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1 SIAM J MATRIX ANAL APPL Vol 31, No 4, pp c 21 Society for Industrial and Applied Matheatics STOCHASTIC GALERKIN MATRICES OLIVER G ERNST AND ELISABETH ULLMANN Abstract We investigate the structural, spectral, and sparsity properties of Stochastic Galerkin atrices as they arise in the discretization of linear differential equations with rando coefficient functions These atrices are characterized as the Galerkin representation of polynoial ultiplication operators In particular, it is shown that the global Galerkin atrix associated with coplete polynoials cannot be diagonalized in the stochastically linear case Key words stochastic Galerkin ethod, stochastic finite eleents, orthogonal polynoials AMS subject classifications 65N3, 65C3, 15A18 DOI 11137/ Introduction As a technique for propagating data uncertainty through the nuerical solution of partial differential equations (PDEs), stochastic finite eleent ethods have received considerable attention in recent years The introduction of a Galerkin discretization schee based on polynoials in rando variables by Ghane and Spanos [13] led to both a wide adoption of this ethod aong practitioners as well as systeatic investigation of the atheatical properties of these schees The basic approach of such Stochastic Galerkin ethods is a variational forulation in which trial and test spaces consist of rando fields rather than deterinistic functions, which are forally described as tensor products of functions of deterinistic variables (usually space and/or tie) on one hand, and functions of a nuber of rando variables on the other For linear partial differential equations with a stochastic differential operator, the resulting Galerkin atrices are sus of tensor products of atrices, of which one factor is associated with the deterinistic function space, and the other with the stochastic function space For nearly all Stochastic Galerkin discretizations, the stochastic factors, which we refer to as Stochastic Galerkin atrices, are of the sae highly structured for, and certain practical questions naturally arise when devising efficient solution algoriths for the Galerkin equations, in particular as the global atrix is typically very large, its diension being the product of the nuber of degrees of freedo in the deterinistic and stochastic function spaces, respectively In this paper we address three issues The first concerns the choice of basis functions in the stochastic space leading to a desirable for of the resulting atrices It is known that, when using tensor product polynoials, a basis can be constructed for which the resulting cobined deterinistic-stochastic Galerkin atrix is block diagonal (cf [5, 6]) We show that this is not the case when the ore popular subspace of coplete polynoials is used Moreover, for the diagonalizable case, we show how this basis can be constructed by solving a sall tridiagonal eigenvalue proble Another issue, arising in the design of iterative solution ethods for Stochastic Galerkin Received by the editors Deceber 1, 28; accepted for publication (in revised for) by J H Brandts January 4, 21; published electronically April 9, 21 This work was supported by grant D/6/12721 of the Geran Acadeic Exchange Service (DAAD) and by grant ER 275/5-1 within the Geran Research Foundation (DFG) Priority Progra Institut für Nuerische Matheatik und Optiierung, TU Bergakadeie Freiberg, D-9596 Freiberg, Gerany (ernst@athtu-freibergde, ullann@athtu-freibergde) 1848 Copyright by SIAM Unauthorized reproduction of this article is prohibited

2 STOCHASTIC GALERKIN MATRICES 1849 equations, is the deterination of the eigenvalues of the Stochastic Galerkin atrices Here we give a partial result covering the stochastically linear case and an additional order in the stochastically nonlinear case in addition to spectral inclusion bounds and syetry properties Finally, we investigate the sparsity structure of the Stochastic Galerkin atrices The paper is organized as follows: in section 2 we describe the Stochastic Galerkin discretization based on both spaces of ultivariate tensor product and coplete polynoials in rando variables, and derive the associated Stochastic Galerkin atrices, displaying their basic Kronecker product structure and how their entries ay be coputed using recurrence relations for orthogonal polynoials Section 3 shows that a well-known diagonalization procedure used in conjunction with tensor product polynoials does not extend to the saller space of coplete polynoials Eigenvalues and eigenvalue bounds are given in section 4, and section 5 gives results on the sparsity structure of Stochastic Galerkin atrices 2 Stochastic Galerkin atrices In this section we describe the two ost coonly occurring types of Stochastic Galerkin atrices, whose entries consist of the expectation of the product of two or three ultivariate polynoials in rando variables, respectively The point of departure is the representation of rando fields (see [2] or [7] for an introduction), which constitute the input data for Stochastic Galerkin discretizations as finite separated expansions (21) a(x, ξ) =a(x)+ a α (x) ψ α (ξ) α I a, α > The deterinistic variable x, which usually represents spatial coordinates and/or tie, lies in a bounded doain D R d of diension d, and the function a : D R denotes the expectation of the rando field a The second variable ξ = ξ(ω) is a vector of a finite nuber M N of independent centered rando variables ξ : Ω R, = 1,, M, with unit variance associated with a probability space (Ω, A, P), consisting of the abstract set Ω of eleentary events, a σ-algebra A on Ω and a probability easure P on A We ake the general assuption that all rando variables that occur have finite second oents and denote the Hilbert space of such rando variables by L 2 P (Ω) Finally, α NM denotes a ulti-index varying in an index set I a with α := α α M,andψ α denotes a polynoial in the variables ξ 1,,ξ M Denoting the range of the th rando variable ξ by Γ := ξ (Ω) and Γ:=Γ 1 Γ M,wehaveξ(ω) Γ for all ω Ω In the following we shall suppress the deterinistic variable x and view a = a(ξ) siply as a rando variable taking values in a finite-diensional space of functions This is sufficient for our purpose of studying the properties of the atrices obtained after Galerkin discretization Moreover, we ention a special case of the expansion (21) which occurs frequently in applications and in which only linear polynoials in ξ appear in the expansion, which then takes on the for (22) a(ξ) =a + M a ξ =1 We refer to this situation as the (stochastically) linear case Copyright by SIAM Unauthorized reproduction of this article is prohibited

3 185 OLIVER G ERNST AND ELISABETH ULLMANN The space of polynoials ψ α is deterined by fixing the ulti-index set I a We distinguish the cases of tensor product polynoials I a = I p := {α N M : α j p j =1,,M} of individual degree at ost p and that of coplete polynoials I a = I C p := {α N M : α p} I p of total degree at ost p Introducing the ultivariate polynoial spaces V p := span{ξ α : α I p } and V C p := span{ξα : α I C p }, where ξ α denotes the M-variate onoial ξ α1 1 ξα2 V p and V C p 2 ξαm, it is easily verified that (23) N p := di V p =(1+p) M and N C p := di V C p = M ( M + p p 21 Galerkin equations The atrices under study arise fro a Galerkin discretization of the equation au = f with given rando functions a and f, resulting in the discrete variational proble of finding u V such that (24) a(ξ)u(ξ)v(ξ) = f(ξ)v(ξ) v V, where V is either V p or Vp C and ξ denotes the expectation of the rando variable ξ We note that (24) is the abstract representation of a Stochastic Galerkin forulation in which the functions of the rando vector ξ can be interpreted to take values either in a function space appropriate for the underlying continuous variational setting or, after discretization in the deterinistic variable, a finite-diensional vector space Assuing further that each of the independent rando variables ξ has a probability density ρ :Γ R +, their joint density ρ :Γ R+ is given by the product ρ = M =1 ρ,andweaywrite u(ξ) = u(ξ) ρ(ξ) dξ = u(ξ 1,,ξ M ) ρ M (ξ M ) dξ M ρ 1 (ξ 1 )dξ 1 Γ Γ 1 Γ M Moreover, instead of L 2 P (Ω), we ay equivalently consider the weighted L2 -space L 2 ρ (Γ) Inserting a representation u = β u βψ β of u with respect to a basis Ψ := {ψ β } of V in (24) and requiring the variational equation to hold for each test function ψ γ, γ I, we arrive at the linear syste of equations Gu = f associated with (24) and the basis Ψ, where the Galerkin atrix G R N N, with N = N p for V = V p or N = Np C for V = V p C, respectively, has the for (25) G = ag + a α G α α > with atrices whose entries associated with each ulti-index pair (β, γ) I I are given by (26) [G ] β,γ = ψ β ψ γ and [G α ] β,γ = ψ α ψ β ψ γ, α I a, α > ) Copyright by SIAM Unauthorized reproduction of this article is prohibited

