Analysis-Based Fast Numerical Algorithms of Applied Mathematics

1995 ◽  
pp. 1460-1467 ◽  
Author(s):  
Vladimir Rokhlin
Keyword(s):  
Applied Mathematics ◽  
Numerical Algorithms

COMPUTATIONAL SOLUTION OF TEMPERATURE DISTRIBUTION IN A THIN ROD OVER A GIVEN INTERVAL I={x├|0

2021 ◽  
Vol 5 (1) ◽  
pp. 608-618
Author(s):  
Falade Kazeen Iyanda ◽  
Ismail Baoku ◽  
Gwanda Yusuf Ibrahim
Keyword(s):  
Temperature Distribution ◽  
Homotopy Perturbation Method ◽  
Initial Temperature ◽  
Applied Mathematics ◽  
Numerical Algorithms ◽  
Initial Temperature Distribution ◽  
Related Field ◽  
Homotopy Perturbation ◽  
Numerical Tools ◽  
New Iterative Method

In this paper, two analytical–numerical algorithms are formulated based on homotopy perturbation method and new iterative method to obtain numerical solution for temperature distribution in a thin rod over a given finite interval. The effects of different parameters such as the coefficient  which accounts for the heat loss and the diffusivity constant  are examined when initial temperature distribution  (trigonometry and algebraic functions) are considered. The error in both algorithms approaches to zero as the computational length  increases. The proposed algorithms have been demonstrated to be quite flexible, robust and accurate. Thus, the algorithms are established as good numerical tools to solve several problems in applied mathematics and other related field of sciences

6. Can you picture that? X‐rays, DNA, and photos

2018 ◽  
pp. 85-101
Author(s):  
Alain Goriely
Keyword(s):  
Magnetic Resonance Imaging ◽  
Computed Tomography ◽  
Magnetic Resonance ◽  
Applied Mathematics ◽  
Numerical Algorithms ◽  
X Rays ◽  
Mathematical Methods ◽  
X Ray Crystallography ◽  
Digital World ◽  
Extract Information

Applied mathematics is also concerned with the manipulation and analysis of signals and data. In our digital world, where huge amounts of data are routinely collected, transmitted, processed, analysed, compressed, encrypted, decrypted, and stored, efficient mathematical methods and numerical algorithms are needed. ‘Can you picture that? X-rays, DNA, and photos’ outlines how applied mathematics is used to extract information from this data using the examples of computed tomography, X-ray crystallography, and the compression of digital image files. Modern developments have seen the discovery of new methods to extract information and process it efficiently. Examples are the theory of wavelets and compressed sensing, which has been implemented in Magnetic Resonance Imaging scanners.

scholarly journals Simulating deformable objects for computer animation: A numerical perspective

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Uri M. Ascher ◽  
Egor Larionov ◽  
Seung Heon Sheen ◽  
Dinesh K. Pai
Keyword(s):  
Numerical Analysis ◽  
Dynamical Systems ◽  
Computer Animation ◽  
Applied Mathematics ◽  
A Priori ◽  
Numerical Algorithms ◽  
Complex Geometries ◽  
Deformable Objects ◽  
Highly Oscillatory ◽  
The Mathematical Model

<p style='text-indent:20px;'>We examine a variety of numerical methods that arise when considering dynamical systems in the context of physics-based simulations of deformable objects. Such problems arise in various applications, including animation, robotics, control and fabrication. The goals and merits of suitable numerical algorithms for these applications are different from those of typical numerical analysis research in dynamical systems. Here the mathematical model is not fixed <i>a priori</i> but must be adjusted as necessary to capture the desired behaviour, with an emphasis on effectively producing lively animations of objects with complex geometries. Results are often judged by how realistic they appear to observers (by the "eye-norm") as well as by the efficacy of the numerical procedures employed. And yet, we show that with an adjusted view numerical analysis and applied mathematics can contribute significantly to the development of appropriate methods and their analysis in a variety of areas including finite element methods, stiff and highly oscillatory ODEs, model reduction, and constrained optimization.</p>

Hypothesis on the Solvability of Parabolic Equations with nonlocal Condition

2002 ◽  
Vol 7 (1) ◽  
pp. 93-104 ◽  
Author(s):  
Mifodijus Sapagovas
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Nonlocal Condition ◽  
Nonlocal Conditions ◽  
Numerical Algorithms ◽  
Uniqueness Of The Solution

Numerous and different nonlocal conditions for the solvability of parabolic equations were researched in many articles and reports. The article presented analyzes such conditions imposed, and observes that the existence and uniqueness of the solution of parabolic equation is related mainly to ”smallness” of functions, involved in nonlocal conditions. As a consequence the hypothesis has been made, stating the assumptions on functions in nonlocal conditions are related to numerical algorithms of solving parabolic equations, and not to the parabolic equation itself.

Topics in Quaternion Linear Algebra

2014 ◽  
Author(s):  
Leiba Rodman
Keyword(s):  
Linear Algebra ◽  
Complex Analysis ◽  
Dimensional Space ◽  
Applied Mathematics ◽  
Three Dimensional ◽  
Number System ◽  
Graduate Course ◽  
Open Problems ◽  
Unpublished Research ◽  
Reference Tool

Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations. Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.

Preface to the special issue, CMAM 2011, no. 3.

2011 ◽  
Vol 11 (3) ◽  
pp. 272
Author(s):  
Ivan Gavrilyuk ◽  
Boris Khoromskij ◽  
Eugene Tyrtyshnikov
Keyword(s):  
Separation Of Variables ◽  
Numerical Algorithms ◽  
Multilevel Methods ◽  
High Dimensional ◽  
Special Issue ◽  
High Dimensions ◽  
Guiding Principle ◽  
Tensor Methods ◽  
Closed World ◽  
The One

Abstract In the recent years, multidimensional numerical simulations with tensor-structured data formats have been recognized as the basic concept for breaking the "curse of dimensionality". Modern applications of tensor methods include the challenging high-dimensional problems of material sciences, bio-science, stochastic modeling, signal processing, machine learning, and data mining, financial mathematics, etc. The guiding principle of the tensor methods is an approximation of multivariate functions and operators with some separation of variables to keep the computational process in a low parametric tensor-structured manifold. Tensors structures had been wildly used as models of data and discussed in the contexts of differential geometry, mechanics, algebraic geometry, data analysis etc. before tensor methods recently have penetrated into numerical computations. On the one hand, the existing tensor representation formats remained to be of a limited use in many high-dimensional problems because of lack of sufficiently reliable and fast software. On the other hand, for moderate dimensional problems (e.g. in "ab-initio" quantum chemistry) as well as for selected model problems of very high dimensions, the application of traditional canonical and Tucker formats in combination with the ideas of multilevel methods has led to the new efficient algorithms. The recent progress in tensor numerical methods is achieved with new representation formats now known as "tensor-train representations" and "hierarchical Tucker representations". Note that the formats themselves could have been picked up earlier in the literature on the modeling of quantum systems. Until 2009 they lived in a closed world of those quantum theory publications and never trespassed the territory of numerical analysis. The tremendous progress during the very recent years shows the new tensor tools in various applications and in the development of these tools and study of their approximation and algebraic properties. This special issue treats tensors as a base for efficient numerical algorithms in various modern applications and with special emphases on the new representation formats.

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