B-SPLINE-BASED MONOTONE MULTIGRID METHODS

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1 SIAM J NUMER ANAL Vo 45, No 3, pp c 2007 Society for Industria and Appied Mathematics B-SPLINE-BASED MONOTONE MULTIGRID METHODS MARKUS HOLTZ AND ANGELA KUNOTH Abstract For the efficient numerica soution of eiptic variationa inequaities on cosed convex sets, mutigrid methods based on piecewise inear finite eements have been investigated over the past decades Essentia to their success is the appropriate approximation of the constraint set on coarser grids which is based on function vaues for piecewise inear finite eements On the other hand, there are a number of probems which profit from higher order approximations Among these are the probem of pricing American options, formuated as a paraboic boundary vaue probem invoving Back Schoes equation with a free boundary In addition to computing the free boundary (the optima exercise price of the option) of particuar importance are accurate pointwise derivatives of the vaue of the stock option up to order two, the so-caed Greek etters In this paper, we propose a monotone mutigrid method for discretizations in terms of B-spines of arbitrary order to sove eiptic variationa inequaities on a cosed convex set In order to maintain monotonicity (upper bound) and quasi optimaity (ower bound) of the coarse grid corrections, we propose an optimized coarse grid correction (OCGC) agorithm which is based on B-spine expansion coefficients We prove that the OCGC agorithm is of optima compexity of the degrees of freedom of the coarse grid and, therefore, the resuting monotone mutigrid method is of asymptoticay optima mutigrid compexity Finay, the method is appied to a standard mode for the vauation of American options In particuar, it is shown that a discretization based on B-spines of order four enabes us to compute the second derivative of the vaue of the stock option to high precision Key words variationa inequaity, inear compementary probem, monotone mutigrid method, cardina higher order B-spine, system of inear inequaities, optimized coarse grid correction agorithm, optima compexity, convergence rates, American option, Greek etters, high precision AMS subject cassifications 65M55, 35J85, 65N30, 65D07 DOI 037/ Introduction The motivation for this paper stems from an appication in Mathematica Finance, the fair pricing of American options In a standard mode, this probem can be formuated as a paraboic boundary vaue probem invoving Back Schoes equation [BS] with a free boundary In addition to computing the free boundary (the optima exercise price of the option), pointwise higher order derivatives of the soution (the vaue of the stock option) are particuary important These socaed Greek etters are needed with high precision as they pay a crucia roe as hedge parameters in the anaysis of market risks Thus, a discretization in terms of higher order basis functions is preferabe On the other hand, for the fast numerica soution of the resuting (semidiscrete) eiptic variationa inequaity, the method of choice is the monotone mutigrid method deveoped in [Ko, Ko2] Mutigrid methods have been proposed previousy for such probems using second order discretizations (ie, standard finite difference stencis or piecewise inear finite eements) in different variants [BC, HM, Ho, Ma] where, however, not a of them have assured, consequenty, that the obstace criterion is met Using piecewise inear finite eement ansatz functions, geometric considerations Received by the editors October 3, 2005; accepted for pubication (in revised form) November 3, 2006; pubished eectronicay May 4, 2007 This work has been supported in part by the Deutsche Forschungsgemeinschaft SFB 6, Universität Bonn Institut für Numerische Simuation, Universität Bonn, Wegeerstr 6, 535 Bonn, Germany (hotz@insuni-bonnde, wwwinsuni-bonnde/ hotz, kunoth@insuni-bonnde, wwwinsunibonnde/ kunoth) 75

2 76 MARKUS HOLTZ AND ANGELA KUNOTH based on point vaues are used in [Ko] to represent the probem-inherent obstaces on coarser grids in such a way that a vioation of the obstace is excuded The difficuty to correcty identifying coarse grid approximations has aso been the motivation for a cascadic mutigrid agorithm for variationa inequaities in [BBS] for which, however, no convergence theory is yet avaiabe In this paper, we generaize the monotone mutigrid (MMG) method from [Ko, Ko2] to discretizations invoving higher order B-spines One of the key ingredients of an MMG method are restrictions of the obstace to coarser grids which satisfy the (upper) bound imposed by the obstace (monotonicity) as we as a ower one which corresponds to the condition of quasi optimaity in [Ko] We formuate the construction of coarse grid approximations as a inear constrained optimization probem with respect to the B-spine expansion coefficients Our construction heaviy profits from properties of B-spines [Bo, Sb] In particuar, we present with our optimized coarse grid correction (OCGC) agorithm a method to construct monotone and quasioptima coarse grid approximations to the obstace function in optima compexity of the coarse grid for B-spine basis functions of any degree Buiding the OCGC scheme into the MMG method, our higher order MMG method is shown to be of optima mutigrid compexity Moreover, foowing the arguments in [Ko], we can prove that our method is gobay convergent and reduces asymptoticay to a inear subspace correction method once the contact set has been identified [HzK] Hence, we can expect particuar robustness of the scheme and fu mutigrid efficiency in the asymptotic range in the numerica experiments This is confirmed by computations for an American option pricing probem in terms of cubic B-spines Detais about the derivation of the probem of fair pricing American options and its formuation as a free boundary vaue probem and corresponding resuts can be found in [WHD, Hz] Of course, once higher order MMG methods are avaiabe, they may be appied to other obstace probems ike Signorini s probem which has been soved using piecewise inear hat functions in [Kr] This paper is structured as foows In section 2 we introduce monotone mutigrid methods, recoect the main features of B-spines, and specify a B-spine-based projected Gauss Seide reaxation as a smoothing component of the scheme In section 3 the crucia ingredients of the higher order MMG schemes, suitabe restriction operators for the obstace function, are presented for B-spine functions of arbitrary degree in the univariate case Their construction for higher spatia dimensions is presented in section 4 using tensor products In section 5 some short remarks concerning the convergence theory for B-spine-based MMG schemes are made Finay, in section 6 we present a numerica exampe of pricing American options The convergence behavior of the projected Gauss Seide and the mutigrid schemes is compared for basis functions of different orders We concude with an estimation of asymptotic mutigrid convergence rates which exhibit fu mutigrid efficiency for the truncated version 2 MMG methods 2 Eiptic variationa inequaities and inear compementary probems Let Ω be a domain in R d and J (v) := 2a(v, v) f(v) a quadratic functiona induced by a continuous, symmetric, and H0 eiptic biinear form a(, ) : H0 (Ω) H0 (Ω) R and a inear functiona f : H0 (Ω) R As usua, H0 (Ω) is the subspace of functions beonging to the Soboev space H (Ω) with zero trace on the boundary We consider the constrained minimization probem (2) find u K: J (u) J(v) for a v K

