Analyticity and Sparsity in Uncertainty Quantification for Pdes with Gaussian Random Field Inputs

Dũng, Dinh, Nguyen, Van Kien, Schwab, Christoph

Analyticity and Sparsity in Uncertainty Quantification for Pdes with Gaussian Random Field Inputs
  • 出版商: Springer
  • 出版日期: 2023-10-14
  • 售價: $2,650
  • 貴賓價: 9.5 折 $2,518
  • 語言: 英文
  • 頁數: 207
  • 裝訂: Quality Paper - also called trade paper
  • ISBN: 3031383834
  • ISBN-13: 9783031383830

    • 海外代購書籍(需單獨結帳)

商品描述

The present book develops the mathematical and numerical analysis of linear, elliptic and parabolic partial differential equations (PDEs) with coefficients whose logarithms are modelled as Gaussian random fields (GRFs), in polygonal and polyhedral physical domains. Both, forward and Bayesian inverse PDE problems subject to GRF priors are considered.
Adopting a pathwise, affine-parametric representation of the GRFs, turns the random PDEs into equivalent, countably-parametric, deterministic PDEs, with nonuniform ellipticity constants. A detailed sparsity analysis of Wiener-Hermite polynomial chaos expansions of the corresponding parametric PDE solution families by analytic continuation into the complex domain is developed, in corner- and edge-weighted function spaces on the physical domain.
The presented Algorithms and results are relevant for the mathematical analysis of many approximation methods for PDEs with GRF inputs, such as model order reduction, neural network and tensor-formatted surrogates of parametric solution families. They are expected to impact computational uncertainty quantification subject to GRF models of uncertainty in PDEs, and are of interest for researchers and graduate students in both, applied and computational mathematics, as well as in computational science and engineering.

商品描述(中文翻譯)

本書探討了具有以高斯隨機場（GRF）模擬的對數係數的線性、橢圓和抛物型偏微分方程（PDE）的數學和數值分析，這些方程出現在多邊形和多面體物理領域中。本書考慮了基於GRF先驗的正向和貝葉斯反向PDE問題。

通過採用GRF的路徑依賴、仿射參數表示，將隨機PDE轉化為等價的可數參數、確定性PDE，具有非均勻橢圓性常數。本書還開展了對相應參數PDE解族的維納-埃爾米特多項式混沌展開的詳細稀疏性分析，通過在物理域上的角和邊權重函數空間中進行解析延拓。

所提出的算法和結果對於使用GRF輸入的PDE的許多近似方法的數學分析具有重要意義，例如模型降階、神經網絡和張量格式的參數解族代理。它們預計對於基於GRF的PDE不確定性計算具有影響力，並且對於應用和計算數學、計算科學和工程領域的研究人員和研究生具有興趣。

作者簡介

Dinh Dũng is professor of applied mathematics at the Vietnam National University, Hanoi. He graduated from the Lomonosov Moscow State University in 1975 (former Soviet Union). There, he received the Ph.D. degree in Mathematics in 1979, and the Dr.Sc. degree in Mathematics in 1985. His research fields are approximation theory and numerical analysis. His recent research interests include computational uncertainty quantification for PDEs with random inputs, and high-dimensional problems: computation complexity, hyperbolic approximation, sampling recovery, numerical weighted integration, deep ReLU network approximation. He delivered plenary talks at the Third Asian Mathematical Conference, October 23-27, 2000, in Manila (Philippines), and at the Tenth Vietnam Mathematical Congress, August 08-12, 2023, Da Nang (Vietnam). He was awarded the Ta Quang Buu Prize for excellent achievement in Information and Computer Sciences in 2015.

Van Kien Nguyen is currently a lecturer in the Department of Mathematical Analysis, University of Transport and Communications, Hanoi, Vietnam. He graduated from University of Science, Vietnam National University Hanoi. He obtained his PhD in Mathematics from Friedrich Schiller University Jena, Germany. His areas of interest are function spaces, approximation theory, and numerical analysis.

Christoph Schwab is professor of mathematics at the Seminar for Applied Mathematics at ETH Zurich. His areas of research encompass Numerical Analysis of Partial Differential Equations, in particular Finite- and Boundary Element Methods, and the mathematical investigation of numerical methods for high-dimensional PDEs, with emphasis on forward and Bayesian inverse problems in numerical Uncertainty Quantification. His results are published in numerous articles in major journals in applied and computational mathematics. He was speaker at ICM2002, and PI of an ERC advanced grant.

Jakob Zech currently holds a position as Assistant Professor for Machine Learning in Scientific Computing at Heidelberg University. His academic path started at TU Wien, where he earned his Bachelor's degree in 2012. He then continued his studies at ETH Zurich, completing his Masters in 2014 and his PhD in Mathematics in 2018. His PhD research revolved around the approximation of high-dimensional parametric PDEs. Upon obtaining his PhD, he received the Early PostdocMobility fellowship from the Swiss National Science Foundation and spent a year as a Postdoc at the Massachusetts Institute of Technology in 2019. Subsequently he joined Heidelberg University in April 2020. His research interests include high-dimensional approximation, deep learning theory, statistical inference, uncertainty quantification, and numerics of PDEs, which resulted in numerous publications in top-tier academic journals.

作者簡介(中文翻譯)

Dinh Dũng是越南國立大學河內分校應用數學教授。他於1975年畢業於莫斯科國立大學（前蘇聯），並於1979年獲得數學博士學位，1985年獲得數學博士學位。他的研究領域包括逼近理論和數值分析。他最近的研究興趣包括用於具有隨機輸入的偏微分方程的計算不確定性量化，以及高維問題：計算複雜性，双曲逼近，抽樣恢復，數值加權積分，深度ReLU網絡逼近。他在2000年10月23日至27日在菲律賓馬尼拉舉行的第三屆亞洲數學會議和2023年8月8日至12日在越南峴港舉行的第十屆越南數學大會上發表了主題演講。他於2015年獲得了塔·奎昂·布獎，以表彰他在信息和計算機科學方面的優秀成就。

Van Kien Nguyen目前是越南河內交通運輸大學數學分析系的講師。他畢業於越南國立大學河內分校理學院，並在德國耶拿弗里德里希·席勒大學獲得數學博士學位。他的研究領域包括函數空間、逼近理論和數值分析。

Christoph Schwab是瑞士聯邦理工學院應用數學研究所的數學教授。他的研究領域包括偏微分方程的數值分析，特別是有限元和邊界元方法，以及高維PDE的數值方法的數學研究，重點放在數值不確定性量化中的正向和貝葉斯反問題上。他的研究成果發表在應用和計算數學的主要期刊上。他曾是2002年國際數學家大會的演講者，並擔任ERC高級研究獎的項目負責人。

Jakob Zech目前在海德堡大學擔任科學計算機機器學習助理教授。他的學術之路始於維也納理工大學，2012年獲得學士學位。然後他在瑞士聯邦理工學院繼續學業，2014年獲得碩士學位，2018年獲得數學博士學位。他的博士研究圍繞高維參數PDE的逼近。在獲得博士學位後，他獲得了瑞士國家科學基金會的早期博士後流動獎學金，並於2019年在麻省理工學院擔任博士後研究員一年。隨後，他於2020年4月加入海德堡大學。他的研究興趣包括高維逼近、深度學習理論、統計推斷、不確定性量化和PDE的數值方法，並在頂級學術期刊上發表了許多論文。

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