An example of ∂̄ problem arising in a finite difference context: Direct and inverse problem for the discrete analog of the equation ψxx+uψ=σψy

1987 ◽  
Vol 28 (4) ◽  
pp. 777-780 ◽  
Author(s):  
O. Ragnisco ◽  
P. M. Santini ◽  
S. Chitlaru‐Briggs ◽  
M. J. Ablowitz
Keyword(s):  
Inverse Problem ◽  
Finite Difference ◽  
Discrete Analog

An Algorithm for the Inverse Solution of Geodesic Sailing without Auxiliary Sphere

2014 ◽  
Vol 67 (5) ◽  
pp. 825-844 ◽  
Author(s):  
Wei-Kuo Tseng
Keyword(s):  
Inverse Problem ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Computational Cost ◽  
Inverse Solution ◽  
Series Expansions ◽  
Trigonometric Functions ◽  
Innovative Method

An innovative algorithm to determine the inverse solution of a geodesic with the vertex or Clairaut constant located between two points on a spheroid is presented. This solution to the inverse problem will be useful for solving problems in navigation as well as geodesy. The algorithm to be described derives from a series expansion that replaces integrals for distance and longitude, while avoiding reliance on trigonometric functions. In addition, these series expansions are economical in terms of computational cost. For end points located at each side of a vertex, certain numerical difficulties arise. A finite difference method together with an innovative method of iteration that approximates Newton's method is presented which overcomes these shortcomings encountered for nearly antipodal regions. The method provided here, which does not involve an auxiliary sphere, was aided by the Computer Algebra System (CAS) that can yield arbitrarily truncated series suitable to the users accuracy objectives and which are limited only by machine precisions.

scholarly journals High-Resolution EEG Source Localization in Segmentation-Free Head Models Based on Finite-Difference Method and Matching Pursuit Algorithm

2021 ◽  
Vol 15 ◽  
Author(s):  
Takayoshi Moridera ◽  
Essam A. Rashed ◽  
Shogo Mizutani ◽  
Akimasa Hirata
Keyword(s):  
Inverse Problem ◽  
High Resolution ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Source Localization ◽  
Difference Method ◽  
Computational Cost ◽  
Head Model ◽  
Eeg Source Localization ◽  
The Brain

Electroencephalogram (EEG) is a method to monitor electrophysiological activity on the scalp, which represents the macroscopic activity of the brain. However, it is challenging to identify EEG source regions inside the brain based on data measured by a scalp-attached network of electrodes. The accuracy of EEG source localization significantly depends on the type of head modeling and inverse problem solver. In this study, we adopted different models with a resolution of 0.5 mm to account for thin tissues/fluids, such as the cerebrospinal fluid (CSF) and dura. In particular, a spatially dependent conductivity (segmentation-free) model created using deep learning was developed and used for more realist representation of electrical conductivity. We then adopted a multi-grid-based finite-difference method (FDM) for forward problem analysis and a sparse-based algorithm to solve the inverse problem. This enabled us to perform efficient source localization using high-resolution model with a reasonable computational cost. Results indicated that the abrupt spatial change in conductivity, inherent in conventional segmentation-based head models, may trigger source localization error accumulation. The accurate modeling of the CSF, whose conductivity is the highest in the head, was an important factor affecting localization accuracy. Moreover, computational experiments with different noise levels and electrode setups demonstrate the robustness of the proposed method with segmentation-free head model.

scholarly journals On the link between finite difference and derivative of polynomials

2017 ◽  
Author(s):  
Kolosov Petro
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Finite Difference ◽  
Differential Calculus ◽  
Divided Difference ◽  
Applied Mathematics ◽  
Numerical Differentiation ◽  
Discrete Analog ◽  
Derivatives Of