3 on the cosed and convex set B-SPLINE-BASED MONOTONE MULTIGRID METHODS 77 K := {v H 0 (Ω) : v(x) g(x) for a x Ω} H 0 (Ω) The function g H0 (Ω) represents an upper obstace for the soution u H0 (Ω) Lower obstaces can be treated in the obvious anaogous way If g satisfies g(x) 0 for a x Ω, then probem (2) admits a unique soution u Kby the Lax Migram theorem It is we-known that (2) can be rewritten as a variationa inequaity; see, eg, [EO, KS]: find u K: a(u, v u) f(v u) for a v Kor, equivaenty, as a inear compementary probem (22) Lu f, u g, (u g)(lu f) =0 amost everywhere in Ω Here L : H 0 (Ω) H (= (H 0 (Ω)) ) is the Riesz operator defined by Lu, v := a(u, v) for a v H 0 (Ω) Discretizing in a finite dimensiona spine space S L of piecewise poynomias on a grid Δ L with uniform grid spacing h L eads to the discrete formuation of (2), (23) find u L K L : J (u L ) J(v L ) for a v L K L on the cosed and convex set K L := {v L S L : v L (x) g L (x) for a x Ω} S L, or, equivaenty, (24) L L u L f L, u L g L, (u L g L )(L L u L f L )=0 In [BHR] reguarity u H 5/2 ɛ (Ω) of the soution u to (22) is shown for arbitrary ɛ>0 Moreover, error estimates u u L H (Ω) = O(h L ) and u u L H (Ω) = O(h 3/2 ɛ L ) are proved in the case of piecewise inear (respectivey, piecewise quadratic) functions, provided the functions f,g are sufficienty reguar 22 The MMG-agorithm For soving (23) numericay, a now-popuar method is the MMG method [Ko] By adding a projection step and empoying specific restriction operators, it can be impemented as a variant of a standard mutigrid scheme Let S S 2 S L H0 (Ω) be a nested sequence of finite dimensiona spaces, and et u ν L S L be the approximation in the νth iteration of the MMG method The basic mutigrid idea is that the error v L := u L u ν, L between the smoothed iterate u ν, L := S (uν L )(S aways being the standard Gauss Seide iteration) and the exact soution u L can be approximated without essentia oss of information on a coarser grid Δ L We expain how this is reaized in the case of a inear compementary probem for two grids Δ L and Δ L Introducing the defect d L := f L L L u ν, L, (24) can be written as (25) L L v L d L, v L g L u ν, L, (v L g L + u ν, L )(L Lv L d L )=0

4 78 MARKUS HOLTZ AND ANGELA KUNOTH On a coarser grid Δ L, the defect probem can now be approximated by L L v L d L, v L g L, (v L g L )(L L v L d L )=0, where d L := rd L and g L := r(g L u ν, L ) with (different) restriction operators r, r : S L S L The soution v L of the coarse grid probem is then used as an approximation to the error v L It is first transported back to the fine grid by a proongation operator p and is then added to the approximation u ν, L It is important that the restriction r is chosen such that the new iterate satisfies the constraint (26) u ν,2 L := uν, L + pv L g L on the fine grid Appying this idea recursivey on severa different grids, one obtains the MMG method for inear compementary probems Agorithm 2 MMG (νth cyce on eve ) Let u ν S be a given approximation A priori smoothing and projection: u ν, := (P S(u ν )) η 2 Coarse grid correction: d := r(f L u ν, ), g := r(g u ν, ), L := rl p If =, sove exacty the inear compementary probem L v d, v g, (v g )(L v d )=0, and set v := v If >, doγ steps of MMG with initia vaue u 0 := 0 and soution v Set u ν,2 := u ν, + pv 3 A posteriori smoothing and projection: u ν,3 := (P S(u ν,2 )) η2 Set u ν+ := u ν,3 The number of a priori and a posteriori smoothing steps is denoted by η and η 2, respectivey For γ = one obtains a V-cyce, for γ = 2 a W-cyce P denotes a projection operator defined in (27) and (2) Condition (26) eads to an inner approximation of the soution set K L and ensures that the mutigrid scheme is robust [Ko] Striving for optima mutigrid efficiency, satisfaction of the constraint shoud not be checked by interpoating v back to the finest grid Instead, specia restriction operators r are needed for the obstace function A corresponding construction for B-spines of genera order k wi be introduced in sections 3 and 4 Next we discuss the projection step for genera order B-spines 23 A B-spine-based projected Gauss Seide scheme Since the operator L is symmetric positive definite and continuous piecewise inear functions are used for discretization, the discrete form (24) can be soved by the projected Gauss Seide scheme; see, eg, [Cr] Given an iterate u ν L, a standard Gauss Seide sweep ū ν L := S (uν L ) is suppemented by a projection uν+ = P ū ν L into the convex set K LIf L

5 B-SPLINE-BASED MONOTONE MULTIGRID METHODS 79 S L consists of hat functions, the projection can be defined for given grid points {θ i } i by (27) P v L (θ i ):=min{v L (θ i ),g L (θ i )} For higher order functions v L, the difficuty arises aready in the univariate case that for given x [θ i,θ i+ ] the estimate (28) min {v L (θ i ),v L (θ i+ )} v L (x) max {v L (θ i ),v L (θ i+ )} is no onger vaid Thus, controing function vaues on grid points is not a sufficient criterion in this case Instead, we propose here a construction using higher order B-spines, which compares B-spine expansion coefficients instead of function vaues and heaviy profits from the fact that B-spines are nonnegative We begin with the univariate case For readers convenience, we reca the reevant facts about B-spine bases from [Bo] Definition 22 (B-spine basis functions) For k N and n N et T := {θ i } i=,,n+k be an expanded knot sequence with uniform grid spacing h L in the interior of the interva I := [a, b] of the form (29) θ = = θ k = a<θ k+ < <θ n <b= θ n+ = = θ n+k Then the B-spine basis functions N i,k of order k are recursivey defined for i =,,n by (20) { if x [θi,θ N i, (x) = i+ ) 0 ese, N i,k (x) = x θ i N i,k (x)+ θ i+k x N i+,k (x) θ i+k θ i θ i+k θ i+ for x I It is known that supp N i,k [θ i,θ i+k ] (oca support), N i,k (x) 0 for a x I (nonnegativity), and N i,k C k 2 (I) (differentiabiity) hods Moreover, the set Σ L := {N,k,,N n,k } constitutes a ocay independent and unconditionay stabe basis with respect to Lp, p for the finite dimensiona space S L = N k,t := span Σ L of the spines of order k Lemma 23 If the B-spine coefficients of v L,g L N k,t = S L satisfy v i g i for a i =,,n, then v L (x) g L (x) hods for a x I Proof Using the representation v L = n i= v i N i,k and g L = n i= g i N i,k and the nonnegativity N i,k (x) 0 for a x I, we deduce that g L (x) v L (x) = n i= (g i v i ) N i,k (x) 0 for a x I Here and in section 5, we use the subscript i in v i =(v L ) i to denote B-spine expansion coefficients The projection can now be defined for B-spine functions of genera order k simiar to (27), but now invoving expansion coefficients by setting (2) P v i := min{v i,g i } Using the same arguments as in [Cr], the resuting projected Gauss Seide scheme sti converges since the discrete soution set {v R n : v i g i for i =,,n} describes a cuboid in R n Moreover, if the probem is nondegenerate, the contact set,

