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An efficient deflation technique for the communication-avoiding conjugate gradient method.

1. Introduction. Krylov subspace methods (KSMs) are a class of iterative algorithms commonly used to solve the linear system Ax = b when A is large and sparse. In each iteration m, updates to the next solution [x.sub.m+1] and residual [r.sub.m+1] consist of one or more sparse matrix-vector multiplications (SpMVs) and vector-vector operations in each iteration. On modern computers, these operations are communication-bound: the movement of data rather than the computation is the limiting factor in performance. Recent efforts have focused on communication-avoiding KSMs (CA-KSMs), which reorder the computations in classical KSMs to perform O(s) computation steps of the algorithm for each communication step; see, e.g., [6, 10, 12, 14, 16, 20, 26, 27, 49, 51]. This formulation allows an O(s) reduction in the total communication costs per iteration, which can translate into significant speedups [36].

In addition to speed per iteration, the performance of iterative methods also depends on the total number of iterations required for convergence. For the conjugate gradient method (CG), the KSM of choice for solving symmetric positive definite (SPD) systems, it is well- known that the rate of convergence in exact arithmetic can be bounded in terms of the eigenvalue distribution. Although these bounds are not always tight and additional complications arise in finite precision computations (see, e.g., [34]), nonetheless, preconditioning techniques, wherein the system's eigenvalue distribution is altered to improve the convergence bounds, have been employed successfully in practice; for a survey of approaches, see [44]. Unfortunately, except for simple examples like (block) Jacobi, polynomial, and sparse approximate inverse preconditioners, the ability to exploit temporal locality across KSM iterations is diminished by preconditioning, and the relative benefits of a CA-KSM, compared to its classical counterpart, decline; see, e.g., [27]. Avoiding communication in a general preconditioned method seems to necessitate significant modifications to the algorithm.

This has stimulated an active area of research in designing practical preconditioners for CA-KSMs with much recent progress. To this end, we propose deflation, which can be viewed as a type of preconditioning with a singular preconditioner, as a feasible technique for improving convergence in CA-KSMs. We derive a communication-avoiding deflated CG (CA-D-CG), based on the deflated CG formulation (which we refer to as 'D-CG') in [45]. Our analysis shows that the additional costs of CA-D-CG over CA-CG are of lower-order, which means that, as in (non-deflated) CA-CG, we still expect a possible O(s) speedup per s steps over the classical implementation. This motivates deflation as a promising preconditioner for CA-KSMs in practice. We give a numerical example and performance modeling results which demonstrate that choosing the number of deflation vectors as well as the blocking factor s result in complex, machine-dependent tradeoffs between the convergence rate and the time per iteration.

2. Related work. Deflation and augmentation techniques have been applied to improve convergence in many KSMs since the mid 1980s; for a survey, see [46, Chapter 9]. Many approaches in the literature can be viewed as instances of more general deflation/augmentation frameworks [21, 25]. Connections have also been drawn between deflation and multilevel preconditioners; see, e.g., [48] and the references therein.

In this work, we consider the case of CG, the first KSM that was modified to perform deflation [17, 39]. We note that the potential for eigenvalue deflation was also known to Rutishauser before such methods gained popularity in the literature [18]. In this work, we study the D-CG formulation as given in [45].

CG is convenient since it allows us to concretely demonstrate our algorithmic reorganization while sidestepping technical issues involving breakdown, i.e., where KSM iterates may not exist in exact arithmetic. However, we see no obstacles beyond breakdown for extending our approach to other KSMs that deflate in a similar manner.

Many authors have reformulated KSMs to reduce communication costs. Our approach is most closely related to the CA-KSMs developed by Hoemmen et al. [27, 36], which in turn are based on s-step KSMs proposed in the 1980s; see [27] for a historical perspective on avoiding communication in KSMs. Works that consider CG in particular include [6, 10, 11, 27, 49, 51]. The derivation of the CA-D-CG algorithm here most closely follows [6].

As mentioned in Section 1, there has been much recent work in the development of practical preconditioners for CA-KSMs. Grigori et al. developed a CA-ILU(0) preconditioner for CA-GMRES [24]. For structured problems, their method exploits a novel mesh ordering to obtain triangular factors that can be applied with less communication. There is also recent work in developing a new "underlapping" technique in communication-avoiding domain decomposition preconditioners for CA-KSMs [5]. For the case of preconditioners with both sparse and low-rank components (e.g., hierarchical semiseparable matrices), applying the low-rank components dominates the communication costs. Techniques in [27, 30] block together several applications of the low-rank components in order to amortize communication costs over several KSM iterations. We also note that there has been recent work in developing a high-performance deflated "pipelined" conjugate gradient method [22].

Previous work has developed efficient deflation techniques for CA-KSMs in order to recover information lost after restarting the Arnoldi process. Wakam and Erhel [41] extended a special case of CA-GMRES [27, 36] with an adaptive augmentation approach. Both their algorithm and ours aim to reduce the frequency of global collectives incurred by deflation/augmentation compared to previous approaches. However, our applications differ: in our case we apply deflation as a more general preconditioning technique for Lanczos-based methods. We note that the algorithm presented in [41] restricts the constructed Krylov bases to Newton polynomials. It may be beneficial to extend their approach to the more general family of polynomials which we consider here.

2.1. The communication-avoiding conjugate gradient method. We briefly review the communication-avoiding conjugate gradient method (CA-CG) for solving Ax = b, where A is SPD and given any vector [x.sub.0] interpreted as an initial approximation to x.

For reference, the classical CG method is shown in Algorithm 2.1. Within the iteration loop, communication occurs in the SpMV [Ap.sub.m] required by lines 3 and 5 as well as in the inner products in lines 3 and 6.

ALGORITHM 2.1. Conjugate Gradient Method (CG).

