Publications related to MATHICSE Technical Report : Discrete least squares polynomial approximation with random evaluations – application to parametric and stochastic elliptic PDES

Controller design and value function approximation for nonlinear dynamical systems

This work considers the infinite-time discounted optimal control problem for continuous time input-affine polynomial dynamical systems subject to polynomial state and box input constraints. We propose a sequence of sum-of-squares (SOS) approximations of th ...

Pergamon-Elsevier Science Ltd2016

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

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

A New Identity for the Least-square Solution of Overdetermined Set of Linear Equations

Afsaneh Asaei, Mohammadjavad Taghizadeh, Saeid Haghighatshoar

In this paper, we prove a new identity for the least-square solution of an over-determined set of linear equation

Ax=b

, where

A

is an

m\times n

full-rank matrix,

b

is a column-vector of dimension

m

, and

m

(the number of equations) is larger tha ...

2015

A New Identity for the Least-square Solution of Overdetermined Set of Linear Equations

Afsaneh Asaei, Mohammadjavad Taghizadeh, Saeid Haghighatshoar

In this paper, we prove a new identity for the least-square solution of an over-determined set of linear equation

Ax=b

, where

A

is an

m\times n

full-rank matrix,

b

is a column-vector of dimension

m

, and

m

(the number of equations) is larger tha ...

Idiap2015

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

From averaging to acceleration, there is only a step-size

Nicolas Henri Bernard Flammarion

We consider the optimization of a quadratic objective function whose gradients are only accessible through a stochastic oracle that returns the gradient at any given point plus a zero-mean finite variance random error. We present the first algorithm that a ...

2015

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

MATHICSE Technical Report : Discrete least squares polynomial approximation with random evaluations – application to parametric and stochastic elliptic PDES

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Controller design and value function approximation for nonlinear dynamical systems

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

Adaptive polynomial approximation by means of random discrete least squares

A New Identity for the Least-square Solution of Overdetermined Set of Linear Equations

A New Identity for the Least-square Solution of Overdetermined Set of Linear Equations

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

From averaging to acceleration, there is only a step-size

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

Controller design and value function approximation for nonlinear dynamical systems

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

Adaptive polynomial approximation by means of random discrete least squares

A New Identity for the Least-square Solution of Overdetermined Set of Linear Equations

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

A New Identity for the Least-square Solution of Overdetermined Set of Linear Equations

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

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

From averaging to acceleration, there is only a step-size

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