Publications by Fabio Nobile | EPFL Graph Search

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The 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

We study the Darcy boundary value problem with log-normal permeability field. We adopt a perturbation approach, expanding the solution in Taylor series around the nominal value of the coefficient, and approximating the expected value of the stochastic solution of the PDE by the expected value of its Taylor polynomial. The recursive deterministic equation satisfied by the expected value of the Taylor polynomial (first moment equation) is formally derived. Well-posedness and regularity results for the recursi

Gaussian random fields are widely used as building blocks for modeling stochastic processes. This paper is concerned with the efficient representation of d-point correlations for such fields, which in turn enables the representation of more general stochas ...

We analyze the stability and accuracy of discrete least squares on multivariate poly- nomial spaces to approximate a given function depending on a multivariate random variable uniformly distributed on a hypercube. The polynomial approximation is calculated starting from pointwise noise-free evaluations of the target function at low- discrepancy point sets. We prove that the discrete least-squares approximation, in a multivariate anisotropic tensor product polynomial space and with evaluations at low-discrep

We study the mixed formulation of the stochastic Hodge-Laplace problem defined on a n-dimensional domain D (

n \geq 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 \geq 1

) of the unique stochastic solution of the stochastic problem. We find stable tensor product finite element discretizations, both full and

In the field of model order reduction for frequency response problems, the minimal rational interpolation (MRI) method has been shown to be quite effective. However, in some cases, numerical instabilities may arise when applying MRI to build a surrogate model over a large frequency range, spanning several orders of magnitude. We propose a strategy to overcome these instabilities, replacing an unstable global MRI surrogate with a union of stable local rational models. The partitioning of the frequency range

We give a new fast method for evaluating sprectral approximations of nonlinear polynomial functionals. We prove that the new algorithm is convergent if the functions considered are smooth enough, under a general assumption on the spectral eigenfunctions that turns out to be satisfied in many cases, including the Fourier and Hermite basis.

In this manuscript, we present a collective multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty, and develop a novel convergence analysis of collective ...

We consider rank-1 lattices for integration and reconstruction of functions with series expansion supported on a finite index set. We explore the connection between the periodic Fourier space and the non-periodic cosine space and Chebyshev space, via tent ...

In this work we consider the problem of approximating the statistics of a given Quantity of Interest (QoI) that depends on the solution of a linear elliptic PDE defined over a random domain parameterized by N random variables. The elliptic problem is remapped on to a corresponding PDE with a fixed deterministic domain. We show that the solution can be analytically extended to a well defined region in CN with respect to the random variables. A sparse grid stochastic collocation method is then used to compute

A kernel method for estimating a probability density function from an independent and identically distributed sample drawn from such density is presented. Our estimator is a linear combination of kernel functions, the coefficients of which are determined b ...

We propose a model order reduction approach for non-intrusive surrogate modeling of parametric dynamical systems. The reduced model over the whole parameter space is built by combining surrogates in frequency only, built at few selected values of the parameters. This, in particular, requires matching the respective poles by solving an optimization problem. We detail how to treat unbalanced cases, where the surrogates to be combined have a different number of poles. If the frequency surrogates are constructe

In this work we introduce the Multi-Index Stochastic Collocation method (MISC) for computing statistics of the solution of a PDE with random data. MISC is a combination technique based on mixed differences of spatial approximations and quadratures over the ...

In this work we deal with the numerical solution of the fluid-structure interaction problem arising in the haemodynamic environment. In particular, we consider BDF and Newmark time discretization schemes, and we study different methods for the treatment of the fluid-structure interface position, focusing on partitioned algorithms for the prescription of the continuity conditions at the fluid-structure interface. We consider explicit and implicit algorithms, and new hybrid methods. We study numerically the p

The majority of problems in aircraft production and operation require decisions made in the presence of uncertainty. For this reason aerodynamic designs obtained with traditional deterministic optimization techniques seeking only optimality in a specific set of conditions may have very poor off-design performances or may even be unreliable. In this work, we present a novel approach for robust and reliability-based design optimization of aerodynamic shapes based on the combination of single and multi-objecti

This work concerns state-space models, in which the state-space is an infinite-dimensional spatial field, and the evolution is in continuous time, hence requiring approximation in space and time. The multilevel Monte Carlo (MLMC) sampling strategy is leveraged in the Monte Carlo step of the ensemble Kalman filter (EnKF), thereby yielding a multilevel ensemble Kalman filter (MLEnKF) for spatio-temporal models, which has provably superior as- ymptotic error/cost ratio. A practically relevant stochastic partia

In this work we present a residual based a posteriori error estimation for a heat equation with a random forcing term and a random diffusion coefficient which is assumed to depend affinely on a finite number of independent random variables. The problem is ...

In this work we consider quasi-optimal versions of the Stochastic Galerkin Method for solving linear elliptic PDEs with stochastic coeffcients. In particular, we consider the case of a finite number N of random inputs and an analytic dependence of the solution of the PDE with respect to the parameters in a polydisc of the complex plane CN. We show that a quasi-optimal approximation is given by a Galerkin projection on a weighted (anisotropic) total degree space and prove a (sub)exponential convergence rate.

In this paper we propose a method for the strong imposition of random Dirichlet boundary conditions in the Dynamical Low Rank (DLR) approximation of parabolic PDEs and, in particular, incompressible Navier Stokes equations. We show that the DLR variational ...

We investigate a one-dimensional model of blood flow in human arteries. In particular, we consider the case when an abrupt variation of mechanical characteristic of artery is caused by the presence of a vascular prosthesis. The derivations of the model and of a numerical scheme adopted for its solution are detailed. Numerical experiments show the effectiveness of the model.

Fabio Nobile

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MATHICSE Technical Report :Convergence analysis of Padé approximations for Helmholtz frequency response problems

MATHICSE Technical Report : Regularity and sparse approximation of the recursive first moment equations for the lognormal Darcy problem

Low-rank tensor approximation for high-order correlation functions of Gaussian random fields

MATHICSE Technical Report : Analysis of discrete least squares on multivariate polynomial spaces with evaluations at low-discrepancy point sets

MATHICSE Technical Report : Moment equations for the mixed formulation of the Hodge Laplacian with stochastic data

MATHICSE Technical Report : A technique for non-intrusive greedy piecewise-rational model reduction of frequency response problems over wide frequency bands

MATHICSE Technical Report : Sparse spectral approximations for computing polynomial functionals

A multigrid method for PDE-constrained optimization with uncertain inputs

Function Integration, Reconstruction And Approximation Using Rank-1 Lattices

MATHICSE Technical Report : Analytic regularity and collocation approximation for PDEs with random domain deformations

Density estimation in RKHS with application to Korobov spaces in high dimensions

MATHICSE Technical Report: Non-intrusive double-greedy parametric model reduction by interpolation of frequency-domain rational surrogates

Multi-index Stochastic Collocation for random PDEs

MATHICSE Technical Report : Time accurate partitioned\ algorithms for the solution of fluid-structure\ interaction problems in haemodynamics

MATHICSE Technical Report : A continuation-multilevel Monte Carlo evolutionary algorithm for robust aerodynamic shape design

MATHICSE Technical Report : Multilevel ensemble Kalman filtering for spatio-temporal processes

A Posteriori Error Estimation for the Stochastic Collocation Finite Element Approximation of the Heat Equation with Random Coefficients

MATHICSE Technical Report : Convergence of quasi-optimal stochastic Galerkin methods for a class of PDES with random coefficients

Dual Dynamically Orthogonal approximation of incompressible Navier Stokes equations with random boundary conditions

A one-dimensional model for blood flow: Application to vascular prosthesis