Require: Approximate solution [x.sub.0] to Ax = b

1: [r.sub.0] = b - [Ax.sub.0], [p.sub.0] = [r.sub.0]
2: for m = 0, 1,... until convergence do
3:     [[alpha].sub.m] = ([r.sup.T.sub.m] [r.sub.m])/([P.sup.T.sub.m]
         [Ap.sub.m])
4:     [x.sub.m+1] = [x.sub.m] + [[alpha].sub.m] [p.sub.m]
5:     [r.sub.m+1] = [r.sub.m] - [[alpha].sub.m] [Ap.sub.m]
6:     [[beta].sub.m+1] = ([r.sup.T.sub.m+1] [r.sub.m+1])/
         ([r.sup.T.sub.m][r.sub.m])
7:     [p.sub.m+1] = [r.sub.m+1] + [[beta].sub.m+1][p.sub.m]
8: end for
9: return [x.sub.m+1]


In CA-CG, iterations are split into an inner loop over 0 [less than or equal to] j < s and an outer loop over k, whose range depends on the number of steps until convergence. We index the iteration m in CG as the iteration m = sk + j in CA-CG. By induction on lines 4, 5, and 7 of Algorithm 2.1, [p.sub.sk+j], [r.sub.sk+j], [x.sub.sk+j] - [x.sub.sk] [member of] [K.sub.s+1] (A, [p.sub.sk]) + [K.sub.s] (A, [r.sub.sk]),

for 0 [less than or equal to] j [less than or equal to] s, where [K.sub.i] (A, v) denotes the i-th Krylov subspace of A with respect to v, i.e.,

[K.sub.i](A, v) = span{v, Av, [A.sup.2]v, ..., [A.sup.i-1]v}.

Therefore, we let length-(2s + 1) vectors [x'.sub.k,j], [r'.sub.k,j], and [p'.sub.k,j] denote the coordinates for [x.sub.sk+j] - [x.sub.sk], [r.sub.sk+j], and [p.sub.sk+j], respectively, in the columns of

(2.1) [V.sub.k] = [[P.sub.k], [R.sub.k]] = [[[rho].sub.0] (A)[p.sub.sk], ..., [[rho].sub.s] (A)[p.sub.sk], [[rho].sub.0](A)[r.sub.sk], ..., [[rho].sub.s-1](A)[r.sub.sk]],

where [[rho].sub.i] is a polynomial of degree i. That is, we have

(2.2) [x.sub.sk+j] [x.sub.sk] = [V.sub.k][x'.sub.k,j], [r.sub.sk+j] = [V.sub.k][r'.sub.k,j], and [p.sub.sk+j] = [V.sub.k][p'.sub.k,j],

for 0 [less than or equal to] j [less than or equal to] s. For brevity, we will refer to [V.sub.k] as a basis, although the columns of [V.sub.k] need not be linearly independent, e.g., for k = 0, [K.sub.s] (A, [r.sub.0]) [subset or equal to] [K.sub.s+1] (A, [p.sub.0]) since [p.sub.0] = [r.sub.0.]

We assume that the polynomials in (2.1) can be computed via a three-term recurrence in terms of the parameters [[gamma].sub.i], [[theta].sub.i], and [[sigma].sub.i], as

(2.3) [[rho].sub.0] (A) = 1, [[rho].sub.1](A) = (A - [[theta].sub.0]I)[[rho].sub.0](A)/[[gamma].sub.0], and [[rho].sub.i+1] (A) = ((A - [[theta].sub.i]I)[[rho].sub.i](A) - [[sigma].sub.i][[rho].sub.i- 1](A))/[[gamma].sub.i],

for 1 [less than or equal to] i < s. This three-term recurrence covers a large class of polynomials including classical orthogonal polynomials. The monomial, Newton, and Chebyshev bases are common choices for generating Krylov bases; see, e.g., [43]. To simplify notation, we assume the basis parameters remain the same throughout the iteration.

The basis [V.sub.k] is generated at the beginning of each outer loop using the current [r.sub.sk] and [p.sub.sk] vectors. Assuming A is sufficiently sparse, these O(s)-dimensional bases can be computed after only reading A (sequential case) or exchanging vector entries with neighbors (parallel case) O(1) times, using the communication-avoiding 'matrix powers kernel' described in [16, 27, 36].

Substituting each polynomial [[rho].sub.i] on the right-hand side of (2.1) by its recursive definition given in (2.3), rearranging terms, and postmultiplying by [p'.sub.k,j], we obtain

(2.4) A[[V.bar].sub.k][P'.sub.k,j] = [V.sub.k][B.sub.k][p'.sub.k,j],

where [[V.bar].sub.k] = [[P.[bar].sub.k], 0, [[R.bar].sub.k], 0], and [[P.bar].sub.k] and [[R.bar].sub.k] are [P.sub.k] and [R.sub.k], respectively, with the last columns omitted, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]'

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] .

Then by (2.2) and (2.4), the multiplication [Ap.sub.sk+j] in the standard basis becomes [B.sub.k][p'.sub.k,j] in the basis [V.sub.k]. Recall that [B.sub.k] and [p'.sub.k,j] are both of dimension O(s), which means that they either fit in fast memory (in the sequential case) or are local to each processor (in the parallel case), and thus the computation [B.sub.k][p'.sub.k,j] does not require data movement.

In each outer loop of Algorithm 2.2 below, we compute the O(s)-by-O(s) Gram matrix [G.sub.k] = [V.sup.T.sub.k][V.sub.k]. Then by (2.2) and (2.4), the inner products in lines 3 and 6 of Algorithm 2.1 can be written as

[r.sup.T.sub.k+j] [r.sub.sk+j] = [r'.sup.T.sub.k,j][G.sub.k][r'.sub.k,j] for 0 [less than or equal to] j [less than or equal to] s and [p.sup.T.sub.sk+j] [Ap.sub.sk+j] = [p'.sup.T.sub.k,j][G.sub.k][B.sub.k][p'.sub.k,j] for 0 [less than or equal to] j < s.

Thus, after [G.sub.k] has been computed in the outer loop, the inner products can be computed without additional communication. Although many details are omitted, this gives the general idea behind avoiding data movement in Lanczos-based KSMs. The resulting CA-CG method is shown below in Algorithm 2.2.

