25,829 results
Search Results
2. Remarks on a recent paper titled: 'On the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings in Banach spaces'
- Author
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Charles E. Chidume
- Subjects
Pure mathematics ,Smoothness (probability theory) ,Applied Mathematics ,lcsh:Mathematics ,Banach space ,Hilbert space ,Regular polygon ,010103 numerical & computational mathematics ,Fixed point ,lcsh:QA1-939 ,01 natural sciences ,Opial property ,010101 applied mathematics ,symbols.namesake ,Accretive ,Uniformly smooth ,Common fixed point ,symbols ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Constant (mathematics) ,Analysis ,Mathematics - Abstract
In a recently published theorem on the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings, Tang et al. (J. Inequal. Appl. 2015:305, 2015) studied a uniformly convex and 2-uniformly smooth real Banach space with the Opial property and best smoothness constant κ satisfying the condition $0 0 < κ < 1 2 , as a real Banach space more general than Hilbert spaces. A well-known example of a uniformly convex and 2-uniformly smooth real Banach space with the Opial property is $E=l_{p}$ E = l p , $2\leq p 2 ≤ p < ∞ . It is shown in this paper that, if κ is the best smoothness constant of E and satisfies the condition $0 0 < κ ≤ 1 2 , then E is necessarily $l_{2}$ l 2 , a real Hilbert space. Furthermore, some important remarks concerning the proof of this theorem are presented.
- Published
- 2021
3. A Note on the Paper 'Optimality Conditions for Vector Optimization Problems with Difference of Convex Maps'
- Author
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Farid Bozorgnia and Allahkaram Shafie
- Subjects
Discrete mathematics ,021103 operations research ,Control and Optimization ,Applied Mathematics ,0211 other engineering and technologies ,Regular polygon ,010103 numerical & computational mathematics ,02 engineering and technology ,Management Science and Operations Research ,Mathematical proof ,01 natural sciences ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Vector optimization ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Metrization theorem ,Theory of computation ,FOS: Mathematics ,0101 mathematics ,Mathematics ,Vector space ,Counterexample - Abstract
In this work, some counterexamples are given to refute some results reported in the paper by Guo and Li [8] (J Optim Theory Appl 162,(2014), 821-844). We correct the faulty in some of their theorems and we present alternative proofs. Moreover, we extend the definition of approximately pseudo-dissipative in the setting of metrizable topological vector spaces.
- Published
- 2019
4. Learning regularization parameters of inverse problems via deep neural networks:Paper
- Author
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Julianne Chung, Matthias Chung, and Babak Maboudi Afkham
- Subjects
FOS: Computer and information sciences ,Computer Science - Machine Learning ,Bilevel optimization ,010103 numerical & computational mathematics ,01 natural sciences ,Regularization (mathematics) ,Theoretical Computer Science ,Machine Learning (cs.LG) ,Bayes' theorem ,Design objective ,Deep neural networks ,Regularization ,FOS: Mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Representation (mathematics) ,Mathematical Physics ,Mathematics ,Applied Mathematics ,Supervised learning ,Deep learning ,Numerical Analysis (math.NA) ,Inverse problem ,Computer Science Applications ,010101 applied mathematics ,Optimal experimental design ,Signal Processing ,Minification ,Algorithm ,Hyperparameter selection - Abstract
In this work, we describe a new approach that uses deep neural networks (DNN) to obtain regularization parameters for solving inverse problems. We consider a supervised learning approach, where a network is trained to approximate the mapping from observation data to regularization parameters. Once the network is trained, regularization parameters for newly obtained data can be computed by efficient forward propagation of the DNN. We show that a wide variety of regularization functionals, forward models, and noise models may be considered. The network-obtained regularization parameters can be computed more efficiently and may even lead to more accurate solutions compared to existing regularization parameter selection methods. We emphasize that the key advantage of using DNNs for learning regularization parameters, compared to previous works on learning via optimal experimental design or empirical Bayes risk minimization, is greater generalizability. That is, rather than computing one set of parameters that is optimal with respect to one particular design objective, DNN-computed regularization parameters are tailored to the specific features or properties of the newly observed data. Thus, our approach may better handle cases where the observation is not a close representation of the training set. Furthermore, we avoid the need for expensive and challenging bilevel optimization methods as utilized in other existing training approaches. Numerical results demonstrate the potential of using DNNs to learn regularization parameters., 27 pages, 16 figures
- Published
- 2021
5. Comment on the paper 'On conservation laws by Lie symmetry analysis for (2+1)-dimensional Bogoyavlensky–Konopelchenko equation in wave propagation' by S. Saha Ray
- Author
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Muhammad Alim Abdulwahhab
- Subjects
Conservation law ,Wave propagation ,010102 general mathematics ,One-dimensional space ,010103 numerical & computational mathematics ,System of linear equations ,01 natural sciences ,Symmetry (physics) ,Computational Mathematics ,symbols.namesake ,Computational Theory and Mathematics ,Modeling and Simulation ,symbols ,0101 mathematics ,Noether's theorem ,Mathematics ,Mathematical physics - Abstract
In a recent paper referred to in the title, the author used the concept of quasi self-adjointness to obtain conservation laws for a system of the ( 2 + 1 ) -dimensional Bogoyavlensky–Konopelchenko equation. Apart from the adjoint system of equations, all the results on the quasi self-adjointness and the subsequent conservation laws obtained are inaccurate. In this comment, we clarify these inaccuracies and also generate conservation laws for a potential form of the underlying equation through Noether theorem.
- Published
- 2018
6. Comment on the paper 'Heat and mass transfer in unsteady MHD slip flow of Casson fluid over a moving wedge embedded in a porous medium in the presence of chemical reaction: Numerical solutions using Keller‐Box method, Imran Ullah, Ilyas Khan, Sharidan Shafie, Numerical Methods for Partial Differential Equations , November 2017, https://doi.org/10.1002/num.22221'
- Author
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Asterios Pantokratoras
- Subjects
Numerical Analysis ,business.product_category ,Applied Mathematics ,010103 numerical & computational mathematics ,Mechanics ,01 natural sciences ,Chemical reaction ,Wedge (mechanical device) ,010101 applied mathematics ,Computational Mathematics ,Ullah ,Mass transfer ,Slip flow ,Casson fluid ,0101 mathematics ,Magnetohydrodynamics ,business ,Porous medium ,Analysis ,Mathematics - Published
- 2018
7. A Note on the Paper 'The Algebraic Structure of the Arbitrary-Order Cone'
- Author
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Yen Chi Roger Lin, Xin-He Miao, and Jein Shan Chen
- Subjects
Pure mathematics ,021103 operations research ,Control and Optimization ,Algebraic structure ,Applied Mathematics ,0211 other engineering and technologies ,Structure (category theory) ,Order (ring theory) ,010103 numerical & computational mathematics ,02 engineering and technology ,Management Science and Operations Research ,01 natural sciences ,Cone (formal languages) ,Combinatorics ,Operator (computer programming) ,Product (mathematics) ,Light cone ,0101 mathematics ,Mathematics ,Counterexample - Abstract
In this short paper, we look into a conclusion drawn by Alzalg (J Optim Theory Appl 169:32---49, 2016). We think the conclusion drawn in the paper is incorrect by pointing out three things. First, we provide a counterexample that the proposed inner product does not satisfy bilinearity. Secondly, we offer an argument why a pth-order cone cannot be self-dual under any reasonable inner product structure on $$\mathbb {R}^n$$Rn. Thirdly, even under the assumption that all elements operator commute, the inner product becomes an official inner product and the arbitrary-order cone can be shown as a symmetric cone, we think this condition is still unreasonable and very stringent so that the result can only be applied to very few cases.
