17 research outputs found
Fully adaptive structure-preserving hyper-reduction of parametric Hamiltonian systems
Model order reduction provides low-complexity high-fidelity surrogate models
that allow rapid and accurate solutions of parametric differential equations.
The development of reduced order models for parametric nonlinear Hamiltonian
systems is still challenged by several factors: (i) the geometric structure
encoding the physical properties of the dynamics; (ii) the slowly decaying
Kolmogorov -width of conservative dynamics; (iii) the gradient structure of
the nonlinear flow velocity; (iv) high variations in the numerical rank of the
state as a function of time and parameters. We propose to address these aspects
via a structure-preserving adaptive approach that combines symplectic dynamical
low-rank approximation with adaptive gradient-preserving hyper-reduction and
parameters sampling. Additionally, we propose to vary in time the dimensions of
both the reduced basis space and the hyper-reduction space by monitoring the
quality of the reduced solution via an error indicator related to the
projection error of the Hamiltonian vector field. The resulting adaptive
hyper-reduced models preserve the geometric structure of the Hamiltonian flow,
do not rely on prior information on the dynamics, and can be solved at a cost
that is linear in the dimension of the full order model and linear in the
number of test parameters. Numerical experiments demonstrate the improved
performances of the resulting fully adaptive models compared to the original
and reduced order models
Gradient-preserving hyper-reduction of nonlinear dynamical systems via discrete empirical interpolation
This work proposes a hyper-reduction method for nonlinear parametric
dynamical systems characterized by gradient fields such as Hamiltonian systems
and gradient flows. The gradient structure is associated with conservation of
invariants or with dissipation and hence plays a crucial role in the
description of the physical properties of the system. Traditional
hyper-reduction of nonlinear gradient fields yields efficient approximations
that, however, lack the gradient structure. We focus on Hamiltonian gradients
and we propose to first decompose the nonlinear part of the Hamiltonian, mapped
into a suitable reduced space, into the sum of d terms, each characterized by a
sparse dependence on the system state. Then, the hyper-reduced approximation is
obtained via discrete empirical interpolation (DEIM) of the Jacobian of the
derived d-valued nonlinear function. The resulting hyper-reduced model retains
the gradient structure and its computationally complexity is independent of the
size of the full model. Moreover, a priori error estimates show that the
hyper-reduced model converges to the reduced model and the Hamiltonian is
asymptotically preserved. Whenever the nonlinear Hamiltonian gradient is not
globally reducible, i.e. its evolution requires high-dimensional DEIM
approximation spaces, an adaptive strategy is performed. This consists in
updating the hyper-reduced Hamiltonian via a low-rank correction of the DEIM
basis. Numerical tests demonstrate the applicability of the proposed approach
to general nonlinear operators and runtime speedups compared to the full and
the reduced models
Physics-based adaptivity of a spectral method for the Vlasov-Poisson equations based on the asymmetrically-weighted Hermite expansion in velocity space
We propose a spectral method for the 1D-1V Vlasov-Poisson system where the
discretization in velocity space is based on asymmetrically-weighted Hermite
functions, dynamically adapted via a scaling and shifting of the
velocity variable. Specifically, at each time instant an adaptivity criterion
selects new values of and based on the numerical solution of the
discrete Vlasov-Poisson system obtained at that time step. Once the new values
of the Hermite parameters and are fixed, the Hermite expansion is
updated and the discrete system is further evolved for the next time step. The
procedure is applied iteratively over the desired temporal interval. The key
aspects of the adaptive algorithm are: the map between approximation spaces
associated with different values of the Hermite parameters that preserves total
mass, momentum and energy; and the adaptivity criterion to update and
based on physics considerations relating the Hermite parameters to the
average velocity and temperature of each plasma species. For the discretization
of the spatial coordinate, we rely on Fourier functions and use the implicit
midpoint rule for time stepping. The resulting numerical method possesses
intrinsically the property of fluid-kinetic coupling, where the low-order terms
of the expansion are akin to the fluid moments of a macroscopic description of
the plasma, while kinetic physics is retained by adding more spectral terms.