6 80 MARKUS HOLTZ AND ANGELA KUNOTH defined by a coefficients for which equaity hods, is identified after a finite number of iterations [Cr, EO] We treat the mutivariate case by taking tensor products Specifying the domain ΩasΩ:= d = [a,b ] R d, the ith d-dimensiona tensor product B-spine of order k on a tensorized extended knot sequence T (d) is defined by d (22) N (d) i,k (x) := N i,k(x ), x Ω, = where i := (i,,i d ) denotes a muti-index Defining S L in anaogy to the univariate case, the resut of Lemma 23 immediatey carries over to the d-dimensiona setting 3 Construction of monotone and quasi-optima obstace approximations In this section, the second essentia ingredient for our B-spine-based MMG methods is provided, the construction of so-caed monotone and quasi-optima coarse grid approximations of the obstace function, which ead to suitabe restriction operators r We begin with the univariate case; the extension to d dimensions foows in section 4 We consider in what foows ony two grids, as the generaization to severa grids is obvious Given an obstace function S which is defined on a fine grid Δ I, we provide an approximation S with respect to a coarser grid T which satisfies S(x) S(x) for a x I; 2 S(x) L k (x) for a x I and a sti-to-be-specified ower barrier L k (x) provided in section 32; 3 S S with respect to a target functiona F k defined beow in (30) The first condition ensures the monotonicity and robustness of the mutigrid scheme, the second an asymptotica reduction of the method to a inear reaxation, and the third an efficient coarse grid correction As the construction is used as a component of the monotone mutigrid scheme striving for optima computationa mutigrid compexity, it aso 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 Specificay, et T be an extended knot sequence with grid spacing H as in (29) and et Δ := { θ i } i=,,ñ+k be a finer knot sequence (3) θ = = θ k = a< θ k+ < < θñ <b= θñ+ = = θñ+k with grid spacing h = 2 H It is defined such that θ i = θ 2i k for i = k,,n+ and 2 (θ i + θ i )= θ 2i k for i = k +,,n+ Then it hods that (32) ñ =2n + k The corresponding spine spaces are N k,δ and N k,t with member functions N i,k,δ and N i,k,t, respectivey Now et the obstace function on the fine grid S N k,δ and its approximation S N k,t be expanded as ñ (33) S = c i N i,k,δ =: c T N k,δ, S = i= n c i N i,k,t =: c T N k,t There is a natura proongation operator p from N k,t to N k,δ for B-spines N i,k,t in terms of their refinement or mask coefficients [Bo, Sb] In the specia case H =2h i=

7 B-SPLINE-BASED MONOTONE MULTIGRID METHODS 8 considered here the refinement reation is given by k (34) N i,k,t = a j N 2i k+j,k,δ j=0 with the subdivision or mask coefficients ( ) k (35) a j := 2 k j for j =0,, k In step 2 of Agorithm 2, we choose the restriction r as the adjoint of p, foowing [Ha] However, for the obstace function the restriction operator r cannot be used since it does not satisfy condition (26) 3 Monotone coarse grid approximations There is a vast amount of iterature, see, eg, [DV, Mv, Pi] especiay from approximation theory, deaing with monotone approximations to a given function g The function ĝ is a monotone (or one-sided) ower approximation to g if ĝ(x) g(x) for a x I There the number n of degrees of freedom of the function ĝ is chosen such that a given approximation accuracy can be reached In contrast to these studies, the question here is different, since the number n of degrees of freedom is given by the mesh size H Definition 3 (monotone coarse grid approximation) For knot sequences T and Δ from (29) and (3), respectivey, we ca S N k,t a monotone ower coarse grid approximation to S N k,δ if S(x) S(x) hods for a x I For hat functions such approximations are constructed in [Ma, Ko] A corresponding construction for higher order functions has, to our knowedge, not been provided so far In view of Lemma 23 we propose here to contro B-spine expansion coefficients Theorem 32 (monotone coarse grid approximation) Let S N k,δ be an upper obstace with S = c T N k,δ for a given order k and the knot sequence Δ from (3) Then S N k,t with S = c T N k,t defined on the knot sequence T from (29) is a monotone ower coarse grid approximation to S if the inequaity system (36) A k c c is satisfied The two-santed matrix A k is defined by a k a k 3 a k a k 2 a k a 0 a k a a2 A k := a k a k a0 a Rñ n with the subdivision coefficients a j from (35) and has maxima rank

8 82 MARKUS HOLTZ AND ANGELA KUNOTH Proof The proof reies on the subdivision property (34) and on the nonnegativity of B-spines We ony consider the case k even as the other case is anaogous Substituting (34) into (33) and sorting according to the basis functions N i,k,δ eads to ñ ( ) S(x) = ak c (i+)/2 + a k 3 c (i+3)/2 + + a c (i+k )/2 Ni,k,Δ (x) i= i odd + ñ i=2 i even ( ak c i/2 + a k 2 c (i+2)/2 + + a 0 c (i+k)/2 ) Ni,k,Δ (x), where a c j with j<orj>nare treated as zero Defining the coefficients { ci ( ) a k c (i+)/2 + a k 3 c (i+3)/2 + + a c (i+k )/2 if i is odd, d i := c i ( ) a k c i/2 + a k 2 c (i+2)/2 + + a 0 c (i+k)/2 if i is even, which can be written in compact matrix/vector form as (37) d i = c i (A k c) i (invoving the ith component of the vector A k c), we obtain (38) S(x) ñ S(x) = d i N i,k,δ (x) i= By Lemma 23 we have S(x) S(x) 0 for a x I, provided d i 0 hods for a i =,,ñ By (37), we obtain the inequaity system (36) Since the B-spines form bases for N k,t and N k,δ, the matrix A k has fu rank for each k Exampe 33 In the specia case of continuous, piecewise inear functions (k = 2), C -smooth; piecewise quadratic (k = 3); and C 2 -smooth, piecewise cubic (k =4) spines, one has 3 A 2 = R (2n ) n, A 3 = A 4 = R (2n 3) n 3 3 R (2n 2) n,

