Class for the reliability methods within DAKOTA/UQ. More...

Inheritance diagram for NonDLocalReliability:

Public Member Functions
	NonDLocalReliability (ProblemDescDB &problem_db, Model &model)
	constructor
	~NonDLocalReliability ()
	destructor
void	derived_init_communicators (ParLevLIter pl_iter)
	derived class contributions to initializing the communicators associated with this Iterator instance
void	derived_set_communicators (ParLevLIter pl_iter)
	derived class contributions to setting the communicators associated with this Iterator instance
void	derived_free_communicators (ParLevLIter pl_iter)
	derived class contributions to freeing the communicators associated with this Iterator instance
void	quantify_uncertainty ()
	performs an uncertainty propagation using analytical reliability methods which solve constrained optimization problems to obtain approximations of the cumulative distribution function of response
void	print_results (std::ostream &s)
	print the approximate mean, standard deviation, and importance factors when using the mean value method or the CDF/CCDF information when using MPP-search-based reliability methods
unsigned short	uses_method () const
	return name of active MPP optimizer
void	method_recourse ()
	perform an MPP optimizer method switch due to a detected conflict
Private Member Functions
void	initial_taylor_series ()
	convenience function for performing the initial limit state Taylor-series approximation
void	mean_value ()
	convenience function for encapsulating the simple Mean Value computation of approximate statistics and importance factors
void	mpp_search ()
	convenience function for encapsulating the reliability methods that employ a search for the most probable point (AMV, AMV+, FORM, SORM)
void	initialize_class_data ()
	convenience function for initializing class scope arrays
void	initialize_level_data ()
	convenience function for initializing/warm starting MPP search data for each response function prior to level 0
void	initialize_mpp_search_data ()
	convenience function for initializing/warm starting MPP search data for each z/p/beta level for each response function
void	update_mpp_search_data (const Variables &vars_star, const Response &resp_star)
	convenience function for updating MPP search data for each z/p/beta level for each response function
void	update_level_data ()
	convenience function for updating z/p/beta level data and final statistics following MPP convergence
void	update_pma_maximize (const RealVector &mpp_u, const RealVector &fn_grad_u, const RealSymMatrix &fn_hess_u)
	update pmaMaximizeG from prescribed probabilities or prescribed generalized reliabilities by inverting second-order integrations
void	update_limit_state_surrogate ()
	convenience function for passing the latest variables/response data to the data fit embedded within uSpaceModel
void	assign_mean_data ()
	update mostProbPointX/U, computedRespLevel, fnGradX/U, and fnHessX/U from ranVarMeansX/U, fnValsMeanX, fnGradsMeanX, and fnHessiansMeanX
void	dg_ds_eval (const RealVector &x_vars, const RealVector &fn_grad_x, RealVector &final_stat_grad)
	convenience function for evaluating dg/ds
Real	dp2_dbeta_factor (Real beta, bool cdf_flag)
	compute factor for derivative of second-order probability with respect to reliability index (from differentiating BREITUNG or HOHENRACK expressions)
Real	signed_norm (const RealVector &mpp_u, const RealVector &fn_grad_u, bool cdf_flag)
	convert norm of mpp_u (u-space solution) to a signed reliability index
Real	signed_norm (Real norm_mpp_u)
	convert norm of u-space vector to a signed reliability index
Real	signed_norm (Real norm_mpp_u, const RealVector &mpp_u, const RealVector &fn_grad_u, bool cdf_flag)
	shared helper function
Real	probability (Real beta)
	Convert reliability to probability using a first-order integration.
Real	probability (bool cdf_flag, const RealVector &mpp_u, const RealVector &fn_grad_u, const RealSymMatrix &fn_hess_u)
	Convert computed reliability to probability using either a first-order or second-order integration.
Real	probability (Real beta, bool cdf_flag, const RealVector &mpp_u, const RealVector &fn_grad_u, const RealSymMatrix &fn_hess_u)
	Convert provided reliability to probability using either a first-order or second-order integration.
Real	reliability (Real p)
	Convert probability to reliability using the inverse of a first-order integration.
Real	reliability (Real p, bool cdf_flag, const RealVector &mpp_u, const RealVector &fn_grad_u, const RealSymMatrix &fn_hess_u)
	Convert probability to reliability using the inverse of a first-order or second-order integration.
bool	reliability_residual (const Real &p, const Real &beta, const RealVector &kappa, Real &res)
	compute the residual for inversion of second-order probability corrections using Newton's method (called by reliability(p))
Real	reliability_residual_derivative (const Real &p, const Real &beta, const RealVector &kappa)
	compute the residual derivative for inversion of second-order probability corrections using Newton's method (called by reliability(p))
void	principal_curvatures (const RealVector &mpp_u, const RealVector &fn_grad_u, const RealSymMatrix &fn_hess_u, RealVector &kappa_u)
	Compute the kappaU vector of principal curvatures from fnHessU.
void	scale_curvature (Real beta, bool cdf_flag, const RealVector &kappa, RealVector &scaled_kappa)
	scale copy of principal curvatures by -1 if needed; else take a view
Static Private Member Functions
static void	RIA_objective_eval (const Variables &sub_model_vars, const Variables &recast_vars, const Response &sub_model_response, Response &recast_response)
