Perform deterministic optimization for MAP before Bayesian calibration

Specification

Alias: none

Argument(s): none

Default: nip pre-solve for emulators

Child Keywords:

Required/Optional

Description of Group

Dakota Keyword

Dakota Keyword Description

Required

(Choose One)

Pre-solve Optimizer (Group 1)

sqp

Uses a sequential quadratic programming method for underlying optimization

nip

Uses a nonlinear interior point method for underlying optimization

none

Deactivates MAP pre-solve prior to initiating the MCMC process.

Description

When specified, Dakota will perform a deterministic derivative-based optimization to maximize the log posterior (minimize the negative log posterior = misfit - log_prior + constant normalization factors). The Markov chain in Bayesian calibration will subsequently be started at the best point found in the optimization (the MAP point), which can eliminate the need for "burn in" of the chain in which some initial portion of the chain is discarded. Note that both optimization methods available (sqp and nip) require derivatives of the negative log posterior, either first-order in the case of SQP (with quasi-Newton Hessians from secant updates) or second-order in the case of full-Newton NIP (with explicit Hessian use). The derivatives will be computed from the same model used for the MCMC process; e.g. if an emulator is used, the emulator derivatives will be used, otherwise they will be based on the user's model specification for the model.

It is important to clarify that the use of the Hessian of the negative log posterior within a full Newton solver does not strictly require Hessians from the model response quantities of interest (QoIs). Rather, the Hessian of the negative log posterior is formed from an exact Hessian of the negative log prior and a misfit Hessian that can be either exact or approximated: the full misfit Hessian can be formed using QoI residuals, gradients, and Hessians or the Gauss-Newton approximate misfit Hessian can be formed using only QoI gradients [4]. This Hessian composition is configured automatically based on MAP solver selection and the emulator's or simulation model's support for derivatives.

Default Behavior The default MAP pre-solve behavior depends on the use of an emulator model within the inference process.

If there is an emulator (for which derivatives are easily computed), then the MAP pre-solve is active by default and a full Newton NIP formulation is selected if OPT++ is available. The default use of a MAP pre-solve can be overridden using "pre_solve none" and the default selection of OPT++ full Newton NIP can be replaced with NPSOL SQP using "pre_solve sqp." Depending on the emulator's support for derivatives of the simulated QoI (gradients for dakota GP and stochastic collocation; gradients and Hessians for surfpack GP and polynomial chaos), the contribution of the misfit Hessian to the Hessian of the negative log posterior will be computed either using the full misfit Hessian or its Gauss-Newton approximation (refer to Bayesian chapter in[4]).

If no emulator model is specified, then the pre-solve is bypassed by default and the MCMC chain is initiated from user-specified (or default) initial value for the prior distributions of the random variables. This default can be overridden by specifying "pre_solve nip" for a full Newton NIP solution or "pre_solve sqp" for an NPSOL SQP solution. Both MAP pre-solve approaches require QoI gradients from the simulation model, and the full Newton approach can further leverage QoI Hessians when available (though not required due to the Gauss-Newton approximation, as explained previously).

Expected Output When pre-solve is enabled, the output will include a deterministic optimization, followed by a Bayesian calibration. The final results will include the MAP point as well as posterior statistics from the MCMC chain. The MAP point that is reported is the point with highest posterior probability spanning both the pre-solve and the subsequent MCMC chain; it will most commonly reflect the end state of the pre-solve, although it can reflect subsequent improvements from the chain evolution, should they occur.

Examples

method
  bayes_calibration queso
    samples = 2000 seed = 348
    delayed_rejection
    emulator
      pce sparse_grid_level = 2
      pre_solve nip # default for emulators