- Published
- 2017
8. On the comparison of inventory replenishment policies with time-varying stochastic demand for the paper industry
- Author
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David Ciprés, Lorena Polo, and David Escuín
- Subjects
0209 industrial biotechnology ,Operations research ,Build to order ,Applied Mathematics ,Supply chain ,Time horizon ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Computational Mathematics ,020901 industrial engineering & automation ,Production planning ,Service level ,Vendor-managed inventory ,Inventory theory ,Perpetual inventory ,0101 mathematics ,Mathematics - Abstract
The aim of this paper is the development of a mathematical model to compute the optimal inventory mix to face stochastic demand at minimum cost in a two-level supply chain. The paper addresses a multi-product dynamic lot-sizing problem under stochastic demand subject to capacity and service level constraints. This model is executed to compare a Make To Order (MTO) strategy to a Vendor Managed Inventory (VMI) partnership between the supplier and their customers. Both strategies provide the demand order to be produced. A schedule of production orders is determined over the planning horizon in order to minimize the inventory holding costs of the supply chain, taking into consideration that the supplier is also responsible of initiating the replenishment orders and deliveries of their customers according to the VMI partnership. The simulation model is illustrated empirically using a real case study: a paper manufacturing company that pursues to improve customer service level and supply chain inventory costs through a proper production planning of their paper machines and a suitable VMI order replenishment schedule.
- Published
- 2017
9. Estimates on the Minimal Stabilizing Horizon Length in Model Predictive Control for the Fokker-Planck Equation**This work was supported by the DFG project Model Predictive Control for the Fokker-Planck Equation, GR 1569/15-1. The paper was written while the second author was visiting the University of Newcastle, Australia
- Author
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Lars Grüne and Arthur Fleig
- Subjects
0209 industrial biotechnology ,Partial differential equation ,Series (mathematics) ,Stochastic process ,010103 numerical & computational mathematics ,02 engineering and technology ,Optimal control ,01 natural sciences ,Controllability ,Model predictive control ,020901 industrial engineering & automation ,Control and Systems Engineering ,Control theory ,Stability theory ,Applied mathematics ,Fokker–Planck equation ,0101 mathematics ,Mathematics - Abstract
In a series of papers by Annunziato and Borzi, Model Predictive Control of the Fokker-Planck equation has been established as a numerically feasible way for controlling stochastic processes via their probability density functions. Numerical simulations suggest that the resulting controller yields an asymptotically stable closed loop system for optimization horizons looking only one time step into the future. In this paper we provide a formal proof of this fact for the Fokker-Planck equation corresponding to the controlled Ornstein-Uhlenbeck process using an L2 cost and control functions that are constant in space. The key step of the proof consists in the verification of an exponential controllability property with respect to the stage cost. Numerical simulations are provided to illustrate our results.
- Published
- 2016
10. Comment on the paper 'Convection from an inverted cone in a porous medium with cross-diffusion effects, F.G. Awad, P. Sibanda, S.S. Motsa, O.D. Makinde, Comput. Math. Appl. 61 (2011) 1431–1441'
- Author
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Asterios Pantokratoras
- Subjects
Convection ,Cross diffusion ,Mathematical analysis ,Thermodynamics ,010103 numerical & computational mathematics ,01 natural sciences ,010305 fluids & plasmas ,Computational Mathematics ,Computational Theory and Mathematics ,Cone (topology) ,Modeling and Simulation ,0103 physical sciences ,0101 mathematics ,Porous medium ,Mathematics - Published
- 2017
11. A note on Lavi’s paper 'A Ganzstellensatz for semi-algebraic sets and a boundedness criterion for rational functions'
- Author
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Tomasz Kowalczyk
- Subjects
Model theory ,Discrete mathematics ,Computer Science::Computer Science and Game Theory ,Algebra and Number Theory ,valuation theory ,010102 general mathematics ,010103 numerical & computational mathematics ,Valuation theory ,Rational function ,01 natural sciences ,Algebra ,model theory ,Real algebraic geometry ,real algebraic geometry ,0101 mathematics ,Algebraic number ,Mathematics - Abstract
This article is intended to indicate and discuss some errors in N. Lavi’s paper “A Ganzstellensatz for semi-algebraic sets and a boundedness criterion for rational functions”.
- Published
- 2018
12. A Note on the Paper 'Optimality Conditions for Optimistic Bilevel Programming Problem Using Convexifactors'
- Author
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Bhawna Kohli
- Subjects
TheoryofComputation_MISCELLANEOUS ,Lemma (mathematics) ,021103 operations research ,Control and Optimization ,Applied Mathematics ,0211 other engineering and technologies ,TheoryofComputation_GENERAL ,010103 numerical & computational mathematics ,02 engineering and technology ,Management Science and Operations Research ,01 natural sciences ,Bilevel optimization ,Algebra ,Bellman equation ,Theory of computation ,0101 mathematics ,GeneralLiterature_REFERENCE(e.g.,dictionaries,encyclopedias,glossaries) ,Mathematics - Abstract
This note concerns about the conclusion of a lemma of a published paper of this journal.
- Published
- 2018
13. The Asymptotic Expansion of Kummer Functions for Large Values of the a-Parameter, and Remarks on a Paper by Olver
- Author
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Hans Volkmer
- Subjects
Pure mathematics ,Logarithm ,media_common.quotation_subject ,Riemann surface ,010102 general mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,Infinity ,01 natural sciences ,Kummer's function ,symbols.namesake ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,Geometry and Topology ,0101 mathematics ,Asymptotic expansion ,Mathematical Physics ,Analysis ,media_common ,Mathematics - Abstract
It is shown that a known asymptotic expansion of the Kummer function $U(a,b,z)$ as $a$ tends to infinity is valid for $z$ on the full Riemann surface of the logarithm. A corresponding result is also proved in a more general setting considered by Olver (1956).
- Published
- 2016
14. A Note on the Paper 'The Asymptotic Behavior of the Composition of Firmly Nonexpansive Mappings'
- Author
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David Ariza-Ruiz, Adriana Nicolae, and Genaro López-Acedo
- Subjects
Discrete mathematics ,021103 operations research ,Control and Optimization ,Applied Mathematics ,0211 other engineering and technologies ,010103 numerical & computational mathematics ,02 engineering and technology ,Management Science and Operations Research ,Composition (combinatorics) ,01 natural sciences ,Theory of computation ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this note we correct an error in the paper (Ariza-Ruiz et al. in J Optim Theory Appl 167:409–429, 2015).
- Published
- 2017
15. Corrigendum to the paper 'Numerical approximation of fractional powers of regularly accretive operators'
- Author
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Andrea Bonito and Joseph E. Pasciak
- Subjects
010101 applied mathematics ,Computational Mathematics ,Numerical approximation ,Applied Mathematics ,General Mathematics ,Calculus ,Applied mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Published
- 2017
16. Remarks on a Paper by Giordano, Laforgia, and Pečarić
- Author
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Mourad E. H. Ismail
- Subjects
symbols.namesake ,Applied Mathematics ,010102 general mathematics ,symbols ,Calculus ,Point (geometry) ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Bessel function ,Analysis ,Mathematics - Abstract
We point out errors and oversights in a paper by Giordano, Laforgia, and Pecaric [3] on inequalities involving Bessel functions.
- Published
- 1997
- Full Text
- View/download PDF
17. Remarks on E. A. Rahmanov's paper 'on the asymptotics of the ratio of orthogonal polynomials'
- Author
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Paul Nevai and Attila Máté
- Subjects
Discrete mathematics ,Mathematics(all) ,Numerical Analysis ,Statement (logic) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Algebra ,Orthogonal polynomials ,0101 mathematics ,Analysis ,Mathematics ,Counterexample - Abstract
It is pointed out that the proof of the basic result of Rahmanov's paper has a serious gap. It is documented by original sources that a statement he relied on in the proof contains a misprint, and it is shown by a counterexample that this statement (with the misprint) is, in fact, false. A somewhat weaker statement is proved true.