9 B-SPLINE-BASED MONOTONE MULTIGRID METHODS 83 Tabe 3 The vaues β k and γ k for orders k =2, 4, 6, 8 k β k γ k Quasi-optima coarse grid approximations Now we can immediatey derive a monotone ower coarse approximation Proposition 34 The spine L k := q T N k,t N k,t with coefficients (39) q i := min { c 2i k,, c 2i } for i =,,n (eaving out c j in the right-hand side if j< or j>ñ) is a monotone ower coarse grid approximation to S = c T N k,δ N k,δ Proof As a row sums of A k are equa to one, the vector q := (q,,q n ) T defined in (39) obviousy satisfies the inequaity system A k q c so that the assertion directy foows from Theorem 32 Remark 35 In the specia case k = 2, the restriction operator ˆr : N 2,Δ N 2,T, S L 2 induced by Proposition 34 coincides with the restriction operator from [Ma] As is iustrated in Figures 3 and 32 for the cases k = 2 and k = 3, the approximation L k can be further improved in many cases This wi be the subject of the next subsections: there q is interpreted as a componentwise ower barrier for the B-spine coefficients c of the desired coarse grid approximation Definition 36 (quasi-optima coarse grid approximation) We ca a monotone ower coarse grid approximation S = c T N k,t to the spine S = c T N k,δ quasi-optima if it is an improvement over L k in the sense that c q hods with q defined in (39) 33 A inear optimization probem Aiming at improving the coarse grid approximation L k from Proposition 34, we define an optima monotone and quasioptima coarse grid approximation S = c T N k,t to a given S = c T N k,δ by formuating a inear optimization probem We choose a target functiona F k which estimates the sum of the distances from approximation to obstace on a coarse grid points, ie, (30) F k (c) := S(θ) S(θ) θ T Lemma 37 The function F k defined in (30) is a inear function R n R of the form (3) F k (c) =ξ T c + η, where (32) ξ := A T k s k R n, s k := (β k,γ k,β k,) T Rñ, and η := s T k c R The vaues β k and γ k can be computed expicity: for odd k we have β k = γ k = 2, and for even k =2, 4, 6, 8 the vaues are dispayed in Tabe 3 Proof By Theorem 32 we have S(x) S(x) = S(x) S(x) for a x I Using (38) we obtain (33) F k (c) = θ T ( ) S(θ) S(θ) = θ T ñ d i N i,k,δ (θ) = i= ñ d i N i,k,δ (θ) i= θ T

10 84 MARKUS HOLTZ AND ANGELA KUNOTH Abbreviating ( s k ) i := θ T N i,k,δ(θ), we next show that s k coincides with s k defined in (32) In fact, θ Δ N i,k,δ(θ) = is easiy shown by induction for k N For odd k we can use a simpe symmetry argument to concude ( s k ) i = 2 For even k two cases must be distinguished according to the position of N i,k,δ Evauating the B-spine on coarse grid points eads to ( s k ) i = β k if θ i+k/2 T and ( s k ) i = γ k in the other case For orders k =2, 4, 6, 8, the concrete vaues β k and γ k are dispayed in Tabe 3 Thus, we have (s k ) i =( s k ) i and empoying (37) in (33) eads to (3), ie, F k (c) = ñ i= (s k) i ( c i (A k c) i )=s T k c st k A k c = ξ T c + η We can now define an optima monotone and quasi-optima coarse grid approximation as the soution of the inear optimization probem (34) Minimize the target functiona F k (c) =ξ T c + η with respect to the constraints A k c c and c q Here A k Rñ n, c Rñ, and q R n are defined as before with ñ =2n k + and ξ R n and η R are given as in (32) The upper inequaity guarantees the monotonicity of the approximation by Theorem 32, whie the second one ensures quasi optimaity by Proposition Soution of the inear optimization probem Via the inear optimization formuation (34) a (with respect to the target functiona F k ) optima monotone and quasi-optima coarse grid approximation may now be obtained, in principe, by the simpex agorithm; see, eg, [Sj] Here the point q R n coud be used as a starting corner by Proposition 34 In a mutigrid scheme, however, the simpex agorithm shoud not be used because the optima compexity O(n) woud be destroyed As shown next, a direct soution for k = 2 can be obtained by the Fourier Motzkin eimination; see, eg, [Sj] For the genera case k > 2 we present afterwards an approximate soution agorithm which can be appied in optima compexity Lemma 38 (direct soution for hat functions) For k =2and given c Rñ, the soution of the inear optimization probem (34) is recursivey given by (35) c := min{ c, 2 c 2 q 2 } c i := min{2 c 2i 2 c i, c 2i, 2 c 2i q i+ } for i =2,,n, c n := min{2 c 2n 2 c n, c 2n } with q i = min { c 2i 2, c 2i, c 2i } for i =,,n defined in (39) In particuar, S = c T N k,t is a monotone and quasi-optima coarse grid approximation to the obstace S = c T N 2,Δ Proof First, the n conditions c q are integrated into the inequaity system A 2 c c from Theorem 32 Then, Fourier Motzkin eimination is appied to the resuting (3n ) n inequaity system so that we obtain the soution range q c min{ c, 2 c 2 q 2 }, q i c i min{2 c 2i 2 c i, c 2i, 2 c 2i q i+ } for i =2,,n, q n c n min{2 c 2n 2 c n, c 2n } Because of (39), q min{ c, 2 c 2 q 2 } hods To minimize the target function F 2 given by Lemma 37, a coefficients c i must be chosen as arge as possibe which eads to (35) Remark 39 The restriction operator r : N 2,Δ N 2,T, S S, impied by Lemma 38, corresponds to the restriction operator from [Ko] which is derived by

