Module Cvodes.Adjoint

module Adjoint: sig .. end

(Adjoint) Sensitivity analysis of ODEs with respect to their parameters.

Provides an alternative to forward sensitivity analysis, which can become prohibitively expensive. This technique does not calculate sensitivities, but rather gradients with respect to the parameters of a relatively few derived functionals of the solution, that is the gradient $\frac{\mathrm{d}G}{\mathrm{d}p}$ of $G(p) = \int_{t_0}^T \! g(t, y, p)\,\mathrm{d}t$. The gradients are evaluated by first calculating forward and checkpointing certain intermediate state values, and then integrating backward to $t_0$.

This documented interface is structured as follows.

  1. Forward solution
  2. Linear solvers
  3. Backward solutions (including Quadrature equations)
  4. Modifying the solver
  5. Querying the solver
  6. Exceptions

type ('data, 'kind) bsession = ('data, 'kind) AdjointTypes.bsession 

A backward session with the CVODES solver. Multiple backward sessions may be associated with a single parent session.

type [> Nvector_serial.kind ] serial_bsession = (Nvector_serial.data, [> Nvector_serial.kind ] as 'a) bsession 

Alias for backward sessions based on serial nvectors.

Forward solution

type interpolation = 
| IPolynomial (*

(CV_POLYNOMIAL)

*)
| IHermite (*

(CV_HERMITE)

*)

Specifies the type of interpolation to use between checkpoints.

val init : ('d, 'k) Cvode.session -> int -> interpolation -> unit

Activates the forward-backward problem. The arguments specify the number of integration steps between consecutive checkpoints, and the type of variable-degree interpolation.

val forward_normal : ('d, 'k) Cvode.session ->
float -> ('d, 'k) Nvector.t -> float * int * Cvode.solver_result

Integrates the forward problem over an interval and saves checkpointing data. The arguments are the next time at which a solution is desired (tout) and a vector to receive the computed result (y). The function returns a triple tret, ncheck, sr: the time reached by the solver, the cumulative number of checkpoints stored, and whether tout was reached. The solver takes internal steps until it has reached or just passed the tout parameter (CV_NORMAL), it then interpolates to approximate y(tout).

val forward_one_step : ('d, 'k) Cvode.session ->
float -> ('d, 'k) Nvector.t -> float * int * Cvode.solver_result

Integrates the forward problem over an interval and saves checkpointing data. The arguments are the next time at which a solution is desired (tout) and a vector to receive the computed result (yret). The function returns a triple tret, ncheck, sr: the time reached by the solver, the cumulative number of checkpoints stored, and whether tout was reached. The solver takes one step (CV_ONE_STEP) and returns the solution reached.

Linear solvers

type ('data, 'kind) session_linear_solver = ('data, 'kind) AdjointTypes.session_linear_solver 

Linear solvers used in backward problems.

type [> Nvector_serial.kind ] serial_session_linear_solver = (Nvector_serial.data, [> Nvector_serial.kind ] as 'a)
session_linear_solver

Alias for linear solvers that are restricted to serial nvectors.

type 'd triple = 'd * 'd * 'd 

Workspaces with three temporary vectors.

type ('t, 'd) jacobian_arg = ('t, 'd) AdjointTypes.jacobian_arg = {
   jac_t : float; (*

The independent variable.

*)
   jac_y : 'd; (*

The forward solution vector.

*)
   jac_yb : 'd; (*

The backward solution vector.

*)
   jac_fyb : 'd; (*

The backward right-hand side function fB.

*)
   jac_tmp : 't; (*

Temporary storage vectors.

*)
}

Arguments common to Jacobian callback functions.

module Diag: sig .. end

Diagonal approximation of Jacobians by difference quotients.

module Dls: sig .. end

Direct Linear Solvers operating on dense, banded, and sparse matrices.

module Spils: sig .. end

Scaled Preconditioned Iterative Linear Solvers.

module Alternate: sig .. end

Alternate Linear Solvers.

Backward solutions

type 'd brhsfn_args = 'd AdjointTypes.brhsfn_args = {
   t : float; (*

The value of the independent variable.

*)
   y : 'd; (*

The vector of dependent-variable values $y(t)$.

*)
   yb : 'd; (*

The vector of backward dependent-variable values $y_B(t)$.

