Fast Methods for Nonlinear Structural Optimization - Collaborations With Glaucio Paulino

Nonlinear structural optimization generally leads to hard and expensive problems. These problems are often defined in terms of an objective function combined with a system of partial differential equations as constraints. Additional constraints may be given as well. The solution of these problems is very expensive, as each optimization step requires the solution of one or more discretized systems of partial differential equations, sometimes a very large number of such systems. In addition to repeatedly evaluating the objective function, additional solves may be needed to compute derivative information. Hence the solution of these problems requires robust optimization methods as well as highly efficient linear and nonlinear solvers. We will survey a range of methods that help solve these problems very efficiently. 

We can use model reduction to turn large problems into small problems, which are much cheaper to solve. Reduced order models can also be used to approximate derivative information. However, the construction of accurate reduced order models often requires expensive initial computations, especially if many parameters are involved. Moreover, for nonlinear problems, we may need to monitor the accuracy of the reduced order model and update the reduced order model if needed. Such updates need to be done as efficiently as possible.

Stochastic methods can be used to estimate gradients, Jacobians, and Hessians rather than computing these exactly. This allows the optimization to proceed with substantially cheaper steps. They can nevertheless be very effective as accurate derivative information is typically not needed to make good progress. Many objective functions and derivatives can be formulated or approximated in terms of estimates of block bilinear and quadratic forms. Moreover, these stochastic techniques can also be used to reduce the cost of computing and updating reduced order models.  

The efficient solution of discretized partial differential equations, computing and updating reduced order models, computing stochastic estimates of objective functions and derivative information and estimating block bilinear and quadratic forms all lead to the solution of long sequences of linear systems. We will consider specialized methods to make the solution of such sequences much more efficient as well.

Finally, in addition to solving these hard optimization problems, we are also interested in assessing the uncertainty in solutions that derive from uncertainties in boundary conditions, forces, material properties, and measurement error.

This talk will survey a range of computational techniques that help solve these problems efficiently and make links to related PDE-constrained optimization problems such as nonlinear inverse problems.