On Rational Krylov and Reduced Basis Methods for Fractional Diffusion

We establish an equivalence between two classes of methods for solving fractional diffusion problems, namely, Reduced Basis Methods (RBM) and Rational Krylov Methods (RKM). In particular, we demonstrate that several recently proposed RBMs for fractional diffusion can be interpreted as RKMs. This changed point of view allows us to give convergence proofs for some methods where none were previously available. We also propose a new RKM for fractional diffusion problems with poles chosen using the best rational approximation of the function 𝑧 −𝑠 with 𝑧 ranging over the spectral interval of the spatial discretization matrix. We prove convergence rates for this method and demonstrate numerically that it is competitive with or superior to many methods from the reduced basis, rational Krylov, and direct rational approximation classes. We provide numerical tests for some elliptic fractional diffusion model problems.

Error estimate of the non-intrusive reduced basis method with finite volume schemes

The context of this paper is the simulation of parameter-dependent partial differential equations (PDEs). When the aim is to solve such PDEs for a large number of parameter values, Reduced Basis Methods (RBM) are often used to reduce computational costs of a classical high fidelity code based on Finite Element Method (FEM), Finite Volume (FVM) or Spectral methods. The efficient implementation of most of these RBM requires to modify this high fidelity code, which cannot be done, for example in an industrial context if the high fidelity code is only accessible as a "black-box" solver. The Non-Intrusive Reduced Basis (NIRB) method has been introduced in the context of finite elements as a good alternative to reduce the implementation costs of these parameter-dependent problems. The method is efficient in other contexts than the FEM one, like with finite volume schemes, which are more often used in an industrial environment. In this case, some adaptations need to be done as the degrees of freedom in FV methods have different meanings. At this time, error estimates have only been studied with FEM solvers. In this paper, we present a generalisation of the NIRB method to Finite Volume schemes and we show that estimates established for FEM solvers also hold in the FVM setting. We first prove our results for the hybrid-Mimetic Finite Difference method (hMFD), which is part the Hybrid Mixed Mimetic methods (HMM) family. Then, we explain how these results apply more generally to other FV schemes. Some of them are specified, such as the Two Point Flux Approximation (TPFA).

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.