Worst-case multi-objective error estimation and adaptivity
It extends goal-oriented error estimation from single to multiple objectives, addressing a known bottleneck in finite-element adaptivity for problems with multiple criteria.
This paper presents a new method for a-posteriori multi-objective error estimation in finite-element approximations, enabling quasi-optimal adaptive refinements for general closed convex subsets of the dual space. Numerical experiments demonstrate the efficacy of the worst-case error estimate in adaptive refinement.
This paper introduces a new computational methodology for determining a-posteriori multi-objective error estimates for finite-element approximations, and for constructing corresponding (quasi-)optimal adaptive refinements of finite-element spaces. As opposed to the classical goal-oriented approaches, which consider only a single objective functional, the presented methodology applies to general closed convex subsets of the dual space and constructs a worst-case error estimate of the finite-element approximation error. This worst-case multi-objective error estimate conforms to a dual-weighted residual, in which the dual solution is associated with an approximate supporting functional of the objective set at the approximation error. We regard both standard approximation errors and data-incompatibility errors associated with incompatibility of boundary data with the trace of the finite-element space. Numerical experiments are presented to demonstrate the efficacy of applying the proposed worst-case multi-objective error in adaptive refinement procedures.