- Published
- 1982
18. Computing isogenies between Jacobians of curves of genus 2 and 3
- Author
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Enea Milio
- Subjects
Pure mathematics ,Algebra and Number Theory ,Mathematics::Number Theory ,Applied Mathematics ,Isotropy ,Prime number ,Theta function ,010103 numerical & computational mathematics ,Paper computing ,Mathematics::Geometric Topology ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Mathematics::Algebraic Geometry ,Genus (mathematics) ,0101 mathematics ,Algebraic number ,Quotient ,Mathematics - Abstract
We present a quasi-linear algorithm to compute isogenies between Jacobians of curves of genus 2 and 3 starting from the equation of the curve and a maximal isotropic subgroup of the l-torsion, for l an odd prime number, generalizing the Velu's formula of genus 1. This work is based from the paper Computing functions on Jacobians and their quotients of Jean-Marc Couveignes and Tony Ezome. We improve their genus 2 case algorithm, generalize it for genus 3 hyperelliptic curves and introduce a way to deal with the genus 3 non-hyperelliptic case, using algebraic theta functions.
- Published
- 2019
19. Implementable tensor methods in unconstrained convex optimization
- Author
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Yurii Nesterov, UCL - SSH/LIDAM/CORE - Center for operations research and econometrics, and UCL - SSH/IMMAQ/CORE - Center for operations research and econometrics
- Subjects
tensor mehtods ,90C06 ,General Mathematics ,0211 other engineering and technologies ,65K05 ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Upper and lower bounds ,90C25 ,Worst-case complexity bounds ,High-order methods ,Tensor methods ,Tensor (intrinsic definition) ,Convergence (routing) ,Applied mathematics ,0101 mathematics ,Mathematics ,021103 operations research ,Full Length Paper ,Regular polygon ,Order (ring theory) ,Function (mathematics) ,Lower complexity bounds ,Convex optimization ,Rate of convergence ,Software - Abstract
In this paper we develop new tensor methods for unconstrained convex optimization, which solve at each iteration an auxiliary problem of minimizing convex multivariate polynomial. We analyze the simplest scheme, based on minimization of a regularized local model of the objective function, and its accelerated version obtained in the framework of estimating sequences. Their rates of convergence are compared with the worst-case lower complexity bounds for corresponding problem classes. Finally, for the third-order methods, we suggest an efficient technique for solving the auxiliary problem, which is based on the recently developed relative smoothness condition (Bauschke et al. in Math Oper Res 42:330–348, 2017; Lu et al. in SIOPT 28(1):333–354, 2018). With this elaboration, the third-order methods become implementable and very fast. The rate of convergence in terms of the function value for the accelerated third-order scheme reaches the level $$O\left( {1 \over k^4}\right) $$ O 1 k 4 , where k is the number of iterations. This is very close to the lower bound of the order $$O\left( {1 \over k^5}\right) $$ O 1 k 5 , which is also justified in this paper. At the same time, in many important cases the computational cost of one iteration of this method remains on the level typical for the second-order methods.
- Published
- 2021
20. Data driven regularization by projection
- Author
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Otmar Scherzer, Andrea Aspri, Yury Korolev, Korolev, Y [0000-0002-6339-652X], Scherzer, O [0000-0001-9378-7452], Apollo - University of Cambridge Repository, Korolev, Yury [0000-0002-6339-652X], and Scherzer, Otmar [0000-0001-9378-7452]
- Subjects
Paper ,Variational regularization ,010103 numerical & computational mathematics ,Gram–Schmidt orthogonalization ,01 natural sciences ,Regularization (mathematics) ,Theoretical Computer Science ,Data-driven ,FOS: Mathematics ,Applied mathematics ,Mathematics - Numerical Analysis ,Gram–Schmidt process ,Schmidt orthogonalization ,Gram– ,0101 mathematics ,data driven regularization ,Mathematical Physics ,Mathematics ,Gram– ,Generality ,Training set ,Radon transform ,inverse problems ,Applied Mathematics ,47A52, 65J20, 65J22, 65F22 ,variational regularization ,regularization by projection ,Numerical Analysis (math.NA) ,Inverse problem ,Computer Science Applications ,010101 applied mathematics ,Signal Processing - Abstract
We study linear inverse problems under the premise that the forward operator is not at hand but given indirectly through some input-output training pairs. We demonstrate that regularization by projection and variational regularization can be formulated by using the training data only and without making use of the forward operator. We study convergence and stability of the regularized solutions in view of Seidman (1980 J. Optim. Theory Appl. 30 535), who showed that regularization by projection is not convergent in general, by giving some insight on the generality of Seidman’s nonconvergence example. Moreover, we show, analytically and numerically, that regularization by projection is indeed capable of learning linear operators, such as the Radon transform.
- Published
- 2022
- Full Text
- View/download PDF
21. On the Role of the Objective in the Optimization of Compartmental Models for Biomedical Therapies
- Author
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Urszula Ledzewicz and Heinz Schättler
- Subjects
Mathematical optimization ,021103 operations research ,Control and Optimization ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1$$\end{document}L1- and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2$$\end{document}L2-type objectives ,49K15 ,Applied Mathematics ,0211 other engineering and technologies ,Nonlinear optimal control ,010103 numerical & computational mathematics ,02 engineering and technology ,Management Science and Operations Research ,Optimal control ,Invited Paper ,01 natural sciences ,Pharmacometrics ,Cancer therapies ,Simple (abstract algebra) ,92C50 ,Theory of computation ,Mathematical modeling ,0101 mathematics ,Heuristics ,Mathematics - Abstract
We review and discuss results obtained through an application of tools of nonlinear optimal control to biomedical problems. We discuss various aspects of the modeling of the dynamics (such as growth and interaction terms), modeling of treatment (including pharmacometrics of the drugs), and give special attention to the choice of the objective functional to be minimized. Indeed, many properties of optimal solutions are predestined by this choice which often is only made casually using some simple ad hoc heuristics. We discuss means to improve this choice by taking into account the underlying biology of the problem.
- Published
- 2020
22. Least squares and maximum likelihood estimation of sufficient reductions in regressions with matrix-valued predictors
- Author
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Daniel Kapla, Ruth M. Pfeiffer, and Efstathia Bura
- Subjects
Maximum likelihood ,010103 numerical & computational mathematics ,01 natural sciences ,Least squares ,Reduction (complexity) ,010104 statistics & probability ,symbols.namesake ,Matrix (mathematics) ,Statistics::Machine Learning ,Dimension (vector space) ,Statistics ,Convergence (routing) ,Regular Paper ,Statistics::Methodology ,0101 mathematics ,Mathematics ,Reduction ,Kronecker product ,Applied Mathematics ,Classification ,Regression ,Computer Science Applications ,Computational Theory and Mathematics ,Modeling and Simulation ,symbols ,Dimension ,Information Systems - Abstract
We propose methods to estimate sufficient reductions in matrix-valued predictors for regression or classification. We assume that the first moment of the predictor matrix given the response can be decomposed into a row and column component via a Kronecker product structure. We obtain least squares and maximum likelihood estimates of the sufficient reductions in the matrix predictors, derive statistical properties of the resulting estimates and present fast computational algorithms with assured convergence. The performance of the proposed approaches in regression and classification is compared in simulations.We illustrate the methods on two examples, using longitudinally measured serum biomarker and neuroimaging data.
- Published
- 2020
23. Convergence analysis of sample average approximation for a class of stochastic nonlinear complementarity problems: from two-stage to multistage
- Author
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Hailin Sun, Bin Zhou, and Jie Jiang
- Subjects
Original Paper ,Class (set theory) ,Applied Mathematics ,Numerical analysis ,Two-stage ,010103 numerical & computational mathematics ,90C33 ,Multistage ,Lipschitz continuity ,Stochastic complementarity problems ,01 natural sciences ,90C15 ,010101 applied mathematics ,Convergence analysis ,Sample average approximation ,Theory of computation ,Convergence (routing) ,Nonlinear complementarity ,Applied mathematics ,Stage (hydrology) ,0101 mathematics ,Mathematics - Abstract
In this paper, we consider the sample average approximation (SAA) approach for a class of stochastic nonlinear complementarity problems (SNCPs) and study the corresponding convergence properties. We first investigate the convergence of the SAA counterparts of two-stage SNCPs when the first-stage problem is continuously differentiable and the second-stage problem is locally Lipschitz continuous. After that, we extend the convergence results to a class of multistage SNCPs whose decision variable of each stage is influenced only by the decision variables of adjacent stages. Finally, some preliminary numerical tests are presented to illustrate the convergence results.
