Title: Model reduction for transport-dominated problems via online adaptive bases and adaptive sampling,
Abstract: This work presents a model reduction approach for problems with coherent
structures that propagate over time such as convection-dominated flows and
wave-type phenomena. Traditional model reduction methods have difficulties with
these transport-dominated problems because propagating coherent structures
typically introduce high-dimensional features that require high-dimensional
approximation spaces. The approach proposed in this work exploits the locality
in space and time of propagating coherent structures to derive efficient
reduced models. First, full-model solutions are approximated locally in time
via local reduced spaces that are adapted with basis updates during time
stepping. The basis updates are derived from querying the full model at a few
selected spatial coordinates. Second, the locality in space of the coherent
structures is exploited via an adaptive sampling scheme that selects at which
components to query the full model for computing the basis updates. Our
analysis shows that, in probability, the more local the coherent structure is
in space, the fewer full-model samples are required to adapt the reduced basis
with the proposed adaptive sampling scheme. Numerical results on benchmark
examples with interacting wave-type structures and time-varying transport
speeds and on a model combustor of a single-element rocket engine demonstrate
the wide applicability of our approach and the significant runtime speedups
compared to full models and traditional reduced models.