	static function used as the objective function in the Reliability Index Approach (RIA) problem formulation. This equality-constrained optimization problem performs the search for the most probable point (MPP) with the objective function of (norm u)^2.
static void	RIA_constraint_eval (const Variables &sub_model_vars, const Variables &recast_vars, const Response &sub_model_response, Response &recast_response)
	static function used as the constraint function in the Reliability Index Approach (RIA) problem formulation. This equality-constrained optimization problem performs the search for the most probable point (MPP) with the constraint of G(u) = response level.
static void	PMA_objective_eval (const Variables &sub_model_vars, const Variables &recast_vars, const Response &sub_model_response, Response &recast_response)
	static function used as the objective function in the Performance Measure Approach (PMA) problem formulation. This equality-constrained optimization problem performs the search for the most probable point (MPP) with the objective function of G(u).
static void	PMA_constraint_eval (const Variables &sub_model_vars, const Variables &recast_vars, const Response &sub_model_response, Response &recast_response)
	static function used as the constraint function in the first-order Performance Measure Approach (PMA) problem formulation. This optimization problem performs the search for the most probable point (MPP) with the equality constraint of (norm u)^2 = (beta-bar)^2.
static void	PMA2_constraint_eval (const Variables &sub_model_vars, const Variables &recast_vars, const Response &sub_model_response, Response &recast_response)
	static function used as the constraint function in the second-order Performance Measure Approach (PMA) problem formulation. This optimization problem performs the search for the most probable point (MPP) with the equality constraint of beta* = beta*-bar.
static void	PMA2_set_mapping (const Variables &recast_vars, const ActiveSet &recast_set, ActiveSet &sub_model_set)
	static function used to augment the sub-model ASV requests for second-order PMA
Private Attributes
Real	computedRespLevel
	output response level calculated
Real	computedRelLevel
	output reliability level calculated for RIA and 1st-order PMA
Real	computedGenRelLevel
	output generalized reliability level calculated for 2nd-order PMA
RealVector	fnGradX
	actual x-space gradient for current function from most recent response evaluation
RealVector	fnGradU
	u-space gradient for current function updated from fnGradX and Jacobian dx/du
RealSymMatrix	fnHessX
	actual x-space Hessian for current function from most recent response evaluation
RealSymMatrix	fnHessU
	u-space Hessian for current function updated from fnHessX and Jacobian dx/du
RealVector	kappaU
	principal curvatures derived from eigenvalues of orthonormal transformation of fnHessU
RealVector	fnValsMeanX
	response function values evaluated at mean x
RealMatrix	fnGradsMeanX
	response function gradients evaluated at mean x
RealSymMatrixArray	fnHessiansMeanX
	response function Hessians evaluated at mean x
RealVector	ranVarMeansU
	vector of means for all uncertain random variables in u-space
bool	initialPtUserSpec
	flag indicating user specification of (any portion of) initialPtU
RealVector	initialPtUSpec
	user specification or default initial guess for local optimization
RealVector	initialPtU
	current starting point for MPP searches in u-space
RealVector	mostProbPointX
	location of MPP in x-space
RealVector	mostProbPointU
	location of MPP in u-space
RealVectorArray	prevMPPULev0
	array of converged MPP's in u-space for level 0. Used for warm-starting initialPtU within RBDO.
RealMatrix	prevFnGradDLev0
	matrix of limit state sensitivities w.r.t. inactive/design variables for level 0. Used for warm-starting initialPtU within RBDO.
RealMatrix	prevFnGradULev0
	matrix of limit state sensitivities w.r.t. active/uncertain variables for level 0. Used for warm-starting initialPtU within RBDO.
RealVector	prevICVars
	previous design vector. Used for warm-starting initialPtU within RBDO.
ShortArray	prevCumASVLev0
	accumulation (using \|=) of all previous design ASV's from requested finalStatistics. Used to detect availability of prevFnGradDLev0 data for warm-starting initialPtU within RBDO.
bool	npsolFlag
	flag representing the optimization MPP search algorithm selection (NPSOL SQP or OPT++ NIP)
bool	warmStartFlag
	flag indicating the use of warm starts
bool	nipModeOverrideFlag
	flag indicating the use of move overrides within OPT++ NIP
bool	curvatureDataAvailable
	flag indicating that sufficient data (i.e., fnGradU, fnHessU, mostProbPointU) is available for computing principal curvatures
bool	kappaUpdated
	track when kappaU requires updating via principal_curvatures()
short	integrationOrder
	integration order (1 or 2) provided by `integration` specification
short	secondOrderIntType
	type of second-order integration: Breitung, Hohenbichler-Rackwitz, or Hong
Real	curvatureThresh
	cut-off value for 1/sqrt() term in second-order probability corrections.
short	taylorOrder
	order of Taylor series approximations (1 or 2) in MV/AMV/AMV+ derived from hessian type
RealMatrix	impFactor
	importance factors predicted by MV
int	npsolDerivLevel
	derivative level for NPSOL executions (1 = analytic grads of objective fn, 2 = analytic grads of constraints, 3 = analytic grads of both).
unsigned short	warningBits
	set of warnings accumulated during execution
Static Private Attributes
static NonDLocalReliability *	nondLocRelInstance
	pointer to the active object instance used within the static evaluator functions in order to avoid the need for static data

