Warm start

Most Krylov methods in this module accept a starting point as argument. The starting point is used as initial approximation to a solution.

solver = CgSolver(n, n, S)
cg!(solver, A, b, itmax=100)
if !issolved(solver)
  cg!(solver, A, b, solver.x, itmax=100) # cg! uses the approximate solution `solver.x` as starting point
end

If the user has an initial guess x0, it can be provided directly.

cg(A, b, x0)

It is also possible to use the warm_start! function to feed the starting point into the solver.

warm_start!(solver, x0)
cg!(solver, A, b)
# the previous two lines are equivalent to cg!(solver, A, b, x0)

If a Krylov method doesn't have the option to warm start, it can still be done explicitly. We provide an example with cg_lanczos!.

solver = CgLanczosSolver(n, n, S)
cg_lanczos!(solver, A, b)
x₀ = solver.x           # Ax₀ ≈ b
r = b - A * x₀          # r = b - Ax₀
cg_lanczos!(solver, A, r)
Δx = solver.x           # AΔx = r
x = x₀ + Δx             # Ax = b

Explicit restarts cannot be avoided in certain block methods, such as TriMR, due to the preconditioners.

# [E  A] [x] = [b]
# [Aᴴ F] [y]   [c]
M = inv(E)
N = inv(F)
x₀, y₀, stats = trimr(A, b, c, M=M, N=N)

# E and F are not available inside TriMR
b₀ = b -  Ex₀ - Ay
c₀ = c - Aᴴx₀ - Fy

Δx, Δy, stats = trimr(A, b₀, c₀, M=M, N=N)
x = x₀ + Δx
y = y₀ + Δy

The storage requierements of Krylov methods based on the Arnoldi process, such as FOM and GMRES, increase as the iteration progresses. For very large problems, the storage costs become prohibitive after only few iterations and restarted variants FOM(k) and GMRES(k) are prefered. In this section, we show how to use warm starts to implement GMRES(k) and FOM(k).

k = 50
solver = GmresSolver(A, b, k)  # FomSolver(A, b, k)
solver.x .= 0                  # solver.x .= x₀ 
nrestart = 0
while !issolved(solver) || nrestart ≤ 10
  solve!(solver, A, b, solver.x, itmax=k)
  nrestart += 1
end

Warm Start

Restarted methods