arXiv subjects

Tuan Anh Nguyen

Publications and source records attributed to Tuan Anh Nguyen.

33 records · Page 2Linked to original sources

arXiv2022

A path-dependent stochastic Gronwall inequality and strong convergence rate for stochastic functional differential equations

We derive a stochastic Gronwall lemma with suprema over the paths in the upper bound of the assumed affine-linear growth assumption. This allows applications to Itô processes with coefficients which depend on earlier time points such as stochastic delay equations or Euler-type approximations of stochastic differential equations. We apply our stochastic Gronwall lemma with path-suprema to stochastic functional differential equations and prove a strong convergence rate for coefficient functions which depend on path-suprema.

math.PR↗

arXiv2022

Multilevel Picard approximations of high-dimensional semilinear partial differential equations with locally monotone coefficient functions

The full history recursive multilevel Picard approximation method for semilinear parabolic partial differential equations (PDEs) is the only method which provably overcomes the curse of dimensionality for general time horizons if the coefficient functions and the nonlinearity are globally Lipschitz continuous and the nonlinearity is gradient-independent. In this article we extend this result to locally monotone coefficient functions. Our results cover a range of semilinear PDEs with polynomial coefficient functions.

math.NA↗

arXiv2022

On the speed of convergence of Picard iterations of backward stochastic differential equations

It is a well-established fact in the scientific literature that Picard iterations of backward stochastic differential equations with globally Lipschitz continuous nonlinearity converge at least exponentially fast to the solution. In this paper we prove that this convergence is in fact at least square-root factorially fast. We show for one example that no higher convergence speed is possible in general. Moreover, if the nonlinearity is $z$-independent, then the convergence is even factorially fast. Thus we reveal a phase transition in the speed of convergence of Picard iterations of backward stochastic differential equations.

math.PR↗

arXiv2022

Multilevel Picard approximations for high-dimensional decoupled forward-backward stochastic differential equations

Backward stochastic differential equations (BSDEs) appear in numeruous applications. Classical approximation methods suffer from the curse of dimensionality and deep learning-based approximation methods are not known to converge to the BSDE solution. Recently, Hutzenthaler et al. (arXiv:2108.10602) introduced a new approximation method for BSDEs whose forward diffusion is Brownian motion and proved that this method converges with essentially optimal rate without suffering from the curse of dimensionality. The central object of this article is to extend this result to general forward diffusions. The main challenge is that we need to establish convergence in temporal-spatial Hölder norms since the forward diffusion cannot be sampled exactly in general.

math.PR↗

arXiv2022

Strong convergence rate of Euler-Maruyama approximations in temporal-spatial Hölder-norms

Classical approximation results for stochastic differential equations analyze the $L^p$-distance between the exact solution and its Euler-Maruyama approximations. In this article we measure the error with temporal-spatial Hölder-norms. Our motivation for this are multigrid approximations of the exact solution viewed as a function of the starting point. We establish the classical strong convergence rate $0.5$ with respect to temporal-spatial Hölder-norms if the coefficient functions have bounded derivatives of first and second order.

math.NA↗

arXiv2021

Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations

Backward stochastic differential equations (BSDEs) belong nowadays to the most frequently studied equations in stochastic analysis and computational stochastics. BSDEs in applications are often nonlinear and high-dimensional. In nearly all cases such nonlinear high-dimensional BSDEs cannot be solved explicitly and it has been and still is a very active topic of research to design and analyze numerical approximation methods to approximatively solve nonlinear high-dimensional BSDEs. Although there are a large number of research articles in the scientific literature which analyze numerical approximation methods for nonlinear BSDEs, until today there has been no numerical approximation method in the scientific literature which has been proven to overcome the curse of dimensionality in the numerical approximation of nonlinear BSDEs in the sense that the number of computational operations of the numerical approximation method to approximatively compute one sample path of the BSDE solution grows at most polynomially in both the reciprocal $1/ \varepsilon$ of the prescribed approximation accuracy $\varepsilon \in (0,\infty)$ and the dimension $d\in \mathbb N=\{1,2,3,\ldots\}$ of the BSDE. It is the key contribution of this article to overcome this obstacle by introducing a new Monte Carlo-type numerical approximation method for high-dimensional BSDEs and by proving that this Monte Carlo-type numerical approximation method does indeed overcome the curse of dimensionality in the approximative computation of solution paths of BSDEs.

math.NA↗

arXiv2021

Multilevel Picard approximations for McKean-Vlasov stochastic differential equations

In the literatur there exist approximation methods for McKean-Vlasov stochastic differential equations which have a computational effort of order $3$. In this article we introduce full-history recursive multilevel Picard (MLP) approximations for McKean-Vlasov stochastic differential equations. We prove that these MLP approximations have computational effort of order $2+$ which is essentially optimal in high dimensions.

math.PR↗

arXiv2020

Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities

The recently introduced full-history recursive multilevel Picard (MLP) approximation methods have turned out to be quite successful in the numerical approximation of solutions of high-dimensional nonlinear PDEs. In particular, there are mathematical convergence results in the literature which prove that MLP approximation methods do overcome the curse of dimensionality in the numerical approximation of nonlinear second-order PDEs in the sense that the number of computational operations of the proposed MLP approximation method grows at most polynomially in both the reciprocal $1/ε$ of the prescribed approximation accuracy $ε>0$ and the PDE dimension $d\in \mathbb{N}=\{1,2,3, \ldots\}$. However, in each of the convergence results for MLP approximation methods in the literature it is assumed that the coefficient functions in front of the second-order differential operator are affine linear. In particular, until today there is no result in the scientific literature which proves that any semilinear second-order PDE with a general time horizon and a non affine linear coefficient function in front of the second-order differential operator can be approximated without the curse of dimensionality. It is the key contribution of this article to overcome this obstacle and to propose and analyze a new type of MLP approximation method for semilinear second-order PDEs with possibly nonlinear coefficient functions in front of the second-order differential operators. In particular, the main result of this article proves that this new MLP approximation method does indeed overcome the curse of dimensionality in the numerical approximation of semilinear second-order PDEs.

