6.1. Introduction
IDAS is part of a software family called SUNDIALS: SUite of Nonlinear and DIfferential/ALgebraic equation Solvers [81]. This suite consists of CVODE, ARKODE, KINSOL, and IDAS, and variants of these with sensitivity analysis capabilities, CVODES and IDAS.
IDAS is a general purpose solver for the initial value problem (IVP) for systems of differential-algebraic equations (DAEs). The name IDAS stands for Implicit Differential-Algebraic solver with Sensitivity capabilities. IDAS is an extension of the IDA solver within SUNDIALS, itself based on on DASPK [26, 27], but is written in ANSI-standard C rather than Fortran77. Its most notable features are that, (1) in the solution of the underlying nonlinear system at each time step, it offers a choice of Newton/direct methods and a choice of Inexact Newton/Krylov (iterative) methods; and (2) it is written in a data-independent manner in that it acts on generic vectors and matrices without any assumptions on the underlying organization of the data. Thus IDAS shares significant modules previously written within CASC at LLNL to support the ordinary differential equation (ODE) solvers CVODE [41, 84] and PVODE [32, 33], and also the nonlinear system solver KINSOL [85].
At present, IDAS may utilize a variety of Krylov methods provided in SUNDIALS that can be used in conjunction with Newton iteration: these include the GMRES (Generalized Minimal RESidual) [122], FGMRES (Flexible Generalized Minimum RESidual) [121], Bi-CGStab (Bi-Conjugate Gradient Stabilized) [157], TFQMR (Transpose-Free Quasi-Minimal Residual) [64], and PCG (Preconditioned Conjugate Gradient) [76] linear iterative methods. As Krylov methods, these require little matrix storage for solving the Newton equations as compared to direct methods. However, the algorithms allow for a user-supplied preconditioner, and, for most problems, preconditioning is essential for an efficient solution.
For very large DAE systems, the Krylov methods are preferable over direct linear solver methods, and are often the only feasible choice. Among the Krylov methods in SUNDIALS, we recommend GMRES as the best overall choice. However, users are encouraged to compare all options, especially if encountering convergence failures with GMRES. Bi-CGFStab and TFQMR have an advantage in storage requirements, in that the number of workspace vectors they require is fixed, while that number for GMRES depends on the desired Krylov subspace size. FGMRES has an advantage in that it is designed to support preconditioners that vary between iterations (e.g. iterative methods). PCG exhibits rapid convergence and minimal workspace vectors, but only works for symmetric linear systems.
IDAS is written with a functionality that is a superset of that of IDA. Sensitivity analysis capabilities, both forward and adjoint, have been added to the main integrator. Enabling forward sensitivity computations in IDAS will result in the code integrating the so-called sensitivity equations simultaneously with the original IVP, yielding both the solution and its sensitivity with respect to parameters in the model. Adjoint sensitivity analysis, most useful when the gradients of relatively few functionals of the solution with respect to many parameters are sought, involves integration of the original IVP forward in time followed by the integration of the so-called adjoint equations backward in time. IDAS provides the infrastructure needed to integrate any final-condition ODE dependent on the solution of the original IVP (in particular the adjoint system).
6.1.1. Changes to SUNDIALS in release 6.7.0
New Features and Enhancements
The default number of stages for the SSP Runge-Kutta methods
ARKODE_LSRK_SSP_S_2 and ARKODE_LSRK_SSP_S_3 in
LSRKStep were changed from 10 and 9, respectively, to their minimum allowable
values of 2 and 4. Users may revert to the previous values by calling
LSRKStepSetNumSSPStages().
Added the optional function ARKodeInit() to ARKODE to enable data
allocation before the first call to ARKodeEvolve() (but after all other
optional input routines have been called), to support users who measure memory
usage before beginning a simulation.
Added the function ARKodeGetStageIndex() that returns the index of the
stage currently being processed, and the total number of stages in the method,
for users who wish to compute auxiliary quantities in their IVP right-hand side
functions during some stages and not others (e.g., in all but the first or last
stage).
Added the functions ARKodeGetLastTime() and ARKodeGetLastState()
to return the last successful time and state achieved by ARKODE, respectively.
ARKODE now allows users to supply functions that will be called before each
internal time step attempt (ARKodeSetPreStepFn()), after each successful
time step (ARKodeSetPostStepFn()), before right-hand side routines are
called on an updated state (ARKodeSetPreRhsFn()), and/or once each
internal step/stage is computed (ARKodeSetPostprocessStepFn()/
ARKodeSetPostprocessStageFn()). These are considered advanced
functions, as they should treat the state vector as read-only, otherwise all
theoretical guarantees of solution accuracy and stability will be lost. As a
result of these new functions, the values of multiple ARKODE return codes (e.g.,
ARK_INTERP_FAIL) have been updated; users who key off of the named constants
will not be affected, but users who rely on the values themselves should update
their codes accordingly.
Note to users utilizing the previously undocumented
ARKodeSetPostprocessStepFn() function, the supplied function is now
called on the newly computed state vector for all step attempts not just
successful steps. To obtain the previous behavior of only calling a function on
successful steps, switch to using ARKodeSetPostStepFn().
