139 research outputs found
Distributed sampling for rational approximation of the acoustic scattering of an airfoil
No full textIn this paper we compute a reduced order model for a time‐harmonic external acoustic scattering problem with parametric frequency. The employed technique is minimal rational interpolation, an explicit moment‐matching method for Hilbert space‐valued meromorphic maps. We study the approximation and stability properties of this technique for different choices of the sample point set, namely fully distributed in the parameter range, and partially and fully confluent. The proposed technique is also compared with an implicit multi moment‐matching method based on Galerkin projection.CSQISpecial Issue: 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM
Convergence analysis of Padé approximations for Helmholtz frequency response problems
No full textThe present work concerns the approximation of the solution map S associated to the parametric Helmholtz boundary value problem, i.e., the map which associates to each (real) wavenumber belonging to a given interval of interest the corresponding solution of the Helmholtz equation. We introduce a least squares rational Padé-type approximation technique applicable to any meromorphic Hilbert space-valued univariate map, and we prove the uniform convergence of the Padé approximation error on any compact subset of the interval of interest that excludes any pole. This general result is then applied to the Helmholtz solution map S, which is proven to be meromorphic in ℂ, with a pole of order one in every (single or multiple) eigenvalue of the Laplace operator with the considered boundary conditions. Numerical tests are provided that confirm the theoretical upper bound on the Padé approximation error for the Helmholtz solution map.</jats:p
Least-Squares Padé approximation of parametric and stochastic Helmholtz maps
No full textThe present work deals with rational model order reduction methods based on the single-point Least-Square (LS) Pade approximation techniques introduced in Bonizzoni et al. (ESAIM Math. Model. Numer. Anal., 52(4), 1261-1284 2018, Math. Comput. 89, 1229-1257 2020). Algorithmical aspects concerning the construction of rational LS-Pade approximants are described. In particular, we show that the computation of the Pade denominator can be carried out efficiently by solving an eigenvalue-eigenvector problem involving a Gramian matrix. The LS-Pade techniques are employed to approximate the frequency response map associated with two parametric time-harmonic acoustic wave problems, namely a transmission-reflection problem and a scattering problem. In both cases, we establish the meromorphy of the frequency response map. The Helmholtz equation with stochastic wavenumber is also considered. In particular, for Lipschitz functionals of the solution and their corresponding probability measures, we establish weak convergence of the measure derived from the LS-Pade approximant to the true one. 2D numerical tests are performed, which confirm the effectiveness of the approximation methods
Moment equations for the mixed formulation of the Hodge Laplacian with stochastic data
No full textWe study the mixed formulation of the stochastic Hodge-Laplace problem defined on a n-dimensional domain D (n ≥ 1), with random forcing term. In particular, we focus on the magnetostatic problem and on the Darcy problem in the three dimensional case. We derive and analyze the moment equations, that is the deterministic equations solved by the m-th moment (m ≥ 1) of the unique stochastic solution of the stochastic problem. We find stable tensor product finite element discretizations, both full and sparse, and provide optimal order of convergence estimates. In particular,
we prove the inf-sup condition for sparse tensor product finite element spaces
Structure Preserving Polytopal Discontinuous Galerkin Methods for the Numerical Modeling of Neurodegenerative Diseases
No full textMany neurodegenerative diseases are connected to the spreading of misfolded prionic proteins. In this paper, we analyse the process of misfolding and spreading of both -synuclein and Amyloid-, related to Parkinson\u27s and Alzheimer\u27s diseases, respectively. We introduce and analyze a positivity-preserving numerical method for the discretization of the Fisher-Kolmogorov equation, modelling accumulation and spreading of prionic proteins. The proposed approximation method is based on the discontinuous Galerkin method on polygonal and polyhedral grids for space discretization and on method time integration scheme. We prove the existence of the discrete solution and a convergence result where the Implicit Euler scheme is employed for time integration. We show that the proposed approach is structure-preserving, in the sense that it guaranteed that the discrete solution is non-negative, a feature that is of paramount importance in practical application. The numerical verification of our numerical model is performed both using a manufactured solution and considering wavefront propagation in two-dimensional polygonal grids. Next, we present a simulation of -synuclein spreading in a two-dimensional brain slice in the sagittal plane. The polygonal mesh for this simulation is agglomerated maintaining the distinction of white and grey matter, taking advantage of the flexibility of PolyDG methods in the mesh construction. Finally, we simulate the spreading of Amyloid- in a patient-specific setting by using a three-dimensional geometry reconstructed from magnetic resonance images and an initial condition reconstructed from positron emission tomography. Our numerical simulations confirm that the proposed method is able to capture the evolution of Parkinson\u27s and Alzheimer\u27s diseases
A cVEM-DG space-time method for the dissipative wave equation
No full textA novel space-time discretization for the (linear) scalar-valued dissipative wave equation is presented. It is a structured approach, namely, the discretization space is obtained tensorizing the Virtual Element (VE) discretization in space with the Discontinuous Galerkin (DG) method in time. As such, it combines the advantages of both the VE and the DG methods. The proposed scheme is implicit and it is proved to be unconditionally stable and accurate in space and time
Tensor-Product Vertex Patch Smoothers for Biharmonic Problems
No full textWe discuss vertex patch smoothers as overlapping domain decomposition methods for fourth order elliptic partial differential equations. We show that they are numerically very efficient and yield high convergence rates. Furthermore, we discuss low rank tensor approximations for their efficient implementation. Our experiments demonstrate that the inexact local solver yields a method which converges fast and uniformly with respect to mesh refinement and polynomial degree. The multiplicative smoother shows superior performance in terms of solution efficiency, requiring fewer iterations in both two- and three-dimensional cases. Additionally, the solver infrastructure supports a mixed-precision approach, executing the multigrid preconditioner in single precision while performing the outer iteration in double precision, thereby increasing throughput by up to 70 %
Discrete tensor product BGG sequences: Splines and finite elements
No full textIn this paper, we provide a systematic discretization of the Bernstein-Gelfand-Gelfand diagrams and complexes over cubical meshes in arbitrary dimension via the use of tensor product structures of one-dimensional piecewise-polynomial spaces, such as spline and finite element spaces. We demonstrate the construction of the Hessian, the elasticity, and div div complexes as examples for our construction
A Tensor-Product Finite Element Cochain Complex with Arbitrary Continuity
No full textWe develop tensor product finite element cochain complexes of arbitrary
smoothness on Cartesian meshes of arbitrary dimension. The first step is the
construction of a one-dimensional -conforming finite element cochain
complex based on a modified Hermite interpolation operator, which is proved to
commute with the exterior derivative by means of a general commutation lemma.
Adhering to a strict tensor product construction we then derive finite element
complexes in higher dimensions
Perturbation analysis for the Darcy problem with log-normal permeability
No full textWe study the single-phase flow in a saturated, bounded heterogeneous porous medium. We model the permeability as a log-normal random field. We perform a perturbation analysis, expanding the solution in Taylor series. The series is directly computable in the case of a random field parametrized by a finite number of random variables. On the other hand, in the case of an infinite dimensional random field, suitable equations satisfied by the derivatives of the stochastic solution can be derived. We give a theoretical upper bound for the norm of the residual of the Taylor expansion which predicts the divergence of the series as the polynomial degree goes to infinity. We provide a formula to compute the optimal degree for the Taylor polynomial and the maximum achievable accuracy of the perturbation approach. Our theoretical findings are confirmed by numerical experiments in the simple case where the permeability field is described using one random variableCSQ
- …
