Generalized Compact Multi-grid

-Extending our recent work, based on the ideas of the multi-grid iteration, we decrease the storage space for a smooth solution of a nonlinear PDF. and, furthermore, for any smooth function on a multi-dimensional grid and on discretization sets other than grids. A new approach to the numerical evaluation and storage of the solutions to a large class of linear part ial differential equations (PDE's ) discretized over a d-dimensional grid G was recently proposed in [1]. The method (which we call Compact Multi.grid, since it follows the framework of the multi-grid i terative process) enables us, in particular, to decrease, by roughly the factor of log N, the t ime and the precision of computing and the storage (memory) space where N is the number of points of the grid G and where the Boolean (bit-) complexity measure is assumed. In practice, N is usually large, so tha t the improvement is significant. In the present paper we again follow the multi-grid iteration scheme but focus on the economization of the memory space. We set a refined and more general framework for this method and arrive at more general results. Our generalized compact multi-grid method enables us equally well to decrease the storage space for the solutions to linear and nonlinear PDE's , and only a routine smoothness assumption is needed for this. Furthermore, the method applies to the compression of the space for the storage of all smooth and many nonsmooth solutions to PDE ' s on multi-dimensional grids and on sets of a more general class in a Euclidean space, which includes many practical cases not covered in [1]. Next, we will formalize our results. Let a, b, c, d and g denote five fixed positive constants, d integer, Gi denotes the d-dimensional grid of IGil Ni -2 di points, Gi = {(jl 2 i , J2 2 i , . . , jd 2i ) , jk = 0, 1 , . . . , 2 i 1; k = 1, . . . , d}, i = 0, 1 , . . . , n, so tha t the projection of Gi into each coordinate edge consists of 2 i equally spaced points of the half-open unit interval {t, 0 _< t < 1}, and the grids Go, G I , . . . , G~ = G recursively refine each other. Suppose that we need to store the approximations u*(x) to a smooth function u(x) given on the finest of these grids, G = G , , within the absolute error bound A = biN e and normalized so tha t lu*(x)l < 1, for x • G, where N = Nn = IGI = IG,~I. We will assume that u*(x) satisfies the following (HSlder's type) smoothness requirement on G: ]u*(x) u*(y)[ ~ a(Ilx yll,) 9 (I) where, say, s = 1,2, or oo, and g is a fixed positive constant, so tha t the fixed point binary representation (and the storage) of each value u* (x) on G requires [c log N l o g b] bits of memory, V. Pan is supported by NSF Grant CCR-9020690 and by PSC CUNY Award ~:662478. J. Reif is supported by DARPA/ISTO Contracts N00014-88-K-0458, DARPA N00014-91-J-1985, N00014-91-C-0114, NASA subcontract 550-63 of prime contract NAS5-30428, US-Israel Binational NSF Grant 88-00282/2, and NSF Grant NSF-IRL9100681. Typeset by .AA~q-TEX 4 V. PAN AND J. RIEF which is O(log N) as N ~ oo. It seems that the order of N log N bits are needed for the storage of the N values of u*(x) at all the N points of G, but actually we have the following result: PROPOSITION 1. O(N) bits of memory sul~ce in order to represent the va/ues of all the approximations to the function n*(x) on a multi-dimensional grid of N points, provided that (1) holds. RgMARK. In the approach of [1], where ui(x) denote the discretizations over the grids Gi of the solution u(x) to a PDE, it is assumed that log 2 In(x) ui(x)[ < a / 3 i for two fixed constants a and/3. This is in the spirit of solving PDE's by the multi-grid methods (see e.g., [2-11]). Now, however, we will show that