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An Introduction to Computational Stochastic PDEs
TLDR
This book offers graduate students and researchers powerful tools for understanding uncertainty quantification for risk analysis and theory is developed in tandem with state-of-the art computational methods through worked examples, exercises, theorems and proofs.Abstract:
Part I. Deterministic Differential Equations: 1. Linear analysis 2. Galerkin approximation and finite elements 3. Time-dependent differential equations Part II. Stochastic Processes and Random Fields: 4. Probability theory 5. Stochastic processes 6. Stationary Gaussian processes 7. Random fields Part III. Stochastic Differential Equations: 8. Stochastic ordinary differential equations (SODEs) 9. Elliptic PDEs with random data 10. Semilinear stochastic PDEs.read more
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Model Reduction and Neural Networks for Parametric PDEs
TL;DR: In this paper, a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces is developed, motivated by the recent successes of neural networks and deep learning, in combination with ideas from model reduction.
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Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients
Ivan G. Graham,Frances Y. Kuo,James A. Nichols,Robert Scheichl,Christoph Schwab,Ian H. Sloan +5 more
TL;DR: A rigorous error analysis for methods constructed from standard continuous and piecewise linear finite element approximation in physical space, truncated Karhunen–Loève expansion for computing realizations of a and lattice-based quasi-Monte Carlo quadrature rules for computing integrals over parameter space which define the expected values.
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On uncertainty quantification in hydrogeology and hydrogeophysics
TL;DR: This review aims at helping hydrogeologists and hydrogeophysicists to identify suitable approaches for UQ that can be applied and further developed to their specific needs and considers hydrogeophysical inversion, where petrophysical uncertainty is often ignored leading to overconfident parameter estimation.
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Low-rank solvers for unsteady Stokes–Brinkman optimal control problem with random data
TL;DR: A new alternating iterative tensor method for an efficient reduction of this system by the low-rank Tensor Train representation and proposes and analyzes a robust Schur complement-based preconditioner for the solution of the saddle-point system.
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Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networks
TL;DR: This work finds strong support for the hypothesis that approximation-theoretical effects heavily influence the practical behavior of learning problems in numerical analysis.
References
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Book
Partial Differential Equations
TL;DR: In this paper, the authors present a theory for linear PDEs: Sobolev spaces Second-order elliptic equations Linear evolution equations, Hamilton-Jacobi equations and systems of conservation laws.
Book
Semigroups of Linear Operators and Applications to Partial Differential Equations
TL;DR: In this article, the authors considered the generation and representation of a generator of C0-Semigroups of Bounded Linear Operators and derived the following properties: 1.1 Generation and Representation.
Book
Real and complex analysis
TL;DR: In this paper, the Riesz representation theorem is used to describe the regularity properties of Borel measures and their relation to the Radon-Nikodym theorem of continuous functions.
Book
A treatise on the theory of Bessel functions
TL;DR: The tabulation of Bessel functions can be found in this paper, where the authors present a comprehensive survey of the Bessel coefficients before and after 1826, as well as their extensions.