ALGORITHM 2.2. Communication-Avoiding Conjugate Gradient (CA-CG).

Require: Approximate solution [x.sub.0] to Ax = b

1: [r.sub.0] = b - [Ax.sub.0], [p.sub.0] = [r.sub.0]
2: for k = 0, 1, ... until convergence do
3:     Compute [P.sub.k], [R.sub.k], let [V.sub.k] = [[P.sub.k],
         [R.sub.k]]; assemble [B.sub.k].
4:     [G.sub.k] = [V.sup.T.sub.k] [V.sub.k]
5:     [p'.sup.T.sub.k,0] = [1, [0.sup.T.sub.2s]], [r'.sup.T.sub.k,0] =
         [[0.sup.T.sub.s+1], 1, [0.sup.T.sub.s-1]], [x'.sup.k,0] =
         [0.sub.2s+1]
6:     for j = 0, ..., s - 1 do
7:         [[alpha].sub.sk+j] =[r'.sup.T.k,j][G.sub.k][r'.sub.k,j])
             /([p'.sup.T.sub.k,j][G.sub.k][B.sub.k][p'.sub.k,j])
8:         [x'.sub.k,j+1] = [x'.sub.k,j] + [[alpha].sub.sk+j]
             [p'.sub.k,j],
9:         [r'.sub.k,j+1] = [r'.sub.k,j] - [[alpha].sub.sk+j][B.sub.k]
             [p.sub.k,j]
10:        [[beta].sub.sk+j+1] = ([r'.sup.T.sub.k,j+1] [G.sub.k]
             [r'.sub.k,j+1])/([r'.sup.T.sub.k,j][G.sub.k][r'.sub.k,j])
11:        [p'.sub.k,j+1] = [r'.sub.k,j+1] + [[beta].sub.sk+j+1]
             [p'.sub.k,j]
12:    end for
13:   [x.sub.sk+s] = [V.sub.k][x'.sub.k,s] + [x.sub.sk],
      [r.sub.sk+s] = [V.sub.k][r'.sub.k,s],
        [p.sub.sk+s] = [V.sub.k][p'.sub.k,s]
14: end for
15: return [x.sub.sk+s]


2.2. Deflated conjugate gradient method. Our CA-D-CG is based on D-CG by Saad et al. [45], shown in Algorithm 2.3 for reference. (As mentioned above, this was not the first appearance of deflated CG in the literature.)

We now summarize the motivation for the use of eigenvalue deflation in the CG method as presented in [45]. It is well-known (see, e.g., [23]) that in exact arithmetic, after m iterations of CG, the error is (loosely) bounded by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with [kappa](A) = [[lambda].sub.n]/[[lambda].sub.1] where [[lambda].sub.1] [less than or equal to] [[lambda].sub.2] [+ or -] ... [less than or equal to] [[lambda].sub.n] are the eigenvalues of the SPD matrix A. The authors of [45] prove that for some set of linearly independent vectors W = [[w.sub.1], [w.sub.2], ..., [w.sub.c]], D-CG applied to Ax = b is mathematically equivalent to CG applied to the positive semidefinite system [H.sup.T] AH[??] = [H.sup.T]b, where H = I--W[([W.sup.T] AW).sup.-1][(AW).sup.T] is the A-orthogonal projection onto [W.sup.[perpendicular to]A] and x = H[??]. When the columns of W are exact eigenvectors associated with [[lambda].sub.1], ..., [[lambda].sub.c], the effective condition number (see [19]) is [[kappa].sub.eff] ([H.sup.T] AH) = [[lambda].sub.n]/[[lambda].sub.c+1]. When the columns of W are approximate eigenvectors associated with [[lambda].sub.1], ..., [[lambda].sub.c], one can expect that [[kappa].sub.eff]([H.sup.T] AH) [approximately equal to] [[lambda].sub.n]/[[lambda].sub.c+1]. Thus if [[lambda].sub.c+1] > [[lambda].sub.1], deflation decreases the effective condition number of the system, thus theoretically improving the bounds on the (exact arithmetic) convergence rate.

We note that it is well-known that in reality, the convergence of the conjugate gradient method is much more accurately described by considering the spacing between eigenvalues of the matrix A, and even these more descriptive bounds do not hold in finite precision [32]. However, the reduction of the effective condition number described above nonetheless remains the motivation behind deflation and in practice can lead to an improved convergence rate in many cases. We also note that there has been recent work in developing tighter bounds on the rate of convergence in the deflated conjugate gradient method; see, e.g., [29].

For consistency, we assume we have an initial guess [x.sub.0] such that [r.sub.0] = b - [Ax.sub.0] [perpendicular to] W. To satisfy this initial requirement, one can choose x0 = x_1 + W (WT AW)-1Wt r-1, where [x.sub.-1] is arbitrary and [r.sub.-1] = b - [Ax.sub.-1] . Note that the selection of the subspace W is out of the scope of this paper. This topic is covered extensively in the literature; see, e.g., [1, 3, 9, 15, 37, 38, 47, 52].

ALGORITHM 2.3. Deflated Conjugate Gradient (D-CG).

