**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Person# Giovanni Migliorati

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related units

Loading

Courses taught by this person

Loading

Related research domains

Loading

Related publications

Loading

People doing similar research

Loading

Related research domains (32)

Courses taught by this person

Related publications (17)

People doing similar research (87)

Least squares

The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by

Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a d

Approximation

An approximation is anything that is intentionally similar but not exactly equal to something else.
Etymology and usage
The word approximation is derived from Latin approximatus, from prox

No results

Loading

Loading

Loading

Related units (4)

Giovanni Migliorati, Fabio Nobile

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 transform and then cosine transform, to transfer known results from the periodic setting into new insights for the non-periodic settings. Fast discrete cosine transform can be applied for the reconstruction phase. To reduce the size of the auxiliary index set in the associated component-by-component (CBC) construction for the lattice generating vectors, we work with a bi-orthonormal set of basis functions, leading to three methods for function reconstruction in the non-periodic settings. We provide new theory and efficient algorithmic strategies for the CBC construction. We also interpret our results in the context of general function approximation and discrete least-squares approximation.

2021Giovanni Migliorati, Fabio Nobile

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 dimensional subspace V_n⊂L^2(Γ,μ) of dimension n, and uses pointwise evaluations of the integrand function φ:Γ→ℝ at m>n independent random points. These points are distributed according to a suitable auxiliary probability measure that depends on V_n. We show that, up to a logarithmic factor, a linear proportionality between m and n with dimension-independent constant ensures stability of the cubature formula with very high probability. We also prove error estimates in probability and in expectation for any n≥1 and m>n, thus covering both pre-asymptotic and asymptotic regimes. Our analysis shows that the expected cubature error decays as √(n/m) times the L^2(Γ,μ)-best approximation error of φ in V_n. On the one hand, for fixed n and m→∞ our cubature formula can be seen as a variance reduction technique for a Monte Carlo estimator, and can lead to enormous variance reduction for smooth integrand functions and subspaces V_n with spectral approximation properties. On the other hand, when we let n,m→∞, our cubature becomes of high order with spectral convergence. Finally we show that, under a more demanding (at least quadratic) proportionality between m and n, the weights of the cubature are positive with very high probability. As an example of application, we discuss the case where the domain Γ has the structure of Cartesian product, μ is a product measure on Γ and the space V_n contains algebraic multivariate polynomials.

2018Giovanni Migliorati, Fabio Nobile

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 transform and then cosine transform, to transfer known results from the periodic setting into new insights for the non-periodic settings. Fast discrete cosine transform can be applied for the reconstruction phase. To reduce the size of the auxiliary index set in the associated component-by-component (CBC) construction for the lattice generating vectors, we work with a bi-orthonormal set of basis functions, leading to three methods for function reconstruction in the non-periodic settings. We provide new theory and efficient algorithmic strategies for the CBC construction. We also interpret our results in the context of general function approximation and discrete least-squares approximation.