4 STOCHASTIC GALERKIN MATRICES 1851 Note that we distinguish the index sets I a for the coefficient field fro that for the basis functions I ; we shall see later that there are reasons for choosing the forer larger than the latter In the stochastically linear case only the M ulti-indices with α = 1 occur in the su and, indexing these with =1, 2,,M, the Galerkin atrices in this case are siply (27) [G ] β,γ = ξ ψ β ψ γ, β, γ I, =1, 2,,M We note that the structure of the atrices G α as given in (26) reveals that these represent ultiplication operators on the ultivariate polynoial spaces V p and Vp C, respectively More precisely, denoting by P V : L 2 ρ(γ) V the orthogonal projection onto either V = V p or V = Vp C, respectively, as well as by M α the linear operator which aps a polynoial ψ V to the product ψ α ψ with a fixed basis eleent ψ α, α I a, the Stochastic Galerkin atrix G α then represents the operator P V M α : V V, ψ P V (ψ α ψ) with respect to the basis Ψ 22 Choice of basis The construction of a basis of ultivariate polynoials is ost easily accoplished using suitable univariate polynoials A convenient approach is based on the univariate orthonoral polynoials {ψ () j (ξ )} j N associated with each of the probability densities ρ as weight functions, such that ( ) ψ () i ψ () j = ψ () i,ψ () j L 2 ρ (Γ) := (ξ)ψ () j (ξ)ρ (ξ) dξ = δ i,j, i,j N ψ () i Γ Proposition 1 Given p N, the (possibly unbounded) intervals Γ =[a,b ] as well as probability densities ρ supported on Γ, then the set of ultivariate polynoials (28) Ψ := {ψ α : α I }, ψ α (ξ) := M =1 ψ () α (ξ ), where {ψ () j } p j= denote the first p +1 orthonoral polynoials with respect to the probability density ρ, for an orthonoral basis of V p for I = I p as well as of Vp C for I = Ip C Proof In view of ψ α ψ β = M =1 ( ) ψ α (),ψ () β = M L 2 ρ (Γ) =1 δ α,β, orthonorality follows directly fro that of the univariate polynoials That the polynoials (28) for a basis is seen by coparing diensions Note that the fact that the ρ are probability densities iplies that (1, 1) L 2 ρ (Γ ) = 1 for all ; hence the orthonoral polynoial of degree zero is always ψ () (ξ ) 1 Moreover, the assuption that each rando variable has zero ean and unit variance iplies (29) (1,ξ ) L 2 ρ (Γ ) = ξ =, (ξ,ξ ) L 2 ρ (Γ ) = ξ 2 =1, Copyright by SIAM Unauthorized reproduction of this article is prohibited

5 1852 OLIVER G ERNST AND ELISABETH ULLMANN and therefore we ust have ψ () 1 (ξ )=ξ for all The construction of the bases for V p and Vp C in Proposition 1 is based on the representation of rando fields in ters of independent rando variables ξ 1,,ξ M A ore general approach, which is beyond the scope of this paper, can be found in [23], where the expansion of rando fields is carried out in ters of statistically dependent rando variables A further alternative approach which eploys wavelet bases in place of polynoials in rando variables has been proposed by Le Maître et al [16, 17] 23 Kronecker structure To fix the atrix representation of the G α,we now introduce an enueration of the basis polynoials, ie, of the index sets I p and Ip C We nuber the ulti-indices (α 1,,α M ) I p associated with the tensor product polynoials lexicographically, with α 1 varying the ost rapidly We derive the ordering of the ulti-indices (α 1,,α M ) Ip C associated with the coplete polynoials fro that of the tensor product polynoials by siply deleting in the forer ordering all ulti-indices in I p \ Ip C An exaple is given in Table 21 Table 21 Enueration of Vp C as derived fro lexicographic ordering of V p for the case M =2, p =3 Note that the polynoials in V p \ Vp C occur in contiguous blocks as α 1 varies α =(α 1,α 2 ) ψ α V3 C? (,) ψ (ξ 1 )ψ (ξ 2 ) (1,) ψ 1 (ξ 1 )ψ (ξ 2 ) (2,) ψ 2 (ξ 1 )ψ (ξ 2 ) (3,) ψ 3 (ξ 1 )ψ (ξ 2 ) (,1) ψ (ξ 1 )ψ 1 (ξ 2 ) (1,1) ψ 1 (ξ 1 )ψ 1 (ξ 2 ) (2,1) ψ 2 (ξ 1 )ψ 1 (ξ 2 ) (3,1) ψ 3 (ξ 1 )ψ 1 (ξ 2 ) (,2) ψ (ξ 1 )ψ 2 (ξ 2 ) (1,2) ψ 1 (ξ 1 )ψ 2 (ξ 2 ) (2,2) ψ 2 (ξ 1 )ψ 2 (ξ 2 ) (3,2) ψ 3 (ξ 1 )ψ 2 (ξ 2 ) (,3) ψ (ξ 1 )ψ 3 (ξ 2 ) (1,3) ψ 1 (ξ 1 )ψ 3 (ξ 2 ) (2,3) ψ 2 (ξ 1 )ψ 3 (ξ 2 ) (3,3) ψ 3 (ξ 1 )ψ 3 (ξ 2 ) The Kronecker product structure of the Stochastic Galerkin atrices (26) follows directly fro definition (28) of the ultivariate orthonoral basis polynoials and is suarized below Given the probability densities {ρ } M =1, the associated sequences {ψ n () } of orthonoral polynoials, and p N,wedenotebyU n () the (p +1) (p + 1) atrices (21) [U n () ] i,j := ψ n () ψ() ψ (), 1 M, i, j p, n N, i j associated with each set of univariate polynoials Proposition 2 The Stochastic Galerkin atrices G α in (26) for α > obtained for the basis (28) of the tensor product polynoial space V = V p are given by (211) G α = U (M) α M U (2) α 2 U (1) α 1 Copyright by SIAM Unauthorized reproduction of this article is prohibited