Require: Approximate solution [x.sub.-1] to Ax = b with residual
[r.sub.-1] = b - [Ax.sub.-1]; n-by-c matrix W of rank c

1: Compute and factorize [W.sup.T] AW
2: [x.sub.0] = [x.sub.-1] + W[([W.sup.T] AW).sup.-1] [W.sup.T]
     [r.sub.-1], [r.sub.0] = b - [Ax.sub.0]
3: [mu] = [([W.sup.T] AW).sup.-1] [W.sup.T] [Ar.sub.0], [p.sub.0] =
     [r.sub.0] - W[mu]
4: for m = 0, 1,... until convergence do
5:     [[alpha].sub.m] = ([r.sup.T.sub.m][r.sub.m])/([p.sup.T.sub.m]
         [Ap.sub.m])
6:     [x.sub.m+1] = [x.sub.m] + [[alpha].sub.m][p.sub.m]
7:     [r.sub.m+1] = [r.sub.m] [[alpha].sub.m][Ap.sub.m]
8:     [[beta].sub.m+1] - ([r.sup.T.sub.m+1][r.sub.m+1])/
         ([r.sup.T.sub.m][r.sub.m])
9:     Solve [W.sup.T] AW [[mu].sub.m+1] = [W.sup.T] [Ar.sub.m+1]
         for [[mu].sub.m+1]
10:    [p.sub.m+1] = [r.sub.m+1] + [[beta].sub.m+1][p.sub.m]
         W[[mu].sub.m+1]
11: end for
12: return [x.sub.m+1]


3. Deflated communication-avoiding conjugate gradient method. We now derive CA-D-CG based on D-CG (Algorithm 2.3). As before, we denote the iteration m in Algorithm 2.3 with m = sk + j to distinguish inner and outer loop iterations. By induction on lines 6, 7, and 10 of Algorithm 2.3, we can write

(3.1) [p.sub.sk+j], [r.sub.sk+j] [member of] [K.sub.s+1] (A, [p.sub.sk]) + [K.sub.s](A, [r.sub.sk]) + [K.sub.s-1] (A, W),

(3.2) [x.sub.sk+j] - [x.sub.sk] [member of] [K.sub.s](A, [p.sub.sk]) + [K.sub.s-1](A, [r.sub.sk]) + [K.sub.s-2](A, W),

for 0 [less than or equal to] j [less than or equal to] s. Deflation also requires the product Arsk+j+1 in the computation of psk+j+1 in line 9. Again, by induction, we can write

(3.3) [Ar.sub.sk+j+1] [member of] [K.sub.s+2](A, [p.sub.sk]) + [K.sub.s+1](A, [r.sub.sk]) + [K.sub.s] (A, W).

As before, we define matrices [P.sub.k] and [R.sub.k] whose columns span the Krylov subspaces [K.sub.s+2](A, [p.sub.sk]) and [K.sub.s+1] (A, [r.sub.sk]), respectively. For deflation, we now also require a basis W for [K.sub.s](A, W). Note that, assuming W does not change throughout the iteration, W needs only be computed once. For the deflated method, we now define the n-by-(2s + 3 + cs) matrix

[V.sub.k] = [[P.sub.k], [R.sub.k], W] = [[[rho].sub.0] (A)[p.sub.sk], ..., [[rho].sub.s+1] (A)[p.sub.sk], [[rho].sub.0](A)[r.sub.sk], ..., [[rho].sub.s](A)[r.sub.sk], [[rho].sub.0](A)W, ..., [[rho].sub.s-1] (A)W],

where [[rho].sub.i] is defined as in (2.1). By (3.2), (3.1), and (3.3), we can then write, 0 [less than or equal to] j [less than or equal to] s, [p.sub.sk+j] = [V.sub.k][p'.sub.k,j], [r.sub.sk+j] = [V.sub.k][r'.sub.k,j,] and [x.sub.sk+j] - [x.sub.sk] [V.sub.k][x'.sub.k,j], i.e., the length-(2s + 3 + cs) vectors [p'.sub.k,j], [r'.sub.k,j], and [x'.sub.k,j] are coordinates for [p.sub.sk+j], [r.sub.sk+j], and [x.sub.sk+j] - [x.sub.sk], respectively, in terms of the columns of [V.sub.k].

As in CA-CG, we can write a recurrence for computing multiplications with A, that is, for 0 [less than or equal to] j < s,

[Ap.sub.sk+j] = A[V.sub.k][p'.sub.k,j] = [V.sub.k] [B.sub.k] [p'.sub.k,j] and [Ar.sub.sk+j+1] = [AV.sub.k] [r'.sub.k,j+1] = [V.sub.k] [B.sub.k], [r'.sub.k,j+1],

where, for the deflated method, we now define block diagonal matrix

[B.sub.k] = [[[C.sub.k,s+2] [0.sub.s+2,1]][[C.sub.k,s+1] [0.sub.s+1,1]] [C.sub.k,s] [cross product] [I.sub.c] [0.sub.cs,1]]].

Thus, a multiplication by [B.sub.k] in the basis [V.sub.k] is equivalent to a multiplication by A in the standard basis.

Assuming the same W is used throughout the iterations, [W.sup.T] AW can be precomputed and factorized offline. The small c-by-c factors of [W.sup.T] AW are assumed to fit in local/fast memory. If we compute the (2s + 3 + cs)-by-(2s + 3 + cs) matrix [G.sub.k] = [V.sup.T.sub.k] [V.sub.k] and extract the c-by-(2s + 3 + cs) submatrix [Z.sub.k] = [W.sup.T] [V.sub.k], then we can form the right-hand side in the solve for [[mu].sub.sk+j+1] in line 9 of Algorithm 2.3 by [W.sup.T] [Ar.sub.sk+j+1] = [Z.sub.k][B.sub.k][r'.sub.k,j+1], replacing a global reduction with a small, local operation. Note that the formulas for computing [[alpha].sub.sk+j] and [[beta].sub.sk+j+1] in Algorithm 2.3 remain the same as in Algorithm 2.1. Thus, using [G.sub.k], we can compute these inner products in CA-D-CG using the same formulas as in CA-CG (lines 7 and 10 of Algorithm 2.2).

Similarly, the formulas for the updates [x.sub.sk+j+1] and [r.sub.sk+j+1] are the same for D-CG and CG, so the formulas for [x'.sub.k,j+1] and [r'.sub.k,j+1] in CA-D-CG remain the same as those in CA-CG (lines 8 and 9). The formula for [p.sub.sk+j+1] in D-CG can be written as

[V.sub.k][P.sub.k,j+1] = [V.sub.k][r'.sub.k,j+1] + [[beta].sub.sk+j+1] [V.sub.k][p'.sub.k,j] - [V.sub.k][[[0.sup.T.sub.2s+3], [[mu].sup.T.sub.k,j+1], [0.sup.T.sub.c(s-1)]].sup.T],

for 0 [less than or equal to] j [less than or equal to] s - 1. Thus, in CA-D-CG, [p'.sub.k,j+1] is updated by

[p'.sub.k,j+1] = [r'.sub.k,j+1] + [[beta].sub.sk+j+1][p'.sub.k,j] - [[[0.sup.T.sub.2s+3], [[mu].sup.T.sub.k,j+1], [0.sup.T.sub.c(s-1])].sup.T].

