1,
Ko2] to discretizations involving higher order B{splines. One of the key ingredients
of an MMG method are restrictions of the obstacle to coarsergrids which satisfy the
(upper) bound imposed by the obstacle (monotonicity) as well as a lower one which
corresponds to the condition of quasi{optimality in [Ko1]. We formulate the con-
struction of coarse grid approximationsas a linear constrained optimization problem
with respect to the B{spline expansion coe(cid:14)cients. Our construction heavily pro(cid:12)ts
from properties of B{splines [Bo, Sb]. In particular, we present with our optimized
coarsegridcorrection(OCGC)algorithmamethodtoconstructmonotoneandquasi{
optimal coarsegrid approximationsto the obstaclefunction in optimal complexity of
the coarse grid for B{spline basis functions of any degree.
Building the OCGC scheme into the MMG method, our higher-order MMG
method is shown to be of optimal multigrid complexity. Moreover, following the
argumentsin[Ko1],wecanprovethatourmethodisgloballyconvergentandreduces
asymptotically to a linear subspace correction method once the contact set has been
identi(cid:12)ed [HzK]. Hence, we can expect particular robustness of the scheme and full
multigrid e(cid:14)ciency in the asymptotic range in the numerical experiments. This is
con(cid:12)rmedbycomputationsforanAmericanoptionpricingproblemin terms of cubic
B{splines. Details about the derivation of the problem of fair prizing American op-
tionsandits formulationasafreeboundaryvalueproblem andcorrespondingresults
can be found in [WHD, Hz2]. Of course, once higher-oder MMG methods are avail-
able, they may be applied to other obstacle problems like Signorini’s problem which
has been solved using piecewise linear hat functions in [Kr].
Thispaperisstructuredasfollows. InSection2weintroducemonotonemultigrid
methods (MMG), recollect the main features of B{splines and specify a B{spline{
based projected Gauss{Seidel relaxation as smoothing component of the scheme. In
Section3thecrucialingredientsofthehigher-orderMMGschemes,suitablerestriction
operators for the obstacle function, are presented for B{spline functions of arbitrary
degreein the univariatecase. Theirconstructionforhigherspatialdimensionsis pre-
sentedinSection4usingtensorproducts. InSection5someshortremarksconcerning
the convergence theory for B{spline{based monotone multigrid schemes are made.
Finally, in Section 6 we present a numerical example of prizing American options.
Theconvergencebehaviorofthe projectedGauss{Seidelandthe multigridschemesis
compared for basis functions of di(cid:11)erent orders. We conclude with an estimation of
asymptotic multigrid convergencerates which exhibit full multigrid e(cid:14)ciency for the
truncated version.
2. Monotone Multigrid Methods.
2.1. Elliptic Variational Inequalities and Linear Complementary Prob-
lems. Let (cid:10) be a domain in Rd and J(v) := 1a(v;v) (cid:0) f(v) a quadratic func-
2
tional induced by a continuous, symmetric and H1{ elliptic bilinear form a((cid:1);(cid:1)) :
0
H1((cid:10))(cid:2)H1((cid:10))!R and a linear functional f :H1((cid:10))!R. As usual, H1((cid:10)) is the
0 0 0 0
subspace of functions belonging to the Sobolev space H1((cid:10)) with zero trace on the
boundary. We consider the constrained minimization problem
(cid:12)nd u2K: J(u)(cid:20)J(v) for all v 2K (2.1)
B{Spline{BasedMonotoneMultigridMethods 3
on the closed and convex set
K:=fv 2H1((cid:10)): v(x)(cid:20)g(x) for all x2(cid:10)g(cid:26)H1((cid:10)):
0 0
The function g 2 H1((cid:10)) represents an upper obstacle for the solution u 2 H1((cid:10)).
