cg_lanczos_shift.jl

# An implementation of the Lanczos version of the conjugate gradient method
# for a family of shifted systems of the form (A + αI) x = b.
#
# The implementation follows
# A. Frommer and P. Maass, Fast CG-Based Methods for Tikhonov-Phillips Regularization,
# SIAM Journal on Scientific Computing, 20(5), pp. 1831--1850, 1999.
#
# C. C. Paige and M. A. Saunders, Solution of Sparse Indefinite Systems of Linear Equations,
# SIAM Journal on Numerical Analysis, 12(4), pp. 617--629, 1975.
#
# Dominique Orban, <dominique.orban@gerad.ca>
# Princeton, NJ, March 2015.

export cg_lanczos_shift, cg_lanczos_shift!

"""
    (x, stats) = cg_lanczos_shift(A, b::AbstractVector{FC}, shifts::AbstractVector{T};
                                  M=I, ldiv::Bool=false,
                                  check_curvature::Bool=false, atol::T=√eps(T),
                                  rtol::T=√eps(T), itmax::Int=0,
                                  timemax::Float64=Inf, verbose::Int=0, history::Bool=false,
                                  callback=solver->false, iostream::IO=kstdout)

`T` is an `AbstractFloat` such as `Float32`, `Float64` or `BigFloat`.
`FC` is `T` or `Complex{T}`.

The Lanczos version of the conjugate gradient method to solve a family
of shifted systems

    (A + αI) x = b  (α = α₁, ..., αₚ)

of size n. The method does _not_ abort if A + αI is not definite.

#### Input arguments

* `A`: a linear operator that models a Hermitian matrix of dimension n;
* `b`: a vector of length n;
* `shifts`: a vector of length p.

#### Keyword arguments

* `M`: linear operator that models a Hermitian positive-definite matrix of size `n` used for centered preconditioning;
* `ldiv`: define whether the preconditioner uses `ldiv!` or `mul!`;
* `check_curvature`: if `true`, check that the curvature of the quadratic along the search direction is positive, and abort if not, unless `linesearch` is also `true`;
* `atol`: absolute stopping tolerance based on the residual norm;
* `rtol`: relative stopping tolerance based on the residual norm;
* `itmax`: the maximum number of iterations. If `itmax=0`, the default number of iterations is set to `2n`;
* `timemax`: the time limit in seconds;
* `verbose`: additional details can be displayed if verbose mode is enabled (verbose > 0). Information will be displayed every `verbose` iterations;
* `history`: collect additional statistics on the run such as residual norms, or Aᴴ-residual norms;
* `callback`: function or functor called as `callback(solver)` that returns `true` if the Krylov method should terminate, and `false` otherwise;
* `iostream`: stream to which output is logged.

#### Output arguments

* `x`: a vector of p dense vectors, each one of length n;
* `stats`: statistics collected on the run in a [`LanczosShiftStats`](@ref) structure.

#### References

* A. Frommer and P. Maass, [*Fast CG-Based Methods for Tikhonov-Phillips Regularization*](https://doi.org/10.1137/S1064827596313310), SIAM Journal on Scientific Computing, 20(5), pp. 1831--1850, 1999.
* C. C. Paige and M. A. Saunders, [*Solution of Sparse Indefinite Systems of Linear Equations*](https://doi.org/10.1137/0712047), SIAM Journal on Numerical Analysis, 12(4), pp. 617--629, 1975.
"""
function cg_lanczos_shift end

"""
    solver = cg_lanczos!(solver::CgLanczosShiftSolver, A, b, shifts; kwargs...)

where `kwargs` are keyword arguments of [`cg_lanczos_shift`](@ref).

See [`CgLanczosShiftSolver`](@ref) for more details about the `solver`.
"""
function cg_lanczos_shift! end

def_args_cg_lanczos_shift = (:(A                        ),
                             :(b::AbstractVector{FC}    ),
                             :(shifts::AbstractVector{T}))

def_kwargs_cg_lanczos_shift = (:(; M = I                        ),
                               :(; ldiv::Bool = false           ),
                               :(; check_curvature::Bool = false),
                               :(; atol::T = √eps(T)            ),
                               :(; rtol::T = √eps(T)            ),
                               :(; itmax::Int = 0               ),
                               :(; timemax::Float64 = Inf       ),
                               :(; verbose::Int = 0             ),
                               :(; history::Bool = false        ),
                               :(; callback = solver -> false   ),
                               :(; iostream::IO = kstdout       ))

def_kwargs_cg_lanczos_shift = mapreduce(extract_parameters, vcat, def_kwargs_cg_lanczos_shift)

args_cg_lanczos_shift = (:A, :b, :shifts)
kwargs_cg_lanczos_shift = (:M, :ldiv, :check_curvature, :atol, :rtol, :itmax, :timemax, :verbose, :history, :callback, :iostream)