The resulting CA-D-CG method is shown in Algorithm 3.1.

ALGORITHM 3.1. Deflated Communication-Avoiding Conjugate Gradient (CA-D-CG).

Require: Approximate solution [x.sub.-1] to Ax = b with residual
[r.sub.-1] = b - [Ax.sub.-1]; n-by-c matrix W of rank c

1: Compute and factorize [W.sup.T] AW
2: Compute W
3: [x.sub.0] = [w.sub.-1] + W[([W.sup.T] AW).sup.-1][W.sup.T]
     [r.sub.-1], [r.sub.0] = b - [Ax.sub.0]
4: [mu] = [([W.sup.T] AW).sup.-1] [W.sup.T] [Ar.sub.0], [p.sub.0] =
     [r.sub.0] - W [mu]
5: for k = 0, 1,... until convergence do
6:     Compute [P.sub.k], [R.sub.k], let [V.sub.k] = [[P.sub.k],
         [R.sub.k], W]; assemble [B.sub.k].
7:     [G.sub.k] = [V.sup.T.sub.k] [V.sub.k]; extract [Z.sub.k] =
         [W.sup.T] [V.sub.k]
8:     [p'.sup.T.sub.sk] = [1, [0.sup.T.sub.2s+2+cs]],
        [r'.sup.T.sub.sk] = [[0.sup.T.sub.s+2], 1, [0.sup.T.sub.s+cs]],
       [x'.sub.sk] = [0.sub.2s+3+cs]
9:     for j = 0, ..., s - 1 do
10:      [[alpha].sub.sk+j] = ([r'.sup.T.sub.k,j][G.sub.k]
         [r'.sub.k,j])/[(p'.sup.T.sub.k,j][G.sub.k][B.sub.k]
           [p'.sub.k,j])
11:      [x'.sub.k,j+1] = [x'.sub.k,j] + [[alpha].sub.k,j] [p'.sub.k,j]
12:      [r'.sub.k,j+1] = [r'.sub.k,j] - [[alpha].sub.k,j]
           [B.sub.k][p'.sub.k,j]
13:      [[beta].sub.sk+j+1] = ([r'.sup.T.sub.k,j+1]
           [G.sub.k][r'.sub.k,j+1])/([r'.sup.T.sub.k,j][G.sub.k]
           [r'.sub.k,j])
14:      Solve [W.sup.T] AW [[mu].sub.k,j+1] = [Z.sub.k] [B.sub.k]
           [r'.sub.k,j+1] for [[mu].sub.k,j+1]
15:      [p'.sub.k,j+1] = [r'.sub.k,j+1] + [[beta].sub.sk+j+1]
           [p'.sub.k,j] - [[[0.sup.T.sub.2s+3], [[mu].sup.T.sub.k,j+1],
           [0.sup.T.sub.c(s-1)]].sup.T]
16:    end for
17:    [x.sub.sk+s] = [V.sub.k][x'.sub.k,s] + [x.sub.sk],
         [r.sub.sk+s] = [V.sub.k][r'.sub.k,s], [p.sub.sk+s] = [V.sub.k]
         [p'.sub.k,s]
18: end for
19: return [x.sub.sk+s]


3.1. Algorithmic extensions. We assume in our derivation that the matrix of the deflation vectors W is constant through the iterations. We could, however, extend CA-D-CG to allow for updating of [W.sub.k] in (some or all) outer loop iterations k; see, e.g., [1, 3, 33, 42, 47] for example applications. (Additional considerations arise when changing the operator during the iterations due to the loss of orthogonality properties [2, Chapter 12]; see also [40].) Updating [W.sub.k] in the outer loop k requires recomputing [W.sub.k], a basis for [K.sub.s] (A, [W.sub.k]). This computation could potentially be fused with the computation of [P.sub.k] and [R.sub.k] such that no extra latency cost is incurred. The quantity [W.sup.T.sub.k] [AW.sub.k] can be recovered from the computation of [G.sub.k], so no additional communication is required. A factorization of the c-by-c matrix [W.sup.T.sub.k] [AW.sub.k] can also be performed locally. Note the number of deflation vectors c could be allowed to vary over outer loop iterations as well. This extension is considered future work.

4. Numerical experiments. For the numerical experiments, our goal is to show that CA-D-CG is competitive with D-CG in terms of the convergence rate. While the approaches are equivalent in exact arithmetic, there is no reason to expect that the CA-D-CG iterates will exactly equal the D-CG iterates in finite precision, given the different sets of floating-point operations performed. There are many open questions about CA-KSMs' finite precision behavior, some of which we hope to address in future work. But in lieu of theoretical results, we will rely on our practical experience that CA-KSMs' iterates deviate from their classical counterparts as the s-step Krylov bases become ill-conditioned (increasingly with s), and this effect can be diminished by picking different polynomial bases [6, 27, 43]. To focus on this potential instability that grows with s, we chose a test problem for which the classical methods are relatively insensitive to rounding errors. Thus, our experiments do not address the possibility that the deviation between the D-CG and CA-D-CG iterates is much larger when the convergence of the classical methods is highly perturbed by rounding errors.

We test the stability of our reformulation on a similar model problem to the one considered in [45] using codes written in a combination of MATLAB and C with linear algebra routines from Intel's Math Kernel Library. We generate a discrete 2D Laplacian by gallery ('poisson' ,512) in MATLAB, so A is an SPD matrix of order n = [512.sup.2]. We pick the right-hand side b equal to A times the vector with entries all [n.sup.-1/2]. Our deflation vectors are the eigenvectors corresponding to the eigenvalues of smallest magnitude computed using MATLAB's eigs. Note that the study in [45] used (known) exact eigenvalues; this difference does not significantly affect the results for this test.

