Solvers

Solvers for linear, nonlinear, and time-dependent problems.

pyiga.solvers.GaussSeidelSmoother(iterations=1, sweep='forward')

Gauss-Seidel smoother.

By default, iterations is 1. The direction to be used is specified by sweep and may be either ‘forward’, ‘backward’, or ‘symmetric’.

exception pyiga.solvers.NoConvergenceError(method, num_iter, last_iterate)
pyiga.solvers.OperatorSmoother(S)

A smoother which applies an arbitrary operator S to the residual and uses the result as an update, i.e.,

\[u \leftarrow u + S(f - Au).\]
pyiga.solvers.SequentialSmoother(smoothers)

Smoother which applies several smoothers in sequence.

pyiga.solvers.crank_nicolson(M, F, J, x, tau, t_end, *, t0=0.0, progress=False)

Solve a time-dependent problem using the Crank-Nicolson method.

Parameters:

  • M (matrix) – the mass matrix

  • F (function) – the right-hand side

  • J (function) – function computing the Jacobian of F

  • x (vector) – the initial value

  • tau0 (float) – the initial time step

  • t_end (float) – the final time up to which to integrate

  • t0 (float) – the initial time; 0 by default

  • progress (bool) – whether to show a progress bar

Returns:

A tuple (times, solutions), where times is a list of increasing times in the interval (t0, t_end), and solutions is a list of vectors which contains the computed solutions at these times.

pyiga.solvers.dirk34(M, F, J, x, tau0, t_end, tol, *, t0=0.0, step_factor=0.9, progress=False)

Solve a time-dependent problem using the DIRK34 Runge-Kutta method.

Parameters:

  • M (matrix) – the mass matrix

  • F (function) – the right-hand side

  • J (function) – function computing the Jacobian of F

  • x (vector) – the initial value

  • tau0 (float) – the initial time step

  • t_end (float) – the final time up to which to integrate

  • tol (float) – error tolerance for choosing the adaptive time step; if None, use constant time steps

  • t0 (float) – the initial time; 0 by default

  • step_factor (float) – the safety factor for choosing the step size

  • progress (bool) – whether to show a progress bar

Returns:

A tuple (times, solutions), where times is a list of increasing times in the interval (t0, t_end), and solutions is a list of vectors which contains the computed solutions at these times.

pyiga.solvers.esdirk23(M, F, J, x, tau0, t_end, tol, *, t0=0.0, step_factor=0.9, progress=False)

Solve a time-dependent problem using the ESDIRK23 Runge-Kutta method.

Parameters:

  • M (matrix) – the mass matrix

  • F (function) – the right-hand side

  • J (function) – function computing the Jacobian of F

  • x (vector) – the initial value

  • tau0 (float) – the initial time step

  • t_end (float) – the final time up to which to integrate

  • tol (float) – error tolerance for choosing the adaptive time step; if None, use constant time steps

  • t0 (float) – the initial time; 0 by default

  • step_factor (float) – the safety factor for choosing the step size

  • progress (bool) – whether to show a progress bar

Returns:

A tuple (times, solutions), where times is a list of increasing times in the interval (t0, t_end), and solutions is a list of vectors which contains the computed solutions at these times.

pyiga.solvers.esdirk34(M, F, J, x, tau0, t_end, tol, *, t0=0.0, step_factor=0.9, progress=False)

Solve a time-dependent problem using the ESDIRK34 Runge-Kutta method.

Parameters:

  • M (matrix) – the mass matrix

  • F (function) – the right-hand side

  • J (function) – function computing the Jacobian of F

  • x (vector) – the initial value

  • tau0 (float) – the initial time step

  • t_end (float) – the final time up to which to integrate

  • tol (float) – error tolerance for choosing the adaptive time step; if None, use constant time steps

  • t0 (float) – the initial time; 0 by default

  • step_factor (float) – the safety factor for choosing the step size

  • progress (bool) – whether to show a progress bar

Returns:

A tuple (times, solutions), where times is a list of increasing times in the interval (t0, t_end), and solutions is a list of vectors which contains the computed solutions at these times.

pyiga.solvers.fastdiag_solver(KM)

The fast diagonalization solver as described in [Sangalli, Tani 2016].

Parameters:

KM – a sequence of length d (dimension of the problem) containing pairs of symmetric matrices (K_i, M_i)

Returns:

A LinearOperator which realizes the inverse of the generalized Laplacian matrix described by the input matrices.

pyiga.solvers.gauss_seidel(A, x, b, iterations=1, indices=None, sweep='forward')

Perform Gauss-Seidel relaxation on the linear system Ax=b, updating x in place.

Parameters:

  • A – the matrix; either sparse or dense, but should be in CSR format if sparse

  • x – the current guess for the solution

  • b – the right-hand side

  • iterations – the number of iterations to perform

  • indices – if given, relaxation is only performed on this list of indices

  • sweep – the direction; either ‘forward’, ‘backward’, or ‘symmetric’

pyiga.solvers.iterative_solve(step, A, f, x0=None, active_dofs=None, tol=1e-08, maxiter=5000)

Solve the linear system Ax=f using a basic iterative method.