0 0
Lower obstacles can be treated in the obvious analogous way. If g satis(cid:12)es g(x) (cid:21) 0
for all x 2 @(cid:10), problem (2.1) admits a unique solution u 2 K by the Lax{Milgram
theorem. It is well{knownthat (2.1)can be rewritten as a variationalinequality, see,
e.g., [EO, KS]: (cid:12)nd u2K: a(u;v(cid:0)u)(cid:21)f(v(cid:0)u) for all v 2K or, equivalently, as
a linear complementary problem
Lu (cid:21) f;
u (cid:20) g; (2.2)
(u(cid:0)g)(Lu(cid:0)f) = 0
almost everywhere in (cid:10). Here L : H1((cid:10)) ! H(cid:0)1(= (H1((cid:10)))0) is the Riesz operator
0 0
de(cid:12)ned by hLu;vi:=a(u;v) for all v 2H1((cid:10)).
0
Discretizingin a(cid:12)nite dimensionalsplinespaceS of piecewise polynomialsona
L
grid (cid:1) with uniform grid spacing h leads to the discrete formulation of (2.1),
L L
(cid:12)nd u 2K : J(u )(cid:20)J(v ) for all v 2K (2.3)
L L L L L L
on the closed and convex set K :=fv 2 S : v (x) (cid:20)g (x) for all x 2 (cid:10)g(cid:26)S ;
L L L L L L
or, equivalently,
L u (cid:21) f ;
L L L
u (cid:20) g ; (2.4)
L L
(u (cid:0)g )(L u (cid:0)f ) = 0:
L L L L L
In[BHR]regularityu2H5=2(cid:0)(cid:15)((cid:10))ofthesolutionuto(2.2)isshownforarbitrary
(cid:15) > 0. Moreover, error estimates ku(cid:0)u k = O(h ) and ku(cid:0)u k =
L H1((cid:10)) L L H1((cid:10))
O(h3=2(cid:0)(cid:15))areprovedinthe caseof piecewiselinear,respectivelypiecewise quadratic,
L
functions, provided the functions f;g are su(cid:14)ciently regular.
2.2. The MMG{algorithm. For solving (2.3) numerically, a by now popular
methodisthemonotonemultigridmethod(MMG)[Ko1]. Byaddingaprojectionstep
and employing speci(cid:12)c restriction operators, it can be implemented as a variant of a
standardmultigridscheme. LetS (cid:26)S (cid:26):::(cid:26)S (cid:26)H1((cid:10))beanestedsequenceof
1 2 L 0
(cid:12)nite{dimensionalspaces,andletu(cid:23) 2S betheapproximationinthe(cid:23){thiteration
L L
of the MMG method. The basic multigrid idea is that the error v := u (cid:0)u(cid:23);1
L L L
between the smoothed iterate u(cid:23);1 := S(u(cid:23)) (S always being the standard Gauss{
L L
Seideliteration)andtheexactsolutionu canbeapproximatedwithoutessentialloss
L
of information on a coarsergrid (cid:1) . We explain how this is realized in the case of
L(cid:0)1
a linear complementary problem for two grids (cid:1) and (cid:1) . Introducing the defect
L L(cid:0)1
d :=f (cid:0)L u(cid:23);1, (2.4) can be written as
L L L L
L v (cid:21) d ;
L L L
v (cid:20) g (cid:0)u(cid:23);1; (2.5)
L L L
(v (cid:0)g +u(cid:23);1)(L v (cid:0)d ) = 0:
L L L L L L
4 MarkusHoltzandAngelaKunoth
On a coarsergrid (cid:1) the defect problem can now be approximated by
L(cid:0)1
L v (cid:21) d ;
L(cid:0)1 L(cid:0)1 L(cid:0)1
v (cid:20) g ;
L(cid:0)1 L(cid:0)1
(v (cid:0)g )(L v (cid:0)d ) = 0;
L(cid:0)1 L(cid:0)1 L(cid:0)1 L(cid:0)1 L(cid:0)1
where d := rd and g := r~(g (cid:0)u(cid:23);1) with (di(cid:11)erent) restriction operators
L(cid:0)1 L L(cid:0)1 L L
r;r~ : S ! S . The solution v of the coarse grid problem is then used as
L L(cid:0)1 L(cid:0)1
an approximation to the error v . It is (cid:12)rst transported back to the (cid:12)ne grid by a
L
prolongationoperatorpand isthen addedto the approximationu(cid:23);1. It isimportant
L
that the restriction r~is chosen such that the new iterate satis(cid:12)es the constraint
u(cid:23);2 :=u(cid:23);1+pv (cid:20)g (2.6)
L L L(cid:0)1 L
on the (cid:12)ne grid. Applying this idea recursivelyon severaldi(cid:11)erent grids,one obtains
the monotone multigrid method (MMG) for linear complementary problems.