In Figures 4.1-4.3, we compare convergence for the model problem using D-CG and CAD-CG with the monomial, Newton, and Chebyshev polynomial basis, respectively, each for a few representative s values. We report the 2-norm of the true residual computed by b - [Ax.sub.m] rather than the recursively updated residual [r.sub.m] and normalize by the 2-norm of the starting residual [r.sub.0] = b (i.e., the starting guess [x.sub.0] is the vector of zeros). We declare convergence after a reduction by a factor of [10.sup.8] in the normalized residual 2-norm. The solid curves correspond to D-CG, and circles correspond to CA-D-CG. We deflate with c [member of] {0, 4, 8} eigenvectors, plotted in black, red, and blue, respectively (when c = 0, D-CG is just CG and CA-D-CG is just CA-CG). Based on the formulas above, this suggests that the condition number [kappa](A) [approximately equal to] 1.07 x [10.sup.5] in the undeflated case (c = 0) should improve to [approximately equal to] 2.13 x [10.sup.4] in the case c = 4 and to [approximately equal to] 1.25 x [10.sup.4] when c =8.

We implemented the Newton basis by choosing parameters in (2.3) as [[sigma].sub.i] =0, [[gamma].sub.i] = 1, and [[theta].sub.i] is the i-th element in a set of Leja-ordered points on the real line segment [[[lambda].sub.c+1], [[lambda].sub.n]]; see, e.g., [43]. We implemented the Chebyshev basis by setting the basis parameters in (2.3) as [[gamma].sub.i] = [absolute value of [[lambda].sub.n] - [[lambda].sub.c+1]] /2 (except [[gamma].sub.0], which is not divided by 2), [[theta].sub.i] = [[lambda].sub.c+1] + [absolute value of [[lambda].sub.n] - [[lambda].sub.c+1]] /2, and [[sigma].sub.i] = [absolute value of [[lambda].sub.n] - [[lambda].sub.c+1]] /8. These recurrence coefficients are based on the bounding ellipse of the spectrum of A, which is, in the present case of a symmetric matrix A, an interval on the real line; see, e.g., [28]. In practice, only a few Ritz values (estimates for the eigenvalues of A) need to be computed up front to sufficiently determine the parameters for the Newton or Chebyshev polynomials. One can also incorporate information about new Ritz values obtained as a byproduct of the iterations to improve the basis conditioning; see [43] for practical details and experiments.

Note that [[lambda].sub.c+1] is used as the smallest eigenvalue in selecting the Newton and Chebyshev parameters above. This is because if the columns of W are the exact eigenvectors of A corresponding to the eigenvalues [[lambda].sub.1], ..., [[lambda].sub.c], then using [[lambda].sub.1] as a basis parameter in the computation of the basis W can cause cancellation and can thus produce a rank-deficient basis. Although this cancellation does not occur in the computation of the bases [P.sub.k] and [R.sub.k], we used the same basis parameters chosen for W (i.e., using [[lambda].sub.c+1]) to compute [P.sub.k] and [R.sub.k] for simplicity with no ill effects.

For the monomial basis (Figure 4.1), convergence is nearly identical for s = 4, but we begin to see a delay in the convergence of CA-D-CG for s = 8 (top-left) and a failure to converge by s = 16. For the Newton basis (Figure 4.2), the two methods have similar convergence behavior past s = 16; only around s = 100 (bottom-right) we begin to notice a significant delay in convergence for CA-D-CG. The situation is similar for the Chebyshev basis (Figure 4.3); only the bottom-right figure now depicts the case s = 220. These results clearly demonstrate that the basis choice plays an important role for the convergence of CAD-CG, at least on this well-behaved model problem. In the next section, we will introduce a coarse performance model to ask about the practical benefits of values as large as s = 220.

5. Performance modeling. In this section, we give a qualitative description of the performance tradeoffs between the four KSMs mentioned above--CG, CA-CG, and their deflated counterparts--on massively parallel machines. For CA-CG and CA-D-CG, we estimate the time for s inner loop iterations and then divide by s to estimate the time per iteration. Note that this ignores relative rates of convergence treated in Section 4. We were motivated to develop CA-D-CG based on the high relative cost of interprocessor communication on large-scale parallel computers. In addition to parallel implementations, CA-KSMs can avoid communication on sequential machines, namely data movement within the memory hierarchy. Indeed, the parallel and sequential approaches naturally compose hierarchically as has been exploited in previous high-performance CA-KSM implementations [36], and we suggest the same for a future CA-D-CG implementation. However, in this section, we will restrict ourselves to a parallel model which ignores sequential communication costs to illustrate the changes in parallel communication costs. We do not claim that this model's predicted speedups are always attainable on real hardware. However, we believe that such models can help to detect the feasibility of a communication-avoiding approach relative to a classical approach as well as to efficiently explore and prune the (machine-specific) parameter tuning space for an optimized implementation.

5.1. Machine model. We model parallel computation as a fully connected network of p homogeneous processors that perform local computations and exchange point-to-point messages. Each processor executes asynchronously and can send or receive at most one message at a time. Passing an n-word message takes [alpha]+[beta]n seconds on both the sending and receiving processors, which cannot perform computation during this time. The constant [alpha] represents a latency cost incurred by every message, while [beta] represents a bandwidth cost linear in the message size. There is no notion of distance on the network, and we assume the network has unbounded buffer capacity. While this simple model's assumptions are not all realistic, similar models are widely used to analyze communication costs on distributed-memory machines; see, e.g., [8]. One could refine this model to obtain, e.g., the LogP model [13], which distinguishes between network latency, software overhead, and network injection bandwidth (blurred between our [alpha] and [beta] terms), allows overlap of communication and computation, and introduces constraints on the message size and the network congestion. We quantify the computation time in terms of the floating-point operations (flops) performed: a processor can only operate on data residing in its local memory (of unbounded capacity), and each flop takes 7 seconds. If a processor performs F flops and sends/receives S messages containing a total of W words, then we model its runtime as [gamma]F + [alpha]S + [beta]W. This is a poor cost model for certain programs like the one where the p processors relay a value from processor 1 to processor p: each processor sends/receives at most 2 words, but the actual runtime grows linearly in p. To count correctly in such situations, one can consider the runtime along critical paths, e.g., in a program activity graph [54]. For our algorithms here, we will only consider certain balanced parallelizations where one processor is always the slowest, so we can simply count F, S, W for that processor to bound the total runtime.

