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

Abstract

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? ? 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 ? of a field variable u(?) 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(?) = ?(u(?)), 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.