11 B-SPLINE-BASED MONOTONE MULTIGRID METHODS 85 5 upper fine grid obstace optima coarse grid approximation quasioptima coarse grid approximation Fig 3 Continuous piecewise inear upper obstace function on the fine grid [0, 4] Z/2 and coarse grid approximations according to Lemma 38 and Proposition 34, respectivey, on the coarse grid [0, 4] Z geometric considerations It is an improvement of the restriction operator ˆr from Remark 35 or [Ma] since r( S) ˆr( S) hods for a S N 2,Δ In Figure 3 a continuous, piecewise inear, upper obstace function, the optima coarse grid approximation according to Lemma 38 and the coarse grid approximation according to Proposition 34 are dispayed The improvement of the simpe approximation L 2 is ceary visibe Since the band width of A k increases with increasing order k, and since the Fourier Motzkin eimination is ony suited for sma matrices or for matrices with mainy zero entries [Sj], a different approach must be found to sove the inear optimization probem in the higher order case k>2 To simpify the notation we define in addition to (35) that a j := 0 for j>kand j<0 Theorem 30 (optimized coarse grid correction (OCGC) scheme) Let S N k,δ be given with S = c T N k,δ Let L k N k,t with L k = q T N k,t be as in (39) and define (j+k)/2 (36) bj = b j (c,,c (j+k)/2 ):= c j a j+k 2ν c ν for j =,,ñ and b j := for j< Letˆb m,i := for m>ñ or m< and (37) ˆbm,i = ˆb i (m+k)/2 m,i (c,,c i ):= c m a m+k 2ν c ν a m+k 2ν q ν ν= ν= ν=i+ for i =,,n and m =2i k +2,,2i, where q j := 0 for j>n Further, et the

12 86 MARKUS HOLTZ AND ANGELA KUNOTH vector c be recursivey defined by (38) c i := min { b2i k a 0, b 2i k+ a, ˆb 2i k+2,i,, ˆb 2i,i a 2 a k } for i =,,n Then S = c T N k,t N k,t is a monotone and quasi-optima coarse grid approximation to S, ie, L k (x) S(x) S(x) for a x I Proof We ony consider the case k odd as the other case is anaogous We first derive conditions which guarantee monotonicity (36) of the approximation Moving a entries a i+k 2j c j of the inequaity system (36) except for the rightmost nonzero ones in each row to the right-hand side eads to a 0 0 b a 0 0 a 0 0 c b2 + b3 (39) 0 a 0 0 a a c +2 c n with := (k )/2 and the new right-hand side coefficients b i defined in (36) From (39) we immediatey obtain that the inequaity system A k c c is satisfied for arbitrary c,,c if } (320) c i min { b2i k a 0, b 2i k+ a b4 bñ bñ for i = +,,n hods Second, we derive conditions which ensure quasi-optimaity c q of the approximation For an arbitrary j { +,,n} the first inequaity of (320) and definition (36) impy a 0 c j b j 2j k = c 2j k a 2j 2ν c ν ν= For every i {,,j }, we therefore obtain the condition a 2j 2i c i c 2j k j a 2j 2ν c ν When we determine c i, we can assume that the c ν s for ν =,,i are aready computed For the c ν, ν = i +,,j, which are yet to be determined, demanding quasi-optimaity c ν q ν eads to ν= ν i i (32) a 2j 2i c i c 2j k a 2j 2ν c ν ν= j a 2j 2ν q ν = ˆb 2j k,i ν=i+

13 B-SPLINE-BASED MONOTONE MULTIGRID METHODS 87 with ˆb j,i defined in (37) Anaogousy we get (322) a 2j 2i+ c i ˆb 2j k+,i for i<jusing the second inequaity of (320) Because of a m = 0 for m>k, the inequaities (32) and (322) ony appy for i + j i + so that we obtain the conditions } (323) c i min {ˆb2i k+2,i a 2,, ˆb 2i,i a k for i =,,n Then both (320) and (323) are satisfied by defining c i, i =,,n, as in (38) which competes the proof Remark 3 If one ony aims at a coarse grid approximation S which is monotone by construction, one coud use the reation (320) and repace the inequaity by an equaity sign However, in many cases the as-arge-as-possibe choice of the components c i according to (320) then has to be baanced to preserve monotonicity by very sma, maybe even negative components c j, j>i, which eads to undesirabe osciations in the soution This is avoided by taking in addition the ower bounds into consideration Exampe 32 In the case k = 2 the recursion (38) recovers the direct soution (324) c i = min{2 c 2i 2 c i, c 2i, 2 c 2i q i+ } from Lemma 38 For k = 3 the recursion (38) simpifies to (325) c i = min { 4 c 2i 3 3 c i, 4 3 c 2i 2 3 c i, 4 3 c 2i 3 q i+, 4 c 2i 3 q i+ } In the case k = 4, one obtains c i := min { 8 c 2i 4 c i 2 6c i, 2 c 2i 3 c i, 4 3 c 2i 2 6 (c i + q i+ ), (326) 2 c 2i q i+, 8 c 2i 6q i+ q i+2 }, where we use the notation that a terms in (324) (326) which invove c j with j< or q j with j>nhave to be omitted Using (32) and expoiting the fact that the number of nonzero terms in each of the sums in the definitions (36) and (37) is bounded by k, the above agorithm works in optima compexity Theorem 33 For fixed k N, the costs of the OCGC agorithm is restricted by O(n) operations Next, we visuaize the effect of our agorithm In Figure 32, one can see a C - smooth, piecewise quadratic upper obstace, the coarse grid approximation obtained by the OCGC agorithm, the coarse grid approximation L 3 N 3,T according to Proposition 34, and the optima coarse grid approximation obtained by the simpex agorithm (Reca, however, that the simpex agorithm does not yied the soution in optima compexity) The improvement of the OCGC approximation over the spine L 3 is ceary visibe There is no difference of our OCGC approximation to the optima coarse grid approximation obtained by the simpex method, except for a sight variation in the interva [0,2] This difference seems to be caused by boundary effects which has been confirmed in further numerica experiments As expected, smooth parts of the obstace are very we approximated, whie variations of the obstace

