Certified Rapid Solution of Partial Differential Equations for Real-Time Parameter Estimation and Optimization

Engineering analysis requires the prediction of selected “outputs” s relevant to ultimate component and system performance; typical outputs include critical stresses or strains, flow rates or pressure drops, and various measures of temperature and heat flux. These outputs are functions of “inputs” μ that serve to identify a particular configuration of the component or system; typical inputs reflect geometry, properties, and boundary conditions and loads.

In many cases, the input-output function is best articulated as a (say) linear functional ℓ of a field variable u(μ) that is the solution to an input-parameterized partial differential equation (PDE); typical field variables and associated PDEs include temperature and steady/unsteady conduction, displacement and equilibrium elasticity/Helmholtz, and velocity and steady incompressible Navier—Stokes. System behavior is thus described by an input-output relation s(μ) = ℓ(u(μ)), the evaluation of which requires solution of the underlying PDE.

Our focus is on “deployed” systems—components or processes in operation in the field—and associated “Assess-Act” scenarios. In the Assess stage we pursue robust parameter estimation (inverse) procedures that map measured-observable outputs to (all) possible system-characteristic and environment-state inputs. In the subsequent Act stage we then pursue adaptive design (optimization) procedures that map mission-objective outputs to best control-variable inputs.