Our assumptions that each processor has unbounded local memory and can execute each flop at the peak rate 1/[gamma] may be unrealistic when the neglected sequential costs are nontrivial. However, when considering large p and small local problems and when performance is dominated by interprocessor communication, we expect that the sequential costs would not significantly increase our models' estimated costs.

We consider two parallel machine models, which we call 'Exa' and 'Grid.' We use [gamma] = 1 x [10.sup.-13] seconds per (double precision) flop in both cases based on predictions for a 'node' of an exascale machine [7, 50]. This flop rate corresponds to a node with a 1024-core processor and its own memory hierarchy with 256 GB of capacity at the last level. However, as discussed above, we ignore this intranode structure. For Exa, the interconnect has parameters [alpha] = 4 x [10.sup.-7] seconds per message and [beta] = 3.7 x [10.sup.-11] seconds per word (4-byte double precision value). For the second machine, Grid, we replace this interconnect by the Internet (via Ethernet) using the parameters [alpha] = [10.sup.-1] and [beta] = 2.5 x [10.sup.-8] given in [35]. In contrast to their predecessors, our models allow an arbitrary number of processors in order to illustrate the asymptotic scaling behavior--we do not claim that every machine configuration modeled is physically realizable.

5.2. Experiments. We assume the same model problem (5-point 2D stencil) and deflation vectors as in the numerical experiments. However, since we are not modeling convergence here, the actual (nonzero) values do not influence the performance model. We assume the [square root of n]-by-[square root of n] mesh is partitioned across a [square root of p]-by-[square root of p] grid of processors, so that each processor owns a contiguous [square root of n/p]-by-[square root of n/p] subsquare, and we assume these fractions are integers. This layout minimizes communication within a factor of [square root of 2] (for this particular stencil, a diamond layout would be asymptotically optimal [4, Chapter 4.8]). We summarize the S, W, F costs for the four algorithms in Appendix A. To simplify the analysis, we restrict s [member of] {1, ..., [square root of n/p]} for the CA-KSMs, which means that the sparse computations only require communication with the (logical) nearest neighbors. The communication-avoiding approach is correct for any s, but the latency cost rises sharply when each processor needs information from a larger neighborhood. We also simplify by using the same blocking parameter s for the sparse and dense computations. In general, one can compute the s-step Krylov bases in smaller blocks and then compute a (larger) Gram matrix, or vice versa, i.e., constructing the Gram matrix blockwise as the s-step bases are computed. In practice, we have observed significant speedups from the Gram matrix construction alone (with no blocking of the SpMV operations) [53], and we suggest tuning the block sizes independently. Also to simplify the analysis, we ignore the preprocessing costs of computing the deflation matrix W (not the algorithmic costs) and computing and factorizing [W.sup.T] AW, assuming that they can be amortized over many iterations. In practice, these costs may not be negligible especially if the number of iterations is small.

We first consider weak scaling in Figure 5.1. We fix n/p = [4.sup.6] and vary the grid parameter p [member of] {[4.sup.x] : x [member of] {2, ..., 14}}. The black curves correspond to the runtime of a single iteration of the classical KSMs: CG (no markers), D-CG with c = 4 (square markers), and D-CG with c = 8 (asterisk markers); the logarithmic dependence on p, due to the collective communications, is evident. The red curves allow us to vary the parameter s [member of] {1, ..., [square root of n/p]}, where s = 1 corresponds to the classical KSMs and s > 1 corresponds to their deflated counterparts (markers mean the same as for the black curves). For comparison with the classical methods, we compute the runtime of one CA-KSM outer loop (with s inner-loop iterations) and then divide by s. On both Exa and Grid, it was beneficial to pick s > 1 for every problem although the optimal s varies as illustrated in Figure 5.3. The best speedups, i.e., the ratio of the runtime with s =1 to the best runtime with s [greater than or equal to] 1, were about 55, 38, and 28 for c = 0, 4, and 8, respectively, on Exa, while the corresponding best speedups on Grid were about 116, 174, and 173.

We now consider strong scaling, presented in Figure 5.2. The curves represent the same algorithms as in the previous figure, except that now we use different problems for the two machines (we use the same range of p as before). Note that the red and black curves coincide for some points on the left of both plots. As the local problem size decreases, so does the range of s values over which the CA-KSMs optimize. For Exa, we fix n = [4.sup.15], so for the largest p, for instance, the processors' subsquares are 2-by-2 and s [member of] {1, 2}; for Grid, we fix n = [4.sup.22]. While all tested KSMs scale when the local problem is large, the CA-KSMs are able to exploit more parallelism than the classical KSMs on both machines. In both cases, the CA-KSM runtime eventually begins to increase, too. The best speedups on Exa were about 49, 42, and 31 for c = 0, 4, and 8, respectively, while the corresponding best speedups on Grid were about 1152, 872, and 673.

Lastly, in Figure 5.3, we demonstrate the benefits of increasing the parameter s for a fixed problem and a varying number c [member of] {0, ..., 50} of deflation vectors. The case c = 0 indicates the non-deflated KSMs and is depicted separately. We plot the CA-KSMs' speedups relative to the classical KSMs, i.e., the points along the line s = 1. For both machines, we fix p = [4.sup.9], but to illustrate the tradeoffs on both, we pick n = [4.sup.14] for Exa and n = [4.sup.20] for Grid. In both cases, we see decreased relative benefits of avoiding communication as c increases as the network bandwidth becomes saturated by the larger reductions. For Exa, for small c it is beneficial to increase s to the maximum [square root of n/p] we consider; for Grid, however, it is never beneficial to increase s to its maximum for any c.