14 88 MARKUS HOLTZ AND ANGELA KUNOTH 9 8 upper fine grid obstace simpex method approximation OCGC--optimized approximation quasioptima approximation Fig 32 Right: C -smooth, quadratic upper obstace function on the fine grid Δ:=[0, 3] Z/2 with OCGC-optimized quadratic restriction, the optima coarse grid approximation obtained by the simpex method and ower quasi optima barrier L 3, a three of which are defined on the coarse grid T := [0, 3] Z of higher frequency can ony be party approximated as it is visibe in the interva [0, 2] In this exampe, the contro poygon of the B-spine coefficients of the OCGC approximation (which is not dispayed here) is party above the contro poygon of the obstace function, athough by construction the OCGC approximation aways ies beow the obstace This indicates that the resut of our OCGC agorithm is superior to aternative methods in which monotone approximations are obtained via monotone restrictions of contro poygons 4 Higher spatia dimensions In the mutivariate case Ω R d, using (22), a d-dimensiona spine S :Ω Rof order k can be represented by (4) S(x) = c i N (d) i,k,t (x) =:ct N (d) k,t (x), x Ω, i I c with coefficients c R nd and indices from I c := {i N d : i m n, m =,,d} The two-scae reation (34) attains the mutivariate refinement reation (42) N (d) i,k,t = j J a (d) j N (d) 2i k+j,k,δ with the index set J := {j N d :0 j m k for m =,,d} and the subdivision coefficients d ( ) (43) a (d) k j := 2 ( k)d for j J ν= The extension of Theorem 32 then reads as foows j ν

15 B-SPLINE-BASED MONOTONE MULTIGRID METHODS 89 Theorem 4 (monotone coarse grid approximation) The spine S = c T N (d) k,t is a monotone coarse grid approximation to the upper obstace S = c T N (d) k,δ if their B-spine expansion coefficients satisfy the inear inequaity system (44) A (d) k c c with the tensor product matrix A (d) k := A k A k R (ñ n)d and A k as in (36) Proof The proof foows by the same arguments as in the univariate case, by using the refinement reation (42) and appying the mutivariate version of Lemma 23 to S(x) S(x) = i If(A (d) k c c) i N (d) i,k,δ (x), where I f := {i N d : i m ñ, m =,,d} using the nonnegativity of (tensor product) B-spines Exampe 42 In the specia case of C 2 -smooth, piecewise cubic (k = 4) spines on a two-dimensiona domain, the system (44) reads 4 8 A4 4 8 A4 8 A4 6 8 A4 8 A4 4 8 A4 4 8 A4 8 A4 6 8 A4 6 8 A4 8 A4 4 8 A4 4 8 A4 c, c,n c n, c n,n As a rows in the system (44) sum to one we immediatey obtain from Theorem 4 the foowing generaization of Proposition 34 Proposition 43 The spine L k := q T N (d) k,t with expansion coefficients c, c,ñ c 2, (45) q i := min{ c j : 2i m k j m 2i m, m =,,d} for i I c (eaving out c j in the right-hand side if j m < or j m > ñ) is a monotone coarse grid approximation to the obstace function S = c T N (d) k,δ In the specia case k = 2 Fourier Motzkin eimination can be appied to the inequaity system (44) with the constraint c q as in the univariate case to obtain Lemma 38 for arbitrary d Lemma 44 (direct soution for hat functions) Define the sum s,i of a neighboring coarse grid coefficients c j to a given fine grid coefficient c except c i by s,i := {j I c: j i, θ j+ θ + <H} c 2,ñ cñ, cñ,ñ c j for I f, i I c, with the Eucidean distance and the mesh size H of the coarse grid Define s,i as s,i with the modification that a coefficients c j in the sum which are not yet known

16 90 MARKUS HOLTZ AND ANGELA KUNOTH are repaced by q j given from (45) Then, for k =2, a monotone and quasi-optima coarse grid approximation is recursivey given by c i := min{2 d m= m (2im ) c s,i :2i m 2 m 2i m for m =,,d} for i I c, eaving out c j in the right-hand side if j m < or j m > ñ To improve the approximation from Proposition 43 in the case k>2, the OCGC agorithm can be appied recursivey with respect to the dimension d as foows Theorem 45 (optimized coarse grid correction (OCGC) scheme for d>) The OCGC agorithm appied dimension-recursivey to the mutivariate inequaity system (44) provides in optima compexity of O(n d ) arithmetic operations a coarse grid approximation S which satisfies the monotonicity and quasi-optimaity condition L k (x) S(x) S(x) for a x Ω Proof We provide the proof for the case k = 3; the other cases foow immediatey by exchanging the system matrix A (d) 3 by A (d) k The (n ñ)d tensor product matrix A (d) 3 can be written as a n ñ matrix of (n ñ) (d ) tensor product matrices A (d) 3 := 3 4 A(d ) 3 4 A(d ) 3 4 A(d ) A(d ) A(d ) 3 4 A(d ) 3 4 A(d ) A(d ) A(d ) 3 4 A(d ) 3 4 A(d ) A(d ) 3 Accordingy, we define c i := (c ij ) i=,,n,j N (d ) : j n Rn(d ) Appying the OCGC agorithm for d = (325) to this formuation, we obtain for i =,n the conditions A (d ) 3 c i min{4 c 2i 3 3A (d ) 3 c i, 4 3 c 2i 2 3 A(d ) 3 c i, 4 3 c 2i 3 A(d ) 3 q i+, 4 c 2i 3A (d ) 3 q i+ } for c i R n(d ) Each of these conditions can again be reduced by one dimension by a further appication of the OCGC agorithm unti the whoe system is soved As the compexity of the OCGC agorithm is O(n), the overa compexity is given by O(n d ) arithmetic operations Exampe 46 In the case k = 3 and d = 2 the mutivariate OCGC agorithm is defined as foows: () For given c Rñ2 determine q R n2 by (45) (2) For i =,,n