Algorithm 2.1. MMG ((cid:23){ th cycle on level ‘(cid:21)1)
‘
Let u(cid:23) 2S be a given approximation.
‘ ‘
1. A priori smoothing and projection : u(cid:23);1 :=(P(cid:14)S(u(cid:23)))(cid:17)1.
‘ ‘
2. Coarse grid correction: d :=r(f (cid:0)L u(cid:23);1),
‘(cid:0)1 ‘ ‘ ‘
g :=r~(g (cid:0)u(cid:23);1),
‘(cid:0)1 ‘ ‘
L :=rL p.
‘(cid:0)1 ‘
If ‘=1, solve exactly the linear complementary problem
L v (cid:21) d ;
‘(cid:0)1 ‘(cid:0)1
v (cid:20) g ;
‘(cid:0)1
(v(cid:0)g )(L v(cid:0)d ) = 0
‘(cid:0)1 ‘(cid:0)1 ‘(cid:0)1
and set v :=v.
‘(cid:0)1
If ‘>1, do (cid:13) steps of MMG with initial value u0 :=0 and solution v .
‘(cid:0)1 ‘(cid:0)1 ‘(cid:0)1
Set u(cid:23);2 :=u(cid:23);1+pv .
‘ ‘ ‘(cid:0)1
3. A posteriori smoothing and projection : u(cid:23);3 :=(P(cid:14)S(u(cid:23);2))(cid:17)2:
‘ ‘
Set u(cid:23)+1 :=u(cid:23);3:
‘ ‘
The number of a priori and a posteriori smoothing steps is denoted by (cid:17) and (cid:17) ,
1 2
respectively. For (cid:13) = 1 one obtains a V{cycle, for (cid:13) = 2 a W{cycle. P denotes a
projection operator de(cid:12)ned in (2.7) and (2.11) below.
Condition(2.6)leadstoaninnerapproximationofthesolutionsetK andensures
L
that the multigrid scheme is robust [Ko1]. Striving for optimal multigrid e(cid:14)ciency,
satisfaction of the constraint should not be checked by interpolating v back to the
‘
(cid:12)nestgrid. Instead,specialrestrictionoperatorsr~areneededfortheobstaclefunction.
A corresponding construction for B{splines of general order k will be introduced in
Sections 3 and 4. Next we discuss the projection step for general order B{splines.
2.3. A B{Spline{Based Projected Gauss{Seidel Scheme. Since the op-
erator L is symmetric positive de(cid:12)nite and continuous piecewise linear functions are
used for discretization,the discrete form (2.4) canbe solvedbythe projected Gauss{
Seidel scheme, see, e.g., [Cr]. Given an iterate u(cid:23), a standard Gauss{Seidel sweep
L
u(cid:22)(cid:23) :=S(u(cid:23))issupplemented byaprojectionu(cid:23)+1 =Pu(cid:22)(cid:23) intothe convexsetK . If
L L L L L
B{Spline{BasedMonotoneMultigridMethods 5
S consists of hat functions, the projection can be de(cid:12)ned for given grid points f(cid:18) g
L i i
by
Pv ((cid:18) ):=minfv ((cid:18) ); g ((cid:18) )g: (2.7)
L i L i L i
Forhigher{orderfunctions v , the di(cid:14)cultyarisesalreadyinthe univariatecasethat
L
for given x2[(cid:18) ;(cid:18) ] the estimate
i i+1
minfv ((cid:18) );v ((cid:18) )g(cid:20)v (x)(cid:20)maxfv ((cid:18) );v ((cid:18) )g (2.8)
L i L i+1 L L i L i+1
is not valid any more. Thus, controlling function values on grid points is not a
su(cid:14)cient criterion in this case. We propose here instead a construction using higher
order B{splines, which compares B{spline expansion coe(cid:14)cients instead of function
values and heavily pro(cid:12)ts from the fact that B{splines are nonnegative. We begin
with the univariate case. For readers’ convenience, we recall the relevant facts about
B{spline bases from [Bo].