- Published
- 2020
24. Numerical methods for static shallow shells lying over an obstacle
- Author
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Paolo Piersanti, Xiaoqin Shen, City University of Hong Kong (CityU), and Xi'an Jiaotong University (Xjtu)
- Subjects
Original Paper ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,010103 numerical & computational mathematics ,Bilinear form ,Half-space ,Obstacle problems · Elliptic variational inequalities ,01 natural sciences ,Finite element method ,010101 applied mathematics ,Elliptic variational inequalities ,Non-conforming finite element method ,Enriching operator ,Obstacle problems ,Shallow shell ,Obstacle ,Theory of computation ,Convergence (routing) ,Obstacle problem ,0101 mathematics ,Nonconforming finite element method ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] ,Mathematics - Abstract
In this paper a finite element analysis to approximate the solution of an obstacle problem for a static shallow shell confined in a half space is presented. First, we rigorously prove some estimates for a suitable enriching operator connecting Morley's triangle to Hsieh-Clough-Tocher triangle. Secondly, we establish an estimate for the approximate bilinear form associated with the problem under consideration. Finally, we conduct an error analysis and we prove the convergence of the proposed numerical scheme.
- Published
- 2020
- Full Text
- View/download PDF
25. A general double-proximal gradient algorithm for d.c. programming
- Author
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Radu Ioan Boț and Sebastian Banert
- Subjects
General Mathematics ,Connection (vector bundle) ,Proximal-gradient algorithm ,0211 other engineering and technologies ,65K05 ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,90C26 ,Convergence analysis ,Convergence (routing) ,FOS: Mathematics ,Point (geometry) ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics ,49M29 ,021103 operations research ,Concave function ,Toland dual ,Full Length Paper ,Regular polygon ,90C26, 90C30, 65K05 ,Numerical Analysis (math.NA) ,Linear map ,Iterated function ,Optimization and Control (math.OC) ,Convex function ,Algorithm ,d.c. programming ,Software ,Kurdyka–Łojasiewicz property - Abstract
The possibilities of exploiting the special structure of d.c. programs, which consist of optimizing the difference of convex functions, are currently more or less limited to variants of the DCA proposed by Pham Dinh Tao and Le Thi Hoai An in 1997. These assume that either the convex or the concave part, or both, are evaluated by one of their subgradients. In this paper we propose an algorithm which allows the evaluation of both the concave and the convex part by their proximal points. Additionally, we allow a smooth part, which is evaluated via its gradient. In the spirit of primal-dual splitting algorithms, the concave part might be the composition of a concave function with a linear operator, which are, however, evaluated separately. For this algorithm we show that every cluster point is a solution of the optimization problem. Furthermore, we show the connection to the Toland dual problem and prove a descent property for the objective function values of a primal-dual formulation of the problem. Convergence of the iterates is shown if this objective function satisfies the Kurdyka--\L ojasiewicz property. In the last part, we apply the algorithm to an image processing model.
- Published
- 2018
26. Waveform relaxation of partial differential equations
- Author
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Zhen Miao and Yao-Lin Jiang
- Subjects
Partial differential equation ,Applied Mathematics ,Numerical analysis ,Short paper ,Relaxation (iterative method) ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Theory of computation ,Convergence (routing) ,Applied mathematics ,Waveform ,0101 mathematics ,Energy (signal processing) ,Mathematics - Abstract
This short paper concludes a general waveform relaxation (WR) method at the PDE level for semi-linear reaction-diffusion equations. For the case of multiple coupled PDE(s), new Jacobi WR and Gauss-Seidel WR are provided to accelerate the convergence result of classical WR. The convergence conditions are proved based on energy estimate. Numerical experiments are demonstrated with several WR methods in parallel to verify the effectiveness of the general WR method.
- Published
- 2018
27. On some previous results for the Drazin inverse of block matrices
- Author
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Jelena Višnjić
- Subjects
Combinatorics ,General Mathematics ,010102 general mathematics ,Drazin inverse ,Short paper ,Block (permutation group theory) ,Block matrix ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
This short paper is motivated by the paper of Bu et al. [C. Bu, C. Feng, P. Dong, A note on computational formulas for the Drazin inverse of certain block matrices, J. Appl. Math. Comput.(38) (2012) 631-640], where the authors gave additive formula for Drazin inverse for matrices under new conditions, and two representations under some specific conditions. Here is shown that the additive formula is not valid for all matrices which satisfy given conditions. Also, here is proved that the representations which were given in mentioned paper do not extend the results given by Hartwig et al. [R. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of a 2 _ 2 block matrix, SIAM J. Matrix. Anal. Appl. (27)(2006) 757-771 ], in fact they are equivalent.
- Published
- 2016
28. Scanning electron diffraction tomography of strain
- Author
-
William R. B. Lionheart, Robert Tovey, Martin Benning, Duncan N. Johnstone, Carola-Bibiane Schönlieb, Paul A. Midgley, Sean M. Collins, Tovey, Robert [0000-0001-5411-2268], Lionheart, William R B [0000-0003-0971-4678], Benning, Martin [0000-0002-6203-1350], Apollo - University of Cambridge Repository, Johnstone, Duncan [0000-0003-3663-3793], and Midgley, Paul [0000-0002-6817-458X]
- Subjects
Paper ,Diffraction ,strain mapping ,math.NA ,FOS: Physical sciences ,010103 numerical & computational mathematics ,01 natural sciences ,transverse ray transform ,Theoretical Computer Science ,Diffraction tomography ,Strain engineering ,FOS: Mathematics ,Precession electron diffraction ,Mathematics - Numerical Analysis ,Tensor ,0101 mathematics ,Mathematical Physics ,cs.NA ,Mathematics ,Condensed Matter - Materials Science ,scanning precession electron diffraction ,Applied Mathematics ,Materials Science (cond-mat.mtrl-sci) ,Infinitesimal strain theory ,computed tomography ,Numerical Analysis (math.NA) ,Inverse problem ,cond-mat.mtrl-sci ,Computer Science Applications ,Computational physics ,010101 applied mathematics ,strain tomography ,Signal Processing ,tensor tomography ,Tomography ,4D-STEM - Abstract
Strain engineering is used to obtain desirable materials properties in a range of modern technologies. Direct nanoscale measurement of the three-dimensional strain tensor field within these materials has however been limited by a lack of suitable experimental techniques and data analysis tools. Scanning electron diffraction has emerged as a powerful tool for obtaining two-dimensional maps of strain components perpendicular to the incident electron beam direction. Extension of this method to recover the full three-dimensional strain tensor field has been restricted though by the absence of a formal framework for tensor tomography using such data. Here, we show that it is possible to reconstruct the full non-symmetric strain tensor field as the solution to an ill-posed tensor tomography inverse problem. We then demonstrate the properties of this tomography problem both analytically and computationally, highlighting why incorporating precession to perform scanning precession electron diffraction may be important. We establish a general framework for non-symmetric tensor tomography and demonstrate computationally its applicability for achieving strain tomography with scanning precession electron diffraction data.
- Published
- 2020
- Full Text
- View/download PDF
29. Romberg extrapolation for Euler summation-based cubature on regular regions
- Author
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Willi Freeden and Christian Gerhards
- Subjects
Discrete mathematics ,Original Paper ,Extrapolation ,010103 numerical & computational mathematics ,01 natural sciences ,Romberg extrapolation ,Cubature ,Mathematics::Numerical Analysis ,010101 applied mathematics ,Trapezoidal rule (differential equations) ,symbols.namesake ,Rate of convergence ,Simple (abstract algebra) ,Modeling and Simulation ,Romberg's method ,symbols ,General Earth and Planetary Sciences ,65D30 ,65B99 ,0101 mathematics ,Remainder ,Cube ,Euler summation ,Mathematics - Abstract
Romberg extrapolation is a long-known method to improve the convergence rate of the trapezoidal rule on intervals. For simple regions such as the cube \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,1]^q$$\end{document}[0,1]q it is directly transferable to cubature in q dimensions. In this paper, we formulate Romberg extrapolation for Euler summation-based cubature on arbitrary q-dimensional regular regions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}\subset \mathbb {R}^q$$\end{document}G⊂Rq and derive an explicit representation for the remainder term.