Parameters:

  • step (callable) – a function which performs the update x_old -> x_new for the iterative method

  • A – matrix or linear operator describing the linear system of equations

  • f (ndarray) – the right-hand side

  • x0 – the starting vector; 0 is used if not specified

  • active_dofs (list or ndarray) – list of active dofs on which the residual is computed. Useful for eliminating Dirichlet dofs without changing the matrix. If not specified, all dofs are active.

  • tol (float) – the desired reduction in the Euclidean norm of the residual relative to the starting residual

  • maxiter (int) – the maximum number of iterations

Returns:

a pair (x, iterations) containing the solution and the number of iterations performed. If maxiter was reached without convergence, the returned number of iterations is infinite.

pyiga.solvers.newton(F, J, x0, atol=1e-06, rtol=1e-06, maxiter=100, freeze_jac=1)

Solve the nonlinear problem F(x) == 0 using Newton iteration.

Parameters:

  • F (function) – function computing the residual of the nonlinear equation

  • J (function) – function computing the Jacobian matrix of F

  • x0 (ndarray) – the initial guess as a vector

  • atol (float) – absolute tolerance for the norm of the residual

  • rtol (float) – relative tolerance with respect to the initial residual

  • maxiter (int) – the maximum number of iterations

  • freeze_jac (int) – if >1, the Jacobian is only updated every freeze_jac steps

Returns:

ndarray – a vector x which approximately satisfies F(x) == 0

pyiga.solvers.rodasp(M, F, J, x, tau0, t_end, tol, *, t0=0.0, step_factor=0.9, progress=False)

Solve a time-dependent problem using the RODASP Rosenbrock method.

Parameters:

  • M (matrix) – the mass matrix

  • F (function) – the right-hand side

  • J (function) – function computing the Jacobian of F

  • x (vector) – the initial value

  • tau0 (float) – the initial time step

  • t_end (float) – the final time up to which to integrate

  • tol (float) – error tolerance for choosing the adaptive time step; if None, use constant time steps

  • t0 (float) – the initial time; 0 by default

  • step_factor (float) – the safety factor for choosing the step size

  • progress (bool) – whether to show a progress bar

Returns:

A tuple (times, solutions), where times is a list of increasing times in the interval (t0, t_end), and solutions is a list of vectors which contains the computed solutions at these times.

pyiga.solvers.ros3p(M, F, J, x, tau0, t_end, tol, *, t0=0.0, step_factor=0.9, progress=False)

Solve a time-dependent problem using the ROS3P Rosenbrock method.

Parameters:

  • M (matrix) – the mass matrix

  • F (function) – the right-hand side

  • J (function) – function computing the Jacobian of F

  • x (vector) – the initial value

  • tau0 (float) – the initial time step

  • t_end (float) – the final time up to which to integrate

  • tol (float) – error tolerance for choosing the adaptive time step; if None, use constant time steps

  • t0 (float) – the initial time; 0 by default

  • step_factor (float) – the safety factor for choosing the step size

  • progress (bool) – whether to show a progress bar

Returns:

A tuple (times, solutions), where times is a list of increasing times in the interval (t0, t_end), and solutions is a list of vectors which contains the computed solutions at these times.

pyiga.solvers.ros3pw(M, F, J, x, tau0, t_end, tol, *, t0=0.0, step_factor=0.9, progress=False)

Solve a time-dependent problem using the ROS3PW Rosenbrock method.

Parameters:

  • M (matrix) – the mass matrix

  • F (function) – the right-hand side

  • J (function) – function computing the Jacobian of F

  • x (vector) – the initial value

  • tau0 (float) – the initial time step

  • t_end (float) – the final time up to which to integrate

  • tol (float) – error tolerance for choosing the adaptive time step; if None, use constant time steps

  • t0 (float) – the initial time; 0 by default

  • step_factor (float) – the safety factor for choosing the step size

  • progress (bool) – whether to show a progress bar

Returns:

A tuple (times, solutions), where times is a list of increasing times in the interval (t0, t_end), and solutions is a list of vectors which contains the computed solutions at these times.

pyiga.solvers.rosi2p1(M, F, J, x, tau0, t_end, tol, *, t0=0.0, step_factor=0.9, progress=False)

Solve a time-dependent problem using the ROSI2P1 Rosenbrock method.

Parameters:

  • M (matrix) – the mass matrix

  • F (function) – the right-hand side

  • J (function) – function computing the Jacobian of F

  • x (vector) – the initial value

  • tau0 (float) – the initial time step

  • t_end (float) – the final time up to which to integrate

  • tol (float) – error tolerance for choosing the adaptive time step; if None, use constant time steps

  • t0 (float) – the initial time; 0 by default

  • step_factor (float) – the safety factor for choosing the step size

  • progress (bool) – whether to show a progress bar

Returns:

A tuple (times, solutions), where times is a list of increasing times in the interval (t0, t_end), and solutions is a list of vectors which contains the computed solutions at these times.

pyiga.solvers.rowdaind2(M, F, J, x, tau0, t_end, tol, *, t0=0.0, step_factor=0.9, progress=False)

Solve a time-dependent problem using the ROWDAIND2 Rosenbrock method.