Definition 2.2 (B{Spline Basis Functions). For k 2 N and n 2 N let T :=
f(cid:18) g be an expanded knot sequence with uniform grid spacing h in the
i i=1;:::;n+k L
interior of the interval I :=[a;b] of the form
(cid:18) =:::=(cid:18) =a<(cid:18) <:::<(cid:18) <b=(cid:18) =:::=(cid:18) : (2.9)
1 k k+1 n n+1 n+k
Then the B{spline basis functions N of order k are recursively de(cid:12)ned for i =
i;k
1;:::;n by
1; if x2[(cid:18) ;(cid:18) )
N (x)= i i+1 ;
i;1 0; else
(cid:26) (2.10)
x(cid:0)(cid:18) (cid:18) (cid:0)x
i i+k
N (x)= N (x)+ N (x)
i;k i;k(cid:0)1 i+1;k(cid:0)1
(cid:18) (cid:0)(cid:18) (cid:18) (cid:0)(cid:18)
i+k(cid:0)1 i i+k i+1
for x2I.
It is known that suppN (cid:18) [(cid:18) ; (cid:18) ] (local support), N (x) (cid:21) 0 for all x 2 I
i;k i i+k i;k
(nonnegativity) and N 2Ck(cid:0)2(I) (di(cid:11)erentiability) holds. Moreoverthe set (cid:6) :=
i;k L
fN ;:::;N g constitutes a locally independent and unconditionally stable basis
1;k n;k
with respect to k(cid:1)k , 1 (cid:20) p (cid:20) 1, for the (cid:12)nite dimensional space S = N :=
Lp L k;T
span(cid:6) of the splines of order k.
L
Lemma 2.3. If the B{spline coe(cid:14)cients of v ;g 2N =S satisfy v (cid:20)g for
L L k;T L i i
all i=1;:::;n, then v (x)(cid:20)g (x) holds for all x2I.
L L
Proof. Usingtherepresentationv = n v N andg = n g N andthe
L i=1 i i;k L i=1 i i;k
nonnegativity N (x) (cid:21) 0 for all x 2 I, we deduce that g (x)(cid:0)v (x) = n (g (cid:0)
i;k P L PL i=1 i
v )N (x)(cid:21)0 for all x2I:
i i;k
P
Here and below in Section 5, we use the subscript i in v = (v ) to denote
i L i
B{spline expansion coe(cid:14)cients.
Theprojectioncannowbede(cid:12)nedforB{splinefunctionsofgeneralorderksimilar
to (2.7) but now involving expansion coe(cid:14)cients by setting
Pv :=minfv ; g g: (2.11)
i i i
Usingthesameargumentsasin[Cr],theresultingprojectedGauss{Seidelschemestill
convergessincethediscretesolutionsetfv2IRn : v (cid:20)g fori=1;:::;ngdescribesa
i i
cuboidinIRn. Moreover,iftheproblemisnon{degenerate,thecontactset,de(cid:12)nedby
6 MarkusHoltzandAngelaKunoth
allcoe(cid:14)cientsforwhichequalityholds,isidenti(cid:12)edaftera(cid:12)nitenumberofiterations
[Cr, EO].
We treat the multivariate case by takingtensor products. Specifying the domain
(cid:10) as (cid:10):= d [a ;b ](cid:26)Rd, the i-th d-dimensional tensor product B{spline of order
‘=1 ‘ ‘
k on a tensorized extended knot sequence T(d) is de(cid:12)ned by
Q
d
N(d)(x):= N (x ); x2(cid:10); (2.12)
i;k i‘;k ‘
‘=1
Y
wherei:=(i ;:::;i )denotesamulti-index. De(cid:12)ningS inanalogytotheunivariate
1 d L
case, the result of Lemma 2.3 immediately carriesover to the d-dimensional setting.
