arXiv subjects

Marcello Longo

Publications and source records attributed to Marcello Longo.

3 recordsLinked to original sources

arXiv2023

The fast reduced QMC matrix-vector product

We study the approximation of integrals $\int_D f(\boldsymbol{x}^\top A) \mathrm{d} \mu(\boldsymbol{x})$, where $A$ is a matrix, by quasi-Monte Carlo (QMC) rules $N^{-1} \sum_{k=0}^{N-1} f(\boldsymbol{x}_k^\top A)$. We are interested in cases where the main cost arises from calculating the products $\boldsymbol{x}_k^\top A$. We design QMC rules for which the computation of $\boldsymbol{x}_k^\top A$, $k = 0, 1, \ldots, N-1$, can be done fast, and for which the error of the QMC rule is similar to the standard QMC error. We do not require that $A$ has any particular structure. For instance, this approach can be used when approximating the expected value of a function with a multivariate normal random variable with a given covariance matrix, or when approximating the expected value of the solution of a PDE with random coefficients. The speed-up of the computation time is sometimes better and sometimes worse than the fast QMC matrix-vector product from [Dick, Kuo, Le Gia, and Schwab, Fast QMC Matrix-Vector Multiplication, SIAM J. Sci. Comput. 37 (2015)]. As in that paper, our approach applies to (polynomial) lattice point sets, but also to digital nets (we are currently not aware of any approach which allows one to apply the fast method from the aforementioned paper of Dick, Kuo, Le Gia, and Schwab to digital nets). Our method does not use FFT, instead we use repeated values in the quadrature points to derive a reduction in the computation time. This arises from the reduced CBC construction of lattice rules and polynomial lattice rules. The reduced CBC construction has been shown to reduce the computation time for the CBC construction. Here we show that it can also be used to also reduce the computation time of the QMC rule.

math.NA↗

arXiv2023

A-posteriori QMC-FEM error estimation for Bayesian inversion and optimal control with entropic risk measure

We propose a novel a-posteriori error estimation technique where the target quantities of interest are ratios of high-dimensional integrals, as occur e.g. in PDE constrained Bayesian inversion and PDE constrained optimal control subject to an entropic risk measure. We consider in particular parametric, elliptic PDEs with affine-parametric diffusion coefficient, on high-dimensional parameter spaces. We combine our recent a-posteriori Quasi-Monte Carlo (QMC) error analysis, with Finite Element a-posteriori error estimation. The proposed approach yields a computable a-posteriori estimator which is reliable, up to higher order terms. The estimator's reliability is uniform with respect to the PDE discretization, and robust with respect to the parametric dimension of the uncertain PDE input.

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arXiv2022

De Rham compatible Deep Neural Network FEM

On general regular simplicial partitions $\mathcal{T}$ of bounded polytopal domains $\Omega \subset \mathbb{R}^d$, $d\in\{2,3\}$, we construct \emph{exact neural network (NN) emulations} of all lowest order finite element spaces in the discrete de Rham complex. These include the spaces of piecewise constant functions, continuous piecewise linear (CPwL) functions, the classical ``Raviart-Thomas element'', and the ``N\'{e}d\'{e}lec edge element''. For all but the CPwL case, our network architectures employ both ReLU (rectified linear unit) and BiSU (binary step unit) activations to capture discontinuities. In the important case of CPwL functions, we prove that it suffices to work with pure ReLU nets. Our construction and DNN architecture generalizes previous results in that no geometric restrictions on the regular simplicial partitions $\mathcal{T}$ of $\Omega$ are required for DNN emulation. In addition, for CPwL functions our DNN construction is valid in any dimension $d\geq 2$. Our ``FE-Nets'' are required in the variationally correct, structure-preserving approximation of boundary value problems of electromagnetism in nonconvex polyhedra $\Omega \subset \mathbb{R}^3$. They are thus an essential ingredient in the application of e.g., the methodology of ``physics-informed NNs'' or ``deep Ritz methods'' to electromagnetic field simulation via deep learning techniques. We indicate generalizations of our constructions to higher-order compatible spaces and other, non-compatible classes of discretizations, in particular the ``Crouzeix-Raviart'' elements and Hybridized, Higher Order (HHO) methods.

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