Added SUNLogger_Set{Error,Warning,Info,Debug}File functions to allow setting
logger output streams with a FILE*.
Updated the Kokkos N_Vector to support Kokkos 5.x versions.
Bug Fixes
Fixed a CMake bug where the SuperLU_MT interface would not be built and
installed without setting the SUPERLUMT_WORKS option to TRUE.
Fixed the embedded coefficients for the ARKODE_TSITOURAS_7_4_5 Butcher
table.
Fixed a bug in LSRKStep where an incorrect state vector could be passed to a
user-supplied dominant eigenvalue function on the first step unless the output
vector passed to ARKodeEvolve() contained the initial condition and when
an eigenvalue estimate is requested on the first step in a subsequent call to
ARKodeEvolve() unless the output vector passed contained the most
recently returned solution.
Fixed a potential bug in LSRKStep’s ARKODE_LSRK_SSP_S_3 method,
where a real number was used instead of an integer, potentially resulting in a
rounding error.
Fixed a bug in MRIStep for estimating the first “slow” time step in an adaptive multirate calculation.
Fixed a bug in MRIStep when using a custom inner integrator that relies on the
input state being the initial condition for the fast integration rather than
retaining the result from the last inner integration or most recent reset call
and the output vector passed to ARKodeEvolve() does not contain the
initial condition on the first call or the last returned solution on subsequent
calls.
Added a missing call to SUNNonlinSolSetup() in MRIStep when using an
IMEX-MRI-SR method.
Fixed a bug in the ARKODE discrete adjoint checkpointing where an incorrect
state would be stored on the first step if the output vector passed to
ARKodeEvolve() did not contain the initial condition on the first call.
Removed extraneous copy of output vector when using ARKODE in ARK_ONE_STEP
mode.
Removed an extraneous copy of the output vector in each step with SplittingStep.
Fixed a bug in logging output from ARKODE, where for some time stepping modules, the current “time” output in the logger was incorrect.
Fixed a bug where passing an empty string to
SUNLogger_Set{Error,Warning,Info,Debug}Filename did not disable the
corresponding logging stream Issue #844.
Deprecation Notices
The CVodeSetMonitorFn and CVodeSetMonitorFrequency functions have been
deprecated and will be removed in the next major release.
Several CMake options have been deprecated in favor of namespaced versions
prefixed with SUNDIALS_ to avoid naming collisions in applications that
include SUNDIALS directly within their CMake builds. Additionally, a consistent
naming convention (SUNDIALS_ENABLE) is now used for all boolean options. The
table below lists the old CMake option names and the new replacements.
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Following the updated CMake options, the macros listed below have been deprecated and replaced with versions that align with the new CMake options.
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For changes in prior versions of SUNDIALS see §18.
6.1.2. Reading this User Guide
The structure of this document is as follows:
In Chapter §6.2, we give short descriptions of the numerical methods implemented by IDAS for the solution of initial value problems for systems of DAEs, along with short descriptions of preconditioning (§6.2.3) and rootfinding (§6.2.4).
The following chapter describes the software organization of the IDAS solver (§6.3).
Chapter §6.4.1 is the main usage document for IDAS for simulation applications. It includes a complete description of the user interface for the integration of DAE initial value problems. Readers that are not interested in using IDAS for sensitivity analysis can then skip the next two chapters.
Chapter §6.4.4 describes the usage of IDAS for forward sensitivity analysis as an extension of its IVP integration capabilities. We begin with a skeleton of the user main program, with emphasis on the steps that are required in addition to those already described in Chapter §6.4.1. Following that we provide detailed descriptions of the user-callable interface routines specific to forward sensitivity analysis and of the additional optional user-defined routines.
Chapter §6.4.5 describes the usage of IDAS for adjoint sensitivity analysis. We begin by describing the IDAS checkpointing implementation for interpolation of the original IVP solution during integration of the adjoint system backward in time, and with an overview of a user’s main program. Following that we provide complete descriptions of the user-callable interface routines for adjoint sensitivity analysis as well as descriptions of the required additional user-defined routines.
Chapter §8 gives a brief overview of the generic
N_Vectormodule shared among the various components of SUNDIALS, as well as details on theN_Vectorimplementations provided with SUNDIALS.Chapter §9 gives a brief overview of the generic
SUNMatrixmodule shared among the various components of SUNDIALS, and details on theSUNMatriximplementations provided with SUNDIALS.Chapter §10 gives a brief overview of the generic
SUNLinearSolvermodule shared among the various components of SUNDIALS. This chapter contains details on theSUNLinearSolverimplementations provided with SUNDIALS. The chapter also contains details on theSUNLinearSolverimplementations provided with SUNDIALS that interface with external linear solver libraries.Chapter §11 describes the
SUNNonlinearSolverAPI and nonlinear solver implementations shared among the various components of SUNDIALS.Finally, in the appendices, we provide detailed instructions for the installation of IDAS, within the structure of SUNDIALS (Appendix §1.1), as well as a list of all the constants used for input to and output from IDAS functions (Appendix §6.5).
6.1.3. SUNDIALS License and Notices
All SUNDIALS packages are released open source, under the BSD 3-Clause license for more details see the LICENSE and NOTICE files provided with all SUNDIALS packages.