mpp_search

Specify which MPP search option to use

Topics

uncertainty_quantification, reliability_methods

Specification

• Alias: None

• Arguments: None

• Default: No MPP search (MV method)

Child Keywords:

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
Required (Choose One)	MPP Approximation	x_taylor_mean	Form Taylor series approximation in “x-space” at variable means
		u_taylor_mean	Form Taylor series approximation in “u-space” at variable means
		x_taylor_mpp	X-space Taylor series approximation with iterative updates
		u_taylor_mpp	U-space Taylor series approximation with iterative updates
		x_two_point	Predict MPP using Two-point Adaptive Nonlinear Approximation in “x-space”
		u_two_point	Predict MPP using Two-point Adaptive Nonlinear Approximation in “u-space”
		x_multi_point	MPP search for local reliability based on QMEA multi-point approximation in u-space
		u_multi_point	MPP search for local reliability based on QMEA multi-point approximation in x-space
		no_approx	Perform MPP search on original response functions (use no approximation)
Optional (Choose One)	Optimization Solver	sqp	Uses a sequential quadratic programming method for underlying optimization
		nip	Uses a nonlinear interior point method for underlying optimization
Optional		integration	Integration approach

Description

The x_taylor_mean MPP search option performs a single Taylor series approximation in the space of the original uncertain variables (“x-space”) centered at the uncertain variable means, searches for the MPP for each response/probability level using this approximation, and performs a validation response evaluation at each predicted MPP. This option is commonly known as the Advanced Mean Value (AMV) method. The u_taylor_mean option is identical to the x_taylor_mean option, except that the approximation is performed in u-space. The x_taylor_mpp approach starts with an x-space Taylor series at the uncertain variable means, but iteratively updates the Taylor series approximation at each MPP prediction until the MPP converges. This option is commonly known as the AMV+ method. The u_taylor_mpp option is identical to the x_taylor_mpp option, except that all approximations are performed in u-space. The order of the Taylor-series approximation is determined by the corresponding responses specification and may be first or second-order. If second-order (methods named \(AMV^2\) and \(AMV^2+\) in [EB06]), the series may employ analytic, finite difference, or quasi Hessians (BFGS or SR1). The x_two_point MPP search option uses an x-space Taylor series approximation at the uncertain variable means for the initial MPP prediction, then utilizes the Two-point Adaptive Nonlinear Approximation (TANA) outlined in [XG98] for all subsequent MPP predictions. The u_two_point approach is identical to x_two_point, but all the approximations are performed in u-space. The x_taylor_mpp and u_taylor_mpp, x_two_point and u_two_point approaches utilize the max_iterations and convergence_tolerance method independent controls to control the convergence of the MPP iterations (the maximum number of MPP iterations per level is limited by max_iterations, and the MPP iterations are considered converged when \(\parallel {\bf u}^{(k+1)} - {\bf u}^{(k)} \parallel_2\) < convergence_tolerance). And, finally, the .no_approx option performs the MPP search on the original response functions without the use of any approximations. The optimization algorithm used to perform these MPP searches can be selected to be either sequential quadratic programming (uses the npsol_sqp optimizer) or nonlinear interior point (uses the optpp_q_newton optimizer) algorithms using the sqp or nip keywords.

In addition to the MPP search specifications, one may select among different integration approaches for computing probabilities at the MPP by using the integration keyword followed by either first_order or second_order. Second-order integration employs the formulation of [HR88] (the approach of [Bre84] and the correction of [Hon99] are also implemented, but are not active). Combining the no_approx option of the MPP search with first- and second-order integrations results in the traditional first- and second-order reliability methods (FORM and SORM). These integration approximations may be subsequently refined using importance sampling. The refinement specification allows the seletion of basic importance sampling ( import), adaptive importance sampling ( adapt_import), or multimodal adaptive importance sampling ( mm_adapt_import), along with the specification of number of samples ( samples) and random seed ( seed). Additional details on these methods are available in [EGC04] and [EB06] and in the main Uncertainty Quantification Capabilities section.