Glossary of Terms
adjoint method
The adjoint method for sensitivity analysis is based on exploiting the adjoint identity given an objective functional to derive and solve a set of adjoint equations. The adjoint solution is then used to compute the functional sensitivity with respect to sensitivity parameters.
BFGS (Broyden–Fletcher–Goldfarb–Shanno)
A specific family of optimization algorithms where updates are made to approximate the inverse of the Hessian matrix.
bounds
An inequality constraint setting lower and upper bounds directly on each control variable degree of freedom.
contributions to objective function
The objective function is a scalar function of the control variables. In the optimization interface, the objective is formed by the summation of contributions from global contributions, probe contributions, and integral contributions to the objective functions.
control variable
The control variables parameterize the optimization or sensitivity problem. The objective function and constraint are expressed in the terms of the control variables. In the mathematical and engineering literature, the control variables are sometimes also referred to as optimization variables, design variables, or decision variable.
design constraint
A constraint which can be evaluated before any multiphysics simulation has been performed. A design constrain can be expressed explicitly in the control variables, without involving the multiphysics problem solution.
design problem
An optimization problem where the objective function quantifies the performance in a multiphysics model. For such problems, the control variable is sometimes referred to as the design variables. Problems of this kind arise in, for example, structural optimization, antenna design, and process optimization.
feasible set
The control variables can be constrained to a feasible set. The feasible set is typically expressed by a set of constraints acting on the control variables. The feasible set can also be implicitly limited by the existence of a solution to a multiphysics problem.
forward method
The forward method for sensitivity analysis is based on solving the equations obtained by applying the chain rule of differentiation, with respect to sensitivity parameters, to the original DAEs.
global inequality constraint
A constraint that sets upper and lower bounds on a general global expression, possibly involving both the control variables and the PDE solution.
integral inequality constraint
A constraint that sets upper and lower bounds on an integral of an expression, possibly involving the PDE solution and control variables, over a set of geometric entities of the same dimension
objective function
A single-valued function of the PDE solution and control variables representing the performance of a multiphysics model or how well a parametric model matches measured data. Alternative terminology used for the objective function is cost function, goal function, or quantity of interest.
optimization problem
The optimization problem is to find values of the control variables, belonging to a given feasible set, such that the objective function attains its minimum (or maximum) value.
parameter estimation problem
An inverse problem where the objective function defines how well a parametric model matches measured data. Replacing the parameters with control variables leads to an optimization problem, which can arise in, for example, geophysical imaging, nondestructive testing, and biomedical imaging.
PDE-constrained optimization problem
An optimization problem where the feasible set is limited by the condition that a given multiphysics model, represented as a PDE, has a unique solution.
PDE solution
The solution to a multiphysics problem in response to specific values of the control variables.
performance constraint
A constraint involving the multiphysics simulation result. Performance constraints in general have the same structure as the objective function, and are as expensive to evaluate.
pointwise inequality constraint
An inequality constraint in a PDE-constrained optimization problem involving an explicit expression in terms of the control variables. The constraint sets lower and upper bounds on the expression for node points in a set of geometric entities of the same dimension.
sensitivity problem
The sensitivity problem determines the gradient of an objective function with respect to the control variables.
solution variables
Designates variables that are not control variables, for example, field variables and global variables.
superbasic variable
A variable is superbasic if it is not currently at one of its bounds.