In many engineering applications, a partial differential equation (PDE) has to be solved very often (“multi-query”) and/or extremely fast (“realtime”) and/or using restricted memory/CPU (“cold computing”). Moreover, the mathematical modeling yields complex systems in the sense that:
(i) each simulation is extremely costly, its CPU time may be in the order of several weeks;
(ii) we are...
One of the most fruitful tasks in data processing is to identify structures in the set where data lie and exploit them to design better models and reliable algorithms.
As a paradigm of this process we show how the cone of positive definite matrices can be endowed with Riemannian geometries alternative to the customary Euclidean geometry. This can provide new tools for data scientists, in...
Numerical homogenization is a methodology for the computational solution of multiscale partial differential equations. It aims at the compression of the corresponding partial differential operators to finite-dimensional sparse surrogate models. The surrogates are valid on a given target scale of interest, thereby accounting for the impact of features on under-resolved scales. This talk shows...
In many engineering applications, a partial differential equation (PDE) has to be solved very often (“multi-query”) and/or extremely fast (“realtime”) and/or using restricted memory/CPU (“cold computing”). Moreover, the mathematical modeling yields complex systems in the sense that:
(i) each simulation is extremely costly, its CPU time may be in the order of several weeks;
(ii) we are...
One of the most fruitful tasks in data processing is to identify structures in the set where data lie and exploit them to design better models and reliable algorithms.
As a paradigm of this process we show how the cone of positive definite matrices can be endowed with Riemannian geometries alternative to the customary Euclidean geometry. This can provide new tools for data scientists, in...
When solving PDEs over tensorized 2D domains, the regularity in the solution often appears in form of an approximate low-rank
structure in the solution vector, if properly reshaped in matrix
form. This enables the use of low-rank methods such as Sylvester solvers (namely, Rational Krylov methods and/or ADI) which allow to treat separable differential operators. We consider the setting where...
Neural networks are a fundamental tool for solving various machine learning tasks, such as supervised and unsupervised classification.
Despite this success, they still have a number of drawbacks, including lack of interpretability and large number of parameters.
In this work, we are particularly interested in learning neural network architectures with flexible activation functions (contrary...
Tensor structured linear operators play an important role in matrix equations and low-rank modelling. Motivated by this we consider the problem of approximating a matrix by a sum of Kronecker products. It is known that an optimal approximation in Frobenius norm can be obtained from the singular value decomposition of a rearranged matrix, but when the goal is to approximate the matrix as a...
Koopman operators are infinite-dimensional operators that globally linearise nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. Their increasing popularity, dubbed “Koopmania”, includes 10,000s of articles over the last decade. However, Koopman operators can have continuous spectra and lack finite-dimensional invariant subspaces, making...
The application of neural networks (NNs) to the numerical solution of PDEs has seen growing popularity in the last five years: NNs have been used as an ansatz space for the solutions, with different training approaches (PINNs, deep Ritz methods, etc.); they have also been used to infer discretization parameters and strategies.
In this talk, I will focus on deep ReLU NN approximation theory. I...
In this talk, I will review several recent results about the sparse optimization of infinite-dimensional variational problems. First, I will focus on the so-called representer theorems that allow to prove, in the case of finite-dimensional data, the existence of a solution given by the linear combination of suitably chosen atoms. In particular, I will try to convey the importance of such...
The talk will be devoted to continuous-time affine control systems and their reachable sets. I will focus on the case when all eigenvalues of the linear part of the system have zero real part. In this case, the reachable sets usually have a non-exponential growth rate as T→∞, and it is usually polynomial. The simplest non-trivial example is the problem of stabilisation (or, conversely,...