@eval begin
  function cg_lanczos_shift($(def_args_cg_lanczos_shift...); $(def_kwargs_cg_lanczos_shift...)) where {T <: AbstractFloat, FC <: FloatOrComplex{T}}
    start_time = time_ns()
    nshifts = length(shifts)
    solver = CgLanczosShiftSolver(A, b, nshifts)
    elapsed_time = ktimer(start_time)
    timemax -= elapsed_time
    cg_lanczos_shift!(solver, $(args_cg_lanczos_shift...); $(kwargs_cg_lanczos_shift...))
    solver.stats.timer += elapsed_time
    return (solver.x, solver.stats)
  end

  function cg_lanczos_shift!(solver :: CgLanczosShiftSolver{T,FC,S}, $(def_args_cg_lanczos_shift...); $(def_kwargs_cg_lanczos_shift...)) where {T <: AbstractFloat, FC <: FloatOrComplex{T}, S <: AbstractVector{FC}}

    # Timer
    start_time = time_ns()
    timemax_ns = 1e9 * timemax

    m, n = size(A)
    (m == solver.m && n == solver.n) || error("(solver.m, solver.n) = ($(solver.m), $(solver.n)) is inconsistent with size(A) = ($m, $n)")
    m == n || error("System must be square")
    length(b) == n || error("Inconsistent problem size")

    nshifts = length(shifts)
    nshifts == solver.nshifts || error("solver.nshifts = $(solver.nshifts) is inconsistent with length(shifts) = $nshifts")
    (verbose > 0) && @printf(iostream, "CG-LANCZOS-SHIFT: system of %d equations in %d variables with %d shifts\n", n, n, nshifts)

    # Tests M = Iₙ
    MisI = (M === I)

    # Check type consistency
    eltype(A) == FC || @warn "eltype(A) ≠ $FC. This could lead to errors or additional allocations in operator-vector products."
    ktypeof(b) <: S || error("ktypeof(b) is not a subtype of $S")

    # Set up workspace.
    allocate_if(!MisI, solver, :v, S, n)
    Mv, Mv_prev, Mv_next = solver.Mv, solver.Mv_prev, solver.Mv_next
    x, p, σ, δhat = solver.x, solver.p, solver.σ, solver.δhat
    ω, γ, rNorms, converged = solver.ω, solver.γ, solver.rNorms, solver.converged
    not_cv, stats = solver.not_cv, solver.stats
    rNorms_history, indefinite = stats.residuals, stats.indefinite
    reset!(stats)
    v = MisI ? Mv : solver.v

    # Initial state.
    ## Distribute x similarly to shifts.
    for i = 1 : nshifts
      x[i] .= zero(FC)  # x₀
    end
    Mv .= b                             # Mv₁ ← b
    MisI || mulorldiv!(v, M, Mv, ldiv)  # v₁ = M⁻¹ * Mv₁
    β = sqrt(@kdotr(n, v, Mv))          # β₁ = v₁ᴴ M v₁
    rNorms .= β
    if history
      for i = 1 : nshifts
        push!(rNorms_history[i], rNorms[i])
      end
    end

    # Keep track of shifted systems with negative curvature if required.
    indefinite .= false

    if β == 0
      stats.niter = 0
      stats.solved = true
      stats.timer = ktimer(start_time)
      stats.status = "x = 0 is a zero-residual solution"
      return solver
    end

    # Initialize each p to v.
    for i = 1 : nshifts
      p[i] .= v
    end

    # Initialize Lanczos process.
    # β₁Mv₁ = b
    @kscal!(n, one(FC) / β, v)           # v₁  ←  v₁ / β₁
    MisI || @kscal!(n, one(FC) / β, Mv)  # Mv₁ ← Mv₁ / β₁
    Mv_prev .= Mv

    # Initialize some constants used in recursions below.
    ρ = one(T)
    σ .= β
    δhat .= zero(T)
    ω .= zero(T)
    γ .= one(T)

    # Define stopping tolerance.
    ε = atol + rtol * β

    # Keep track of shifted systems that have converged.
    for i = 1 : nshifts
      converged[i] = rNorms[i] ≤ ε
      not_cv[i] = !converged[i]
    end
    iter = 0
    itmax == 0 && (itmax = 2 * n)