math.NA↗

arXiv2020

Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations

For a long time it is well-known that high-dimensional linear parabolic partial differential equations (PDEs) can be approximated by Monte Carlo methods with a computational effort which grows polynomially both in the dimension and in the reciprocal of the prescribed accuracy. In other words, linear PDEs do not suffer from the curse of dimensionality. For general semilinear PDEs with Lipschitz coefficients, however, it remained an open question whether these suffer from the curse of dimensionality. In this paper we partially solve this open problem. More precisely, we prove in the case of semilinear heat equations with gradient-independent and globally Lipschitz continuous nonlinearities that the computational effort of a variant of the recently introduced multilevel Picard approximations grows polynomially both in the dimension and in the reciprocal of the required accuracy.

math.PR↗

arXiv2019

A Liouville principle for the random conductance model under degenerate conditions

We consider a random conductance model on the $d$-dimensional lattice, $d\in[2,\infty)\cap\mathbb{N}$, where the conductances take values in $(0,\infty)$ and are however not assumed to be bounded from above and below. We assume that the law of the random conductances is stationary and ergodic with respect to translations on $\mathbb{Z}^d$ and invariant with respect to reflections on $\mathbb{Z}^d$ and satisfies a similar moment bound as that by Andres, Deuschel, and Slowik (2015), under which a quenched FCLT holds. We prove a first-order Liouville theorem. In the proof we construct the sublinear correctors in the discrete and adapt boundary estimates for harmonic extensions from the work in the continuum done by Bella, Fehrman, and Otto (2018) to the discrete.

math.AP↗

arXiv2019

A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations

Deep neural networks and other deep learning methods have very successfully been applied to the numerical approximation of high-dimensional nonlinear parabolic partial differential equations (PDEs), which are widely used in finance, engineering, and natural sciences. In particular, simulations indicate that algorithms based on deep learning overcome the curse of dimensionality in the numerical approximation of solutions of semilinear PDEs. For certain linear PDEs this has also been proved mathematically. The key contribution of this article is to rigorously prove this for the first time for a class of nonlinear PDEs. More precisely, we prove in the case of semilinear heat equations with gradient-independent nonlinearities that the numbers of parameters of the employed deep neural networks grow at most polynomially in both the PDE dimension and the reciprocal of the prescribed approximation accuracy. Our proof relies on recently introduced multilevel Picard approximations of semilinear PDEs.

math.NA↗

arXiv2019

An $L^p$-comparison, $p\in (1,\infty)$, on the finite differences of a discrete harmonic function at the boundary of a discrete box

It is well-known that for a harmonic function $u$ defined on the unit ball of the $d$-dimensional Euclidean space, $d\geq 2$, the tangential and normal component of the gradient $\nabla u$ on the sphere are comparable by means of the $L^p$-norms, $p\in(1,\infty)$, up to multiplicative constants that depend only on $d,p$. This paper formulates and proves a discrete analogue of this result for discrete harmonic functions defined on a discrete box on the $d$-dimensional lattice with multiplicative constants that do not depend on the size of the box.

math.NA↗

arXiv2019

Deep Learning Approach for Receipt Recognition

Inspired by the recent successes of deep learning on Computer Vision and Natural Language Processing, we present a deep learning approach for recognizing scanned receipts. The recognition system has two main modules: text detection based on Connectionist Text Proposal Network and text recognition based on Attention-based Encoder-Decoder. We also proposed pre-processing to extract receipt area and OCR verification to ignore handwriting. The experiments on the dataset of the Robust Reading Challenge on Scanned Receipts OCR and Information Extraction 2019 demonstrate that the accuracies were improved by integrating the pre-processing and the OCR verification. Our recognition system achieved 71.9% of the F1 score for detection and recognition task.

cs.CV↗

arXiv2017

Quenched invariance principles for the random conductance model on a random graph with degenerate ergodic weights

We consider a stationary and ergodic random field $\{ω(e) : e \in E_d\}$ that is parameterized by the edge set of the Euclidean lattice $\mathbb{Z}^d$, $d \geq 2$. The random variable $ω(e)$, taking values in $[0, \infty)$ and satisfying certain moment bounds, is thought of as the conductance of the edge $e$. Assuming that the set of edges with positive conductances give rise to a unique infinite cluster $\mathcal{C}_{\infty}(ω)$, we prove a quenched invariance principle for the continuous-time random walk among random conductances under relatively mild conditions on the structure of the infinite cluster. An essential ingredient of our proof is a new anchored relative isoperimetric inequality.

math.PR↗

arXiv2007

Spontaneous decay of an excited atom placed near a rectangular plate

Using the Born expansion of the Green tensor, we consider the spontaneous decay rate of an excited atom placed in the vicinity of a rectangular plate. We discuss the limitations of the commonly used simplifying assumption that the plate extends to infinity in the lateral directions and examine the effects of the atomic dipole moment orientation, atomic position, and plate boundary and thickness on the atomic decay rate. In particular, it is shown that in the boundary region, the spontaneous decay rate can be strongly modified.

quant-ph↗