ETNA - Electronic Transactions on Numerical Analysis最新文献

Characterizations of adjoint Sobolev embedding operators with applications in inverse problems 伴随Sobolev嵌入算子的性质及其在逆问题中的应用

ETNA - Electronic Transactions on Numerical Analysis Pub Date : 2022-02-10 DOI: 10.1553/etna_vol59s116

Simon Hubmer, Ekaterina Sherina, R. Ramlau

{"title":"Characterizations of adjoint Sobolev embedding operators with applications in inverse problems","authors":"Simon Hubmer, Ekaterina Sherina, R. Ramlau","doi":"10.1553/etna_vol59s116","DOIUrl":"https://doi.org/10.1553/etna_vol59s116","url":null,"abstract":"We consider the Sobolev embedding operator $E_s : H^s(Omega) to L_2(Omega)$ and its role in the solution of inverse problems. In particular, we collect various properties and investigate different characterizations of its adjoint operator $E_s^*$, which is a common component in both iterative and variational regularization methods. These include variational representations and connections to boundary value problems, Fourier and wavelet representations, as well as connections to spatial filters. Moreover, we consider characterizations in terms of Fourier series, singular value decompositions and frame decompositions, as well as representations in finite dimensional settings. While many of these results are already known to researchers from different fields, a detailed and general overview or reference work containing rigorous mathematical proofs is still missing. Hence, in this paper we aim to fill this gap by collecting, introducing and generalizing a large number of characterizations of $E_s^*$ and discuss their use in regularization methods for solving inverse problems. The resulting compilation can serve both as a reference as well as a useful guide for its efficient numerical implementation in practice.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130219619","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Nonequispaced Fast Fourier Transform Boost for the Sinkhorn Algorithm Sinkhorn算法的非均衡快速傅里叶变换增强

ETNA - Electronic Transactions on Numerical Analysis Pub Date : 2022-01-19 DOI: 10.1553/etna_vol58s289

Rajmadan Lakshmanan, A. Pichler, D. Potts

引用次数: 5

Sparse mixture models inspired by ANOVA decompositions 基于方差分析分解的稀疏混合模型

ETNA - Electronic Transactions on Numerical Analysis Pub Date : 2021-05-31 DOI: 10.1553/etna_vol55s142

J. Hertrich, F. Ba, G. Steidl

引用次数: 1

On the tangential cone condition for electrical impedance tomography 电阻抗层析成像的切锥条件

ETNA - Electronic Transactions on Numerical Analysis Pub Date : 2021-05-06 DOI: 10.1553/etna_vol57s17

S. Kindermann

引用次数: 4

Computing the matrix fractional power with the double exponential formula 用双指数公式计算矩阵的分数次幂

ETNA - Electronic Transactions on Numerical Analysis Pub Date : 2020-12-03 DOI: 10.1553/etna_vol54s558

Fuminori Tatsuoka, T. Sogabe, Y. Miyatake, T. Kemmochi, Shaoliang Zhang

{"title":"Computing the matrix fractional power with the double exponential formula","authors":"Fuminori Tatsuoka, T. Sogabe, Y. Miyatake, T. Kemmochi, Shaoliang Zhang","doi":"10.1553/etna_vol54s558","DOIUrl":"https://doi.org/10.1553/etna_vol54s558","url":null,"abstract":"Two quadrature-based algorithms for computing the matrix fractional power $A^alpha$ are presented in this paper. These algorithms are based on the double exponential (DE) formula, which is well-known for its effectiveness in computing improper integrals as well as in treating nearly arbitrary endpoint singularities. The DE formula transforms a given integral into another integral that is suited for the trapezoidal rule; in this process, the integral interval is transformed to the infinite interval. Therefore, it is necessary to truncate the infinite interval into an appropriate finite interval. In this paper, a truncation method, which is based on a truncation error analysis specialized to the computation of $A^alpha$, is proposed. Then, two algorithms are presented -- one computes $A^alpha$ with a fixed number of abscissas, and the other computes $A^alpha$ adaptively. Subsequently, the convergence rate of the DE formula for Hermitian positive definite matrices is analyzed. The convergence rate analysis shows that the DE formula converges faster than the Gaussian quadrature when $A$ is ill-conditioned and $alpha$ is a non-unit fraction. Numerical results show that our algorithms achieved the required accuracy and were faster than other algorithms in several situations.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"29 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120915523","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 3

A comparison of reduced-order modeling approaches using artificial neural networks for PDEs with bifurcating solutions 具有分岔解的偏微分方程的人工神经网络降阶建模方法的比较

ETNA - Electronic Transactions on Numerical Analysis Pub Date : 2020-10-14 DOI: 10.1553/etna_vol56s52

M. Hess, A. Quaini, G. Rozza

引用次数: 4

Analysis of BDDC algorithms for Stokes problems with hybridizable discontinuous Galerkin discretizations 杂化不连续Galerkin离散Stokes问题的BDDC算法分析

ETNA - Electronic Transactions on Numerical Analysis Pub Date : 2020-09-28 DOI: 10.1553/etna_vol52s553

Xuemin Tu, Bin Wang, Jinjin Zhang

引用次数: 4

Estimating the time-dependent contact rate of SIR and SEIR models in mathematical epidemiology using physics-informed neural networks 利用物理信息神经网络估计数学流行病学中SIR和SEIR模型的时变接触率

ETNA - Electronic Transactions on Numerical Analysis Pub Date : 2020-09-22 DOI: 10.1553/etna_vol56s1

Viktor Grimm, Alexander Heinlein, A. Klawonn, M. Lanser, J. Weber

{"title":"Estimating the time-dependent contact rate of SIR and SEIR models in mathematical epidemiology using physics-informed neural networks","authors":"Viktor Grimm, Alexander Heinlein, A. Klawonn, M. Lanser, J. Weber","doi":"10.1553/etna_vol56s1","DOIUrl":"https://doi.org/10.1553/etna_vol56s1","url":null,"abstract":"The course of an epidemic can be often successfully described mathematically using compartment models. These models result in a system of ordinary diﬀerential equations. Two well-known examples are the SIR and the SEIR models. The transition rates between the different compartments are deﬁned by certain parameters which are speciﬁc for the respective virus. Often, these parameters can be taken from the literature or can be determined from statistics. However, the contact rate or the related eﬀective reproduction number are in general not constant and thus cannot easily be determined. Here, a new machine learning approach based on physics-informed neural networks is presented that can learn the contact rate from given data for the dynamical systems given by the SIR and SEIR models. The new method generalizes an already known approach for the identiﬁcation of constant parameters to the variable or time-dependent case. After introducing the new method, it is tested for synthetic data generated by the numerical solution of SIR and SEIR models. Here, the case of exact and perturbed data is considered. In all cases, the contact rate can be learned very satisfactorily. Finally, the SEIR model in combination with physics-informed neural networks is used to learn the contact rate for COVID-19 data given by the course of the epidemic in Germany. The simulation of the number of infected individuals over the course of the epidemic, using the learned contact rate, is very promising.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"80 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116351337","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 8

On the regularization effect of stochastic gradient descent applied to least-squares 应用于最小二乘的随机梯度下降的正则化效果

ETNA - Electronic Transactions on Numerical Analysis Pub Date : 2020-07-27 DOI: 10.1553/etna_vol54s610

S. Steinerberger

引用次数: 5

Algebraic analysis of two-level multigrid methods for edge elements 边缘元两级多重网格法的代数分析

ETNA - Electronic Transactions on Numerical Analysis Pub Date : 2019-11-13 DOI: 10.1553/ETNA_VOL51S387

Artem Napov, R. Perrussel

引用次数: 0