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Dinh Dũng

Publications and source records attributed to Dinh Dũng.

At least 19 recordsLinked to original sources
arXiv2026

Sampling recovery in Bochner spaces and applications to parametric PDEs

We prove convergence rates of linear sampling recovery of functions in abstract Bochner spaces satisfying weighted summability of their generalized polynomial chaos expansion coefficients. The underlying algorithm is a function-valued extension of the least squares method widely used and thoroughly studied in scalar-valued function recovery. We apply our theory to two core problems in Computational Uncertainty Quantification. First, we address non-intrusive approximations of solutions to parametric elliptic or parabolic PDEs with log-normal or affine inputs using a finite set of particular solvers. Second, we consider approximating infinite-dimensional holomorphic functions that arise as solutions to more general parametric PDEs with Gaussian random field inputs. Our framework allows a unified treatment of the log-normal and affine input models and yields substantial improvements to the state of the art of these problems. More specifically, we obtain convergence rates that improve known results by a polynomial factor in the log-normal case and by a logarithmic factor in the affine case.

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arXiv2026

Local Gevrey regularity and Quasi-Monte Carlo quadrature for PDEs parameterized on non-compact domains

We establish local Gevrey regularity for the weak solution to parametric divergence-form diffusion elliptic PDEs, assuming the diffusion coefficient itself possesses local Gevrey parametric regularity over a non-compact domain. Here "local Gevrey regularity" means that the regularity is determined in a neighborhood of each parametric point and depends on that point. Explicit bounds on parametric derivatives of the weak solution are proved in the $H^1$-norm. Building on this local Gevrey framework, we develop a novel theoretical treatment of the dimension truncation error for infinite-dimensional integration. We prove convergence rates of quasi-Monte Carlo quadrature for the finite-dimensional Gaussian-weighted integration of integrands having local Gevrey regularity. Together with the finite element discretization error, the errors arising from dimension truncation and quasi-Monte Carlo quadrature yield a complete error analysis and convergence rates for the fully discrete approximation of a bounded linear functional of the weak solution. Numerical experiments confirming the theoretical convergence rates are presented.

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arXiv2026

Simultaneous spatial-parametric collocation approximation for parametric PDEs with log-normal random inputs

We establish convergence rates for a fully discrete, multi-level, linear collocation method solving parametric elliptic PDEs on bounded polygonal domains with log-normal inputs. The method uses a finite set of function evaluations in the spatial-parametric domain. Compared with the best-known fully discrete collocation rates, these rates are significantly improved and, up to logarithmic factors, match the rates of best n-term approximations. The results follow from applying general multi-level linear sampling recovery theory in abstract Bochner spaces -- via extended least-squares -- to infinite-dimensional holomorphic functions. The abstract multi-level recovery in Bochner spaces guarantees yield the improved rates when specialized to the parametric PDE setting.

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arXiv2026

Elliptic PDEs on log-Gaussian Shapes: Sparsity and Finite Element Discretization

In this article, we consider the solution to elliptic diffusion problems on a class of random domains obtained by log-Gaussian random homothety of the unit disk respectively an annulus. We model the problem under consideration and verify the existence and uniqueness of the random solution by path-wise pullback to the nominal unit disk respectively annulus. We prove the analytic regularity of the solution with respect to the random input parameter. We consider the numerical approximation of the random diffusion problem by means of continuous, piecewise linear Lagrangian Galerkin Finite Elements with numerical quadrature in the nominal domain, and by sparse grid interpolation and quadrature of Gauss-Hermite Smolyak and Quasi-Monte Carlo type in the parameter domain. The theoretical findings are complemented by numerical results.

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arXiv2026

Piecewise-polynomial interpolations and quadratures for parametric PDEs with log-Laplace random inputs

We establish a sparsity in terms of $\ell_p$-summability and weighted $\ell_2$-summability for the coefficients of the Laguerre generalized piecewise-polynomial chaos expansion of solutions to parametric elliptic PDEs with log-Laplace random inputs. From the sparsity, we derive convergence rates for semi-discrete approximations with respect to parametric variables. These rates are valid for sparse-grid, piecewise-polynomial interpolations and the generated quadratures, and to related extended least-squares approximations and generated quadratures.