Moreover, the scheme features conservation of total mass, momentum and energy
associated in the discrete, for periodic boundary conditions. A set of
numerical experiments confirms that the adaptive method outperforms the
non-adaptive one in terms of accuracy and stability of the numerical solution
Rank-adaptive structure-preserving reduced basis methods for Hamiltonian systems
This work proposes an adaptive structure-preserving model order reduction
method for finite-dimensional parametrized Hamiltonian systems modeling
non-dissipative phenomena. To overcome the slowly decaying Kolmogorov width
typical of transport problems, the full model is approximated on local reduced
spaces that are adapted in time using dynamical low-rank approximation
techniques. The reduced dynamics is prescribed by approximating the symplectic
projection of the Hamiltonian vector field in the tangent space to the local
reduced space. This ensures that the canonical symplectic structure of the
Hamiltonian dynamics is preserved during the reduction. In addition, accurate
approximations with low-rank reduced solutions are obtained by allowing the
dimension of the reduced space to change during the time evolution. Whenever
the quality of the reduced solution, assessed via an error indicator, is not
satisfactory, the reduced basis is augmented in the parameter direction that is
worst approximated by the current basis. Extensive numerical tests involving
wave interactions, nonlinear transport problems, and the Vlasov equation
demonstrate the superior stability properties and considerable runtime speedups
of the proposed method as compared to global and traditional reduced basis
approaches
Structure-Preserving Reduced Basis Methods for Hamiltonian Systems with a State-dependent Poisson Structure
We develop structure-preserving reduced basis methods for a large class of nondissipative problems by resorting to their formulation as Hamiltonian dynamical systems. With this perspective, the phase space is naturally endowed with a Poisson manifold structure which encodes the physical properties, symmetries, and conservation laws of the dynamics. The goal is to design reduced basis methods for the general state-dependent degenerate Poisson structure based on a two-step approach. First, via a local approximation of the Poisson tensor, we split the Hamiltonian dynamics into an "almost symplectic" part and the trivial evolution of the Casimir invariants. Second, canonically symplectic reduced basis techniques are applied to the nontrivial component of the dynamics, preserving the local Poisson tensor kernel exactly. The global Poisson structure and the conservation properties of the phase flow are retained by the reduced model in the constant-valued case and up to errors in the Poisson tensor approximation in the state-dependent case. A priori error estimates for the solution of the reduced system are established. A set of numerical simulations is presented to corroborate the theoretical findings
The multi-dimensional Hermite-discontinuous Galerkin method for the Vlasov-Maxwell equations
We discuss the development, analysis, implementation, and numerical
assessment of a spectral method for the numerical simulation of the
three-dimensional Vlasov-Maxwell equations. The method is based on a spectral
expansion of the velocity space with the asymmetrically weighted Hermite
functions. The resulting system of time-dependent nonlinear equations is
discretized by the discontinuous Galerkin (DG) method in space and by the
method of lines for the time integration using explicit Runge-Kutta
integrators. The resulting code, called Spectral Plasma Solver (SPS-DG), is
successfully applied to standard plasma physics benchmarks to demonstrate its
accuracy, robustness, and parallel scalability
Dynamical reduced basis methods for Hamiltonian systems
We consider model order reduction of parameterized Hamiltonian systems describing nondissipative phenomena, like wave-type and transport dominated problems. The development of reduced basis methods for such models is challenged by two main factors: the rich geometric structure encoding the physical and stability properties of the dynamics and its local low-rank nature. To address these aspects, we propose a nonlinear structure-preserving model reduction where the reduced phase space evolves in time. In the spirit of dynamical low-rank approximation, the reduced dynamics is obtained by a symplectic projection of the Hamiltonian vector field onto the tangent space of the approximation manifold at each reduced state. A priori error estimates are established in terms of the projection error of the full model solution onto the reduced manifold. For the temporal discretization of the reduced dynamics we employ splitting techniques. The reduced basis satisfies an evolution equation on the manifold of symplectic and orthogonal rectangular matrices having one dimension equal to the size of the full model. We recast the problem on the tangent space of the matrix manifold and develop intrinsic temporal integrators based on Lie group techniques together with explicit Runge-Kutta (RK) schemes. The resulting methods are shown to converge with the order of the RK integrator and their computational complexity depends only linearly on the dimension of the full model, provided the evaluation of the reduced flow velocity has a comparable cost
Dynamical Reduced Basis Methods for Hamiltonian Systems
We consider model order reduction of parameterized Hamiltonian systems describing nondissipative phenomena, like wave-type and transport dominated problems. The development of reduced basis methods for such models is challenged by two main factors: the rich geometric structure encoding the physical and stability properties of the dynamics and its local low-rank nature. To address these aspects, we propose a nonlinear structure-preserving model reduction where the reduced phase space evolves in time. In the spirit of dynamical low-rank approximation, the reduced dynamics is obtained by a symplectic projection of the Hamiltonian vector field onto the tangent space of the approximation manifold at each reduced state. A priori error estimates are established in terms of the projection error of the full model solution onto the reduced manifold. For the temporal discretization of the reduced dynamics we employ splitting techniques. The reduced basis satisfies an evolution equation on the manifold of symplectic and orthogonal rectangular matrices having one dimension equal to the size of the full model. We recast the problem on the tangent space of the matrix manifold and develop intrinsic temporal integrators based on Lie group techniques together with explicit Runge–Kutta (RK) schemes. The resulting methods are shown to converge with the order of the RK integrator and their computational complexity depends only linearly on the dimension of the full model, provided the evaluation of the reduced flow velocity has a comparable cost
Splitting-Based Structure Preserving Discretizations for Magnetohydrodynamics
We start from the splitting of the equations of single-fluid magnetohydrodynamics (MHD) into a magnetic induction part and a fluid part. We design novel numerical methods for the MHD system based on the coupling of Galerkin schemes for the electromagnetic fields via finite element exterior calculus (FEEC) with finite volume methods for the conservation laws of fluid mechanics. Using a vector potential based formulation, the magnetic induction problem is viewed as an instance of a generalized transient advection problem of differential forms. For the latter, we rely on an Eulerian method of lines with explicit Runge–Kutta timestepping and on structure preserving spatial upwind discretizations of the Lie derivative in the spirit of finite element exterior calculus. The balance laws for the fluid constitute a system of conservation laws with the magnetic induction field as a space and time dependent coefficient, supplied at every time step by the structure preserving discretization of the magnetic induction problem. We describe finite volume schemes based on approximate Riemann solvers adapted to accommodate the electromagnetic contributions to the momentum and energy conservation. A set of benchmark tests for the two-dimensional planar ideal MHD equations provide numerical evidence that the resulting lowest order coupled scheme has excellent conservation properties, is first order accurate for smooth solutions, conservative and stable.ISSN:2426-839