Shape Holomorphy of the stationary Navier-Stokes Equations *

We consider the stationary Stokes and Navier-Stokes Equations for viscous, incompressible flow in parameter dependent bounded domains D T , subject to homogeneous Dirichlet (" no-slip ") boundary conditions on ∂D T. Here, D T is the image of a given fixed nominal Lipschitz domainˆDdomainˆ domainˆD ⊆ R d , d ∈ {2, 3}, under a map T : R d → R d. We establish shape holomorphy of Leray solutions which is to say, holomorphy of the map T → (ˆ u T , ˆ p T) where (ˆ u T , ˆ p T) ∈ H 1 0 (ˆ D) d ×L 2 (ˆ D) denotes the pullback of the corresponding weak solutions and T varies in W k,∞ with k ∈ {1, 2}, depending on the type of pullback. We consider in particular parametrized families {T y : y ∈ U } ⊆ W 1,∞ of domain mappings, with parameter domain U = [−1, 1] N and with affine dependence of T y on y. The presently obtained shape holomorphy implies summability results and n-term approximation rate bounds for gpc (" generalized polynomial chaos ") expansions for the corresponding parametric solution map y → (ˆ u y , ˆ p y) ∈ H 1 0 (ˆ D) d × L 2 (ˆ D)

Shape analyticity and singular perturbations for layer potential operators

We study the effect of regular and singular domain perturbations on layer potential operators for the Laplace equation. First, we consider layer potentials supported on a diffeomorphic image (Ω) of a reference set Ω and we present some real analyticity results for the dependence upon the map. Then we introduce a perforated domain Ω(ϵ) with a small hole of size ϵ and we compute power series expansions that describe the layer potentials on Ω(ϵ) when the parameter ϵ approximates the degenerate value ϵ = 0

Higher-order Quasi-Monte Carlo Training of Deep Neural Networks

We present a novel algorithmic approach and an error analysis leveraging Quasi-Monte Carlo points for training deep neural network (DNN) surrogates of Data-to-Observable (DtO) maps in engineering design. Our analysis reveals higher-order consistent, deterministic choices of training points in the input data space for deep and shallow Neural Networks with holomorphic activation functions such as tanh. These novel training points are proved to facilitate higher-order decay (in terms of the number of training samples) of the underlying generalization error, with consistency error bounds that are free from the curse of dimensionality in the input data space, provided that DNN weights in hidden layers satisfy certain summability conditions. We present numerical experiments for DtO maps from elliptic and parabolic PDEs with uncertain inputs that confirm the theoretical analysis

Uncertainty quantification for random domains using periodic random variables

We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently introduced by Kaarnioja, Kuo, and Sloan (SIAM J. Numer. Anal., 2020) in which a countably infinite number of independent random variables enter the random field as periodic functions. We develop lattice quasi-Monte Carlo (QMC) cubature rules for computing the expected value of the solution to the Poisson problem subject to domain uncertainty. These QMC rules can be shown to exhibit higher order cubature convergence rates permitted by the periodic setting independently of the stochastic dimension of the problem. In addition, we present a complete error analysis for the problem by taking into account the approximation errors incurred by truncating the input random field to a finite number of terms and discretizing the spatial domain using finite elements. The paper concludes with numerical experiments demonstrating the theoretical error estimates.Comment: 38 pages, 3 figure

Uncertainty quantification for random domains using periodic random variables

We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently introduced by Kaarnioja et al. (SIAM J. Numer. Anal., 2020) in which a countably infinite number of independent random variables enter the random field as periodic functions. We develop lattice quasi-Monte Carlo (QMC) cubature rules for computing the expected value of the solution to the Poisson problem subject to domain uncertainty. These QMC rules can be shown to exhibit higher order cubature convergence rates permitted by the periodic setting independently of the stochastic dimension of the problem. In addition, we present a complete error analysis for the problem by taking into account the approximation errors incurred by truncating the input random field to a finite number of terms and discretizing the spatial domain using finite elements. The paper concludes with numerical experiments demonstrating the theoretical error estimates

Optimal approximation of infinite-dimensional holomorphic functions

Over the last decade, approximating functions in infinite dimensions from samples has gained increasing attention in computational science and engineering, especially in computational uncertainty quantification. This is primarily due to the relevance of functions that are solutions to parametric differential equations in various fields, e.g. chemistry, economics, engineering, and physics. While acquiring accurate and reliable approximations of such functions is inherently difficult, current benchmark methods exploit the fact that such functions often belong to certain classes of holomorphic functions to get algebraic convergence rates in infinite dimensions with respect to the number of (potentially adaptive) samples $m$. Our work focuses on providing theoretical approximation guarantees for the class of $(\boldsymbol{b},\varepsilon)$-holomorphic functions, demonstrating that these algebraic rates are the best possible for Banach-valued functions in infinite dimensions. We establish lower bounds using a reduction to a discrete problem in combination with the theory of $m$-widths, Gelfand widths and Kolmogorov widths. We study two cases, known and unknown anisotropy, in which the relative importance of the variables is known and unknown, respectively. A key conclusion of our paper is that in the latter setting, approximation from finite samples is impossible without some inherent ordering of the variables, even if the samples are chosen adaptively. Finally, in both cases, we demonstrate near-optimal, non-adaptive (random) sampling and recovery strategies which achieve close to same rates as the lower bounds

Sparse approximation of triangular transports. Part I: the finite dimensional case

For two probability measures $\rho$ and $\pi$ with analytic densities on the $d$-dimensional cube $[-1,1]^d$, we investigate the approximation of the unique triangular monotone Knothe-Rosenblatt transport $T:[-1,1]^d\to [-1,1]^d$, such that the pushforward $T_\sharp\rho$ equals $\pi$. It is shown that for $d\in\mathbb{N}$ there exist approximations $\tilde T$ of $T$, based on either sparse polynomial expansions or deep ReLU neural networks, such that the distance between $\tilde T_\sharp\rho$ and $\pi$ decreases exponentially. More precisely, we prove error bounds of the type $\exp(-\beta N^{1/d})$ (or $\exp(-\beta N^{1/(d+1)})$ for neural networks), where $N$ refers to the dimension of the ansatz space (or the size of the network) containing $\tilde T$; the notion of distance comprises the Hellinger distance, the total variation distance, the Wasserstein distance and the Kullback-Leibler divergence. Our construction guarantees $\tilde T$ to be a monotone triangular bijective transport on the hypercube $[-1,1]^d$. Analogous results hold for the inverse transport $S=T^{-1}$. The proofs are constructive, and we give an explicit a priori description of the ansatz space, which can be used for numerical implementations.Comment: The original manuscript arXiv:2006.06994v1 has been split into two parts; the present paper is the first par

Shape Holomorphy of the Stationary Navier--Stokes Equations

ISSN:0036-1410ISSN:1095-715