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arXiv2026

Sparsity for parametric PDEs with log-gamma random inputs and applications

We propose a novel method for establishing the sparsity of the coefficients of the Laguerre generalized polynomial chaos expansion of solutions to parametric elliptic PDEs with log-gamma inputs on $\mathbb{R}_+^\infty$. The established sparsity is quantified by $\ell_p$-summability and weighted $\ell_2$-summability of the coefficients. Building on these sparsity results, we derive convergence rates for semi-discrete approximations in the parametric variables. These rates apply to sparse-grid polynomial interpolations, extended least-squares approximations and the associated semi-discrete quadrature rules. Moreover, a counterpart of our method for parametric elliptic PDEs with log-normal inputs yields a significant improvement in the sufficient condition for $\ell_p$-summability when the component functions in the log-normal representation of the parametric diffusion coefficients have global support, compared with results obtained in prior works.

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arXiv2026

Approximation of PDE solution manifolds: Sparse-grid interpolation and quadrature

We study fully-discrete approximations and quadratures of infinite-variate functions in abstract Bochner spaces associated with a Hilbert space $X$ and an infinite-tensor-product Jacobi measure. For target infinite-variate functions taking values in $X$ which admit absolutely convergent Jacobi generalized polynomial chaos expansions, with suitable weighted summability conditions for the coefficient sequences, we generalize and improve prior results on construction of sequences of finite sparse-grid tensor-product polynomial interpolation approximations and quadratures, based on the univariate Chebyshev points. For a generic stable discretization of $X$ in terms of a dense sequence $(V_m)_{m \in \mathbb{N}_0}$ of finite-dimensional subspaces, we obtain fully-discrete, linear approximations in terms of so-called sparse-grid tensor-product projectors, with convergence rates of approximations as well as of sparse-grid tensor-product quadratures of the target functions. We verify the abstract assumptions in two fundamental application settings: first, a linear elliptic diffusion equation with affine-parametric coefficients and second, abstract holomorphic maps between separable Hilbert spaces with affine-parametric input data encoding. For these settings, as in [37,20], cancellation of anti-symmetric terms in ultra-spherical Jacobi generalized polynomial chaos expansion coefficients implies crucially improved convergence rates of sparse-grid tensor-product quadrature with respect to the infinite-tensor-product Jacobi weight, free from the ``curse-of-dimension". Largely self-contained proofs of all results are developed. Approximation convergence rate results in the present setting which are based on construction of neural network surrogates, for unbounded parameter ranges with Gaussian measures, will be developed in extensions of the present work.

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arXiv2026

Sparse-grid polynomial interpolation approximation and integration for parametric and stochastic elliptic PDEs with lognormal inputs

By combining a certain approximation property in the spatial domain, and weighted $\ell_2$-summability of the Hermite polynomial expansion coefficients in the parametric domain obtained in [M. Bachmayr, A. Cohen, R. DeVore and G. Migliorati, ESAIM Math. Model. Numer. Anal. $\bf 51$(2017), 341-363] and [M. Bachmayr, A. Cohen, D. Dũng and C. Schwab, SIAM J. Numer. Anal. $\bf 55$(2017), 2151-2186], we investigate linear non-adaptive methods of fully discrete polynomial interpolation approximation as well as fully discrete weighted quadrature methods of integration for parametric and stochastic elliptic PDEs with lognormal inputs. We explicitly construct such methods and prove corresponding convergence rates in $n$ of the approximations by them, where $n$ is a number characterizing computation complexity. The linear non-adaptive methods of fully discrete polynomial interpolation approximation are sparse-grid collocation methods. Moreover, they generate in a natural way discrete weighted quadrature formulas for integration of the solution to parametric and stochastic elliptic PDEs and its linear functionals, and the error of the corresponding integration can be estimated via the error in the Bochner space $L_1({\mathbb R}^\infty,V,γ)$ norm of the generating methods where $γ$ is the Gaussian probability measure on ${\mathbb R}^\infty$ and $V$ is the energy space. We also briefly consider similar problems for parametric and stochastic elliptic PDEs with affine inputs, and by-product problems of non-fully discrete polynomial interpolation approximation and integration. In particular, the convergence rate of non-fully discrete obtained in this paper improves the known one.