*)
}

Arguments common to Cvodes.Adjoint.brhsfn_no_sens and Cvodes.Adjoint.brhsfn_with_sens.

type 'd brhsfn_no_sens = 'd brhsfn_args -> 'd -> unit 

Backward functions without forward sensitivities. They are passed the arguments:

Within the function, raising a Sundials.RecoverableFailure exception indicates a recoverable error. Any other exception is treated as an unrecoverable error.

The vectors in the function arguments should not be accessed after the function returns.

type 'd brhsfn_with_sens = 'd brhsfn_args -> 'd array -> 'd -> unit 

Backward functions with forward sensitivities. They are passed the arguments:

Within the function, raising a Sundials.RecoverableFailure exception indicates a recoverable error. Any other exception is treated as an unrecoverable error.

The vectors in the function arguments should not be accessed after the function returns.

type 'd brhsfn = 
| NoSens of 'd brhsfn_no_sens (*

No dependency on forward sensitivities.

*)
| WithSens of 'd brhsfn_with_sens (*

Dependency on forward sensitivities.

*)

Functions that evaluate the right-hand side of a backward ODE system with or without forward sensitivities.

type ('d, 'k) tolerance = 
| SStolerances of float * float (*

(rel, abs) : scalar relative and absolute tolerances.

*)
| SVtolerances of float * ('d, 'k) Nvector.t (*

(rel, abs) : scalar relative and vector absolute tolerances.

*)

Tolerance specifications.

type ('data, 'kind) iter = 
| Newton of ('data, 'kind) session_linear_solver (*

Newton iteration with a given linear solver

*)
| Functional (*

Functional iteration (non-stiff systems only)

*)

Choice of method for solving non-linear systems that arise in solver formulas.

val init_backward : ('d, 'k) Cvode.session ->
Cvode.lmm ->
('d, 'k) iter ->
('d, 'k) tolerance ->
'd brhsfn ->
float -> ('d, 'k) Nvector.t -> ('d, 'k) bsession

Creates and initializes a backward session attached to an existing (forward) session. The call

init_backward s lmm iter tol fb tb0 yb0

has as arguments:

This function does everything necessary to initialize a backward session, i.e., it makes the calls referenced below. The Cvodes.Adjoint.backward_normal and Cvodes.Adjoint.backward_one_step functions may be called directly.

module Quadrature: sig .. end

Support for backward quadrature equations that may or may not depend on forward sensitivities.

val backward_normal : ('d, 'k) Cvode.session -> float -> unit

Integrates a backward ODE system over an interval. The solver takes internal steps until it has reached or just passed the specified value.

val backward_one_step : ('d, 'k) Cvode.session -> float -> unit

Like Cvodes.Adjoint.backward_normal but returns after one internal solver step.

val get : ('d, 'k) bsession -> ('d, 'k) Nvector.t -> float

Fills the given vector with the solution of the backward ODE problem at the returned time, interpolating if necessary.

val get_dky : ('d, 'k) bsession ->
('d, 'k) Nvector.t -> float -> int -> unit

Returns the interpolated solution or derivatives. get_dky s dkyb t k computes the kth derivative of the backward function at time t, i.e., $\frac{d^\mathtt{k}y_B(\mathtt{t})}{\mathit{dt}^\mathtt{k}}$, and stores it in dkyb. The arguments must satisfy $t_n - h_u \leq \mathtt{t} \leq t_n$—where $t_n$ denotes Cvodes.Adjoint.get_current_time and $h_u$ denotes Cvodes.Adjoint.get_last_step,— and $0 \leq \mathtt{k} \leq q_u$—where $q_u$ denotes Cvodes.Adjoint.get_last_order.

This function may only be called after a successful return from either Cvodes.Adjoint.backward_normal or Cvodes.Adjoint.backward_one_step.

val get_y : ('d, 'k) Cvode.session -> ('d, 'k) Nvector.t -> float -> unit

Fills the vector with the interpolated forward solution at the given time during a backward simulation.

val reinit : ('d, 'k) bsession ->
?iter:('d, 'k) iter -> float -> ('d, 'k) Nvector.t -> unit

Reinitializes the backward problem with new parameters and state values. The values of the independent variable, i.e., the simulation time, and the state variables must be given. It is also possible to change the solution method (and linear solver).

