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Open AccessFeature PaperArticle

Contrast-Independent, Partially-Explicit Time Discretizations for Nonlinear Multiscale Problems

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Mathematics 2021, 9(23), 3000; https://doi.org/10.3390/math9233000 - 23 Nov 2021

Cited by 6 | Viewed by 1011

Abstract

This work continues a line of work on developing partially explicit methods for multiscale problems. In our previous works, we considered linear multiscale problems where the spatial heterogeneities are at the subgrid level and are not resolved. In these works, we have introduced [...] Read more.

This work continues a line of work on developing partially explicit methods for multiscale problems. In our previous works, we considered linear multiscale problems where the spatial heterogeneities are at the subgrid level and are not resolved. In these works, we have introduced contrast-independent, partially explicit time discretizations for linear equations. The contrast-independent, partially explicit time discretization divides the spatial space into two components: contrast dependent (fast) and contrast independent (slow) spaces defined via multiscale space decomposition. Following this decomposition, temporal splitting was proposed, which treats fast components implicitly and slow components explicitly. The space decomposition and temporal splitting are chosen such that they guarantees stability, and we formulated a condition for the time stepping. This condition was formulated as a condition on slow spaces. In this paper, we extend this approach to nonlinear problems. We propose a splitting approach and derive a condition that guarantees stability. This condition requires some type of contrast-independent spaces for slow components of the solution. We present numerical results and show that the proposed methods provide results similar to implicit methods with a time step that is independent of the contrast. Full article

(This article belongs to the Special Issue Computational Mathematics, Algorithms, and Data Processing)

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Open AccessArticle

Learning Algorithms for Coarsening Uncertainty Space and Applications to Multiscale Simulations

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Mathematics 2020, 8(5), 720; https://doi.org/10.3390/math8050720 - 04 May 2020

Cited by 7 | Viewed by 1788

Abstract

In this paper, we investigate and design multiscale simulations for stochastic multiscale PDEs. As for the space, we consider a coarse grid and a known multiscale method, the generalized multiscale finite element method (GMsFEM). In order to obtain a small dimensional representation of [...] Read more.

In this paper, we investigate and design multiscale simulations for stochastic multiscale PDEs. As for the space, we consider a coarse grid and a known multiscale method, the generalized multiscale finite element method (GMsFEM). In order to obtain a small dimensional representation of the solution in each coarse block, the uncertainty space needs to be partitioned (coarsened). This coarsenining collects realizations that provide similar multiscale features as outlined in GMsFEM (or other method of choice). This step is known to be computationally demanding as it requires many local solves and clustering based on them. In this work, we take a different approach and learn coarsening the uncertainty space. Our methods use deep learning techniques in identifying clusters (coarsening) in the uncertainty space. We use convolutional neural networks combined with some techniques in adversary neural networks. We define appropriate loss functions in the proposed neural networks, where the loss function is composed of several parts that includes terms related to clusters and reconstruction of basis functions. We present numerical results for channelized permeability fields in the examples of flows in porous media. Full article

(This article belongs to the Special Issue Computational Mathematics, Algorithms, and Data Processing)

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Open AccessArticle

Angular Correlation Using Rogers-Szegő-Chaos

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Christine Schmid

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Kyle J. DeMars

Mathematics 2020, 8(2), 171; https://doi.org/10.3390/math8020171 - 01 Feb 2020

Cited by 2 | Viewed by 2083

Abstract

Polynomial chaos expresses a probability density function (pdf) as a linear combination of basis polynomials. If the density and basis polynomials are over the same field, any set of basis polynomials can describe the pdf; however, the most logical choice of polynomials is [...] Read more.

Polynomial chaos expresses a probability density function (pdf) as a linear combination of basis polynomials. If the density and basis polynomials are over the same field, any set of basis polynomials can describe the pdf; however, the most logical choice of polynomials is the family that is orthogonal with respect to the pdf. This problem is well-studied over the field of real numbers and has been shown to be valid for the complex unit circle in one dimension. The current framework for circular polynomial chaos is extended to multiple angular dimensions with the inclusion of correlation terms. Uncertainty propagation of heading angle and angular velocity is investigated using polynomial chaos and compared against Monte Carlo simulation. Full article

(This article belongs to the Special Issue Computational Mathematics, Algorithms, and Data Processing)

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Open AccessArticle

Bivariate Thiele-Like Rational Interpolation Continued Fractions with Parameters Based on Virtual Points

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Mathematics 2020, 8(1), 71; https://doi.org/10.3390/math8010071 - 02 Jan 2020

Cited by 8 | Viewed by 2127

Abstract

The interpolation of Thiele-type continued fractions is thought of as the traditional rational interpolation and plays a significant role in numerical analysis and image interpolation. Different to the classical method, a novel type of bivariate Thiele-like rational interpolation continued fractions with parameters is [...] Read more.

