Optimization Module Glossary

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 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 may 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 may also be implicitly limited by the existence of a solution to a multiphysics problem.

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 parameterized 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 parameterized model matches measured data. Replacing the parameters with control variables leads to an optimization problem. Such problems 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.

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.