Publications by Giovanni Migliorati

Function Integration, Reconstruction And Approximation Using Rank-1 Lattices

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 ...

2021

MATHICSE Technical Report : Function integration, reconstruction and approximation using rank-1 lattices

Fabio Nobile, Giovanni Migliorati

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 ...

MATHICSE2019

Stable high-order randomized cubature formulae in arbitrary dimension

Fabio Nobile, Giovanni Migliorati

We propose and analyse randomized cubature formulae for the numerical integration of functions with respect to a given probability measure μ defined on a domain Γ⊆ℝ^d, in any dimension d. Each cubature formula is conceived to be exact on a given finite dim ...

2018

Discrete least-squares approximations over optimized downward closed polynomial spaces in arbitrary dimension

Fabio Nobile, Giovanni Migliorati

We analyze the accuracy of the discrete least-squares approximation of a function

u

in multivariate polynomial spaces

P_\Lambda:=span\{y\mapsto y^\nu \,: \, \nu\in \Lambda\}

with

\Lambda\subset N_0^d

over the domain

\Gamma:=[-1,1]^d

, based on the s ...

2017

Adaptive polynomial approximation by means of random discrete least squares

Giovanni Migliorati

We address adaptive multivariate polynomial approximation by means of the discrete least-squares method with random evaluations, to approximate in the L2 probability sense a smooth function depending on a random variable distributed according to a given pr ...

Springer2015

Discrete least squares polynomial approximation with random evaluations − application to parametric and stochastic elliptic PDEs

Fabio Nobile, Giovanni Migliorati

Motivated by the numerical treatment of parametric and stochastic PDEs, we analyze the least-squares method for polynomial approximation of multivariate functions based on random sampling according to a given probability measure. Recent work has shown that ...

2015

MATHICSE Technical Report : Discrete least-squares approximations over optimized downward closed polynomial spaces in arbitrary dimension

Fabio Nobile, Giovanni Migliorati

We analyze the accuracy of the discrete least-squares approximation of a function u in multivariate polynomial spaces

P_\Lambda := span\{y \mapsto y^\nu | \nu \in \Lambda\}

with

\Lambda\subset N_0^d

over the domain

\Gamma := [-1,1]^d

, based on the sa ...

MATHICSE2015

Convergence estimates in probability and in expectation for discrete least squares with noisy evaluations at random points

Fabio Nobile, Giovanni Migliorati

We study the accuracy of the discrete least-squares approximation on a finite-dimensional space of a real-valued target function from noisy pointwise evaluations at independent random points distributed according to a given sampling probability measure. Th ...

2015

MATHICSE Technical Report : Convergence estimates in probability and in expectation for discrete least squares with noisy evaluations at random points

Fabio Nobile, Giovanni Migliorati

We study the accuracy of the discrete least-squares approximation on a finite dimensional space of a real-valued target function from noisy pointwise evaluations at independent random points distributed according to a given sampling probability measure. Th ...

MATHICSE2015

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

Fabio Nobile, Giovanni Migliorati

We analyze the stability and accuracy of discrete least squares on multivariate polynomial spaces to approximate a given function depending on a multivariate random variable uniformly distributed on a hypercube. The polynomial approximation is calculated s ...

2015

Giovanni Migliorati

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Function Integration, Reconstruction And Approximation Using Rank-1 Lattices

MATHICSE Technical Report : Function integration, reconstruction and approximation using rank-1 lattices

Stable high-order randomized cubature formulae in arbitrary dimension

Discrete least-squares approximations over optimized downward closed polynomial spaces in arbitrary dimension

Adaptive polynomial approximation by means of random discrete least squares

Discrete least squares polynomial approximation with random evaluations − application to parametric and stochastic elliptic PDEs

MATHICSE Technical Report : Discrete least-squares approximations over optimized downward closed polynomial spaces in arbitrary dimension

Convergence estimates in probability and in expectation for discrete least squares with noisy evaluations at random points

MATHICSE Technical Report : Convergence estimates in probability and in expectation for discrete least squares with noisy evaluations at random points

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

MATHICSE Technical Report : Discrete least-squares approximations over optimized downward closed polynomial spaces in arbitrary dimension

Function Integration, Reconstruction And Approximation Using Rank-1 Lattices

Adaptive polynomial approximation by means of random discrete least squares

Convergence estimates in probability and in expectation for discrete least squares with noisy evaluations at random points

MATHICSE Technical Report : Convergence estimates in probability and in expectation for discrete least squares with noisy evaluations at random points

Discrete least-squares approximations over optimized downward closed polynomial spaces in arbitrary dimension

Stable high-order randomized cubature formulae in arbitrary dimension

Discrete least squares polynomial approximation with random evaluations − application to parametric and stochastic elliptic PDEs

MATHICSE Technical Report : Function integration, reconstruction and approximation using rank-1 lattices

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