3. Constructionof Monotoneand Quasi{optimalObstacle Approxima-
tions. In this section, the second essential ingredient for our B{spline{based mono-
tonemultigridmethodsisprovided,theconstructionofso{calledmonotoneandquasi{
optimal coarse grid approximations of the obstacle function, which lead to suitable
restriction operators r~. We begin with the univariate case; the extension to d di-
mensions follows in Section 4. We consider in the following only two grids, as the
generalization to several grids is obvious. Given an obstacle function S~ which is de-
(cid:12)ned on a (cid:12)ne grid (cid:1)(cid:26)I, we provide an approximation S with respect to a coarser
grid T which satis(cid:12)es
1. S(x)(cid:20)S~(x) for all x2I;
2. S(x) (cid:21) L (x) for all x 2 I and a still to be speci(cid:12)ed lower barrier L (x)
k k
provided below in Section 3.2;
3. S (cid:25)S~ with respect to a target functional F de(cid:12)ned below in (3.10).
k
The (cid:12)rst condition ensuresthe monotonicity androbustnessof the multigridscheme,
thesecondanasymptoticalreductionofthemethodtoalinearrelaxationandthethird
an e(cid:14)cient coarsegrid correction. As the construction is used as a component of the
monotonemultigridscheme,strivingforoptimalcomputationalmultigridcomplexity,
it also has to satisfy
4. the number of arithmetic operations must be of order O(n) where n denotes
the number of degrees of freedom on the coarse grid.
Speci(cid:12)cally, let T be an extended knot sequence with grid spacing H as in (2.9) and
let (cid:1):=f(cid:18)~g be a (cid:12)ner knot sequence
i i=1;:::;n~+k
(cid:18)~ =:::=(cid:18)~ =a<(cid:18)~ <:::<(cid:18)~ <b=(cid:18)~ =:::=(cid:18)~ (3.1)
1 k k+1 n~ n~+1 n~+k
with grid spacing h= 1H. It is de(cid:12)ned such that (cid:18) =(cid:18)~ for i=k;:::;n+1 and
2 i 2i(cid:0)k
1((cid:18) +(cid:18) )=(cid:18)~ for i=k+1;:::;n+1. Then it holds
2 i(cid:0)1 i 2i(cid:0)k(cid:0)1
n~ =2n+1(cid:0)k: (3.2)
The corresponding spline spaces are N and N with member functions N
k;(cid:1) k;T i;k;(cid:1)
andN ,respectively. Letnowthe obstaclefunction onthe (cid:12)negrid S~2N and
i;k;T k;(cid:1)
its approximationS 2N be expanded as
k;T
n~ n
S~= c~ N =:~cT N ; S = c N =:cT N : (3.3)
i i;k;(cid:1) k;(cid:1) i i;k;T k;T
i=1 i=1
X X
There is a natural prolongation operator p from N to N for B{splines N in
k;T k;(cid:1) i;k;T
terms of their re(cid:12)nement or mask coe(cid:14)cients [Bo, Sb]. In the special case H = 2h
B{Spline{BasedMonotoneMultigridMethods 7
considered here the re(cid:12)nement relation is given by
k
N = a N (3.4)
i;k;T j 2i(cid:0)k+j;k;(cid:1)
j=0
X
with the subdivision or mask coe(cid:14)cients
k
a :=21(cid:0)k for j =0;:::; k: (3.5)
j
j
(cid:18) (cid:19)
In Step 2 of Algorithm 2.1, we choose the restriction r as the adjoint of p, following
[Ha]. However, for the obstacle function the restriction operator r cannot be used
since it does not satisfy condition (2.6).