The interpolation of Thiele-type continued fractions is thought of as the traditional rational interpolation and plays a significant role in numerical analysis and image interpolation. Different to the classical method, a novel type of bivariate Thiele-like rational interpolation continued fractions with parameters is proposed to efficiently address the interpolation problem. Firstly, the multiplicity of the points is adjusted strategically. Secondly, bivariate Thiele-like rational interpolation continued fractions with parameters is developed. We also discuss the interpolant algorithm, theorem, and dual interpolation of the proposed interpolation method. Many interpolation functions can be gained through adjusting the parameter, which is flexible and convenient. We also demonstrate that the novel interpolation function can deal with the interpolation problems that inverse differences do not exist or that there are unattainable points appearing in classical Thiele-type continued fractions interpolation. Through the selection of proper parameters, the value of the interpolation function can be changed at any point in the interpolant region under unaltered interpolant data. Numerical examples are given to show that the developed methods achieve state-of-the-art performance. Full article

(This article belongs to the Special Issue Computational Mathematics, Algorithms, and Data Processing)

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Open AccessArticle

Mixed Generalized Multiscale Finite Element Method for Darcy-Forchheimer Model

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Mathematics 2019, 7(12), 1212; https://doi.org/10.3390/math7121212 - 10 Dec 2019

Cited by 4 | Viewed by 2011

Abstract

In this paper, the solution of the Darcy-Forchheimer model in high contrast heterogeneous media is studied. This problem is solved by a mixed finite element method (MFEM) on a fine grid (the reference solution), where the pressure is approximated by piecewise constant elements; [...] Read more.

In this paper, the solution of the Darcy-Forchheimer model in high contrast heterogeneous media is studied. This problem is solved by a mixed finite element method (MFEM) on a fine grid (the reference solution), where the pressure is approximated by piecewise constant elements; meanwhile, the velocity is discretized by the lowest order Raviart-Thomas elements. The solution on a coarse grid is performed by using the mixed generalized multiscale finite element method (mixed GMsFEM). The nonlinear equation can be solved by the well known Picard iteration. Several numerical experiments are presented in a two-dimensional heterogeneous domain to show the good applicability of the proposed multiscale method. Full article

(This article belongs to the Special Issue Computational Mathematics, Algorithms, and Data Processing)

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Open AccessArticle

Scattered Data Interpolation and Approximation with Truncated Exponential Radial Basis Function

by

Qiuyan Xu

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Zhiyong Liu

Mathematics 2019, 7(11), 1101; https://doi.org/10.3390/math7111101 - 14 Nov 2019

Cited by 5 | Viewed by 2343

Abstract

Surface modeling is closely related to interpolation and approximation by using level set methods, radial basis functions methods, and moving least squares methods. Although radial basis functions with global support have a very good approximation effect, this is often accompanied by an ill-conditioned [...] Read more.

Surface modeling is closely related to interpolation and approximation by using level set methods, radial basis functions methods, and moving least squares methods. Although radial basis functions with global support have a very good approximation effect, this is often accompanied by an ill-conditioned algebraic system. The exceedingly large condition number of the discrete matrix makes the numerical calculation time consuming. The paper introduces a truncated exponential function, which is radial on arbitrary n-dimensional space

R^{n}

and has compact support. The truncated exponential radial function is proven strictly positive definite on

R^{n}

while internal parameter l satisfies

l \geq ⌊ \frac{n}{2} ⌋ + 1

. The error estimates for scattered data interpolation are obtained via the native space approach. To confirm the efficiency of the truncated exponential radial function approximation, the single level interpolation and multilevel interpolation are used for surface modeling, respectively. Full article

(This article belongs to the Special Issue Computational Mathematics, Algorithms, and Data Processing)

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Open AccessEditor’s ChoiceArticle

Universal Function Approximation by Deep Neural Nets with Bounded Width and ReLU Activations

by

Boris Hanin

Mathematics 2019, 7(10), 992; https://doi.org/10.3390/math7100992 - 18 Oct 2019

Cited by 116 | Viewed by 9484

Abstract

This article concerns the expressive power of depth in neural nets with ReLU activations and a bounded width. We are particularly interested in the following questions: What is the minimal width

w_{\min} (d)

so that ReLU nets of width [...] Read more.