6. Future work and conclusions. In this work, we have demonstrated that deflation can be performed in a communication-avoiding way and is thus suitable for the use as a preconditioner for CA-KSMs. We derived CA-D-CG, which is equivalent to D-CG in exact arithmetic but can be implemented such that parallel latency is reduced by a factor of O(s) over a fixed number of iterations. Performance modeling shows predicted speedups of CA-D-CG over D-CG for a number of n/p ratios on two model architectures for s values constrained to s [less than or equal to] [square root of n/p]. We performed numerical experiments for a model problem to illustrate the benefits of deflation for the convergence rate. Our results also demonstrate that by using better conditioned bases of Newton and Chebyshev polynomials, s can be made very large before the convergence behavior of CA-D-CG deviates significantly from D-CG. However, for more difficult problems to be studied in future work, we expect the practical range of s to be more restricted.

We also point out that, as in the classical case, our CA-D-CG method is mathematically equivalent to applying CA-CG (Algorithm 2.2) to the transformed system [H.sup.T] AH[??] = [H.sup.T]b. If we were to instead perform deflation in this way, communication-avoiding techniques related to blocking covers for linear relaxation [30, 31] and other ideas from those works could be applied in the Krylov basis computation.

The communication-avoiding reorganization applied here can also be applied to many other deflated KSMs including adaptive deflation approaches (see, e.g., [1, 3, 33, 42, 47]), where the matrix W is allowed to change. Our future work will address these applications, as well as a distributed-memory implementation to evaluate the performance of our approaches on real parallel machines.

Acknowledgments. This material is based upon work supported by the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Numbers DE-SC0004938, DE-SC0003959, and DE-SC0010200, by the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, X-Stack program under Award Numbers DE-SC0005136, DE-SC0008699, DE-SC0008700, and AC02-05CH11231, by DARPA Award HR0011-12-20016, as well as contributions from Intel, Oracle, and MathWorks.

Appendix A: Complexity analysis. We can identify the elements of the vector iterates with vertices on a [square root of n]-by-[square root of n] 2D mesh. As explained above, each processor is assigned a [square root of n/p]-by-[square root of n/p] subsquare. The matrix A for the model problem is a stencil with constant coefficients and can be represented in O(1) words. In the case of variable coefficients, we would partition A in a overlapping block rowwise fashion as explained in [35]. The number of flops, words moved, and messages required for s steps of CG, CA-CG, D-CG, and CA-D-CG are as follows:

[Flops.sub.CG] = s(19n/p + 2 [log.sub.2] p)

[Words.sub.CG] = s(4[square root of n/p] + 4 [log.sub.2] p)

[Mess.sub.CG] = s(4 [log.sub.2] p + 4)

[Flops.sub.CA-CG] = 18(n/p)s + s(20s + 3(2s + 1)(4s + 1) + 10) + [12s.sup.3] + 2(n/p)(4s + 1) + (n/p)(4s + 3) + 36 [square root of n/[ps.sup.2]] + ((2s + 1)(2s + 2)(2(n/p) + [log.sub.2]p - 1))/2

[Words.sub.CA-CG] = 8[square root of n/ps] + [4s.sup.2] + [log.sub.2] p(2s + 1)(2s + 2)

[Mess.sub.CA-CG] = 2 [log.sub.2] p + 8

[Flops.sub.D-CG] = s(30(n/p) + 2[log.sub.2]p + c(2(n/p) + [log.sub.2]p - 1) + [2c.sup.2] + (n/p)(2c - 1))

[Words.sub.D-CG] = s((4 + 2c) [log.sub.2] p + 8[square root of n/p])

[Mess.sub.D-CG] = s(6 [log.sub.2] p + 8)

[Flops.sub.CA-D-CG] = 12 [(s + 1).sup.3] + 2(n/p)(4s + 2 cs + 5) + (n/p)(4s + 2cs + 7) + 36 [square root of n/p][(s + 1).sup.2] + 18(n/p)(s + 1) + s(24s + c(4s + 2cs + 5) + 4(2s + cs + 3)(4s + 2cs + 5) + 12cs + [2c.sub.2] + 36) + (2s + 3)(s + cs + 2)(2(n/p) + [log.sub.2] p - 1)

[Words.sub.CA-D-CG] = 4[(s + 1).sup.2] + 8[square root of n/p](s + 1) + 2 [log.sub.2] p(2s + 3)(s + cs + 2)

[Mess.sub.CA-D-CG] = 2 [log.sub.2] p + 8

For the CA-KSMs, we exploit neither the symmetry of [G.sub.k] nor the nonzero structure of [B.sub.k] and the length-O(s) coefficient vectors.

For D-CG, we note that one can compute AW offline (in line 1) and avoid the SpMV [Ar.sub.m+1] in line 9. While this may improve some constant factors by up to 2, it does not avoid the global reduction in the subsequent application of [(AW).sup.T], which our performance modeling suggests is often the dominant cost.

We note that the [log.sub.2] p terms in the computation and bandwidth costs can often be reduced by exploiting efficient collectives based on recursive halving/doubling approaches; see [8] for a survey. These approaches require that the number of words in the collective is at least p, which was not always true in our experiments, hence our use of simpler tree- based collectives.

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ERIN CARSON ([dagger]), NICHOLAS KNIGHT ([dagger]), JAMES DEMMEL ([dagger]) ([double dagger])

* Received October 13, 2013. Accepted November 2, 2014. Published online on December 12, 2014. Recommended by K. Jbilou.

([dagger]) Department of EECS, University of California, Berkeley ({ecc2z, knight, demmel}@cs .Berkeley. edu).

([double dagger]) Department of Mathematics, University of California, Berkeley.
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Author:Carson, Erin; Knight, Nicholas; Demmel, James
Publication:Electronic Transactions on Numerical Analysis
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Date:Dec 1, 2014
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