Multiresolutional Analysis 327 the solution x and d,is its fine-scale component.Formally eliminating d,from (7.5)by substituting d.=Asd-Bs s.)yields Ts,-Cs As Bs,)s,=s1-Cs,As dr (7.7) This equation is called a reduced equation,while the operator Rs,=Ts,-Cs As,Bs (7.8) is a one step reduction of the operator S;also known as the Schur complement of As the block matrix Bsy Cs,Ts, Note that the solution s,of the reduced equation is exactly P,where P is the projection onto V andx is the solution of (7.4).Note that the reduced equation is not the same as the averaged equation,which is given by Ts 5:=51 (7.9) Once we have obtained the reduced equation,it may formally be reduced again to produce an equation on Vand the solution of this equation is just the V2 component of the solution for (7.4).Likewise,we may reduce these equations recursively n times (assuming that,if the multiresolution analysis is on a bounded domain,then j+ns)to produce an equation on V the solution of which is the projection of the solution of (7.4)on V We note that in the finite-dimensional case,if we are considering a multiresolution analysis defined on a domain in R,the reduced equation(7.5)has half as many unknowns as the original equation(7.4).If the domain is in R2,then the reduced equations have one-fourth as many unknowns as the original equation. Reduction,therefore,preserves the coarse-scale behaviour of solutions while reducing the number of unknowns. In order to iterate the reduction step over many scales,we need to preserve the form of the equation as a way of deriving a recurrence relation.In (7.4)and(7.5), both S,and Rs,are matrices,and thus the procedure may be repeated.However, identification of the matrix structure is usually not sufficient.In particular,even though the matrix A for ODEs and PDEs is sparse,the component As term may become dense,changing the equation from a local one to a global one.It is important to know under what circumstances the local nature of the differentialMultiresolutional Analysis 327 the solution x and x d is its fine-scale component. Formally eliminating x d from (7.5) by substituting ( ) x S f S x d A d B s j j = − −1 yields ( ) S S S S x f CS AS d f T C A B s s j j j j j j −1 −1 − = − (7.7) This equation is called a reduced equation, while the operator S j S j S j S j S j R T C A B−1 = − (7.8) is a one step reduction of the operator S j also known as the Schur complement of the block matrix ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ j j j j S S S S C T A B . Note that the solution x s of the reduced equation is exactly P x j+1 , where Pj+1 is the projection onto Vj+1 and x is the solution of (7.4). Note that the reduced equation is not the same as the averaged equation, which is given by S x f T s s j = ~ (7.9) Once we have obtained the reduced equation, it may formally be reduced again to produce an equation on Vj+2 and the solution of this equation is just the Vj+2 component of the solution for (7.4). Likewise, we may reduce these equations recursively n times (assuming that, if the multiresolution analysis is on a bounded domain, then j+n§0) to produce an equation on Vj+n , the solution of which is the projection of the solution of (7.4) on Vj+n . We note that in the finite-dimensional case, if we are considering a multiresolution analysis defined on a domain in R, the reduced equation (7.5) has half as many unknowns as the original equation (7.4). If the domain is in R2 , then the reduced equations have one-fourth as many unknowns as the original equation. Reduction, therefore, preserves the coarse-scale behaviour of solutions while reducing the number of unknowns. In order to iterate the reduction step over many scales, we need to preserve the form of the equation as a way of deriving a recurrence relation. In (7.4) and (7.5), both S j and S j R are matrices, and thus the procedure may be repeated. However, identification of the matrix structure is usually not sufficient. In particular, even though the matrix A for ODEs and PDEs is sparse, the component −1 j AS term may become dense, changing the equation from a local one to a global one. It is important to know under what circumstances the local nature of the differential