- Published
- 2017
30. The Four-Parameter PSS Method for Solving the Sylvester Equation
- Author
-
Yan-Ran Li, Xin-Hui Shao, and Hai-Long Shen
- Subjects
Iterative method ,General Mathematics ,lcsh:Mathematics ,Positive and skew-Hermitian iterative method ,Value (computer science) ,020206 networking & telecommunications ,010103 numerical & computational mathematics ,02 engineering and technology ,Paper based ,lcsh:QA1-939 ,01 natural sciences ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Sylvester equation ,FPPSS iterative method ,0202 electrical engineering, electronic engineering, information engineering ,Computer Science (miscellaneous) ,Applied mathematics ,Order (group theory) ,Computer Science::Programming Languages ,0101 mathematics ,Coefficient matrix ,Engineering (miscellaneous) ,Mathematics - Abstract
In order to solve the Sylvester equations more efficiently, a new four parameters positive and skew-Hermitian splitting (FPPSS) iterative method is proposed in this paper based on the previous research of the positive and skew-Hermitian splitting (PSS) iterative method. We prove that when coefficient matrix A and B satisfy certain conditions, the FPPSS iterative method is convergent in the parameter&rsquo, s value region. The numerical experiment results show that compared with previous iterative method, the FPPSS iterative method is more effective in terms of iteration number IT and runtime.
- Published
- 2019
- Full Text
- View/download PDF
31. The Daugavet equation in Banach spaces with alternatively convex-smooth duals
- Author
-
Paweł Wójcik
- Subjects
Pure mathematics ,Mathematics::Functional Analysis ,010102 general mathematics ,Short paper ,Banach space ,Regular polygon ,Daugavet equation ,010103 numerical & computational mathematics ,Characterization (mathematics) ,01 natural sciences ,Linear subspace ,acs spaces ,46B20 ,affine subspaces ,47A62 ,luacs spaces ,Dual polyhedron ,Affine transformation ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Abstract
This short paper gives a necessary and sufficient condition for the Daugavet equation $\|I+T\|=1+\|T\|$ . A new characterization of the solution of the Daugavet equation in terms of invariant affine subspaces is given. We also study the notions of alternatively convex or smooth (acs) and locally uniformly alternatively convex or smooth (luacs).
- Published
- 2018
32. Structure aware Runge–Kutta time stepping for spacetime tents
- Author
-
Joachim Schöberl, Jay Gopalakrishnan, and Christoph Wintersteiger
- Subjects
Original Paper ,Partial differential equation ,65M20 ,Spacetime ,Numerical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,Mathematics::Numerical Analysis ,Causality ,010101 applied mathematics ,Runge–Kutta methods ,Nonlinear system ,Discontinuous Galerkin method ,Local time stepping ,Applied mathematics ,0101 mathematics ,Galerkin method ,Hyperbolic partial differential equation ,65M60 ,Mathematics - Abstract
We introduce a new class of Runge–Kutta type methods suitable for time stepping to propagate hyperbolic solutions within tent-shaped spacetime regions. Unlike standard Runge–Kutta methods, the new methods yield expected convergence properties when standard high order spatial (discontinuous Galerkin) discretizations are used. After presenting a derivation of nonstandard order conditions for these methods, we show numerical examples of nonlinear hyperbolic systems to demonstrate the optimal convergence rates. We also report on the discrete stability properties of these methods applied to linear hyperbolic equations.
- Published
- 2020
33. Global optimization in Hilbert space
- Author
-
Benoît Chachuat, Boris Houska, Engineering & Physical Science Research Council (EPSRC), and Commission of the European Communities
- Subjects
Technology ,Optimization problem ,Mathematics, Applied ,0211 other engineering and technologies ,CONVEX COMPUTATION ,010103 numerical & computational mathematics ,02 engineering and technology ,ELLIPSOIDS ,01 natural sciences ,90C26 ,93B40 ,Convergence analysis ,0102 Applied Mathematics ,Branch-and-lift ,CUT ,Mathematics ,65K10 ,021103 operations research ,Full Length Paper ,Operations Research & Management Science ,0103 Numerical and Computational Mathematics ,Bounded function ,Physical Sciences ,symbols ,49M30 ,Calculus of variations ,INTEGRATION ,SET ,Complexity analysis ,Complete search ,Operations Research ,General Mathematics ,APPROXIMATIONS ,Set (abstract data type) ,symbols.namesake ,Applied mathematics ,ALGORITHM ,0101 mathematics ,INTERSECTION ,Global optimization ,0802 Computation Theory and Mathematics ,Science & Technology ,Infinite-dimensional optimization ,Hilbert space ,Computer Science, Software Engineering ,Constraint (information theory) ,Computer Science ,Software - Abstract
We propose a complete-search algorithm for solving a class of non-convex, possibly infinite-dimensional, optimization problems to global optimality. We assume that the optimization variables are in a bounded subset of a Hilbert space, and we determine worst-case run-time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set. Because these run-time bounds are independent of the number of optimization variables and, in particular, are valid for optimization problems with infinitely many optimization variables, we prove that the algorithm converges to an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}ε-suboptimal global solution within finite run-time for any given termination tolerance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document}ε>0. Finally, we illustrate these results for a problem of calculus of variations.
- Published
- 2017
34. A generalized preconditioned parameterized inexact Uzawa method for singular saddle point problems
- Author
-
Zhen Chao and Guoliang Chen
- Subjects
Generalization ,Applied Mathematics ,Short paper ,Parameterized complexity ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Saddle point ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Calculus ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this short paper, we introduce and analyze a generalized preconditioned parameterized inexact Uzawa method for solving singular saddle point problems, this method is a generalization of the PPIU method (see, Gao and Kong, 2010), then we give the semi-convergent conditions of this method, which are more practical for the applications.
- Published
- 2016
35. Some Discrete Functional Analysis Tools
- Author
-
Thierry Gallouët
- Subjects
Functional analysis ,010102 general mathematics ,Short paper ,Mathematics::Analysis of PDEs ,Stefan problem ,010103 numerical & computational mathematics ,01 natural sciences ,Parabolic partial differential equation ,Physics::Fluid Dynamics ,Convergence (routing) ,Compressibility ,Applied mathematics ,0101 mathematics ,Navier–Stokes equations ,Mathematics - Abstract
The objective of this short paper is to present discrete functional analysis tools for proving the convergence of numerical schemes, mainly for elliptic and parabolic equations (Stefan problem and incompressible and compressible Navier–Stokes equations, for instance). The main part of these results are given in some papers coauthored with several coworkers.