    # Build format strings for printing.
    (verbose > 0) && (fmt = Printf.Format("%5d" * repeat("  %8.1e", nshifts) * "  %.2fs\n"))
    kdisplay(iter, verbose) && Printf.format(iostream, fmt, iter, rNorms..., ktimer(start_time))

    solved = !reduce(|, not_cv)
    tired = iter ≥ itmax
    status = "unknown"
    user_requested_exit = false
    overtimed = false

    # Main loop.
    while ! (solved || tired || user_requested_exit || overtimed)
      # Form next Lanczos vector.
      # βₖ₊₁Mvₖ₊₁ = Avₖ - δₖMvₖ - βₖMvₖ₋₁
      mul!(Mv_next, A, v)                  # Mvₖ₊₁ ← Avₖ
      δ = @kdotr(n, v, Mv_next)            # δₖ = vₖᴴ A vₖ
      @kaxpy!(n, -δ, Mv, Mv_next)          # Mvₖ₊₁ ← Mvₖ₊₁ - δₖMvₖ
      if iter > 0
        @kaxpy!(n, -β, Mv_prev, Mv_next)   # Mvₖ₊₁ ← Mvₖ₊₁ - βₖMvₖ₋₁
        @. Mv_prev = Mv                    # Mvₖ₋₁ ← Mvₖ
      end
      @. Mv = Mv_next                      # Mvₖ ← Mvₖ₊₁
      MisI || mulorldiv!(v, M, Mv, ldiv)   # vₖ₊₁ = M⁻¹ * Mvₖ₊₁
      β = sqrt(@kdotr(n, v, Mv))           # βₖ₊₁ = vₖ₊₁ᴴ M vₖ₊₁
      @kscal!(n, one(FC) / β, v)           # vₖ₊₁  ←  vₖ₊₁ / βₖ₊₁
      MisI || @kscal!(n, one(FC) / β, Mv)  # Mvₖ₊₁ ← Mvₖ₊₁ / βₖ₊₁

      # Check curvature: vₖᴴ(A + sᵢI)vₖ = vₖᴴAvₖ + sᵢ‖vₖ‖² = δₖ + ρₖ * sᵢ with ρₖ = ‖vₖ‖².
      # It is possible to show that σₖ² (δₖ + ρₖ * sᵢ - ωₖ₋₁ / γₖ₋₁) = pₖᴴ (A + sᵢ I) pₖ.
      MisI || (ρ = @kdotr(n, v, v))
      for i = 1 : nshifts
        δhat[i] = δ + ρ * shifts[i]
        γ[i] = 1 / (δhat[i] - ω[i] / γ[i])
      end
      for i = 1 : nshifts
        indefinite[i] |= γ[i] ≤ 0
      end

      # Compute next CG iterate for each shifted system that has not yet converged.
      # Stop iterating on indefinite problems if requested.
      for i = 1 : nshifts
        not_cv[i] = check_curvature ? !(converged[i] || indefinite[i]) : !converged[i]
        if not_cv[i]
          @kaxpy!(n, γ[i], p[i], x[i])
          ω[i] = β * γ[i]
          σ[i] *= -ω[i]
          ω[i] *= ω[i]
          @kaxpby!(n, σ[i], v, ω[i], p[i])

          # Update list of systems that have not converged.
          rNorms[i] = abs(σ[i])
          converged[i] = rNorms[i] ≤ ε
        end
      end

      if length(not_cv) > 0 && history
        for i = 1 : nshifts
          not_cv[i] && push!(rNorms_history[i], rNorms[i])
        end
      end

      # Is there a better way than to update this array twice per iteration?
      for i = 1 : nshifts
        not_cv[i] = check_curvature ? !(converged[i] || indefinite[i]) : !converged[i]
      end
      iter = iter + 1
      kdisplay(iter, verbose) && Printf.format(iostream, fmt, iter, rNorms..., ktimer(start_time))

      user_requested_exit = callback(solver) :: Bool
      solved = !reduce(|, not_cv)
      tired = iter ≥ itmax
      timer = time_ns() - start_time
      overtimed = timer > timemax_ns
    end
    (verbose > 0) && @printf(iostream, "\n")

    # Termination status
    tired               && (status = "maximum number of iterations exceeded")
    solved              && (status = "solution good enough given atol and rtol")
    user_requested_exit && (status = "user-requested exit")
    overtimed           && (status = "time limit exceeded")

    # Update stats. TODO: Estimate Anorm and Acond.
    stats.niter = iter
    stats.solved = solved
    stats.timer = ktimer(start_time)
    stats.status = status
    return solver
  end
end