Parameters:

  • M (matrix) – the mass matrix

  • F (function) – the right-hand side

  • J (function) – function computing the Jacobian of F

  • x (vector) – the initial value

  • tau0 (float) – the initial time step

  • t_end (float) – the final time up to which to integrate

  • tol (float) – error tolerance for choosing the adaptive time step; if None, use constant time steps

  • t0 (float) – the initial time; 0 by default

  • step_factor (float) – the safety factor for choosing the step size

  • progress (bool) – whether to show a progress bar

Returns:

A tuple (times, solutions), where times is a list of increasing times in the interval (t0, t_end), and solutions is a list of vectors which contains the computed solutions at these times.

pyiga.solvers.sdirk21(M, F, J, x, tau0, t_end, tol, *, t0=0.0, step_factor=0.9, progress=False)

Solve a time-dependent problem using the SDIRK21 (Ellsiepen) Runge-Kutta method.

Parameters:

  • M (matrix) – the mass matrix

  • F (function) – the right-hand side

  • J (function) – function computing the Jacobian of F

  • x (vector) – the initial value

  • tau0 (float) – the initial time step

  • t_end (float) – the final time up to which to integrate

  • tol (float) – error tolerance for choosing the adaptive time step; if None, use constant time steps

  • t0 (float) – the initial time; 0 by default

  • step_factor (float) – the safety factor for choosing the step size

  • progress (bool) – whether to show a progress bar

Returns:

A tuple (times, solutions), where times is a list of increasing times in the interval (t0, t_end), and solutions is a list of vectors which contains the computed solutions at these times.

pyiga.solvers.sdirk3(M, F, J, x, tau, t_end, *, t0=0.0, progress=False)

Solve a time-dependent problem using the SDIRK3 Runge-Kutta method.

Parameters:

  • M (matrix) – the mass matrix

  • F (function) – the right-hand side

  • J (function) – function computing the Jacobian of F

  • x (vector) – the initial value

  • tau0 (float) – the initial time step

  • t_end (float) – the final time up to which to integrate

  • t0 (float) – the initial time; 0 by default

  • progress (bool) – whether to show a progress bar

Returns:

A tuple (times, solutions), where times is a list of increasing times in the interval (t0, t_end), and solutions is a list of vectors which contains the computed solutions at these times.

pyiga.solvers.sdirk3_b(M, F, J, x, tau, t_end, *, t0=0.0, progress=False)

Solve a time-dependent problem using the SDIRK3 (alternate) Runge-Kutta method.

Parameters:

  • M (matrix) – the mass matrix

  • F (function) – the right-hand side

  • J (function) – function computing the Jacobian of F

  • x (vector) – the initial value

  • tau0 (float) – the initial time step

  • t_end (float) – the final time up to which to integrate

  • t0 (float) – the initial time; 0 by default

  • progress (bool) – whether to show a progress bar

Returns:

A tuple (times, solutions), where times is a list of increasing times in the interval (t0, t_end), and solutions is a list of vectors which contains the computed solutions at these times.

pyiga.solvers.solve_hmultigrid(hs, A, f, strategy='cell_supp', smoother='gs', smooth_steps=2, tol=1e-08, maxiter=5000)

Solve a linear scalar problem in a hierarchical spline space using local multigrid.

Parameters:

  • hs – the HSpace which describes the HB- or THB-spline space

  • A – the matrix describing the discretization of the problem

  • f – the right-hand side vector

  • strategy (string) –

    how to choose the smoothing sets. Valid options are

    • "new": only the new dofs per level

    • "trunc": all dofs which interact via the truncation operator

    • "cell_supp": all dofs whose support intersects that of the new ones (support extension)

    • "func_supp"

  • smoother (string) –

    the multigrid smoother to use. Valid options are

    • "gs": forward Gauss-Seidel for pre-smoothing, backward Gauss-Seidel for post-smoothing

    • "forward_gs": always use forward Gauss-Seidel

    • "backward_gs": always use backward Gauss-Seidel

    • "symmetric_gs": use complete symmetric Gauss-Seidel sweep for both pre- and post-smoothing

    • "exact": use an exact direct solver as a pre-smoother (no post-smoothing)

  • smooth_steps (int) – the number of pre- and post-smoothing steps

  • tol (float) – the desired reduction in the residual

  • maxiter (int) – the maximum number of iterations

Returns:

a pair (x, iterations) containing the solution and the number of iterations performed. If maxiter was reached without convergence, the returned number of iterations is infinite.

pyiga.solvers.twogrid(A, f, P, smoother, u0=None, tol=1e-08, smooth_steps=2, maxiter=1000)

Generic two-grid method with arbitrary smoother.

Parameters:

  • A – stiffness matrix on fine grid

  • f – right-hand side

  • P – prolongation matrix from coarse to fine grid

  • smoother – a function with arguments (A,u,f) which applies one smoothing iteration in-place to u

  • u0 – starting value; 0 if not given

  • tol – desired reduction relative to initial residual

  • smooth_steps – number of smoothing steps

  • maxiter – maximum number of iterations

Returns:

ndarray – the computed solution to the equation Au = f