3.1. Monotone Coarse Grid Approximations. There is a vast amount of
literature, see, e.g., [DV, Mv, Pi] especially from approximation theory, dealing with
monotone approximations to a given function g. The function g^ is a monotone (or
one{sided) lower approximation to g if g^(x) (cid:20) g(x) for all x 2 I. There the number
n of degrees of freedom of the function g^ is chosen such that a given approximation
accuracy can be reached. In contrast to these studies, the question here is di(cid:11)erent,
since the number n of degrees of freedom is given by the mesh size H.
Definition 3.1 (Monotone CoarseGrid Approximation). For knot sequences T
and (cid:1) from (2.9) and (3.1), respectively, we call S 2 N a monotone lower coarse
k;T
grid approximation to S~2N if S(x)(cid:20)S~(x) holds for all x2I.
k;(cid:1)
For hat functions such approximations are constructed in [Ma, Ko1]. A cor-
responding construction for higher{order functions has to our knowledge not been
providedsofar. In viewof Lemma2.3weproposeheretocontrolB{splineexpansion
coe(cid:14)cients.
Theorem 3.2 (Monotone Coarse Grid Approximation). Let S~ 2 N be an
k;(cid:1)
upper obstacle with S~ = ~cT N for a given order k and the knot sequence (cid:1) from
k;(cid:1)
(3.1). Then S 2 N with S = cT N de(cid:12)ned on the knot sequence T from (2.9)
k;T k;T
is a monotone lower coarse grid approximation to S~ if the inequality system
A c(cid:20)~c (3.6)
k
is satis(cid:12)ed. The two{slanted matrix A is de(cid:12)ned by
k
a a
0 ak(cid:0)1 ak(cid:0)3 ... 1
k k(cid:0)2
BB ak(cid:0)1 a0 CC
B C
BB ak a1 CC
B .. C
B . a C
Ak :=BBB ...2 CCC2Rn~(cid:2)n
BBB a ... CCC
B k(cid:0)1 C
B a C
B k C
BB ... a CC
B 0 C
B C
@ a1 A
with the subdivision coe(cid:14)cients a from (3.5) and has maximal rank.
j
8 MarkusHoltzandAngelaKunoth
Proof. The proof relies on the subdivision property (3.4) and on the nonnega-
tivity of B{splines. We only consider the case k even as the other case is analogous.
Substituting (3.4) into (3.3) and sortingaccordingto the basisfunctions N leads
i;k;(cid:1)
to
n~
S(x) = a c +a c +:::+a c N (x)
k(cid:0)1 (i+1)=2 k(cid:0)3 (i+3)=2 1 (i+k(cid:0)1)=2 i;k;(cid:1)
i=1
iXodd(cid:0) (cid:1)
n~(cid:0)1
+ a c +a c +:::+a c N (x);
k i=2 k(cid:0)2 (i+2)=2 0 (i+k)=2 i;k;(cid:1)
i=2
iXeven(cid:0) (cid:1)
where all c with j <1 or j >n are treated as zero. De(cid:12)ning the coe(cid:14)cients
j
c~ (cid:0) a c +a c +:::+a c ; if i is odd;
i k(cid:0)1 (i+1)=2 k(cid:0)3 (i+3)=2 1 (i+k(cid:0)1)=2
d :=
i
( c~i(cid:0)(cid:0)akci=2+ak(cid:0)2c(i+2)=2+:::+a0c(i+k)=2 ; (cid:1) if i is even;
which can be wr(cid:0)itten in compact matrix/vectorform as (cid:1)
d =c~ (cid:0)(A c) (3.7)
i i k i
(involving the ith component of the vector A c), we obtain
k
n~
S~(x)(cid:0)S(x)= d N (x): (3.8)
i i;k;(cid:1)
i=1
X
By Lemma 2.3 we have S~(x)(cid:0)S(x) (cid:21) 0 for all x 2 I, provided d (cid:21) 0 holds for all
i
i=1;:::;n~. By(3.7),weobtaintheinequalitysystem(3.6). SincetheB-splinesform
bases for N and N , the matrix A has full rank for each k.
