1,602 research outputs found
A Dynamically Adaptive Sparse Grid Method for Quasi-Optimal Interpolation of Multidimensional Analytic Functions
In this work we develop a dynamically adaptive sparse grids (SG) method for
quasi-optimal interpolation of multidimensional analytic functions defined over
a product of one dimensional bounded domains. The goal of such approach is to
construct an interpolant in space that corresponds to the "best -terms"
based on sharp a priori estimate of polynomial coefficients. In the past, SG
methods have been successful in achieving this, with a traditional construction
that relies on the solution to a Knapsack problem: only the most profitable
hierarchical surpluses are added to the SG. However, this approach requires
additional sharp estimates related to the size of the analytic region and the
norm of the interpolation operator, i.e., the Lebesgue constant. Instead, we
present an iterative SG procedure that adaptively refines an estimate of the
region and accounts for the effects of the Lebesgue constant. Our approach does
not require any a priori knowledge of the analyticity or operator norm, is
easily generalized to both affine and non-affine analytic functions, and can be
applied to sparse grids build from one dimensional rules with arbitrary growth
of the number of nodes. In several numerical examples, we utilize our
dynamically adaptive SG to interpolate quantities of interest related to the
solutions of parametrized elliptic and hyperbolic PDEs, and compare the
performance of our quasi-optimal interpolant to several alternative SG schemes
Oblique projection for scalable rank-adaptive reduced-order modeling of nonlinear stochastic PDEs with time-dependent bases
Time-dependent basis reduced order models (TDB ROMs) have successfully been
used for approximating the solution to nonlinear stochastic partial
differential equations (PDEs). For many practical problems of interest,
discretizing these PDEs results in massive matrix differential equations (MDEs)
that are too expensive to solve using conventional methods. While TDB ROMs have
the potential to significantly reduce this computational burden, they still
suffer from the following challenges: (i) inefficient for general
nonlinearities, (ii) intrusive implementation, (iii) ill-conditioned in the
presence of small singular values, and (iv) error accumulation due to fixed
rank. To this end, we present a scalable method based on oblique projections
for solving TDB ROMs that is computationally efficient, minimally intrusive,
robust in the presence of small singular values, rank-adaptive, and highly
parallelizable. These favorable properties are achieved via low-rank
approximation of the time discrete MDE. Using the discrete empirical
interpolation method (DEIM), a low-rank decomposition is computed at each
iteration of the time stepping scheme, enabling a near-optimal approximation at
a fraction of the cost. We coin the new approach TDB-CUR since it is equivalent
to a CUR decomposition based on sparse row and column samples of the MDE. We
also propose a rank-adaptive procedure to control the error on-the-fly.
Numerical results demonstrate the accuracy, efficiency, and robustness of the
new method for a diverse set of problems
Energy preserving model order reduction of the nonlinear Schr\"odinger equation
An energy preserving reduced order model is developed for two dimensional
nonlinear Schr\"odinger equation (NLSE) with plane wave solutions and with an
external potential. The NLSE is discretized in space by the symmetric interior
penalty discontinuous Galerkin (SIPG) method. The resulting system of
Hamiltonian ordinary differential equations are integrated in time by the
energy preserving average vector field (AVF) method. The mass and energy
preserving reduced order model (ROM) is constructed by proper orthogonal
decomposition (POD) Galerkin projection. The nonlinearities are computed for
the ROM efficiently by discrete empirical interpolation method (DEIM) and
dynamic mode decomposition (DMD). Preservation of the semi-discrete energy and
mass are shown for the full order model (FOM) and for the ROM which ensures the
long term stability of the solutions. Numerical simulations illustrate the
preservation of the energy and mass in the reduced order model for the two
dimensional NLSE with and without the external potential. The POD-DMD makes a
remarkable improvement in computational speed-up over the POD-DEIM. Both
methods approximate accurately the FOM, whereas POD-DEIM is more accurate than
the POD-DMD
- …