In both domain decomposition (DD) and multigrid (MG) methods, one may compose the application of the block updates or coarse corrections as either additive or multiplicative. For pointwise solvers, this is the difference between the Jacobi and Gauss-Seidel iterations. The multiplicative smoother for $Ax = b$ acting as $S(x^{old}, b) = x^{new}$ is applied as$ x_{i+1} = S_n(S_{n-1}( ..., S_1(x_i, b) ..., b), b) $and the additive smoother is applied as$ x_{i+1} = x_{i} + \displaystyle\sum_{\ell = 0}^{n}\lambda_\ell(S_\ell(x_i, b) - x_i) $for some damping $\lambda_i$.  The general consensus appears to be that multiplicative smoothers have much more rapid convergence properties, but I was wondering: under what situations is the performance of the additive variants of these algorithms better?More specifically, Does anyone have any use cases where the additive variant should and/or does perform significantly better than the multiplicative variant?  Are there theoretical reasons for this?  Most literature on multigrid is fairly pessimistic about the Additive method, but it's used so much in the DD context as additive Schwarz.  This also extends to the much more general issue of composing linear and nonlinear solvers, and which type of constructions will perform well and perform well in parallel.
In what application cases are additive preconditioning schemes superior to multiplicative ones?
linear algebra;performance;multigrid;domain decomposition
Additive methods expose more concurrency. They are generally only faster than multiplicative methods if you can use that concurrency. For example, coarse levels of multigrid are typically latency-limited. If you move coarse levels to smaller subcommunicators, then they could be solved independently from the finer levels. With a multiplicative scheme, all the procs have to wait while the coarse levels are solved.Also, if the algorithm needs reductions on every level, the additive variant may be able to coalesce them where as the multiplicative method is forced to perform them sequentially.