This article concerns the expressive power of depth in neural nets with ReLU activations and a bounded width. We are particularly interested in the following questions: What is the minimal width

w_{\min} (d)

so that ReLU nets of width

w_{\min} (d)

(and arbitrary depth) can approximate any continuous function on the unit cube

{[0, 1]}^{d}

arbitrarily well? For ReLU nets near this minimal width, what can one say about the depth necessary to approximate a given function? We obtain an essentially complete answer to these questions for convex functions. Our approach is based on the observation that, due to the convexity of the ReLU activation, ReLU nets are particularly well suited to represent convex functions. In particular, we prove that ReLU nets with width

d + 1

can approximate any continuous convex function of d variables arbitrarily well. These results then give quantitative depth estimates for the rate of approximation of any continuous scalar function on the d-dimensional cube

{[0, 1]}^{d}

by ReLU nets with width

d + 3

. Full article

(This article belongs to the Special Issue Computational Mathematics, Algorithms, and Data Processing)

Open AccessArticle

Prediction of Discretization of GMsFEM Using Deep Learning

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Mathematics 2019, 7(5), 412; https://doi.org/10.3390/math7050412 - 08 May 2019

Cited by 6 | Viewed by 3438

Abstract

In this paper, we propose a deep-learning-based approach to a class of multiscale problems. The generalized multiscale finite element method (GMsFEM) has been proven successful as a model reduction technique of flow problems in heterogeneous and high-contrast porous media. The key ingredients of [...] Read more.

In this paper, we propose a deep-learning-based approach to a class of multiscale problems. The generalized multiscale finite element method (GMsFEM) has been proven successful as a model reduction technique of flow problems in heterogeneous and high-contrast porous media. The key ingredients of GMsFEM include mutlsicale basis functions and coarse-scale parameters, which are obtained from solving local problems in each coarse neighborhood. Given a fixed medium, these quantities are precomputed by solving local problems in an offline stage, and result in a reduced-order model. However, these quantities have to be re-computed in case of varying media (various permeability fields). The objective of our work is to use deep learning techniques to mimic the nonlinear relation between the permeability field and the GMsFEM discretizations, and use neural networks to perform fast computation of GMsFEM ingredients repeatedly for a class of media. We provide numerical experiments to investigate the predictive power of neural networks and the usefulness of the resultant multiscale model in solving channelized porous media flow problems. Full article

(This article belongs to the Special Issue Computational Mathematics, Algorithms, and Data Processing)

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Open AccessFeature PaperEditor’s ChoiceArticle

The Multivariate Theory of Connections

by

Daniele Mortari

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Carl Leake

Mathematics 2019, 7(3), 296; https://doi.org/10.3390/math7030296 - 22 Mar 2019

Cited by 29 | Viewed by 3929

Abstract

This paper extends the univariate Theory of Connections, introduced in (Mortari, 2017), to the multivariate case on rectangular domains with detailed attention to the bivariate case. In particular, it generalizes the bivariate Coons surface, introduced by (Coons, 1984), by providing analytical expressions, called [...] Read more.

This paper extends the univariate Theory of Connections, introduced in (Mortari, 2017), to the multivariate case on rectangular domains with detailed attention to the bivariate case. In particular, it generalizes the bivariate Coons surface, introduced by (Coons, 1984), by providing analytical expressions, called constrained expressions, representing all possible surfaces with assigned boundary constraints in terms of functions and arbitrary-order derivatives. In two dimensions, these expressions, which contain a freely chosen function,

g (x, y)

, satisfy all constraints no matter what the

g (x, y)

is. The boundary constraints considered in this article are Dirichlet, Neumann, and any combinations of them. Although the focus of this article is on two-dimensional spaces, the final section introduces the Multivariate Theory of Connections, validated by mathematical proof. This represents the multivariate extension of the Theory of Connections subject to arbitrary-order derivative constraints in rectangular domains. The main task of this paper is to provide an analytical procedure to obtain constrained expressions in any space that can be used to transform constrained problems into unconstrained problems. This theory is proposed mainly to better solve PDE and stochastic differential equations. Full article

(This article belongs to the Special Issue Computational Mathematics, Algorithms, and Data Processing)

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Review

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Open AccessReview

Trigonometrically-Fitted Methods: A Review

by

Changbum Chun

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Beny Neta

Mathematics 2019, 7(12), 1197; https://doi.org/10.3390/math7121197 - 06 Dec 2019

Cited by 1 | Viewed by 1907

Abstract

Numerical methods for the solution of ordinary differential equations are based on polynomial interpolation. In 1952, Brock and Murray have suggested exponentials for the case that the solution is known to be of exponential type. In 1961, Gautschi came up with the idea [...] Read more.

Numerical methods for the solution of ordinary differential equations are based on polynomial interpolation. In 1952, Brock and Murray have suggested exponentials for the case that the solution is known to be of exponential type. In 1961, Gautschi came up with the idea of using information on the frequency of a solution to modify linear multistep methods by allowing the coefficients to depend on the frequency. Thus the methods integrate exactly appropriate trigonometric polynomials. This was done for both first order systems and second order initial value problems. Gautschi concluded that “the error reduction is not very substantial unless” the frequency estimate is close enough. As a result, no other work was done in this direction until 1984 when Neta and Ford showed that “Nyström’s and Milne-Simpson’s type methods for systems of first order initial value problems are not sensitive to changes in frequency”. This opened the flood gates and since then there have been many papers on the subject. Full article

(This article belongs to the Special Issue Computational Mathematics, Algorithms, and Data Processing)

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