- Published
- 2017
36. A numerical study of different projection-based model reduction techniques applied to computational homogenisation
- Author
-
Reza Zabihyan, Julia Mergheim, Dominic Soldner, Benjamin Brands, and Paul Steinmann
- Subjects
Mathematical optimization ,Constitutive equation ,Computational Mechanics ,Ocean Engineering ,010103 numerical & computational mathematics ,01 natural sciences ,Applied mathematics ,Boundary value problem ,Computational homogenisation ,0101 mathematics ,Galerkin method ,Mathematics ,Model order reduction ,Original Paper ,Reduced-order modelling ,Geometrically nonlinear ,Applied Mathematics ,Mechanical Engineering ,Hyperelasticity ,Tangent ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,Computational Theory and Mathematics ,Hyperelastic material ,Hyper-reduction - Abstract
Computing the macroscopic material response of a continuum body commonly involves the formulation of a phenomenological constitutive model. However, the response is mainly influenced by the heterogeneous microstructure. Computational homogenisation can be used to determine the constitutive behaviour on the macro-scale by solving a boundary value problem at the micro-scale for every so-called macroscopic material point within a nested solution scheme. Hence, this procedure requires the repeated solution of similar microscopic boundary value problems. To reduce the computational cost, model order reduction techniques can be applied. An important aspect thereby is the robustness of the obtained reduced model. Within this study reduced-order modelling (ROM) for the geometrically nonlinear case using hyperelastic materials is applied for the boundary value problem on the micro-scale. This involves the Proper Orthogonal Decomposition (POD) for the primary unknown and hyper-reduction methods for the arising nonlinearity. Therein three methods for hyper-reduction, differing in how the nonlinearity is approximated and the subsequent projection, are compared in terms of accuracy and robustness. Introducing interpolation or Gappy-POD based approximations may not preserve the symmetry of the system tangent, rendering the widely used Galerkin projection sub-optimal. Hence, a different projection related to a Gauss-Newton scheme (Gauss-Newton with Approximated Tensors- GNAT) is favoured to obtain an optimal projection and a robust reduced model.
- Published
- 2017
37. The domain interface method in non-conforming domain decomposition multifield problems
- Author
-
Javier Oliver, M. Cafiero, A. Ferrer, J. C. Cante, and Oriol Lloberas-Valls
- Subjects
Discretization ,Interface (Java) ,Multiphysics ,Computational Mechanics ,Ocean Engineering ,010103 numerical & computational mathematics ,Mixed formulations ,01 natural sciences ,Domain decomposition methods ,symbols.namesake ,Non-conforming interface ,Polygon mesh ,0101 mathematics ,Mortar methods ,Mathematics ,Original Paper ,Delaunay triangulation ,Applied Mathematics ,Mechanical Engineering ,Mathematical analysis ,010101 applied mathematics ,Computational Mathematics ,Computational Theory and Mathematics ,Weak coupling techniques for non-matching meshes ,Lagrange multiplier ,symbols - Abstract
The Domain Interface Method (DIM) is extended in this contribution for the case of mixed fields as encountered in multiphysics problems. The essence of the non-conforming domain decomposition technique consists in a discretization of a fictitious zero-thickness interface as in the original methodology and continuity of the solution fields across the domains is satisfied by incorporating the corresponding Lagrange Multipliers. The multifield DIM inherits the advantages of its irreducible version in the sense that the connections between non-matching meshes, with possible geometrically non-conforming interfaces, is accounted by the automatic Delaunay interface discretization without considering master and slave surfaces or intermediate surface projections as done in many established techniques, e.g. mortar methods. The multifield enhancement identifies the Lagrange multiplier field and incorporates its contribution in the weak variational form accounting for the corresponding consistent stabilization term based on a Nitsche method. This type of constraint enforcement circumvents the appearance of instabilities when the Ladyzhenskaya---Babuska---Brezzi (LBB) condition is not fulfilled by the chosen discretization. The domain decomposition framework is assessed in a large deformation setting for mixed displacement/pressure formulations and coupled thermomechanical problems. The continuity of the mixed field is studied in well selected benchmark problems for both mixed formulations and the objectivity of the response is compared to reference monolithic solutions. Results suggest that the presented strategy shows sufficient potential to be a valuable tool in situations where the evolving physics at particular domains require the use of different spatial discretizations or field interpolations.
- Published
- 2016
38. d-Hermite rings and skew $$\textit{PBW}$$ PBW extensions
- Author
-
Oswaldo Lezama and Claudia Gallego
- Subjects
Hermite polynomials ,Rank (linear algebra) ,General Mathematics ,010102 general mathematics ,Short paper ,Skew ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,symbols.namesake ,Computational Theory and Mathematics ,Kronecker delta ,symbols ,Kronecker's theorem ,Finitely-generated abelian group ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this short paper we study the d-Hermite condition about stably free modules for skew $$\textit{PBW}$$ extensions. For this purpose, we estimate the stable rank of these non-commutative rings. In addition, and closely related with these questions, we will prove Kronecker’s theorem about the radical of finitely generated ideals for some particular types of skew $$\textit{PBW}$$ extensions.
- Published
- 2015
39. The non-locality of Markov chain approximations to two-dimensional diffusions
- Author
-
Christoph Reisinger
- Subjects
Numerical Analysis ,General Computer Science ,Markov chain ,Applied Mathematics ,Probability (math.PR) ,Short paper ,Finite difference ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,01 natural sciences ,Theoretical Computer Science ,010101 applied mathematics ,Quantum nonlocality ,Modeling and Simulation ,Lattice (order) ,FOS: Mathematics ,Markov decision process ,Statistical physics ,Mathematics - Numerical Analysis ,0101 mathematics ,Anisotropy ,Mathematics - Probability ,Mathematics - Abstract
In this short paper, we consider discrete-time Markov chains on lattices as approximations to continuous-time diffusion processes. The approximations can be interpreted as finite difference schemes for the generator of the process. We derive conditions on the diffusion coefficients which permit transition probabilities to match locally first and second moments. We derive a novel formula which expresses how the matching becomes more difficult for larger (absolute) correlations and strongly anisotropic processes, such that instantaneous moves to more distant neighbours on the lattice have to be allowed. Roughly speaking, for non-zero correlations, the distance covered in one timestep is proportional to the ratio of volatilities in the two directions. We discuss the implications to Markov decision processes and the convergence analysis of approximations to Hamilton-Jacobi-Bellman equations in the Barles-Souganidis framework., Corrected two errata from previous and journal version: definition of R in (5) and summations in (7)
- Published
- 2017
40. Theoretical and empirical analysis of trading activity
- Author
-
Ludovic Tangpi, Walter Schachermayer, Mathias Pohl, and Alexander Ristig
- Subjects
021103 operations research ,Quantitative Finance - Trading and Market Microstructure ,Full Length Paper ,General Mathematics ,Financial market ,0211 other engineering and technologies ,Sigma ,Time scaling ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Trading and Market Microstructure (q-fin.TR) ,Universality (dynamical systems) ,FOS: Economics and business ,Stock exchange ,91G80 ,0101 mathematics ,Volatility (finance) ,Empirical evidence ,Mathematical economics ,Scaling ,Software ,Mathematics - Abstract
Understanding the structure of financial markets deals with suitably determining the functional relation between financial variables. In this respect, important variables are the trading activity, defined here as the number of trades N, the traded volume V, the asset price P, the squared volatility \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^2$$\end{document}σ2, the bid-ask spread S and the cost of trading C. Different reasonings result in simple proportionality relations (“scaling laws”) between these variables. A basic proportionality is established between the trading activity and the squared volatility, i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \sim \sigma ^2$$\end{document}N∼σ2. More sophisticated relations are the so called 3/2-law \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N^{3/2} \sim \sigma P V /C$$\end{document}N3/2∼σPV/C and the intriguing scaling \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \sim (\sigma P/S)^2$$\end{document}N∼(σP/S)2. We prove that these “scaling laws” are the only possible relations for considered sets of variables by means of a well-known argument from physics: dimensional analysis. Moreover, we provide empirical evidence based on data from the NASDAQ stock exchange showing that the sophisticated relations hold with a certain degree of universality. Finally, we discuss the time scaling of the volatility \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}σ, which turns out to be more subtle than one might naively expect.