17 B-SPLINE-BASED MONOTONE MULTIGRID METHODS 9 upper fine grid obstace OCGC coarse grid approximation quasi-optima approximation Fig 4 Two-dimensiona C -smooth, quadratic, upper obstace function defined on fine grid [0, 6] 2 (Z/2) 2 and coarse grid approximations from Proposition 43 (quasi-optima) and Theorem 45 (OCGC) on the coarse grid [0, 6] 2 Z 2 define g R n by g j := q i,j and f Rñ by f j := min{4 c 2i 3,j 3(A 3 c i ) j, 4 3 c 2i 2,j 3 (A 3c i ) j, 4 3 c 2i,j 3 (A 3q i+ ) j, 4 c 2i,j 3(A 3 q i+ ) j }, sove the univariate probem A 3 e f, e g by the d-ocgc Agorithm, set c i,j := e j The spines which correspond to the coefficient vector q and c from Proposition 43 and Theorem 45, respectivey, are dispayed in Figure 4 for a given upper obstace function defined on the fine grid [0, 6] 2 (Z/2) 2 The resuting MMG method in the mutivariate case can now be impemented by adding the projection operator (2) and the obstace approximation from Theorem 45 to a standard mutigrid method The standard mutigrid method for tensor products of higher order B-spines is described, eg, in [Hö, HRW] for the case d> 5 Convergence theory for B-spine-based MMG methods It is shown in [Ko] that MMG methods are gobay convergent and asymptoticay reducing to a inear subspace correction method, provided noda basis functions and monotone and quasi-optima restriction operators r are used Because of the ack of such restriction operators for smooth functions, the MMG method has so far been restricted to hat functions Using B-spines as basis functions, we have aready transferred the scheme to functions of genera smoothness in section 2 Suitabe restriction operators have been constructed in sections 3 and 4 We have estabished in the extended version of this paper [HzK] that a convergence resuts from [Ko] can be transferred to B-spine basis functions, using their expansion coefficients instead of function vaues 6 Numerica exampe To present a numerica exampe from the area of Mathematica Finance, we choose the domain Ω L := R + [0,T), the differentia

18 92 MARKUS HOLTZ AND ANGELA KUNOTH Fig 6 Soution V (S, t) of the inear compementary probem (62) operator (6) L := t + 2 σ2 S 2 2 S + rs 2 S r, and the function H(S) :=(K S) + We consider the inear compementary probem to find V = V (S, t) H (Ω L ), such that (62) [(L V )(S, t)] (V (S, t) H(S)) = 0, (L V )(S, t) 0, V (S, t) H(S) hods for a (S, t) Ω L, with boundary data V (S, t) =0forS, V (S, t) =H(S) for S 0 and fina data V (S, T )=H(S) for S R + As it is shown in [WHD], the soution V describes the fair vaue of an American put option with strike price K and maturity T which depends on an underying stock with vaue S and voatiity σ No anaytica soution is known for the probem (62) so that one has to resort to numerica soution schemes In the numerica experiments we used for the inear compementary probem (62), the parameters K = 0 for the strike price, T = for maturity, σ =06 for voatiity, and r =25% for the interest rate The numerica soution V and the obstace function H are dispayed in Figure 6 in the case of M = N = 64 grid points in space and time If the obstace function is set to minus infinity, the soution V describes the fair vaue of a European put option (see [WHD]) In that case an anaytica soution is known and given by the famous Back Schoes formua; see [BS] Using a Crank Nicoson finite difference scheme for the time discretization and at east continuous piecewise finite eements for the space discretization, the method converges quadraticay Empoying higher order finite eement functions, the derivatives of the soution V which provide important hedge parameters in the option pricing context can be determined by direct differentiation of the basis functions Using B-spine

19 B-SPLINE-BASED MONOTONE MULTIGRID METHODS k=2 k=3 k=4 00 pointwise error stock price S 0006 k=2 k=3 k= pointwise error stock price S Fig 62 Comparison of pointwise error for Deta and Gamma at time t = 0 for orders k =2, 3, 4 and N = M = 275 bases of order k we obtain a derivatives up to the (k 2)th derivative in quadratic convergence In particuar, pointwise derivatives, the so-caed Greek etters, can be computed up to high accuracy These resuts, as we as extensive discussion, can be found in [Hz] As an iustration of the impressibe difference a variabe order k may offer, we dispay in Figure 62 ony the pointwise errors of Deta:= V S and Gamma:= 2 V S 2 In view of this appication, we woud ike to point out that our higher order MMG method coud aso be appied to the vauation of basket options, at east for sma

20 94 MARKUS HOLTZ AND ANGELA KUNOTH 0 k=2 k=3 k= L2-error 0000 e-005 e-006 e-007 e-008 e number of iterations Fig 63 PSOR iteration history of one time step with M = 256 baskets with d = 2ord = 3 Simiar to the univariate case, the mutivariate Back Schoes equation can be transformed into a mutivariate heat diffusion probem, as shown in [RW] 6 Convergence behavior of Gauss Seide and MMG schemes In the foowing, ony one time step of probem (62) is considered to anayze the performance of the mutigrid scheme In Figure 63, the iteration errors of the projected Gauss Seide scheme are dispayed for different orders k The impact of the order k is ceary visibe Next we compare the convergence behavior of the foowing methods: PSOR Projected Gauss Seide scheme, MMG Monotone mutigrid method with optimized approximation of the obstace according to Lemma 38 and Theorem 30, TrMMG Truncated version of the monotone mutigrid method [Ko] with optimized approximation of the obstace according to Lemma 38 and Theorem 30, MMG (q-opt) Monotone mutigrid method with simpe approximation of the obstace according to Proposition 34, TrMMG (q-opt) Truncated version of the monotone mutigrid method with simpe approximation of the obstace according to Proposition 34, MG Linear mutigrid method appied to the unrestricted probem To anayze the infuence of the order k on the convergence behavior, the case k = 2 is systematicay compared to the case k =3 Fork>3simiar resuts are expected In the experiments of the finance parameters used in the previous section, the finest eve L = 7 and a random initia guess have been chosen To make sure that the iteration does not terminate too eary, we have seected independenty of the discretization error the stopping criterion u ν+ L u ν L 0 2,

21 B-SPLINE-BASED MONOTONE MULTIGRID METHODS PSOR MMG (q-opt) TrMMG (q-opt) MMG TrMMG MG 0000 L2-error e-006 e-008 e-00 e number of iterations 00 PSOR MMG (q-opt) TrMMG (q-opt) MMG TrMMG MG 0000 L2-error e-006 e-008 e-00 e number of iterations Fig 64 Iteration history for hat functions (k =2) (top) and for C -smooth basis functions (k =3) (bottom) where u ν L denotes the νth iterate on the finest grid L The numerica resuts are summarized in Figure 64 and Tabe 6 In the third coumn in Tabe 6, the number ν 0 of iterations needed to identify the contact set K (u L ) is dispayed In the next coumn It, we ist the number of iterations which are needed to sove the probem up to machine accuracy To compare the costs of the schemes, we empoy the definition of a work unit (WU) from [BC] A work unit WU = WU L denotes the costs of one iteration step of the projected Gauss Seide scheme on the finest grid L The costs WU of one iteration step on eve L is then given