Modifying the solver (optional input functions)

val set_no_sensitivity : ('d, 'k) Cvode.session -> unit

Cancels the storage of sensitivity checkpointing data during forward solution (with Cvodes.Adjoint.forward_normal or Cvodes.Adjoint.forward_one_step).

val set_tolerances : ('d, 'k) bsession -> ('d, 'k) tolerance -> unit

Sets the integration tolerances for the backward problem.

val set_max_ord : ('d, 'k) bsession -> int -> unit

Specifies the maximum order of the linear multistep method.

val set_max_num_steps : ('d, 'k) bsession -> int -> unit

Specifies the maximum number of steps taken in attempting to reach a given output time.

val set_init_step : ('d, 'k) bsession -> float -> unit

Specifies the initial step size.

val set_min_step : ('d, 'k) bsession -> float -> unit

Specifies a lower bound on the magnitude of the step size.

val set_max_step : ('d, 'k) bsession -> float -> unit

Specifies an upper bound on the magnitude of the step size.

val set_stab_lim_det : ('d, 'k) bsession -> bool -> unit

Indicates whether the BDF stability limit detection algorithm should be used.

Querying the solver (optional output functions)

val get_work_space : ('d, 'k) bsession -> int * int

Returns the real and integer workspace sizes.

val get_num_steps : ('d, 'k) bsession -> int

Returns the cumulative number of internal steps taken by the solver.

val get_num_rhs_evals : ('d, 'k) bsession -> int

Returns the number of calls to the backward right-hand side function.

val get_num_lin_solv_setups : ('d, 'k) bsession -> int

Returns the number of calls made to the linear solver's setup function.

val get_num_err_test_fails : ('d, 'k) bsession -> int

Returns the number of local error test failures that have occurred.

val get_last_order : ('d, 'k) bsession -> int

Returns the integration method order used during the last internal step.

val get_current_order : ('d, 'k) bsession -> int

Returns the integration method order to be used on the next internal step.

val get_last_step : ('d, 'k) bsession -> float

Returns the integration step size taken on the last internal step.

val get_current_step : ('d, 'k) bsession -> float

Returns the integration step size to be attempted on the next internal step.

val get_actual_init_step : ('d, 'k) bsession -> float

Returns the the value of the integration step size used on the first step.

val get_current_time : ('d, 'k) bsession -> float

Returns the the current internal time reached by the solver.

val get_num_stab_lim_order_reds : ('d, 'k) bsession -> int

Returns the number of order reductions dictated by the BDF stability limit detection algorithm.

val get_tol_scale_factor : ('d, 'k) bsession -> float

Returns a suggested factor by which the user's tolerances should be scaled when too much accuracy has been requested for some internal step.

val get_err_weights : ('d, 'k) bsession -> ('d, 'k) Nvector.t -> unit

Returns the solution error weights at the current time.

val get_est_local_errors : ('d, 'k) bsession -> ('d, 'k) Nvector.t -> unit

Returns the vector of estimated local errors.

val get_integrator_stats : ('d, 'k) bsession -> Cvode.integrator_stats

Returns the integrator statistics as a group.

val print_integrator_stats : ('d, 'k) bsession -> Stdlib.out_channel -> unit

Prints the integrator statistics on the given channel.

val get_num_nonlin_solv_iters : ('d, 'k) bsession -> int

Returns the number of nonlinear (functional or Newton) iterations performed.

val get_num_nonlin_solv_conv_fails : ('d, 'k) bsession -> int

Returns the number of nonlinear convergence failures that have occurred.

val get_nonlin_solv_stats : ('d, 'k) bsession -> int * int

Returns both the numbers of nonlinear iterations performed nniters and nonlinear convergence failures nncfails.

Exceptions

exception AdjointNotInitialized

Adjoint sensitivity analysis was not initialized.

exception NoForwardCall

Neither Cvodes.Adjoint.forward_normal nor Cvodes.Adjoint.forward_one_step has been called.

exception ForwardReinitFailure

Reinitialization of the forward problem failed at the first checkpoint (corresponding to the initial time of the forward problem).

exception ForwardFailure

An error occured when integrating the forward problem from a checkpoint.

exception NoBackwardProblem

No backward problem has been created.

exception BadFinalTime

The final time was outside the interval over which the forward problem was solved.