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arXiv2026

Weighted approximate sampling recovery and integration based on B-spline interpolation and quasi-interpolation

We propose novel methods for approximate sampling recovery and integration of functions in the Freud-weighted Sobolev space $W^r_{p,w}(\mathbb{R})$. The approximation error of sampling recovery is measured in the norm of the Freud-weighted Lebesgue space $L_{q,w}(\mathbb{R})$. Namely, we construct equidistant compact-supported B-spline quasi-interpolation and interpolation sampling algorithms $Q_{ρ,m}$ and $P_{ρ,m}$ which are asymptotically optimal in terms of the sampling $n$-widths $\varrho_n(\boldsymbol{W}^r_{p,w}(\mathbb{R}), L_{q,w}(\mathbb{R}))$ for every pair $p,q \in [1,\infty]$, and prove the right convergence rate of these sampling $n$-widths, where $\boldsymbol{W}^r_{p,w}(\mathbb{R})$ denotes the unit ball in $W^r_{p,w}(\mathbb{R})$. The algorithms $Q_{ρ,m}$ and $P_{ρ,m}$ are based on truncated scaled B-spline quasi-interpolation and interpolation, respectively. We also prove the asymptotical optimality and right convergence rate of the equidistant quadratures generated from $Q_{ρ,m}$ and $P_{ρ,m}$, for Freud-weighted numerical integration of functions in $W^r_{p,w}(\mathbb{R})$.

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arXiv2025

Laguerre- and Laplace-weighted integration of mixed-smoothness functions

We investigate the approximation of generalized Laguerre- or Laplace-weighted integrals over $\mathbb{R}^d_+$ or $\mathbb{R}^d$ of functions from generalized Laguerre- or Laplace-weighted Sobolev spaces of mixed smoothness, respectively. We prove upper and lower bounds of the convergence rate of optimal quadratures with respect to $n$ integration nodes for functions from these spaces. The upper bound is performed by sparse-grid quadratures with integration nodes on step hyperbolic corners or hyperbolic crosses in the function domain $\mathbb{R}^d_+$ or $\mathbb{R}^d$, respectively.

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arXiv2025

Weighted sampling recovery of functions with mixed smoothness

We studied linear weighted sampling algorithms and their optimality for approximate recovery of functions with mixed smoothness on $\mathbb{R}^d$ from a set of $n$ their sampled values. Functions to be recovered are in weighted Sobolev spaces $W^r_{p,w}(\mathbb{R}^d)$ of mixed smoothness, and the approximation error is measured by the norm of the weighted Lebesgue space $L_{q,w}(\mathbb{R}^d)$. Here, the weight $w$ is a tensor-product Freud-type weight. The optimality of linear sampling algorithms is investigated in terms of sampling $n$-widths. We constructed linear sampling algorithms on sparse grids of sampled points which form a step hyperbolic cross in the function domain, and which give upper bounds for the corresponding sampling $n$-widths. We proved that in the one-dimensional case, these algorithms realize the exact convergence rate of the $n$-sampling widths.

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arXiv2024

Weighted hyperbolic cross polynomial approximation

We study linear polynomial approximation of functions in weighted Sobolev spaces $W^r_{p,w}(\mathbb{R}^d)$ of mixed smoothness $r \in \mathbb{N}$, and their optimality in terms of Kolmogorov and linear $n$-widths of the unit ball $\boldsymbol{W}^r_{p,w}(\mathbb{R}^d)$ in these spaces. The approximation error is measured by the norm of the weighted Lebesgue space $L_{q,w}(\mathbb{R}^d)$. The weight $w$ is a tensor-product Freud weight. For $1\le p,q \le \infty$ and $d=1$, we prove that the polynomial approximation by de la Vallée Poussin sums of the orthonormal polynomial expansion of functions with respect to the weight $w^2$, is asymptotically optimal in terms of relevant linear $n$-widths $λ_n\big(\boldsymbol{W}^r_{p,w}(\mathbb{R}, L_{q,w}(\mathbb{R})\big)$ and Kolmogorov $n$-widths $d_n\big(\boldsymbol{W}^r_{p,w}(\mathbb{R}), L_{q,w}(\mathbb{R})\big)$ for $1\le q \le p <\infty$. For $1\le p,q \le \infty$ and $d\ge 2$, we construct linear methods of hyperbolic cross polynomial approximation based on tensor product of successive differences of dyadic-scaled de la Vallée Poussin sums, which are counterparts of hyperbolic cross trigonometric linear polynomial approximation, and give some upper bounds of the error of these approximations for various pair $p,q$ with $1 \le p, q \le \infty$. For some particular weights $w$ and $d \ge 2$, we prove the right convergence rate of $λ_n\big(\boldsymbol{W}^r_{2,w}(\mathbb{R}^d), L_{2,w}(\mathbb{R}^d)\big)$ and $d_n\big(\boldsymbol{W}^r_{2,w}(\mathbb{R}^d), L_{2,w}(\mathbb{R}^d)\big)$ which is performed by a constructive hyperbolic cross polynomial approximation.

