# 6.1. Introduction

IDAS is part of a software family called SUNDIALS: SUite of Nonlinear and DIfferential/ALgebraic equation Solvers [70]. 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
[23, 24], 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
[38, 73] and PVODE [29, 30], and also the
nonlinear system solver KINSOL [74].

At present, IDAS may utilize a variety of Krylov methods provided in SUNDIALS that can be used in conjuction with Newton iteration: these include the GMRES (Generalized Minimal RESidual) [106], FGMRES (Flexible Generalized Minimum RESidual) [105], Bi-CGStab (Bi-Conjugate Gradient Stabilized) [131], TFQMR (Transpose-Free Quasi-Minimal Residual) [56], and PCG (Preconditioned Conjugate Gradient) [65] 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 6.1.0

**Bug Fixes**

Fixed a bug in v7.1.0 with the SYCL N_Vector `N_VSpace`

function.

For changes in prior versions of SUNDIALS see §15.

## 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 additonal 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_Vector`

module shared among the various components of SUNDIALS, as well as details on the`N_Vector`

implementations provided with SUNDIALS.Chapter §9 gives a brief overview of the generic

`SUNMatrix`

module shared among the various components of SUNDIALS, and details on the`SUNMatrix`

implementations provided with SUNDIALS.Chapter §10 gives a brief overview of the generic

`SUNLinearSolver`

module shared among the various components of SUNDIALS. This chapter contains details on the`SUNLinearSolver`

implementations provided with SUNDIALS. The chapter also contains details on the`SUNLinearSolver`

implementations provided with SUNDIALS that interface with external linear solver libraries.Chapter §11 describes the

`SUNNonlinearSolver`

API 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.