Technical Area: Numerical Optimization

Numerical optimization is an outer-loop problem used in many applications to select parameters to minimize or maximize a quality of interest. Examples include training problems in machine learning and sparse regression, parameter and state estimation, inverse problems, design optimization, and optimal control of partial differential equations. SciDAC and the DOE ASCR Applied Mathematics Program have funded research on methods for solving PDE-constrained optimization, derivative-free optimization, discrete optimization, robust optimization and optimization under uncertainty, bilevel optimization and adversarial games, multi-objective optimization, and sensitivity analysis.

Our focus in FASTMath is to develop methods for:

• Large-scale nonlinear- and PDE-constrained optimization problems that may include noisy functions, discrete variables, and multiple objectives

• Structured derivative-free optimization problems and sensitivity analysis using surrogate models

• Bilevel optimization problems and adversarial games

• Computing adjoint and forward sensitivities

We assist application teams with modeling and solving their optimization problems using these methods to, for example:

• Construct physics-constrained ML approximations to expensive operators

• Determine parameters for ice sheet models from available data that has been collected

• Design electromagnetic cloaks via PDE-constrained discrete optimization methods

• Calculate Gaussian process models in cosmology simulations

• Solve optimization problems in quantum circuit synthesis