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arXiv2023

Analyticity and sparsity in uncertainty quantification for PDEs with Gaussian random field inputs

We establish sparsity and summability results for coefficient sequences of Wiener-Hermite polynomial chaos expansions of countably-parametric solutions of linear elliptic and parabolic divergence-form partial differential equations with Gaussian random field inputs. The novel proof technique developed here is based on analytic continuation of parametric solutions into the complex domain. It differs from previous works that used bootstrap arguments and induction on the differentiation order of solution derivatives with respect to the parameters. The present holomorphy-based argument allows a unified, ``differentiation-free'' proof of sparsity (expressed in terms of $\ell^p$-summability or weighted $\ell^2$-summability) of sequences of Wiener-Hermite coefficients in polynomial chaos expansions in various scales of function spaces. The analysis also implies corresponding analyticity and sparsity results for posterior densities in Bayesian inverse problems subject to Gaussian priors on uncertain inputs from function spaces. Our results furthermore yield dimension-independent convergence rates of various \emph{constructive} high-dimensional deterministic numerical approximation schemes such as single-level and multi-level versions of Hermite-Smolyak anisotropic sparse-grid interpolation and quadrature in both forward and inverse computational uncertainty quantification.

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arXiv2023

Optimal numerical integration and approximation of functions on $\mathbb{R}^d$ equipped with Gaussian measure

We investigate the numerical approximation of integrals over $\mathbb{R}^d$ equipped with the standard Gaussian measure $γ$ for integrands belonging to the Gaussian-weighted Sobolev spaces $W^α_p(\mathbb{R}^d, γ)$ of mixed smoothness $α\in \mathbb{N}$ for $1 < p < \infty$. We prove the asymptotic order of the convergence of optimal quadratures based on $n$ integration nodes and propose a novel method for constructing asymptotically optimal quadratures. As for related problems, we establish by a similar technique the asymptotic order of the linear, Kolmogorov and sampling $n$-widths in the Gaussian-weighted space $L_q(\mathbb{R}^d, γ)$ of the unit ball of $W^α_p(\mathbb{R}^d, γ)$ for $1 \leq q < p < \infty$ and $q=p=2$.

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arXiv2023

Numerical weighted integration of functions having mixed smoothness

We investigate the approximation of weighted integrals over $\mathbb{R}^d$ for integrands from weighted Sobolev spaces of mixed smoothness. We prove upper and lower bounds of the convergence rate of optimal quadratures with respect to $n$ integration nodes for functions from these spaces. In the one-dimensional case $(d=1)$, we obtain the right convergence rate of optimal quadratures. For $d \ge 2$, the upper bound is performed by sparse-grid quadratures with integration nodes on step hyperbolic crosses in the function domain $\mathbb{R}^d$.

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arXiv2022

Deep ReLU neural network approximation in Bochner spaces and applications to parametric PDEs

We investigate non-adaptive methods of deep ReLU neural network approximation in Bochner spaces $L_2({\mathbb U}^\infty, X, μ)$ of functions on ${\mathbb U}^\infty$ taking values in a separable Hilbert space $X$, where ${\mathbb U}^\infty$ is either ${\mathbb R}^\infty$ equipped with the standard Gaussian probability measure, or ${\mathbb I}^\infty:= [-1,1]^\infty$ equipped with the Jacobi probability measure. Functions to be approximated are assumed to satisfy a certain weighted $\ell_2$-summability of the generalized chaos polynomial expansion coefficients with respect to the measure $μ$. We prove the convergence rate of this approximation in terms of the size of approximating deep ReLU neural networks. These results then are applied to approximation of the solution to parametric elliptic PDEs with random inputs for the lognormal and affine cases.

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arXiv2022

Collocation approximation by deep neural ReLU networks for parametric elliptic PDEs with lognormal inputs

We obtained convergence rates of the collocation approximation by deep ReLU neural networks of solutions to elliptic PDEs with lognormal inputs, parametrized by $\boldsymbol{y}$ from the non-compact set $\mathbb{R}^\infty$. The approximation error is measured in the norm of the Bochner space $L_2(\mathbb{R}^\infty, V, γ)$, where $γ$ is the infinite tensor product standard Gaussian probability measure on $\mathbb{R}^\infty$ and $V$ is the energy space. We also obtained similar results for the case when the lognormal inputs are parametrized on $\mathbb{R}^M$ with very large dimension $M$, and the approximation error is measured in the $\sqrt{g_M}$-weighted uniform norm of the Bochner space $L_\infty^{\sqrt{g}}(\mathbb{R}^M, V)$, where $g_M$ is the density function of the standard Gaussian probability measure on $\mathbb{R}^M$.

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