6 STOCHASTIC GALERKIN MATRICES 1853 For α =we have G α = I N Moreover, for α =1with α =1we obtain (212) G α = G = I p+1 I p+1 U () 1 I p+1 I p+1, in which the univariate atrix U () 1 fro (21) is the th factor (fro right to left) of the Kronecker product Proof Relation (211) follows fro (26) due to the independence of the rando variables {ξ } M =1, as a result of which all integrals decouple to products of onediensional integrals The order of the Kronecker product in (211) results fro the lexicographic ordering we have fixed for the ulti-index set I p, in which α 1 varies ost rapidly Relation (212) follows fro the orthonorality of the univariate polynoials ψ () j, as a result of which U () = I p+1 When passing fro tensor product polynoials to the space of coplete polynoials V = Vp C, the atrices G α lose their Kronecker product structure since Vp C in contrast to V p is not a tensor product space One can, however, describe the atrix structure in ters of the tensor product case by aking use of the fact that, since the ultivariate polynoial basis (28) of Vp C is a subset of the corresponding basis of V p, the Stochastic Galerkin atrix G α associated with Vp C is a principal subatrix of that obtained for V p The following characterization of the structure of the Stochastic Galerkin atrices obtained for coplete polynoials in the stochastically linear case generalizes a result given in [21, Lea 31] Lea 3 For the stochastically linear case, denote the Stochastic Galerkin atrices obtained for the ultivariate polynoial basis (28) of the space of coplete polynoials Vp C by G C, =, 1,,MThen (a) G C = I N ; (b) for =1,,M each atrix G C is perutation-siilar to a block diagonal atrix consisting of ( ) M 1+p p diagonal blocks, each of which is a leading principal subatrix of the univariate atrix U () 1 given in (21) The first block, and only this, contains the entire atrix U () 1 Proof Assertion (a) follows by orthonorality of the basis polynoials ψ β We next consider the case = 1 and show that G1 C is block diagonal with blocks as described in assertion (b) Indeed, G1 C is a principal subatrix of the corresponding atrix G 1 fro the tensor product polynoial basis, which is block diagonal (cf (212)), with a diagonal block U (1) 1 R (p+1) (p+1) repeated (p +1) M 1 ties along the diagonal: (213) G 1 = I p+1 I p+1 U (1) 1 = I (p+1) M 1 U (1) 1 We obtain G1 C fro G 1 by deleting the rows and coluns associated with ultiindices in I p \ Ip C, ie, to basis polynoials with total degree exceeding p Notethat each diagonal block U (1) 1 of (213) corresponds to a range of ulti-indices (α 1, α) I p with a fixed subindex α N 1 and α 1 ranging fro to p We distinguish two cases: if α >p, then all ulti-indices of this block lie outside Ip C and therefore all associated rows and coluns of G 1 are deleted If, on the other hand, α p, then the ulti-indices (α 1, α) with α 1 p α are retained According to the ordering we have introduced for the basis polynoials of V p and Vp C (cf Table 21), this is a contiguous set of rows and coluns fro the beginning of a diagonal block, thus leaving a leading principal subatrix of U (1) 1 in the corresponding diagonal block Copyright by SIAM Unauthorized reproduction of this article is prohibited

7 1854 OLIVER G ERNST AND ELISABETH ULLMANN of G1 C Insuary,inpassingfroG 1 to G1 C, soe diagonal blocks are deleted and those reaining are replaced by leading principal subatrices with order ranging fro to p Order p is obtained only for α =, ie, the first block The nuber of reaining blocks is given by the nuber of ulti-indices α of length M 1such that α p, whichis ( ) M 1+p p The reaining cases α =1,α =1with>1 follow by the sae arguent after peruting G and G C in such a way that the ulti-indices α 1 and α are interchanged This corresponds to a reordering of the basis polynoials resulting fro the exchange of the independent variables ξ 1 and ξ 24 Matrix entries Since the Stochastic Galerkin atrices G α are built up fro the univariate atrices U n (), consisting of Kronecker products (211) of these in case of the tensor product polynoial space V p and principal subatrices of these in case of the coplete polynoial space Vp C, analysis of their entries leads us to investigate the atrices U n (), ie, the triple products (214) ψ() ψ (), i, j p, =1,,M, n N ψ () n i j The product ψ () i ψ () j of two orthonoral polynoials fro which the basis (28) is constructed is a polynoial of exact degree i + j Therefore there exist coefficients such that g () kij i+j (215) ψ () i ψ () j = g () kij ψ() k k= By orthonorality we ust have g () kij = ψ () k ψ () i ψ () () j In particular, g kij = whenever k>i+ j Proposition 4 The univariate Galerkin atrices U n () with respect to the orthonoral polynoials with degree p associated with the weight function ρ are identically zero for n>2p As a consequence of Proposition 4, the appropriate ulti-index set I a fro which the expansion of the input rando field a is constructed in (21) is given by I a = I 2p for the tensor product case (I = I p ) and I a = I2p C for coplete polynoials (I = Ip C ) For this choice one obtains the full Galerkin projection in (24) even when the expansion of the input rando field consists of an infinite nuber of ters This fact was, to the best of the authors knowledge, first observed by Matthies and Keese in [19] The task of coputing the coefficients g () kij in (215) is known in the orthogonal polynoials literature as the linearization of products proble; see [4, Lecture 5] The linearization coefficients for the orthonoral polynoials associated with coon probability density functions can be found in Appendix A, where several explicit forulas for these coefficients are collected We note that this proble has quite a long history; see, eg, [1, 9] We single out the special case of the univariate atrices (21) obtained for n =1, naely [U () 1 ] i,j = ξψ () i (ξ)ψ () j (ξ) = ξψ () i (ξ)ψ () j (ξ) ρ (ξ) dξ, Γ i, j, p Copyright by SIAM Unauthorized reproduction of this article is prohibited