- Published
- 2018
41. Automatised selection of load paths to construct reduced-order models in computational damage micromechanics: from dissipation-driven random selection to Bayesian optimization
- Author
-
Stéphane Pierre Bordas, Pierre Kerfriden, David Amsallem, Wing Kam Liu, Olivier Goury, Deformable Robots Simulation Team (DEFROST ), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Stanford University, University of Luxembourg [Luxembourg], Northwestern University [Evanston], School of Engineering [Cardiff], Cardiff University, and EPSRC funding under grant EP/J01947X/1ERC Stg grant agreement No. 279578AFOSR grant No. FA9550-14-1-0032
- Subjects
Multiscale ,Mathematical optimization ,Materials science & engineering [C09] [Engineering, computing & technology] ,Computational Mechanics ,Empirical Interpolation Method ,Ocean Engineering ,010103 numerical & computational mathematics ,Parameter space ,01 natural sciences ,reduced basis ,Homogeneisation ,[PHYS.MECA.STRU]Physics [physics]/Mechanics [physics]/Structural mechanics [physics.class-ph] ,Damage mechanics ,Nonlinear fracture mechanics ,computational homogenisation ,Computational homogenisation ,0101 mathematics ,Mathematics ,Model order reduction ,Original Paper ,MOR ,Applied Mathematics ,Mechanical Engineering ,Bayesian optimization ,Hyperreduction ,Micromechanics ,damage mechanics ,Dissipation ,010101 applied mathematics ,Science des matériaux & ingénierie [C09] [Ingénierie, informatique & technologie] ,Computational Mathematics ,TA ,Computational Theory and Mathematics ,multiscale ,model order reduction ,[PHYS.MECA.STRU]Physics [physics]/Mechanics [physics]/Mechanics of the structures [physics.class-ph] ,Representative elementary volume ,Snapshot (computer storage) ,Algorithm - Abstract
International audience; In this paper, we present new reliable model order reduction strategies for computational micromechanics. The difficulties rely mainly upon the high dimensionality of the parameter space represented by any load path applied onto the representative volume element (RVE). We take special care of the challenge of selecting an exhaustive snapshot set. This is treated by first using a random sampling of energy dissipating load paths and then in a more advanced way using Bayesian optimization associated with an interlocked division of the parameter space. Results show that we can insure the selection of an exhaustive snapshot set from which a reliable reduced-order model (ROM) can be built.
- Published
- 2016
42. The Squared Slacks Transformation in Nonlinear Programming
- Author
-
Dominique Orban, Paul Armand, DMI (XLIM-DMI), XLIM (XLIM), Université de Limoges (UNILIM)-Centre National de la Recherche Scientifique (CNRS)-Université de Limoges (UNILIM)-Centre National de la Recherche Scientifique (CNRS), GERAD, and École Polytechnique de Montréal (EPM)
- Subjects
Squared slacks transformation, Nonlinear programming ,Mathematical optimization ,021103 operations research ,Short paper ,0211 other engineering and technologies ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Nonlinear programming ,Transformation (function) ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,0101 mathematics ,lcsh:Science (General) ,Implementation ,Squared slacks transformation ,Mathematics ,Sequential quadratic programming ,lcsh:Q1-390 - Abstract
International audience; In this short paper, we recall the use of squared slacks used to transform inequality constraints into equalities and several reasons why their introduction may be harmful in many algorithmic frameworks routinely used in nonlinear programming. Numerical examples performed with the sequential quadratic programming method illustrate those reasons. Our results are reproducible with state-of-the-art implementations of the methods concerned and mostly serve a pedagogical purpose, which we believe will be useful not only to practitioners and students, but also to researchers.
- Published
- 2012
43. A stabilized finite element method for finite-strain three-field poroelasticity
- Author
-
Rafel Bordas, David Kay, Simon Tavener, and Lorenz Berger
- Subjects
Original Paper ,Applied Mathematics ,Mechanical Engineering ,Mathematical analysis ,Bandwidth (signal processing) ,Poromechanics ,Fluid flux ,Computational Mechanics ,Ocean Engineering ,010103 numerical & computational mathematics ,01 natural sciences ,Finite element method ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Computational Theory and Mathematics ,Lagrange multiplier ,Finite strain theory ,Compressibility ,symbols ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
We construct a stabilized finite-element method to compute flow and finite-strain deformations in an incompressible poroelastic medium. We employ a three-field mixed formulation to calculate displacement, fluid flux and pressure directly and introduce a Lagrange multiplier to enforce flux boundary conditions. We use a low order approximation, namely, continuous piecewise-linear approximation for the displacements and fluid flux, and piecewise-constant approximation for the pressure. This results in a simple matrix structure with low bandwidth. The method is stable in both the limiting cases of small and large permeability. Moreover, the discontinuous pressure space enables efficient approximation of steep gradients such as those occurring due to rapidly changing material coefficients or boundary conditions, both of which are commonly seen in physical and biological applications.
- Published
- 2017
44. Building a completely positive factorization
- Author
-
Immanuel M. Bomze
- Subjects
Original Paper ,Copositive optimization ,Euclidean space ,Dimension (graph theory) ,0211 other engineering and technologies ,021107 urban & regional planning ,010103 numerical & computational mathematics ,02 engineering and technology ,cp-rank ,Management Science and Operations Research ,01 natural sciences ,Orthant ,Combinatorics ,Matrix (mathematics) ,Factorization ,Symmetric matrix ,Embedding ,Schur complement ,0101 mathematics ,Gramian matrix ,Mathematics - Abstract
A symmetric matrix of order n is called completely positive if it has a symmetric factorization by means of a rectangular matrix with n columns and no negative entries (a so-called cp factorization), i.e., if it can be interpreted as a Gram matrix of n directions in the positive orthant of another Euclidean space of possibly different dimension. Finding this factor therefore amounts to angle packing and finding an appropriate embedding dimension. Neither the embedding dimension nor the directions may be unique, and so many cp factorizations of the same given matrix may coexist. Using a bordering approach, and building upon an already known cp factorization of a principal block, we establish sufficient conditions under which we can extend this cp factorization to the full matrix. Simulations show that the approach is promising also in higher dimensions.
- Published
- 2017
45. The Origins of the Alternating Schwarz Method
- Author
-
Gerhard Wanner and Martin J. Gander
- Subjects
Pure mathematics ,Mathematics::Complex Variables ,010102 general mathematics ,Short paper ,Riemann mapping theorem ,010103 numerical & computational mathematics ,16. Peace & justice ,01 natural sciences ,symbols.namesake ,Riemann hypothesis ,Dirichlet's principle ,symbols ,0101 mathematics ,Mathematics - Abstract
The origins of the alternating Schwarz method lie in the difficulty to prove the Dirichlet principle. This principle was evoked by Riemann in the proof of what is now the well known Riemann Mapping Theorem. We tell in this short paper the story of this exciting journey through the world of research mathematicians, up to the first computational Schwarz methods.
- Published
- 2014
46. The domain interface method: a general-purpose non-intrusive technique for non-conforming domain decomposition problems
- Author
-
Oriol Lloberas-Valls, M. Cafiero, J. C. Cante, Javier Oliver, Universitat Politècnica de Catalunya. Departament d'Enginyeria Civil i Ambiental, Universitat Politècnica de Catalunya. Departament de Física, and Universitat Politècnica de Catalunya. RMEE - Grup de Resistència de Materials i Estructures en l'Enginyeria
- Subjects
Engineering, Civil ,Discretization ,Interface (Java) ,Computational Mechanics ,Engineering, Multidisciplinary ,Ocean Engineering ,010103 numerical & computational mathematics ,01 natural sciences ,Domain decomposition methods ,Matemàtiques i estadística::Anàlisi numèrica [Àrees temàtiques de la UPC] ,symbols.namesake ,Non-conforming interface ,COMPDESMAT Project ,Polygon mesh ,Engineering, Ocean ,Decomposition method ,0101 mathematics ,Engineering, Aerospace ,Engineering, Biomedical ,Mortar methods ,Mathematics ,Original Paper ,Delaunay triangulation ,Mechanical Engineering ,Applied Mathematics ,Computer Science, Software Engineering ,Finite element method ,Engineering, Marine ,010101 applied mathematics ,Engineering, Manufacturing ,Engineering, Mechanical ,Computational Mathematics ,Computational Theory and Mathematics ,Descomposició, Mètode de ,Lagrange multiplier ,Weak coupling techniques for non-matching meshes ,COMP-DES-MAT Project ,Engineering, Industrial ,symbols ,Algorithm - Abstract
A domain decomposition technique is proposed which is capable of properly connecting arbitrary non-conforming interfaces. The strategy essentially consists in considering a fictitious zero-width interface between the non-matching meshes which is discretized using a Delaunay triangulation. Continuity is satisfied across domains through normal and tangential stresses provided by the discretized interface and inserted in the formulation in the form of Lagrange multipliers. The final structure of the global system of equations resembles the dual assembly of substructures where the Lagrange multipliers are employed to nullify the gap between domains. A new approach to handle floating subdomains is outlined which can be implemented without significantly altering the structure of standard industrial finite element codes. The effectiveness of the developed algorithm is demonstrated through a patch test example and a number of tests that highlight the accuracy of the methodology and independence of the results with respect to the framework parameters. Considering its high degree of flexibility and non-intrusive character, the proposed domain decomposition framework is regarded as an attractive alternative to other established techniques such as the mortar approach.
