The capabilities in this area are spread across several Trilinos packages, which are independently developed software libraries. However, it is a high priority of the software developers to make the codes interoperable wherever possible, and we are well on our way to unifying the interface to the applications around a interface class called the ModelEvaluator. The ModelEvaluator is found in the Thyra and EpetraExt packages, and examples of its implementation can be found in many example problems in the analysis code packages.
A pair of packages also summarized in this section are the Sacado Automatic Differentiation capability and the Stokhos library for polynomial chaos expansions for embedded Uncertainty Quantification. The interface between Sacado and Stokhos and a C++ application code requires templating of the relevant pieces of code on a Scalar type.
All the capabilities in this area rely heavily on the lowerlevel (inner loop) algorithms such as scalable linear solvers and preconditioners, and Trilinos utilities and interfaces. The algorithm development in this area is particularly well aligned with several of the Trilinos Strategic Goals, including Full Vertical Coverage, Scalability, and Hardened Solvers.
If you have questions on the algorithm, software, or vision of this capability area, please contact Andy Salinger or any of the package leads listed below.
Package Name  Quick Description  Point of Contact 

NOX  Nonlinear Solver with Globalized Newton's methods  Roger Pawlowski 
LOCA  Parameter Continuation, Bifurcation Tracking  Eric Phipps 
Rythmos  Time integration algorithms  Todd Coffey 
MOOCHO  Embedded (PDEconstrained) Optimization, rSQP  Roscoe Bartlett 
Aristos  Fullspace intrusive optimization (not yet released)  Dennis Ridzal 
Sacado  Automatic Differentiation using Expression Templates  Eric Phipps 
Stokhos  StochasticGalerkin Uncertainty Quantification Tools  Eric Phipps 
TriKota  Interface to Dakota toolkit for a Trilinos application (not yet released)  Andy Salinger 
Dakota  Dakota is a mature and widelyused software toolkit at Sandia, independent from Trilinos, that delivers many related analysis capabilities using nonintrusive (black box) methods. These include numerous algorithms in the areas of Optimization, Uncertainty Quantification, NonlinearLeastSquares, and Reliability. A adapter package called TriKota is under development for the Trilinos 10.0 release to make these capabilities accessible through the same interface as the above anaysis codes. 
DemoApps  The DemoApps code project is building a prototype PDE code primarily from Trilinos packages. The code will demonstrate the use of Embedded Nonlinear Analysis Tools as well as cutting edge algorithms from all other Trilinos capability areas (not yet externally released).

The Nonlinear Solver NOX solves a set of nonlinear algebraic equations for The methods included Newtonbased algorithms such as inexact Newton, matrixfree NewtonKrylov, linesearch methods, trustregion methods, tensor methods, and homotopy methods. Most of these methods ask the user to supply a Jacobian matrix and include the computational step of approximating .
The Time Integration algorithms in Rythmos solve ODEs and DAEs of the form . The algorithms include explicit and implicit methods with adaptive step size control and integration order control, including BDF and RungeKutta methods. Given an initial conditions , the methods find a series of solutions at time points . For systems with parameter dependence, , a sensitivity analysis capability is now available to solve for simultaneously. A checkpointing interface for adjoint integrations is under development.
A set of Bifurcation Tracking algorithms have been implemented in LOCA. These algorithms augment the steadystate system of equations with extra distinguishing conditions that find a parameter value where there is an exchange of stability (). This augmented system is then solved by the nonlinear solver package. Specifically, turning point (folds), pitchfork bifurcations, and Hopf bifurcations can be located and tracked. Similarly, this formulation can be used for Constraint Enforcement, where the extra equations can be supplied by the user, and the same number of parameters can be freed to be part of the solution.
An initial implementation of a capability for solving for SpaceTime and Periodic Solutions called "4D" in LOCA has been developed. With this approach, analysis algorithms designed for steady problems can be applied to transient problems. The allatonce methodology greatly expands the size of the system, which is in part mitigated by the ability of "4D" to parallelize computations over the time axes.
Steadystate (or equilibrium) solutions, including bifurcations points, can be tracked through parameter space with Parameter Continuation algorithms in LOCA. An curve of points are calculated in this manner. The code can perform Natural Parameter or Pseudo Arclength continuation methods, with a variety of predictors and steps size control algorithms. Homotopy algorithms have been preprogrammed using this interface as a globalized nonlinear solver.
A driver for performing Stability Analysis is included in the parameter continuation library. This implements spectral transformations, and then calls an eigensolver (typically Trilinos' Anasazi package). For generalized eigenvalue problems of the form , the spectral transformations include ShiftandInvert and the generalized Cayley transformation
PDEConstrained Optimization algorithms have been implemented in MOOCHO and Aristos packages to take advantage of the efficiencies accessible by embedded algorithms. These have the form: The algorithms use QuasiNewton approaches to simultaneously minimize the objective function and satisfy the constraint equations. The algorithms can make use if adjoint information if available to most efficiently solve systems with large design spaces. Both reducedspace (MOOCHO) and fullspace (Aristos) algorithms have been developed.
Automatic Differentiation tools for C++ codes have been developed in SACADO to automatically extract analytic derivative information from codes. This capability is implemented with expression templates that essentially inline code that performs the chain rule. The application code must template a key part of their code (such as the single element fill portion in a finite element code) on the <scalar> type.
Embedded UQ methods are under active development. This includes the Stokhos tools to automate the propagation of random variables through codes, such as stochastic finite element formulations, leveraging the same templated interfaces as the automatic differentiation capability. Other pieces include the subsequent nonlinear solution, transient propoagation, and linear solution of the full stochastic system.