8 STOCHASTIC GALERKIN MATRICES 1855 Fro the well-known three-ter recurrence satisfied by real orthonoral polynoials (216) βj+1 ψ j+1 (ξ) =(ξ α j )ψ j (ξ) β j ψ j 1 (ξ), j =, 1,, ψ 1, (see, eg, [11, Definition 13 and Theore 127]), where we have oitted the superscript (), we observe that α β1 (217) U () 1 = β1 α 1 βp βp is the Jacobi atrix of recurrence coefficients of the orthonoral polynoials associated with the weight function ρ It is, in particular, well known that the coefficients βj are positive nubers and that the eigenvalues of U () 1 are the distinct real zeros of the orthonoral polynoial of degree p + 1 associated with weight function ρ Moreover, these zeros are contained in the support of ρ 3 Diagonalization in the stochastically linear case When Stochastic Galerkin discretizations are applied to PDEs with rando data (see, eg, [13,24,5,19]),theN stochastic degrees of freedo are coupled in a tensor product fashion to, say, N x deterinistic degrees of freedo, resulting in global stochasticdeterinistic Galerkin atrices of potentially very large diension N x N Since the solution process entails solving linear systes of equations with this coefficient atrix, decoupling these equations will result in substantial coputational savings For the stochastically linear case (27) with tensor product polynoials, a change of basis under which the Stochastic Galerkin atrices becoe diagonal was introduced in [5, section 7] In coupled stochastic-deterinistic forulations this results in block diagonal coefficient atrices, each block being of the size of one deterinistic proble, which is reiniscent of Monte-Carlo siulation In this section we recall the diagonalizing change of basis, give a siplification for the eigenvalue calculations involved in its construction, and show that diagonalization is not possible for coplete polynoial spaces Throughout this section we consider only the stochastically linear case and eploy the notation G, =, 1,,M, introduced in (27) To diagonalize the atrices G we pass fro the orthonoral basis Ψ given in (28)toanewbasis Ψ ofv p or Vp C, respectively, and denote the resulting Stochastic Galerkin atrices by Ĝ,=,,MInorderthatĜ, whichisjustthegraian atrix of Ψ with respect to the L 2 ρ (Γ)-inner product, be diagonal, we see that Ψ ust again consist of orthogonal ultivariate polynoials, and we again take the to be noralized Denoting by Ψ =[ψ 1,ψ 2,,ψ N ]and Ψ =[ ψ 1, ψ 2,, ψ N ] the basis functions arranged as row vectors, the change of basis between the orthonoral bases Ψand Ψ is effected by an orthogonal atrix V R N N such that Ψ = ΨV As a result, the Stochastic Galerkin atrices of the two bases are related by (31) Ĝ = V G V, =, 1,,M We thus arriveatthe probleoffinding anorthogonalatrixv which siultaneously diagonalizes the atrices {G } M =,ataskknownassiultaneous diagonalization by orthogonal congruence; see [14, section 45] Since (31) represents a spectral decoposition, this proble is equivalent to finding a coon syste of orthogonal eigenvectors of the atrices G α p Copyright by SIAM Unauthorized reproduction of this article is prohibited

9 1856 OLIVER G ERNST AND ELISABETH ULLMANN 31 Tensor product polynoials For the tensor product polynoial space V p with basis (28) it is apparent fro the structure of the atrices G as given in (212) that the atrix (32) V = V M V M 1 V 1 siultaneously diagonalizes all atrices G if and only if the orthogonal atrices {V } M =1 satisfy (33) U () 1 V = V Λ, =1,,M, with diagonal atrices {Λ } M =1, ie, if the coluns of V are the orthonoral eigenvectors of U () 1 Fro their definition in (21) the atrices U () 1 are given by [U () 1 ] i,j = ξ ψ () i ψ () j, i,j =, 1,,p, =1,,M In other words, for each of the M rando variables ξ 1,,ξ M we seek a set of polynoials { of degree p with the properties (34a) (34b) ψ () j } p j= ψ() i ξ ψ () i ψ () j = Γ ψ () j = Γ ψ() () i (ξ) ψ j (ξ) ρ (ξ) dξ = δ i,j, ψ() i (ξ) () ψ j (ξ) ξρ (ξ) dξ = δ i,j λ () i for i, j =,,pand =1,,MIn[5]Babuška, Tepone, and Zouraris suggested the nae double-orthogonal polynoials for polynoials satisfying (34), since they are siultaneously orthogonal with respect to the weight functions w(ξ) =ρ (ξ) and w(ξ) =ξρ (ξ) To copute such double-orthogonal polynoials in general, an iediate approach is to represent the polynoials with respect to the onoial basis {1,ξ,,ξ p }, such that (dropping the superscripts and subscripts referring to for now) [ ψ, ψ 1,, ψ p ]=[1,ξ,,ξ p ]V Inserting this representation into (34) reveals that V is the solution of the generalized eigenvalue proble in ters of the oent atrices M 1 V = M V Λ [M ] i,j = ξ i+j, [M 1 ] i,j = ξ i+j+1, i,j =, 1,,p A sipler (standard) eigenvalue proble is obtained if we start fro the basis of orthonoral polynoials {ψ j } p j= instead of the onoials In this case the oent atrix is replaced with the Graian atrix of this basis, ie, the identity and the atrix M 1 is replaced by the atrix U 1 =[ ξψ i (ξ)ψ j (ξ) ] p i,j=, which we identified in (217) as the syetric tridiagonal Jacobi atrix associated Copyright by SIAM Unauthorized reproduction of this article is prohibited

10 STOCHASTIC GALERKIN MATRICES 1857 with the weight function ρ In suary, constructing the double-orthogonal polynoial basis ΨofV p requires only the Jacobi atrices U () 1 up to degree p for each of the probability density functions ρ, =1, 2,,M, or, equivalently, the coefficients {α j,β j } p j= for the three-ter recurrence of the orthogonal polynoials generated by ρ These can be obtained either fro the literature or generated using the well-known Stieltjes procedure, the polynoial equivalent of the Lanczos process The coluns of the atrices V in (32) and (33) are then obtained as the noralized eigenvectors of the Jacobi atrices U () 1 with the corresponding eigenvalues foring the diagonal of the atrices Λ in (33) Theore 5 For the space of tensor product polynoials V p, the Stochastic Galerkin atrices {G } M = are siultaneously diagonalized by the Heritian atrix V = V M V 1 V 1 in (32), the Kronecker factors of which contain the orthonoral eigenvectors of the univariate atrices U () 1, resulting in the M diagonal atrices Ĝ = V G V = I p+1 I p+1 Λ I p+1 I p+1, =1, 2,,M, as well as Ĝ = I The diagonal atrices Λ contain the eigenvalues of the atrices U () 1, respectively, ie, the zeros of the orthogonal polynoial of degree p +1 associated with the weight function ρ The atrix V effects a change of basis fro the orthogonal polynoials {ψ j } p j= to the basis of double-orthogonal polynoials { ψ j } p j= satisfying (34) We note that the sae arguent also applies when different polynoial degrees p are used in each rando variable and ephasize that each rando variable ay have a different probability density ρ Moreover, the distributions need not possess a density function; all that is needed is the recurrence coefficients {α () j,β () j } p j=, ie, the existence of the distribution s oents What is crucial to the diagonalization is the tensor product structure of the polynoial space, which is a direct consequence of the independence of the rando variables ξ 32 Coplete polynoials Turning now to the space Vp C of coplete polynoials, we assue M>1andp> to avoid trivial cases Lea 6 For the stochastically linear case with M>1rando variables and polynoial degree p 1, the Stochastic Galerkin atrices {G} C M =1 associated with the space of coplete polynoials Vp C are singular Proof For each of the atrices {G C}M =1, we find a nonzero basis function which is apped to zero by the operator represented by G C For each fixed, our assuptions M>1andp> iply that the polynoial basis Ψ in (28) contains at least one polynoial ψ(ξ) =ψ (k) p (ξ k ), k of exact degree p depending only on the kth rando variable ξ k When applied to ψ, the ultiplication operator associated with G C yields the orthogonal projection of the polynoial ξ ψ(ξ) ontovp C We verify that ξ ψ(ξ) is orthogonal to Vp C Indeed, Copyright by SIAM Unauthorized reproduction of this article is prohibited