- Published
- 2016
47. Fixing and extending some recent results on the ADMM algorithm
- Author
-
Sebastian Banert, Radu Ioan Boţ, and Ernö Robert Csetnek
- Subjects
0211 other engineering and technologies ,65K05 ,Positive semidefinite operators ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,90C25 ,symbols.namesake ,Convergence (routing) ,FOS: Mathematics ,Point (geometry) ,Mathematics - Numerical Analysis ,0101 mathematics ,ADMM algorithm ,Mathematics - Optimization and Control ,Lagrangian ,Mathematics ,Variable (mathematics) ,Original Paper ,021103 operations research ,47H05 ,Applied Mathematics ,Numerical analysis ,Hilbert space ,Computational mathematics ,Numerical Analysis (math.NA) ,Saddle points ,Optimization and Control (math.OC) ,Theory of computation ,Convex optimization ,symbols ,Variable metrics ,Algorithm - Abstract
We investigate the techniques and ideas used in the convergence analysis of two proximal ADMM algorithms for solving convex optimization problems involving compositions with linear operators. Besides this, we formulate a variant of the ADMM algorithm that is able to handle convex optimization problems involving an additional smooth function in its objective, and which is evaluated through its gradient. Moreover, in each iteration we allow the use of variable metrics, while the investigations are carried out in the setting of infinite dimensional Hilbert spaces. This algorithmic scheme is investigated from the point of view of its convergence properties., Comment: Updates in Section 2 concerning the derivation of the convergence rates + a unifying convergence theorem for the sequence of iterates
- Published
- 2016
- Full Text
- View/download PDF
48. Adjoint multi-start-based estimation of cardiac hyperelastic material parameters using shear data
- Author
-
Joakim Sundnes, Marie E. Rognes, Martin Sandve Alnæs, and Gabriel Balaban
- Subjects
Quantitative Biology::Tissues and Organs ,0206 medical engineering ,Physics::Medical Physics ,Finite Element Analysis ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Modelling and Simulation ,Parameter estimation ,Applied mathematics ,Humans ,0101 mathematics ,Mathematics ,Original Paper ,Multi-start optimization ,Ogden ,Estimation theory ,Mechanical Engineering ,Hyperelasticity ,Models, Cardiovascular ,Reproducibility of Results ,Heart ,Numerical Analysis, Computer-Assisted ,020601 biomedical engineering ,Finite element method ,Elasticity ,Simple shear ,Nonlinear system ,Cardiac mechanics ,Adjoint equation ,Modeling and Simulation ,Hyperelastic material ,Relaxation (approximation) ,Stress, Mechanical ,Algorithms ,Biotechnology - Abstract
Cardiac muscle tissue during relaxation is commonly modeled as a hyperelastic material with strongly nonlinear and anisotropic stress response. Adapting the behavior of such a model to experimental or patient data gives rise to a parameter estimation problem which involves a significant number of parameters. Gradient-based optimization algorithms provide a way to solve such nonlinear parameter estimation problems with relatively few iterations, but require the gradient of the objective functional with respect to the model parameters. This gradient has traditionally been obtained using finite differences, the calculation of which scales linearly with the number of model parameters, and introduces a differencing error. By using an automatically derived adjoint equation, we are able to calculate this gradient more efficiently, and with minimal implementation effort. We test this adjoint framework on a least squares fitting problem involving data from simple shear tests on cardiac tissue samples. A second challenge which arises in gradient-based optimization is the dependency of the algorithm on a suitable initial guess. We show how a multi-start procedure can alleviate this dependency. Finally, we provide estimates for the material parameters of the Holzapfel and Ogden strain energy law using finite element models together with experimental shear data.
- Published
- 2015
49. New Lagrangian function for nonconvex primal-dual decomposition
- Author
-
H. Mukai and Akio Tanikawa
- Subjects
0209 industrial biotechnology ,Mathematical optimization ,Optimization problem ,Short paper ,Structure (category theory) ,MathematicsofComputing_NUMERICALANALYSIS ,0211 other engineering and technologies ,Mathematics::Optimization and Control ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Separable space ,symbols.namesake ,020901 industrial engineering & automation ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,Modelling and Simulation ,Decomposition (computer science) ,0101 mathematics ,Mathematics ,021103 operations research ,Primal dual ,Computational Mathematics ,Computational Theory and Mathematics ,Lagrangian relaxation ,Modeling and Simulation ,symbols ,Lagrangian - Abstract
In this paper, a new Lagrangian function is reported which is particularly suited for large-scale nonconvex optimization problems with separable structure. Our modification convexifies the standard Lagrangian function without destroying its separable structure so that the primal-dual decomposition technique can be applied even to nonconvex optimization problems. Furthermore, the proposed Lagrangian results in two levels of iterative optimization as compared with the three levels needed for techniques recently proposed for nonconvex primal-dual decomposition.
- Full Text
- View/download PDF
50. Immersed boundary-conformal isogeometric method for linear elliptic problems
- Author
-
Pablo Antolin, Xiaodong Wei, Benjamin Marussig, and Annalisa Buffa
- Subjects
Discretization ,Computational Mechanics ,Stabilized method ,Boundary (topology) ,Conformal boundary/interface ,Ocean Engineering ,Conformal map ,010103 numerical & computational mathematics ,Isogeometric analysis ,01 natural sciences ,symbols.namesake ,Boolean operations ,FOS: Mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics ,Original Paper ,Applied Mathematics ,Mechanical Engineering ,Mathematical analysis ,Numerical Analysis (math.NA) ,Immersed boundary method ,010101 applied mathematics ,Computational Mathematics ,Boundary layer ,Boundary representation ,Computational Theory and Mathematics ,Immersed method ,Dirichlet boundary condition ,symbols - Abstract
We present a novel isogeometric method, namely the Immersed Boundary-Conformal Method (IBCM), that features a layer of discretization conformal to the boundary while employing a simple background mesh for the remaining domain. In this manner, we leverage the geometric flexibility of the immersed boundary method with the advantages of a conformal discretization, such as intuitive control of mesh resolution around the boundary, higher accuracy per degree of freedom, automatic satisfaction of interface kinematic conditions, and the ability to strongly impose Dirichlet boundary conditions. In the proposed method, starting with a boundary representation of a geometric model, we extrude it to obtain a corresponding conformal layer. Next, a given background B-spline mesh is cut with the conformal layer, leading to two disconnected regions: an exterior region and an interior region. Depending on the problem of interest, one of the two regions is selected to be coupled with the conformal layer through Nitsche’s method. Such a construction involves Boolean operations such as difference and union, which therefore require proper stabilization to deal with arbitrarily cut elements. In this regard, we follow our precedent work called the minimal stabilization method (Antolin et al in SIAM J Sci Comput 43(1):A330–A354, 2021). In the end, we solve several 2D benchmark problems to demonstrate improved accuracy and expected convergence with IBCM. Two applications that involve complex geometries are also studied to show the potential of IBCM, including a spanner model and a fiber-reinforced composite model. Moreover, we demonstrate the effectiveness of IBCM in an application that exhibits boundary-layer phenomena.
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