22 96 MARKUS HOLTZ AND ANGELA KUNOTH Scheme smoothing step 2 smoothing steps ν 0 It WU ν 0 It WU PSOR MMG (q-opt) k = 2 TrMMG (q-opt) MMG TrMMG PSOR MMG (q-opt) k = 3 TrMMG (q-opt) MMG TrMMG Tabe 6 Number of iterations needed to identify the contact set and to compute the soution up to machine accuracy and the cost in work units by WU =2 L WU L The number of work units which is needed to reach the stop criteria is dispayed in the ast coumn WU in Tabe 6 The numerica resuts show that aready one or two smoothing steps are sufficient with regard to cost and accuracy In comparison to the Gauss Seide reaxation, the cost is substantiay reduced in the mutigrid schemes The truncated versions TrMMG and TrMMG (q-opt) converge in a cases faster than the standard versions MMG or MMG (q-opt) Moreover, mutigrid methods with an optimized approximation of the obstace according to Lemma 38 or Theorem 30 converge faster than the simpe approximations according to Proposition 34 For hat functions, this corresponds to the resuts in [Ko] For the higher order case, this indicates the quaity of the OCGC approximations from section 34 The contact set is identified correcty by a methods within ony a few iterations Considering the above resuts within the time discretization when soving the instationary probem, we wish to point out that the average number of iterations per time step is much smaer This is due to the fact that the soution of the previous time step serves as a good initia guess Therefore, we can expect that the asymptotic phase dominates the convergence behavior of the mutigrid scheme The asymptotic mutigrid rates are discussed in the foowing section 62 Mutigrid convergence rates The convergence rate ρ of a mutigrid scheme with + eves is given by u ν+ u 2 ρ u ν u 2 Here u S denotes the exact soution and u ν S the approximate soution in the νth iteration step A scheme is said to have mutigrid convergence if ρ is bounded independenty of the grid size by a constant ρ < The asymptotic convergence rates are estimated for the V-cyce of the truncated version TrMMG with + eves according to ρ uν + u ν 2 u ν u ν 2

23 B-SPLINE-BASED MONOTONE MULTIGRID METHODS smoothing step 2 smoothing steps 3 smoothing steps 4 smoothing steps 03 convergence rates number of unknowns smoothing step 2 smoothing steps 3 smoothing steps 4 smoothing steps 03 convergence rates number of unknowns Fig 65 Asymptotic convergence rates for the case k =2(eft) and k =3(right) depending on the number M of unknowns Here ν is chosen such that u ν + u ν In Figure 65 the resuts are dispayed on the eft-hand side for continuous, piecewise inear basis functions and on the right-hand side for C -smooth, piecewise quadratic basis functions We recover the favorabe convergence rates of standard mutigrid schemes which are bounded in our case by ρ 03 (k = 2) and ρ 027 (k = 3) in the case of ony one smoothing step on each refinement eve

24 98 MARKUS HOLTZ AND ANGELA KUNOTH Acknowedgments We woud ike to thank Michae Griebe for pointing out the probem of deriving MMG methods based on higher order basis functions and their possibe appication to the computation of American options We aso want to thank Rof Krause for hepfu discussions on MMG methods REFERENCES [BC] A Brandt and C W Cryer, Mutigrid agorithms for the soution of inear compementarity probems arising from free boundary probems, SIAM J Sci Statist Comput, 4 (983), pp [BHR] F Brezzi, W W Hager, and P A Raviart, Error estimates for the finite eement soution of variationa inequaities, Numer Math, 28 (977), pp [BBS] H Bum, D Braess, and F T Suttmeier, A cascadic mutigrid agorithm for variationa inequaities, Comput Vis Sci, 7 (2004), pp [Bo] C de Boor, A Practica Guide to Spines, revised edition, App Math Sci 27, Springer- Verag, New York, 200 [BS] F Back and M Schoes, The pricing of options and corporate iabiities, J Poit Econ, 8 (973), pp [Cr] C W Cryer, The soution of a quadratic programming probem using systematic overreaxation, SIAM J Contro, 9 (97), pp [DV] R DeVore, One-sided approximation of functions, J Approximation Theory, (968), pp 25 [EO] C M Eiott and J K Ockendon, Weak and Variationa Methods for Moving Boundary Probems, Res Notes in Math 59, Pitman, Boston, 982 [Ha] W Hackbusch, Mutigrid Methods and Appications, Springer Ser Comput Math 4, Springer-Verag, Berin, 985 [Hö] K Höig, Finite Eement Methods with B-Spines, Frontiers App Math 26, SIAM, Phiadephia, 2003 [HRW] K Höig, U Reif, and J Wipper, Mutigrid methods with web-spines, Numer Math, 9 (2002), pp [HM] W Hackbusch and H-D Mittemann, On mutigrid methods for variationa inequaities, Numer Math, 42 (983), pp [Hz] M Hotz, The Computation of American Option Price Sensitivities using a Monotone Mutigrid Method for Higher Order B-spine Discretizations, manuscript, 2004, submitted, in revision [HzK] M Hotz and A Kunoth, B-Spine-based Monotone Mutigrid Methods Extended Version, manuscript, 2005, avaiabe onine at kunoth/papers/papershtm [Ho] R H W Hoppe, Mutigrid agorithms for variationa inequaities, SIAM J Numer Ana, 24 (987), pp [Ko] R Kornhuber, Monotone mutigrid methods for eiptic variationa inequaities, I, Numer Math, 69 (994), pp [Ko2] R Kornhuber, Adaptive Monotone Mutigrid Methods for Noninear Variationa Probems, B G Teubner, Stuttgart, Germany, 997 [Kr] R Krause, Monotone Mutigrid Methods for Signorini s Probem with Friction, Dissertation, FU Berin, Berin, Germany, 200 [KS] D Kinderehrer and G Stampacchia, An Introduction to Variationa Inequaities and their Appications, Academic Press, New York, 980 [Ma] J Mande, A mutieve iterative method for symmetric, positive definite inear compementarity probems, App Math Optim, (984), pp [Mv] E R Matveev, On a super one-sided spine approximation of functions of severa variabes, Izvestiya VUZ Matematika, 32 (988), pp [Pi] A M Pinkus, On L -Approximation, Cambridge Tracts in Math 93, Cambridge University Press, Cambridge, UK, 989 [RW] C Reisinger and G Wittum, On mutigrid for anisotropic equations and variationa inequaities: Pricing muti-dimensiona European and American options, Comput Vis Sci, 7 (2004), pp 89 97