11 1858 OLIVER G ERNST AND ELISABETH ULLMANN for any basis polynoial ψ α, α I C p,weobtain M ξ ψ(ξ)ψ α (ξ) = ξ ψ p (k) (ξ k) = ξ ψ () α (ξ ) l=1 ψ (l) α l (ξ l ) ψ (k) p (ξ M k)ψ α (k) k (ξ k ) l=1 l,k ψ α (l) l (ξ l ) and assert that one of the first two factors on the right-hand side of the last equality ust vanish Otherwise, a nonzero first factor iplies α = 1, and for the second factor not to vanish it is necessary that α k = p, which together iply α p +1,a contradiction to ψ α Vp C Theore 7 For the stochastically linear case with M>1rando variables and polynoial degree p 1, the Stochastic Galerkin atrices {G} C M = associated with the space of coplete polynoials Vp C are not siultaneously diagonalizable Proof In the proof of Lea 6 it was shown that for each atrix G C there exists an index k such that ψ p (k) lies in the null space of the operator represented by G, C ie, ψ p (k) is an eigenfunction associated with eigenvalue zero We show that this polynoial is not an eigenfunction of the operator represented by Gk C, a necessary condition for siultaneous diagonalizability Otherwise, we would have ξ k ψ p (k) (ξ k ) ψ α (ξ) = λ ψ p (k) (ξ k ) ψ α (ξ) ψ α Ψ In particular, choosing ψ α = ψ (k) p 1,weobtain ξ k ψ (k) p by orthogonality of ψ (k) p 1 (ξ k ) ψ (k) p 1 (ξ k) = λ ψ (k) p (ξ k ) ψ (k) p 1 (ξ k) = and ψ(k) p The ter on the left-hand side, however, is just the last entry on the first subdiagonal of the Jacobi atrix U (k) 1 associated with the orthonoral polynoials generated by the weight function ρ k Since this quantity is always positive (see [11, Definition 13 and Theore 127]), we have arrived at a contradiction Many authors use the space of coplete polynoials V p for the stochastic discretization (see, eg, [13, 19, 22, 21]) to avoid the so-called curse of diensionality since the nuber of degrees of freedo in V p grows exponentially with the nuber of rando variables M in contrast to Vp C where this growth is only algebraic (cf (23)) On the other hand, Theore 7 shows that an uncoupling of the equations is not possible for this saller space By consequence, in Stochastic Galerkin discretizations of coupled deterinistic-stochastic probles, using the saller space of coplete polynoials of degree p requires the solution of large linear systes of equations in which several instances of the deterinistic proble are coupled; see [12, 2, 21, 1, 8] for ethods for solving the fully coupled syste 4 Eigenvalues of Stochastic Galerkin atrices In this section we present eigenvalue location results for Stochastic Galerkin atrices Such results, besides being of interest in their own right, are necessary in the analysis of preconditioning schees for Stochastic Galerkin discretizations (cf [21, 1, 8]) Copyright by SIAM Unauthorized reproduction of this article is prohibited

12 STOCHASTIC GALERKIN MATRICES Tensor product polynoials We begin with the sipler situation of tensor product polynoials The Kronecker product structure of the Stochastic Galerkin atrices in this case iediately yields the following eigenvalue bounds Theore 8 The eigenvalues of the Stochastic Galerkin atrices G α for the space V p of tensor product polynoials are given by { M } (41) Λ(G α )= λ () : λ () Λ(U α () ), α =(α 1,,α M ) I 2 p =1 Proof Assertion (41) follows fro the well-known expression for the eigenvalues of a Kronecker product (211), as can be found, eg, in [15, Theore 4212] Corollary 9 In the stochastically linear case the eigenvalues of the Stochastic Galerkin atrices {G } M =1 obtained for the space of tensor product polynoials V p are given by Λ(G )=Λ(U () 1 ), =1,,M In other words, the eigenvalues consist of the roots of the orthogonal polynoial of degree p +1 generated by the weight function ρ Proof This follows iediately fro (212) and the discussion at the end of section 2 In the stochastically nonlinear case, a coplete characterization of the eigenvalues of the atrices G α requires the eigenvalues of the atrices U n () associated with the univariate weight function ρ for n 2 Soe first results on this topic are presented in section 44 In general, however, the coplete inforation on the eigenvalues of these atrices is not available We therefore show how inclusion bounds on the spectru of the Stochastic Galerkin atrices ay be obtained with the help of suitable Gaussian quadrature rules To this end, let (η,i,w,i ) δ i=1 denote the nodes and weights, respectively, of the δ -point Gaussian quadrature rule associated with the probability density function ρ, =1,,MThatis, ψ = ψ(ξ) ρ (ξ) dξ Γ δ i=1 ψ () (η,i )w,i Each quadrature rule above is exact for polynoials ψ span{1,ξ,,ξ 2δ 1 }Furtherore, we define the tensor product grid of quadrature nodes, (42) Ξ δ := =1{η,1,η,2,,η,δ }, M in which the coponents δ of the ulti-index δ N M denote the nuber of nodes in the th quadrature rule Theore 1 The eigenvalues of the Stochastic Galerkin atrices G α for the space V p of tensor product polynoials with the basis {ψ α } introduced in (28) are bounded by Λ(G α ) [θ α, Θ α ],where (43) θ α := in{ψ α (η) :η Ξ δ }, Θ α := ax{ψ α (η) :η Ξ δ }, α I 2p, where Ξ δ is an M-diensional grid of quadrature nodes as defined in (42), inwhich the nuber of nodes in the th rule is at least δ := p + α +1 2, =1,,M Copyright by SIAM Unauthorized reproduction of this article is prohibited

