SNESConvergedReason#

reason a SNESSolve() was determined to have converged or diverged

Synopsis#

typedef enum {                       /* converged */
  SNES_CONVERGED_FNORM_ABS      = 2, /* ||F|| < atol */
  SNES_CONVERGED_FNORM_RELATIVE = 3, /* ||F|| < rtol*||F_initial|| */
  SNES_CONVERGED_SNORM_RELATIVE = 4, /* Newton computed step size small; || delta x || < stol || x ||*/
  SNES_CONVERGED_ITS            = 5, /* maximum iterations reached */
  SNES_BREAKOUT_INNER_ITER      = 6, /* Flag to break out of inner loop after checking custom convergence. */
                                     /* it is used in multi-phase flow when state changes */
  /* diverged */
  SNES_DIVERGED_FUNCTION_DOMAIN      = -1, /* the new x location passed the function is not in the domain of F */
  SNES_DIVERGED_FUNCTION_COUNT       = -2,
  SNES_DIVERGED_LINEAR_SOLVE         = -3, /* the linear solve failed */
  SNES_DIVERGED_FNORM_NAN            = -4,
  SNES_DIVERGED_MAX_IT               = -5,
  SNES_DIVERGED_LINE_SEARCH          = -6,  /* the line search failed */
  SNES_DIVERGED_INNER                = -7,  /* inner solve failed */
  SNES_DIVERGED_LOCAL_MIN            = -8,  /* || J^T b || is small, implies converged to local minimum of F() */
  SNES_DIVERGED_DTOL                 = -9,  /* || F || > divtol*||F_initial|| */
  SNES_DIVERGED_JACOBIAN_DOMAIN      = -10, /* Jacobian calculation does not make sense */
  SNES_DIVERGED_TR_DELTA             = -11,
  SNES_CONVERGED_TR_DELTA_DEPRECATED = -11,

  SNES_CONVERGED_ITERATING = 0
} SNESConvergedReason;

Values#

SNES_CONVERGED_FNORM_ABS - 2-norm(F) <= abstol
SNES_CONVERGED_FNORM_RELATIVE - 2-norm(F) <= rtol*2-norm(F(x_0)) where x_0 is the initial guess
SNES_CONVERGED_SNORM_RELATIVE - The 2-norm of the last step <= stol * 2-norm(x) where x is the current
SNES_DIVERGED_FUNCTION_COUNT - The user provided function has been called more times than the maximum set in SNESSetTolerances()
SNES_DIVERGED_DTOL - The norm of the function has increased by a factor of divtol set with SNESSetDivergenceTolerance()
SNES_DIVERGED_FNORM_NAN - the 2-norm of the current function evaluation is not-a-number (NaN), this is usually caused by a division of 0 by 0.
SNES_DIVERGED_MAX_IT - SNESSolve() has reached the maximum number of iterations requested
SNES_DIVERGED_LINE_SEARCH - The line search has failed. This only occurs for SNES solvers that use a line search
SNES_DIVERGED_LOCAL_MIN - the algorithm seems to have stagnated at a local minimum that is not zero.
SNES_CONERGED_ITERATING - this only occurs if SNESGetConvergedReason() is called during the SNESSolve()

Notes#

The two most common reasons for divergence are an incorrectly coded or computed Jacobian or failure or lack of convergence in the linear system (in this case we recommend testing with -pc_type lu to eliminate the linear solver as the cause of the problem).

SNES_DIVERGED_LOCAL_MIN can only occur when using the line-search variant of SNES. The line search wants to minimize Q(alpha) = 1/2 || F(x + alpha s) ||^2_2 this occurs at Q’(alpha) = s^T F’(x+alpha s)^T F(x+alpha s) = 0. If s is the Newton direction - F’(x)^(-1)F(x) then you get Q’(alpha) = -F(x)^T F’(x)^(-1)^T F’(x+alpha s)F(x+alpha s); when alpha = 0 Q’(0) = - ||F(x)||^2_2 which is always NEGATIVE if F’(x) is invertible. This means the Newton direction is a descent direction and the line search should succeed if alpha is small enough.

If F’(x) is NOT invertible AND F’(x)^T F(x) = 0 then Q’(0) = 0 and the Newton direction is NOT a descent direction so the line search will fail. All one can do at this point is change the initial guess and try again.

An alternative explanation: Newton’s method can be regarded as replacing the function with its linear approximation and minimizing the 2-norm of that. That is F(x+s) approx F(x) + F’(x)s so we minimize || F(x) + F’(x) s ||^2_2; do this using Least Squares. If F’(x) is invertible then s = - F’(x)^(-1)F(x) otherwise F’(x)^T F’(x) s = -F’(x)^T F(x). If F’(x)^T F(x) is NOT zero then there exists a nontrivial (that is F’(x)s != 0) solution to the equation and this direction is s = - [F’(x)^T F’(x)]^(-1) F’(x)^T F(x) so Q’(0) = - F(x)^T F’(x) [F’(x)^T F’(x)]^(-T) F’(x)^T F(x) = - (F’(x)^T F(x)) [F’(x)^T F’(x)]^(-T) (F’(x)^T F(x)). Since we are assuming (F’(x)^T F(x)) != 0 and F’(x)^T F’(x) has no negative eigenvalues Q’(0) < 0 so s is a descent direction and the line search should succeed for small enough alpha.

Note that this RARELY happens in practice. Far more likely the linear system is not being solved (well enough?) or the Jacobian is wrong.

SNES_DIVERGED_MAX_IT means that the solver reached the maximum number of iterations without satisfying any convergence criteria. SNES_CONVERGED_ITS means that SNESConvergedSkip() was chosen as the convergence test; thus the usual convergence criteria have not been checked and may or may not be satisfied.