Detailed Description

Class for the reliability methods within DAKOTA/UQ.

The NonDLocalReliability class implements the following reliability methods through the support of different limit state approximation and integration options: mean value (MVFOSM/MVSOSM), advanced mean value method (AMV, AMV^2) in x- or u-space, iterated advanced mean value method (AMV+, AMV^2+) in x- or u-space, two-point adaptive nonlinearity approximation (TANA) in x- or u-space, first order reliability method (FORM), and second order reliability method (SORM). All options except mean value employ an optimizer (currently NPSOL SQP or OPT++ NIP) to solve an equality-constrained optimization problem for the most probable point (MPP). The MPP search may be formulated as the reliability index approach (RIA) for mapping response levels to reliabilities/probabilities or as the performance measure approach (PMA) for performing the inverse mapping of reliability/probability levels to response levels.

Member Function Documentation

void RIA_objective_eval	(	const Variables &	sub_model_vars,
		const Variables &	recast_vars,
		const Response &	sub_model_response,
		Response &	recast_response
	)		`[static, private]`

static function used as the objective function in the Reliability Index Approach (RIA) problem formulation. This equality-constrained optimization problem performs the search for the most probable point (MPP) with the objective function of (norm u)^2.

This function recasts a G(u) response set (already transformed and approximated in other recursions) into an RIA objective function.

References Response::active_set_request_vector(), Variables::continuous_variables(), Response::function_gradient_view(), Response::function_hessian_view(), and Response::function_value().

Referenced by NonDLocalReliability::mpp_search().

void RIA_constraint_eval	(	const Variables &	sub_model_vars,
		const Variables &	recast_vars,
		const Response &	sub_model_response,
		Response &	recast_response
	)		`[static, private]`

static function used as the constraint function in the Reliability Index Approach (RIA) problem formulation. This equality-constrained optimization problem performs the search for the most probable point (MPP) with the constraint of G(u) = response level.

This function recasts a G(u) response set (already transformed and approximated in other recursions) into an RIA equality constraint.

References Response::active_set_request_vector(), Response::function_gradient(), Response::function_gradient_view(), Response::function_hessian(), Response::function_value(), NonDLocalReliability::nondLocRelInstance, NonDReliability::requestedTargetLevel, and NonDReliability::respFnCount.

Referenced by NonDLocalReliability::mpp_search().

void PMA_objective_eval	(	const Variables &	sub_model_vars,
		const Variables &	recast_vars,
		const Response &	sub_model_response,
		Response &	recast_response
	)		`[static, private]`

static function used as the objective function in the Performance Measure Approach (PMA) problem formulation. This equality-constrained optimization problem performs the search for the most probable point (MPP) with the objective function of G(u).