13 186 OLIVER G ERNST AND ELISABETH ULLMANN Proof Since the Stochastic Galerkin atrices G α, α I 2p,aresyetric,the largest eigenvalue of G α satisfies v G α v ψα ψ 2 β,γ I λ ax (G α )= ax v R Np \{ } v = ax v ψ V p\{} ψ 2 =ax p c β c γ ψ α ψ β ψ γ β,γ I p c β c γ ψ β ψ γ M ι β,γ I =ax p c β c γ =1 i=1 ψ() α (η,i )ψ () β (η,i )ψ γ () (η,i )w,i M ι β,γ I p c β c γ =1 i=1 ψ() β (η,i )ψ γ () (η,i )w,i for 2ι 1 α +2p, or, equivalently, ι p + α +1 2, =1,,M In addition, the nuerator in the last expression above can be estiated as follows: β,γ I p c β c M ι γ =1 i=1 ψ α () (η,i )ψ () β (η,i )ψ γ () (η,i )w,i { M } ax =1 ψ() α (η,i ):i =1,,ι,=1,,M β,γ I p c β c M ι γ =1 i=1 ψ () β (η,i )ψ γ () (η,i )w,i Thus, defining the ulti-index δ N M with δ := p + α +1 2, =1,,M,we obtain the desired result for the largest eigenvalue of G α : { M } λ ax (G α ) ax =1 ψ() α (η,i ):i =1,,δ,=1,,M =ax{ψ α (η) :η Ξ δ } The lower bound for the sallest eigenvalue of G α follows analogously Corollary 11 In the stochastically linear case, the inclusion bound in (43) is sharp Proof For α =1,α =1,wehaveδ = p +1, =1,,M, in Theore 1 Noting that ψ α (ξ) =ψ () (ξ )=ξ, the spectral bounds for G α in (43) read θ α = in {η,i}, Θ α = ax {η,i} i=1,,p+1 i=1,,p+1 Finally, since the quadrature nodes η,i, i =1,,p+ 1, are the zeros of ψ () p+1,the assertion follows fro Corollary 9 42 Coplete polynoials For the coplete polynoial spaces Vp C, the eigenvalues of the associated Stochastic Galerkin atrices Gα C ay be bounded by those of their tensor product polynoial counterparts Corollary 12 The eigenvalues of the Stochastic Galerkin atrices Gα C for the of coplete polynoials are bounded by space V C p (44) Λ(G C α ) [λ in(g α ),λ ax (G α )], α I C 2p, where for each ulti-index α I2p C the atrix G α =[ ψ α ψ β ψ γ ] β,γ Ip denotes the Stochastic Galerkin atrix with the sae ulti-index α where the polynoials ψ β and ψ γ vary over V p rather than the saller space Vp C Proof The inclusion (44) follows by a Rayleigh quotient arguent because for each α I2p C the atrix GC α is a principal subatrix of G α (see [14, Theore 4315]) Copyright by SIAM Unauthorized reproduction of this article is prohibited

14 STOCHASTIC GALERKIN MATRICES 1861 Corollary 13 The eigenvalues of the Stochastic Galerkin atrices Gα C space Vp C of coplete polynoials are bounded by for the Λ(G C α ) [θ α, Θ α ], α I C 2p, where θ α and Θ α are defined in (43) Proof This follows fro Corollary 12 and Theore 1 In the stochastically linear case for coplete polynoials we can characterize the eigenvalues copletely (See also [21, Lea 31] for the sae observation in a ore restricted context) Theore 14 In the stochastically linear case the eigenvalues of the Stochastic Galerkin atrices {G C}M C =1 for the space Vp of coplete polynoials are given by p+1 (45) Λ(G C )= j=1 Λ(U () 1,j ), where U () 1,j denotes the jth leading principal subatrix of the Jacobi atrix U () 1 R (p+1) (p+1) associated with the weight function ρ In other words, the eigenvalues of G C consist of the union of all zeros of the orthonoral polynoials {ψ() j } p+1 j=1 generated by the weight function ρ Proof This assertion follows fro the fact that, as shown in the proof of Lea 3, the atrices G C are perutation-siilar to a block diagonal atrix whose diagonal blocks are Jacobi atrices of diension 1, 2,,p+ 1 associated with the orthonoral polynoials generated by the weight function ρ, where all Jacobi atrices occur Corollary 15 In the stochastically linear case the inclusion bound (44) is sharp Proof This follows fro Theore 14 and the interlacing property of the zeros of real orthogonal polynoials (cf [11, Theore 12]) 43 Even weight functions and the ultivariate case We shall assue throughout the reainder of this section that the weight functions ρ, and therefore also their product, are even functions of ξ, ie, that (46) ρ ( ξ )=ρ (ξ ) ξ Γ In the priary case of interest where the weight functions ρ are probability density functions, (46) is satisfied for any coonly occurring probability distributions, notably the centered Gaussian distribution as well as a centered unifor distribution Proposition 16 For even weight functions the associated ultivariate orthonoral basis polynoials ψ α in (28) are even or odd functions according to whether their total degree α is even or odd, respectively, ie, there holds (47) ψ α (ξ) =( 1) α ψ α ( ξ) α N M Proof In [11, Theore 117] this relation is established for onic univariate orthogonal polynoials After noralization we have ψ () j (ξ )=( 1) j ψ () j ( ξ ), j, =1,,M Copyright by SIAM Unauthorized reproduction of this article is prohibited

15 1862 OLIVER G ERNST AND ELISABETH ULLMANN Hence, for the ultivariate orthonoral polynoials in (28) we obtain ψ α (ξ) = M =1 ψ () α (ξ )= M ( 1) α ψ α () ( ξ )=( 1) α ψ α ( ξ) =1 Proposition 17 For even weight functions the entries of the Stochastic Galerkin atrices G α satisfy the relation (48) ψ α ψ β ψ γ =( 1) α + β + γ ψ α ψ β ψ γ Proof Relation (48) follows fro (47) by substituting ξ for ξ in the M integrals into which ψ α ψ β ψ γ decouples For ulti-indices α of odd degree the syetry property (47) of the basis polynoials results in a syetric spectru for G α Theore 18 For even weight functions the eigenvalues of the Stochastic Galerkin atrices G α lie syetric with respect to the origin Proof To show that λ Λ(G α ) iplies λ Λ(G α ), denote by [v] β, β I, where I denotes either I p or Ip C, the coponents of the eigenvector v associated with an eigenvalue λ Setting ṽ to be the vector obtained by replacing the coponent [v] β of v with ( 1) β [v] β, there holds [G α ṽ] β = γ I ψ α ψ β ψ γ [ṽ] γ = γ I( 1) α + β + γ ψ α ψ β ψ γ ( 1) γ [v] γ =( 1) α + β λ[v] β = λ[ṽ] β 44 Even weight functions and the univariate case In the reainder of this section we present soe first results on the eigenvalues of the Stochastic Galerkin atrices in the univariate case, ie, the atrices U n () For clarity of presentation, we oit the superscript () Note that, for typographical reasons, we shall represent finite sequences of orthogonal polynoials as colun vectors in this section Since all atrices of recurrence coefficients are syetric this should not cause confusion We first recall that for even weight functions the diagonal recurrence coefficients α k in (216) and (217) vanish, ie, the three-ter recurrence (216) siplifies to (49) βk+1 ψ k+1 (ξ) =ξψ k (ξ) β k ψ k 1 (ξ), k =, 1,, ψ 1, and the associated tridiagonal Jacobi atrices U 1 have a vanishing diagonal The connection of eigenvalues of the atrices U 1 =: U 1,p R (p+1) (p+1) to the zeros of ψ p+1 is revealed by collecting the recurrence relations (49) for k =, 1,,p, yielding (41) ξ ψ (ξ) ψ 1 (ξ) ψ p 1 (ξ) ψ p (ξ) = U 1,p ψ (ξ) ψ 1 (ξ) ψ p 1 (ξ) ψ p (ξ) + βp+1 ψ p+1 (ξ) For each of the p + 1 distinct zeros {μ p+1,l } p+1 l=1 of ψ p+1, setting ξ = μ p+1,l in (41) represents an eigenvalue-eigenvector relation for the atrix U 1,p Copyright by SIAM Unauthorized reproduction of this article is prohibited