k;T k;(cid:1) k
Example 3.3. In the special case of continuous, piecewise linear functions
(k =2), C1{smooth, piecewise quadratic (k =3) and C2{smooth, piecewise cubic
(k =4) splines one has
1 3 1
0 1 1 1 0 1 3 1
2 2
A2 =BBBBBBBBBBBBBB .112.. 12 11212 21CCCCCCCCCCCCCC2R(2n(cid:0)1)(cid:2)n; A3 = 14BBBBBBBBBBBBB 31...13 31...133 1 CCCCCCCCCCCCC2R(2n(cid:0)2)(cid:2)n;
@ 1A @ 1 3 A
4 4
0 1 6 1 1
B 4 4 C
B C
B 1 6 1 C
A4 = 81BBBB ... ... CCCC2R(2n(cid:0)3)(cid:2)n:
B 1 6 1 C
B C
B 4 4 C
B C
B 1 6 1 C
B C
@ 4 4 A
B{Spline{BasedMonotoneMultigridMethods 9
3.2. Quasi{optimal Coarse Grid Approximations. Now we can immedi-
ately derive a monotone lower coarse approximation.
Proposition 3.4. The spline L :=qT N 2N with coe(cid:14)cients
k k;T k;T
q :=minfc~ ; :::; c~ g for i=1;:::;n (3.9)
i 2i(cid:0)k 2i
(leaving out c~ in the right hand side if j < 1 or j > n~) is a monotone lower coarse
j
grid approximation to S~=~cT N 2N .
k;(cid:1) k;(cid:1)
Proof. AsallrowsumsofA areequaltoone,thevectorq:=(q ;:::;q )T de(cid:12)ned
k 1 n
in(3.9)obviouslysatis(cid:12)estheinequalitysystemA q(cid:20)~csothattheassertiondirectly
k
follows from Theorem 3.2.
Remark3.5. Inthespecialcasek =2,therestrictionoperatorr^: N !N ,
2;(cid:1) 2;T
S~7!L inducedby Proposition 3.4 coincides with therestrictionoperator from [Ma].
2
As it is illustrated in Figure 3.1 and 3.2 for the cases k = 2 and k = 3, the
approximationL canbefurther improvedin manycases. Thiswill bethe subjectof
k
the next subsections: there q is interpreted as a componentwise lower barrier for the
B{spline coe(cid:14)cients c of the desired coarse grid approximation.
Definition 3.6 (Quasi{optimalCoarseGrid Approximation). We call a mono-
tone lower coarse grid approximation S =cT N to the spline S~=~cT N quasi{
k;T k;(cid:1)
optimal if it is an improvement over L in the sense that c(cid:21)q holds with q de(cid:12)ned
k
in (3.9).
3.3. A Linear Optimization Problem. Aiming at improving the coarse grid
approximation L from Proposition 3.4, we de(cid:12)ne an optimal monotone and quasi{
k
optimalcoarsegridapproximationS =cT N toagiven S~=~cT N byformulat-
k;T k;(cid:1)
ing a linear optimization problem. We choose a target functional F which estimates
k
thesumofthedistancesfromapproximationtoobstacleonallcoarsegridpoints,i.e.,
F (c):= jS~((cid:18))(cid:0)S((cid:18))j: (3.10)
k
(cid:18)2T
X
Lemma 3.7. The function F de(cid:12)ned in (3.10) is a linear function Rn ! R of
k
the form
F (c)=(cid:24)Tc+(cid:17) (3.11)
k
where
(cid:24) :=(cid:0)ATs 2Rn; s :=((cid:12) ;(cid:13) ;(cid:12) ;:::)T 2Rn~ and (cid:17) :=sT~c2R: (3.12)
k k k k k k k
The values (cid:12) and (cid:13) can be computed explicitly: for odd k we have (cid:12) =(cid:13) = 1, and
k k k k 2
for even k=2;4;6;8 the values are displayed in Table 3.1.
k 2 4 6 8
(cid:12) 1 2 17 166
k 3 30 315
(cid:13) 0 1 13 149
k 3 30 315
Table 3.1
The values (cid:12)k and (cid:13)k fororders k=2;4;6;8.