This work proposes a windowed least-squares (WLS) approach for model reduction of dynamical systems. The proposed sequentially minimizes the time-continuous full-order-model residual within low-dimensional space–time trial subspace over time windows. comprises generalization existing approaches, as particular instances methodology recover Galerkin, Petrov–Galerkin (LSPG), and LSPG projection. I...

We derive an adaptive solver for random elliptic boundary value problems, using techniques from adaptive wavelet methods. Substituting wavelets by polynomials of the random parameters leads to a modular solver for the parameter dependence, which combines with any discretization on the spatial domain. We show optimality properties of this solver, and present numerical computations. Introduction ...

An efficient and accurate numerical scheme is presented for the three-dimensional Navier–Stokes equations in primitive variables in a cylinder. The scheme is based on a spectral-Galerkin approximation for the space variables and a second-order projection scheme for time. The new spectral-projection scheme is implemented to simulate unsteady incompressible flows in a cylinder. c © 2002 Elsevier ...

The stochastic Galerkin and collocation methods are used to solve an advectiondiffusion equation with uncertain and spatially varying viscosity. We investigate wellposedness, monotonicity and stability for the extended system resulting from the Galerkin projection of the advection-diffusion equation onto the stochastic basis functions. High-order summation-by-parts operators and weak imposition...

We consider the bilinear Schrödinger equation with several controls and simplespectrum drift. Under some regularity assumptions on the control operators and generic conditions on the controllability of the Galerkin approximations we show exact controllability in projection on the first n given eigenstates, n ∈ N arbitrary. Our methods rely on Lie-algebraic control techniques applied to the Gale...

We present a numerical algorithm for solving the Hamilton-Jacobi Bellman equation using a successive Galerkin-wavelet projection scheme. According to this scheme, the so-called Generalized-HamiltonJacobi-Bellman (GHJB) equation is solved iteratively starting from a stabilizing solution. As basis function for the Galerkin projections we consider the antiderivatives of the well-known Daubechies’ ...

We present an adaptive spacetime discontinuous Galerkin (SDG) method for linearized elastodynamics. The SDG method uses a simple Bubnov-Galerkin projection that delivers stable and oscillation–free solutions, with O (N) complexity and exact momentum balance on every spacetime element. An extended version of the Tent Pitcher algorithm generates unstructured spacetime grids that support simultane...

Abstract Established reduced‐order modeling (ROM) methods, for instance, Galerkin‐projection, approximate the solution by linearly projecting high‐dimensional spaces to a lower‐dimensional space spanned reduced basis. However, accuracy of these methods may be insufficient complex and multiscale simulations due restriction linear space. Alternatively, autoencoders (AEs) can used nonlinear dimens...

An adaptive method to accelerate time-dependent solvers for PDEs has been proposed [1,2,3] that adaptively combines on the fly a standard numerical solver and a low-dimensional system. The latter, in turn, combines POD (to identify modes) and Galerkin projection. To improve the computational efficiency of the method, it is essential to use a reduced number of points (instead of the whole comput...