16 STOCHASTIC GALERKIN MATRICES 1863 To deterine the eigenvalues of the atrices U 2,p, we first establish a recurrence relation siilar to (49) for the product ψ 2 ψ k Lea 19 For even weight functions the polynoial sequence {ψ 2 ψ k } k satisfies the five-ter recurrence (411) ψ 2 ψ k = c 2 βk+2 β k+1 ψ k+2 +[c 2 (β k+1 + β k )+c ]ψ k + c 2 βk β k 1 ψ k 2, k =, 1, 2,, where ψ 2 (ξ) =c 2 ξ 2 + c,andψ k (ξ) for k = 1, 2 Proof Recalling fro the proof of Proposition 16 that the orthonoral polynoial ψ 2 is an even function, we ay factor it as (412) ψ 2 (ξ) =c 2 (ξ λ)(ξ + λ), where λ denotes the positive root Fro the recurrence relation (49) we deduce (413) (414) (ξ + λ)ψ k (ξ) = β k+1 ψ k+1 (ξ)+λψ k (ξ)+ β k ψ k 1 (ξ), (ξ λ)ψ k (ξ) = β k+1 ψ k+1 (ξ) λψ k (ξ)+ β k ψ k 1 (ξ) Utilizing (412) together with (413) and (414), we obtain ψ 2 (ξ)ψ k (ξ) =c 2 (ξ λ)(ξ + λ)ψ k (ξ) ( = c 2 (ξ λ) βk+1 ψ k+1 (ξ)+λψ k (ξ)+ ) β k ψ k 1 (ξ) ( = c 2 βk+1 βk+2 ψ k+2 (ξ) λψ k+1 (ξ)+ ) β k+1 ψ k (ξ) ( + c 2 λ βk+1 ψ k+1 (ξ) λψ k (ξ)+ ) β k ψ k 1 (ξ) ( + c 2 βk βk ψ k (ξ) λψ k 1 (ξ)+ ) β k 1 ψ k 2 (ξ) = c 2 βk+2 β k+1 ψ k+2 (ξ)+c 2 (β k+1 + β k λ 2 )ψ k (ξ)+c 2 βk β k 1 ψ k 2 (ξ) Substituting c = ψ 2 () = c 2 λ 2 establishes (411) Collecting the five-ter recurrence (411) for k =, 1,,pin a anner analogous to (41) now yields ψ 2 (ξ) ψ (ξ) ψ 1 (ξ) ψ p 1 (ξ) ψ p (ξ) = U 2,p ψ (ξ) ψ 1 (ξ) ψ p 1 (ξ) ψ p (ξ) + c 2 βp+1 β p ψ p+1 (ξ) c 2 βp+2 β p+1 ψ p+2 (ξ) To obtain an eigenvalue-eigenvector relation it is necessary that both the last two entries vanish in the last vector on the right-hand side for soe values of ξ By the interlacing property of real orthogonal polynoials, however, it follows that ψ p+1 and ψ p+2 have no coon zeros However, if the basis polynoials are ordered in an odd-even fashion, ie, ψ (ξ),ψ 2 (ξ),ψ 4 (ξ),,ψ p (ξ),ψ 1 (ξ),ψ 3 (ξ),,ψ p 1 (ξ), Copyright by SIAM Unauthorized reproduction of this article is prohibited

17 1864 OLIVER G ERNST AND ELISABETH ULLMANN assuing p is even, the following block 2 2 structure eerges: ψ (ξ) ψ (ξ) ψ 2 (ξ) ψ 2 (ξ) [ ] (415) ψ 2 (ξ) ψ p (ξ) U even 2,p O ψ 1 (ξ) = ψ p (ξ) O U2,p odd ψ 1 (ξ) + c 2 βp+2 β p+1 ψ p+2 (ξ) ψ 3 (ξ) ψ 3 (ξ) ψ p 1 (ξ) ψ p 1 (ξ) c 2 βp+1 β p ψ p+1 (ξ) The reordered atrix U 2,p is block diagonal because the recurrence (411) only couples polynoials of even index with polynoials with even index, as well as odd with odd, respectively The block diagonal structure in (415) now reveals that the eigenvalues of U 2,p are those of U2,p even together with those of U2,p odd These are obtained by considering the two uncoupled recurrences ψ (ξ) ψ (ξ) and ψ 2 (ξ) ψ 2 (ξ) = U even ψ 2 (ξ) 2,p + ψ p (ξ) ψ p (ξ) ψ 2 (ξ) ψ 1 (ξ) ψ 3 (ξ) ψ p 1 (ξ) = U 2,p odd ψ 1 (ξ) ψ 3 (ξ) ψ p 1 (ξ) c 2 βp+2 β p+1 ψ p+2 (ξ) + c 2 βp+1 β p ψ p+1 (ξ) Turning to the first, we observe that ψ p+2 has p + 2 roots which lie syetrically about the origin In view of the fact that ψ 2 (ξ) is an even function of ξ, we conclude that all eigenvalues of U2,p even are obtained by inserting the positive roots of ψ p+2 into ψ 2 Analogously, in the odd recurrences we obtain the eigenvalues of U2,p odd by inserting the positive roots of ψ p+1 into ψ 2 Noting that the case of p odd works in an analogous anner, we suarize our findings in the following theore Theore 2 (a) In case p =2k, k N, assuing even weight functions, the eigenvalues of the univariate Stochastic Galerkin atrix U 2,p are given by Λ(U 2,p )={ψ 2 (μ p+2,l )} k+1 l=1 {ψ 2(μ p+1,l )} k l=1, where {μ p+2,l } k+1 l=1 denote the k +1 positive roots of ψ p+2, and{μ p+1,l } k l=1 denote the k positive roots of ψ p+1 (b) In case p =2k +1, k N, assuing even weight functions, the eigenvalues of U 2,p are given by Λ(U 2,p )={ψ 2 (μ p+2,l )} k+1 l=1 {ψ 2(μ p+1,l )} k+1 l=1, where {μ p+2,l } k+1 l=1 denote the k +1 positive roots of ψ p+2, and{μ p+1,l } k+1 l=1 denote the k +1 positive roots of ψ p+1 Copyright by SIAM Unauthorized reproduction of this article is prohibited