even the weaker assumption (1) is sufficient to make our compact scheme work. To arrive at Proposition 1, let g(v) denote the number of bits in the fraction of the floating point representation of a binary number v, and define the auxiliary functions u~(x) on Gi that minimize £(u[(x)) subject to lu[(x) u*(x)l < a2 -ai for x E Gi and for i = 0, 1 , . . . ,n 1, that is, u*(x) is obtained by rounding off u*(x) to £(u*(x)) = [g i l o g z a] bits, thus ignoring the bits of u*(x) that represent the values less than 2-t(uT(x)) -k. Denote ei(x) = , ' ( , , ) ULl(X). for x e G,-1. i = 1 . . . . , , . (2) Further, for every point y of Gi G i 1 , fix some (say, northwestern on a 2-dimensional grid) nearest neighbor x = xi (y) on Gi-1 such that I ly xl l = 2 i , (3) and define ei(y) = u*(y) U*_l(X ). (4) Equation (2) and the definiton of uT(x) imply that le,(x)l < a(2-a' + 2 -a¢ ' -o ) = a 2-a( i -1) (1 + 2 -a) . (5) Since the bits of ei(x) corresponding to the values less than 2-fa i-log2 a] are ignored, we obtain that g(ei(x)) < [g i log 2 a] [g i g log2(a(1 + 2-9))J, and therefore, t (e , (x)) < 2g + 3. (6) Furthermore, apply (2) and (4) and deduce that ei(y) = ( u T ( y ) uT(x) )+ ei(x). Now apply (1), (3) and (5) and deduce that lei(Y)l < a 2-~¢~-x)(1 + 2-2~), and therefore, t (e i (y)) _< 3g + 3, if y E Gi. (7) Now, according to the compact multi-grid storage scheme, we first store the value u*(x) at the point x of Go [by using [clog N log b] = O(log N) memory bits] and then, recursively for i = 1 , . . . , n, store the values e i (y ) at all the Ni points y of Gi [by using at most (3g+3)Ni N i 1 n n I memory bits due to (6) and (7)]. The overall storage of at most (3g + 3) )"~i=1 Ni )-']~i=0 Ni : (3g + 3) )-~n=l 2di ~"~i=0n-1 2di < 2(3g + 3)N = O(N) bits suffice to store eT(y) for all y E Gi and fo i = 1 , . . . , n . This is a compact representation of u*(x) on G, since if we need, we may recover u*(x) for any point x of G by using the saved values u*(x) on Go and e i (x ) on Gi for i = 1 , . . . ,n and by recursively applying (2) and (4) for i = 1 , . . . , n. For each x of G, this recovery takes at most O(logn) bit-operations; furthermore, in many applications, we may store the function u*(x) in the above compressed form and only very rarely need to decompress it (see [1], the end of Section 1.4). The above approach and the results of Proposition 1 can be extended in the two following directions. 1. Instead of the above functions u~(x), obtained by the truncation of u(x), we may use any auxiliary functions ui(x) on the grids Gi for i = 0, 1 , . . . , n, such that u*(x) = u~(x) on G, and Ui(X) ]9/ U i _ I ( X ) JrC/(X) , X ~ G i , Compact multi-grid 5 where Pi is a prolongation operator and Pi u i l (x ) is a prolongation of Uil(y) from Gi-1 to Gi obtained by means of interpolation (typically, by averaging) of the values of ui(y) taken at a certain set of points y of Gi-1 that lie near x, and where for each x E Gi and for each i, the fraction of the floating point binary representation of ei(x) contains at most g bits for a fixed cons tant g. Then Propos i t ion 1 is extended as long as the prolongat ion opera tors Pi enforce tha t g ( e i (x ) ) = O(1), for all i. 2. Propos i t ion 1 can be extended to the case where the grids G o , . . . , Gn are replaced by any rapidly expanding sets S o , . . . , Sn (such tha t ]Sn[ = N, IS01 = O(1), and for every point y of Si there is its neighbor x of Si-1 such tha t [lui(y) Ui_l(X)[ I < 7 i , and fur thermore, [Nil >_ OIS i l [ , for two constants 7 > 1 and 0 > 1, and So C $1 C . . . C S , -S). Note t h a t this extension enables us to t reat many nonsmoo th funct ions u(x) too, since we may increase the densi ty of S where u(x) is not smooth .