This function recasts a G(u) response set (already transformed and approximated in other recursions) into an PMA objective function.

References Response::active_set_request_vector(), Variables::continuous_variables(), NonDLocalReliability::curvatureDataAvailable, Response::function_gradient(), Response::function_gradient_view(), Response::function_hessian(), Response::function_hessian_view(), Response::function_value(), NonDLocalReliability::integrationOrder, NonDLocalReliability::kappaUpdated, NonDReliability::mppSearchType, NonDLocalReliability::nondLocRelInstance, NonDReliability::pmaMaximizeG, NonDReliability::respFnCount, and NonDLocalReliability::update_pma_maximize().

Referenced by NonDLocalReliability::mpp_search().

void PMA_constraint_eval	(	const Variables &	sub_model_vars,
		const Variables &	recast_vars,
		const Response &	sub_model_response,
		Response &	recast_response
	)		`[static, private]`

static function used as the constraint function in the first-order Performance Measure Approach (PMA) problem formulation. This optimization problem performs the search for the most probable point (MPP) with the equality constraint of (norm u)^2 = (beta-bar)^2.

This function recasts a G(u) response set (already transformed and approximated in other recursions) into a first-order PMA equality constraint on reliability index beta.

References Response::active_set_request_vector(), Variables::continuous_variables(), Response::function_gradient_view(), Response::function_hessian_view(), Response::function_value(), NonDLocalReliability::nondLocRelInstance, and NonDReliability::requestedTargetLevel.

Referenced by NonDLocalReliability::mpp_search().

void PMA2_constraint_eval	(	const Variables &	sub_model_vars,
		const Variables &	recast_vars,
		const Response &	sub_model_response,
		Response &	recast_response
	)		`[static, private]`

static function used as the constraint function in the second-order Performance Measure Approach (PMA) problem formulation. This optimization problem performs the search for the most probable point (MPP) with the equality constraint of beta* = beta*-bar.

This function recasts a G(u) response set (already transformed and approximated in other recursions) into a second-order PMA equality constraint on generalized reliability index beta-star.

Referenced by NonDLocalReliability::mpp_search().

void initial_taylor_series ( ) [private]

convenience function for performing the initial limit state Taylor-series approximation

An initial first- or second-order Taylor-series approximation is required for MV/AMV/AMV+/TANA or for the case where momentStats (from MV) are required within finalStatistics for subIterator usage of NonDLocalReliability.

References Response::active_set_request_vector(), Iterator::activeSet, Model::component_parallel_mode(), Model::compute_response(), Model::continuous_variables(), Model::current_response(), NonD::finalStatistics, NonDLocalReliability::fnGradsMeanX, NonDLocalReliability::fnHessiansMeanX, NonDLocalReliability::fnValsMeanX, Response::function_gradients(), Response::function_hessians(), Response::function_values(), Model::hessian_type(), Iterator::iteratedModel, NonD::momentStats, NonDReliability::mppSearchType, NonD::natafTransform, Analyzer::numFunctions, NonD::numUncertainVars, ActiveSet::request_vector(), NonD::requestedGenRelLevels, NonD::requestedProbLevels, NonD::requestedRelLevels, NonD::requestedRespLevels, Iterator::subIteratorFlag, NonDLocalReliability::taylorOrder, and NonDReliability::uSpaceModel.

Referenced by NonDLocalReliability::mean_value(), and NonDLocalReliability::mpp_search().

void initialize_class_data ( ) [private]

convenience function for initializing class scope arrays

Initialize class-scope arrays and perform other start-up activities, such as evaluating median limit state responses.

References Response::active_set_derivative_vector(), NonD::finalStatistics, NonDReliability::importanceSampler, NonD::initialize_random_variables(), NonDReliability::integrationRefinement, Iterator::iterator_rep(), NonDReliability::mppModel, NonD::natafTransform, Analyzer::numFunctions, NonDReliability::numRelAnalyses, NonD::numUncertainVars, NonDLocalReliability::prevCumASVLev0, NonDLocalReliability::prevFnGradDLev0, NonDLocalReliability::prevFnGradULev0, NonDLocalReliability::prevMPPULev0, NonDLocalReliability::ranVarMeansU, Iterator::subIteratorFlag, Model::update_from_subordinate_model(), and NonDLocalReliability::warmStartFlag.

