The Adjoint Petrov–Galerkin method for non-linear model reduction

Abstract

We formulate a new projection-based reduced-ordered modeling technique for non-linear dynamical systems. The proposed technique, which we refer to as the Adjoint Petrov-Galerkin (APG) method, is derived by decomposing the generalized coordinates of a dynamical system into a resolved coarse-scale set and an unresolved fine-scale set. A Markovian finite memory assumption within the Mori-Zwanzig formalism is then used to develop a reduced-order representation of the coarse-scales. This procedure leads to a closed reduced-order model that displays commonalities with the adjoint stabilization method used in finite elements. The formulation is shown to be equivalent to a Petrov-Galerkin method with a non-linear, time-varying test basis, thus sharing some similarities with the Least-Squares Petrov-Galerkin method. Theoretical analysis examining a priori error bounds and computational cost is presented. Numerical experiments on the compressible Navier-Stokes equations demonstrate that the proposed method can lead to improvements in numerical accuracy, robustness, and computational efficiency over the Galerkin method on problems of practical interest. Improvements in numerical accuracy and computational efficiency over the Least-Squares Petrov-Galerkin method are observed in most cases.(G ROM) has been used successfully in a variety of problems. When applied to general non-self-adjoint and non-linear problems, however, theoretical analysis and numerical experiments have shown that Galerkin ROM lacks a priori guarantees of stability, accuracy, and convergence [9]. This last issue is particularly challenging as it demonstrates that enriching a ROM basis does not necessarily improve the solution [10]. The development of stable and accurate reduced-order modeling techniques for complex non-linear systems is the motivation for the current work.A significant body of research aimed at producing accurate and stable ROMs for complex non-linear problems exists in the literature. These efforts include, but are not limited to, "energy-based" inner products [9,11], symmetry transformations [12], basis adaptation [13,14], L 1 -norm minimization [15], projection subspace rotations [16], and least-squares residual minimization approaches [17,18,19,20,21,22,23,24]. The Least-Squares Petrov-Galerkin (LSPG) [22] method comprises a particularly popular leastsquares residual minimization approach and has been proven to be an effective tool for non-linear model reduction. Defined at the fully-discrete level (i.e., after spatial and temporal discretization), LSPG relies on least-squares minimization of the FOM residual at each time-step. While the method lacks a priori stability guarantees for general non-linear systems, it has been shown to be effective for complex problems of interest [24,23,25]. Additionally, as it is formulated as a minimization problem, physical constraints such as conservation can be naturally incorporated into the ROM formulation [26]. At the fully-discrete level, LSPG is sensitive to both the time integration scheme as ...

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