18 STOCHASTIC GALERKIN MATRICES 1865 Corollary 21 Assuing even weight functions, the upper inclusion bound in (43) for the univariate Stochastic Galerkin atrix U 2,p is sharp Proof ForM =1andα =(2)wehaveδ =(p + 2) in Theore 1 Thus, for the atrix U 2,p, the upper spectral bound in (43) reads Θ α =ax{ψ 2 (η) :ψ p+2 (η) =} The assertion follows fro Theore 2 and the interlacing property of the zeros of real orthogonal polynoials (cf [11, Theore 12]) Can we proceed in the sae way in order to copute the eigenvalues of the next atrix in turn, ie, the Stochastic Galerkin atrix U 3,p? Following the lines of the proof of Lea 19, it is easy to establish a seven-ter recurrence relation for the product ψ 3 ψ k In fact, as we will see in the proof of Lea 23, the product ψ n ψ k of any two orthonoral polynoials generated by an even weight function can be expressed in ters of the polynoials ψ l, with degree l satisfying the relation n k l n + k and n + k + l is an even integer Thus, the product ψ n ψ k satisfies a(2n + 1)-ter recurrence relation Lea 22 For even weight functions the polynoial sequence {ψ 3 ψ k } k satisfies the seven-ter recurrence relation (416) ψ 3 (ξ)ψ k (ξ) =c 3 βk+3 β k+2 β k+1 ψ k+3 (ξ)+c 3 βk β k 1 β k 2 ψ k 3 (ξ) +(c 3 (β k+2 + β k+1 + β k )+c 1 ) ψ k+1 (ξ) +(c 3 (β k+1 + β k + β k 1 )+c 1 ) ψ k 1 (ξ), k =, 1, 2,, where ψ 3 (ξ) =c 3 ξ 3 + c 1 ξ,andψ k (ξ) for k = 1, 2, 3 We oit the technical proof and utilize the recurrence relation (416) in the by now well-established way: ψ (ξ) ψ (ξ) ψ 1 (ξ) ψ 1 (ξ) (417) ψ 3 (ξ) = U 3,p + ψ p 2 (ξ) ψ p 2 (ξ) ψ p 1 (ξ) ψ p (ξ) ψ p 1 (ξ) ψ p (ξ) c 3 βp+1 β p β p 1 ψ p+1 (ξ) c 3 βp+2 β p+1 β p ψ p+2 (ξ) c 3 βp+3 β p+2 β p+1 ψ p+3 (ξ)+r(ξ) where r(ξ) =(c 3 (β p+2 + β p+1 + β p )+c 1 )ψ p+1 (ξ) To answer the question above, it is not obvious in which way, if at all, one could use the atrix equation (417) for the eigenvalue coputation of U 3,p The idea of reordering, which was applied successfully for the eigenvalue coputation of U 2,p, does not lead to a decoupling of the atrix U 3,p, resulting instead in a block antidiagonal atrix Moreover, when using (417), we assue the eigenvectors of U 3,p to have a special structure, an approach which ight be isleading 5 Sparsity of Stochastic Galerkin atrices In this section we investigate the sparsity pattern of the Stochastic Galerkin atrices G α in (26), where we assue even probability density functions ρ ; see (46) In particular, we derive upper bounds on the nuber of nonzero entries of the univariate Stochastic Galerkin atrices U n in (21), and of the atrices G α where α = 1 for the space of coplete polynoials V = Vp C Inforation on the sparsity pattern is useful for the efficient ipleentation of atrix-vector ultiplication with Stochastic Galerkin atrices 51 The univariate case We first specialize the linearization forula (215) for even weight functions Again we oit the sub- and superscripts which distinguish the rando variables and weight functions, Copyright by SIAM Unauthorized reproduction of this article is prohibited

19 1866 OLIVER G ERNST AND ELISABETH ULLMANN Lea 23 The product of any two orthonoral polynoials ψ i and ψ j associated with an even weight function has the representation (51) ψ i (ξ)ψ j (ξ) = i+j k= i j, i+j+k is even g kij ψ k (ξ) Proof Following an idea given by Markett [18, section 1], we deduce a recurrence relation for the product ψ i ψ j utilizing the reordered three-ter recurrence relation in (49): hence (52) ψ i ψ j = =ξψ i (ξ)ψ j (ξ) ξψ j (ξ)ψ i (ξ) = β i ψ i 1 (ξ)ψ j (ξ)+ β i+1 ψ i+1 (ξ)ψ j (ξ) β j ψ j 1 (ξ)ψ i (ξ) β j+1 ψ j+1 (ξ)ψ i (ξ); βi βi+1 βj 1 ψ i 1 ψ j 1 + ψ i+1 ψ j 1 ψ i ψ j 2 βj βj βj For j = 1,,i the product ψ i ψ j can be traced back to the initial products ψ k ψ = ψ k, k = i j, i j +2,,i+ j 2,i+ j; cf Figure 51 In addition the polynoials ψ k on the right-hand side of (51) ust have the sae parity as the product ψ i ψ j ; hence i + j + k is even in all cases j (i,j) i i j i j+2 i+j 2 i+j Fig 51 Visualization of recurrence relation (52) We note that by orthonorality we have g kij = ψ k ψ i ψ j = [U k ] i,j in (51); cf section 24 Thus, for even weight functions we obtain the following necessary condition for vanishing entries of U n Corollary 24 For even weight functions, [U n ] i,j iplies i j n i + j and i + j + n is even The nuber of nonzero entries of U n ay be bounded as follows Lea 25 (a) In case n =2k, k N, (p n +1)(n +1)+k 2, n p, nnz(u n ) (p k +1) 2, p+1 n 2p,, n > 2p Copyright by SIAM Unauthorized reproduction of this article is prohibited

20 STOCHASTIC GALERKIN MATRICES 1867 (b) In case n =2k +1, k N, (p n +1)(n +1)+k 2 + k, n p, nnz(u n ) (p k +1)(p k), p+1 n 2p,, n > 2p Proof We distinguish three cases: (1) n>2p: byproposition4wehaveu n = O, ie, nnz(u n )= (2) p +1 n 2p: i, j p iplies i j p; hence the necessary condition in Corollary 24 reduces to n i + j and i + j + n is even Note that atrix positions with i + j = n are adissible since i + j + n = n + n =2n is always even We count the nuber of adissible atrix positions running along the antidiagonals of the atrix If n =2k, k N, iseven,soisi + j, eaning that entries on the ain diagonal of U n are adissible; see Figure 52(a): p k nnz(u n ) 2s +1=(p k +1)+(p k)(p k +1)=(p k +1) 2 s= If n =2k +1, k N, is odd, so is i + j, eaning that entries on the ain diagonal of U n are not adissible; see Figure 52(b): p k nnz(u n ) 2s =(p k)(p k +1) s=1 (3) n p: As in case (2) atrix positions with i + j = n are adissible In addition positions that satisfy i j = n are adissible since i + j + n = j ± n + j + n =2j + n ± n is always even Again we count the nuber of adissible atrix positions running along the antidiagonals of the atrix U n If n = 2k, k N, is even, entries on the ain diagonal of U n are adissible; see Figure 53(a): k 1 nnz(u n ) (p +1 n)(n +1)+ 2s +1 s= =(p +1 n)(n +1)+k +(k 1) k =(p +1 n)(n +1)+k 2 If n =2k +1, k N, is odd, entries on the ain diagonal of U n are not adissible; see Figure 53(b): nnz(u n ) (p +1 n)(n +1)+ k 2s s=1 =(p +1 n)(n +1)+k 2 + k The bounds given in Lea 25 are sharp, since we count the exact nuber of nonzero entries provided that every adissible entry in U n is not zero This is true, for exaple, for the standard Gaussian probability density function, cf (A1) in Appendix A1 and Figure 54 Copyright by SIAM Unauthorized reproduction of this article is prohibited