Referenced by NonDLocalReliability::mpp_search().

void initialize_level_data ( ) [private]

convenience function for initializing/warm starting MPP search data for each response function prior to level 0

For a particular response function prior to the first z/p/beta level, initialize/warm-start optimizer initial guess (initialPtU), expansion point (mostProbPointX/U), and associated response data (computedRespLevel, fnGradX/U, and fnHessX/U).

Referenced by NonDLocalReliability::mpp_search().

void initialize_mpp_search_data ( ) [private]

convenience function for initializing/warm starting MPP search data for each z/p/beta level for each response function

For a particular response function at a particular z/p/beta level, warm-start or reset the optimizer initial guess (initialPtU), expansion point (mostProbPointX/U), and associated response data (computedRespLevel, fnGradX/U, and fnHessX/U).

Referenced by NonDLocalReliability::mpp_search().

void update_mpp_search_data	(	const Variables &	vars_star,
		const Response &	resp_star
	)		`[private]`

convenience function for updating MPP search data for each z/p/beta level for each response function

Includes case-specific logic for updating MPP search data for the AMV/AMV+/TANA/NO_APPROX methods.

Referenced by NonDLocalReliability::mpp_search().

void update_level_data ( ) [private]

convenience function for updating z/p/beta level data and final statistics following MPP convergence

Updates computedRespLevels/computedProbLevels/computedRelLevels, finalStatistics, warm start, and graphics data.

Referenced by NonDLocalReliability::mpp_search().

void dg_ds_eval	(	const RealVector &	x_vars,
		const RealVector &	fn_grad_x,
		RealVector &	final_stat_grad
	)		`[private]`

convenience function for evaluating dg/ds

Computes dg/ds where s = design variables. Supports potentially overlapping cases of design variable augmentation and insertion.

References Response::active_set_derivative_vector(), Iterator::activeSet, Model::all_continuous_variable_ids(), Model::component_parallel_mode(), Model::compute_response(), Dakota::contains(), Model::continuous_variable_ids(), Model::continuous_variables(), Dakota::copy_data(), Model::current_response(), ActiveSet::derivative_vector(), NonD::finalStatistics, Response::function_gradient_copy(), Response::function_gradients(), Model::inactive_continuous_variable_ids(), Iterator::iteratedModel, NonDReliability::mppSearchType, NonD::natafTransform, Iterator::primaryACVarMapIndices, ActiveSet::request_value(), ActiveSet::request_values(), NonDReliability::respFnCount, Iterator::secondaryACVarMapTargets, and NonDReliability::uSpaceModel.

Referenced by NonDLocalReliability::mean_value(), NonDLocalReliability::mpp_search(), and NonDLocalReliability::update_level_data().

Real dp2_dbeta_factor	(	Real	beta,
		bool	cdf_flag
	)		`[private]`

compute factor for derivative of second-order probability with respect to reliability index (from differentiating BREITUNG or HOHENRACK expressions)

Compute sensitivity of second-order probability w.r.t. beta for use in derivatives of p_2 or beta* w.r.t. auxilliary parameters s (design, epistemic) or derivatives of beta* w.r.t. u in PMA2_constraint_eval().

References Dakota::abort_handler(), NonDLocalReliability::curvatureDataAvailable, NonDLocalReliability::curvatureThresh, NonDLocalReliability::kappaU, NonD::numUncertainVars, NonDLocalReliability::probability(), NonDLocalReliability::scale_curvature(), NonDLocalReliability::secondOrderIntType, and NonDLocalReliability::warningBits.

Referenced by NonDLocalReliability::PMA2_constraint_eval(), and NonDLocalReliability::update_level_data().

Real probability	(	Real	beta,
		bool	cdf_flag,
		const RealVector &	mpp_u,
		const RealVector &	fn_grad_u,
		const RealSymMatrix &	fn_hess_u
	)		`[private]`

Convert provided reliability to probability using either a first-order or second-order integration.

Converts beta into a probability using either first-order (FORM) or second-order (SORM) integration. The SORM calculation first calculates the principal curvatures at the MPP (using the approach in Ch. 8 of Haldar & Mahadevan), and then applies correction formulations from the literature (Breitung, Hohenbichler-Rackwitz, or Hong).

The documentation for this class was generated from the following files:

NonDLocalReliability.hpp
NonDLocalReliability.cpp

Public Member Functions

Private Member Functions

Static Private Member Functions

Private Attributes

Static Private Attributes

Detailed Description

Member Function Documentation