The main aim of this paper to establish the relations between forward, backward and central finite (divided) differences (that is discrete analog of the derivative) and partial & ordinary high-order derivatives of the polynomials.MSC 2010: 46G05, 30G25, 39-XXarXiv:1608.00801Keywords: Finite difference, Derivative, Divided difference, Ordinary differential equation, Partial differential equation, Partial derivative, Differential calculus, Difference Equations, Numerical Differentiation, Finite difference coefficient, Polynomial, Power function, Monomial, Exponential function, Exponentiation, arXiv, Preprint, Calculus, Mathematics, Mathematical analysis, Numerical methods, Applied Mathematics

scholarly journals AN INVERSE BOUNDARY VALUE PROBLEM FOR A SECOND ORDER ELLIPTIC EQUATION IN A RECTANGLE

2014 ◽  
Vol 19 (2) ◽  
pp. 241-256 ◽  
Author(s):  
Yashar T. Mehraliyev ◽  
Fatma Kanca
Keyword(s):  
Inverse Problem ◽  
Elliptic Equation ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Second Order ◽  
Order Elliptic Equation ◽  
Inverse Boundary ◽  
Numerical Tests ◽  
Second Order Elliptic Equation

In this paper, the inverse problem of finding a coefficient in a second order elliptic equation is investigated. The conditions for the existence and uniqueness of the classical solution of the problem under consideration are established. Numerical tests using the finite-difference scheme combined with an iteration method is presented and the sensitivity of this scheme with respect to noisy overdetermination data is illustrated.

scholarly journals New schemes for a two-dimensional inverse problem with temperature overspecification

2001 ◽  
Vol 7 (3) ◽  
pp. 283-297 ◽  
Author(s):  
Mehdi Dehghan
Keyword(s):  
Inverse Problem ◽  
Finite Difference ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Central Processor ◽  
Two Dimensional ◽  
Alternating Direction Implicit ◽  
Fully Implicit ◽  
Implicit Schemes ◽  
Alternating Direction

Two different finite difference schemes for solving the two-dimensional parabolic inverse problem with temperature overspecification are considered. These schemes are developed for indentifying the control parameter which produces, at any given time, a desired temperature distribution at a given point in the spatial domain. The numerical methods discussed, are based on the (3,3) alternating direction implicit (ADI) finite difference scheme and the (3,9) alternating direction implicit formula. These schemes are unconditionally stable. The basis of analysis of the finite difference equation considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett [17]. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. These schemes use less central processor times than the fully implicit schemes for two-dimensional diffusion with temperature overspecification. The alternating direction implicit schemes developed in this report use more CPU times than the fully explicit finite difference schemes, but their unconditional stability is significant. The results of numerical experiments are presented, and accuracy and the Central Processor (CPU) times needed for each of the methods are discussed. We also give error estimates in the maximum norm for each of these methods.

An Inverse Problem for Determination of Transient Surface Temperature From Piezoelectric Sensor Measurement

1998 ◽  
Vol 65 (2) ◽  
pp. 367-373 ◽  
Author(s):  
Fumihiro Ashida ◽  
T. R. Tauchert
Keyword(s):  
Inverse Problem ◽  
Finite Difference ◽  
Electric Potential ◽  
Piezoelectric Sensor ◽  
Analytical Form ◽  
Potential Difference ◽  
Transfer Coefficients ◽  
Disk Thickness ◽  
Electric Potential Difference ◽  
Symmetry Properties

The time-varying ambient temperature on the face of a piezoelectric disk is inferred from a knowledge of the thermally induced electric potential difference across the disk thickness. The temperature-sensing disk has a circular planform, possesses hexagonal material symmetry properties, and is constrained by a rigid, thermally insulated, electrically charge-free ring. A potential function approach, together with Laplace transforms, is employed to solve the inverse problem for a particular form of electric potential difference. Also presented is a finite difference formulation which does not require specification of an analytical form for the potential difference. Numerical results are given for the predicted transient ambient temperatures corresponding to various combinations of disk thickness-to-radius ratios and surface heat transfer coefficients. Through-thickness distributions of temperature, stresses, and electric field intensities are shown, and a comparison of the exact and finite difference results is provided.

scholarly journals Comparative analysis of the numerical methods for 3D continuation problem for parabolic equation with data on the part of the boundary

2021 ◽  
Vol 2092 (1) ◽  
pp. 012010
Author(s):  
Aleksei Prikhodko ◽  
Maxim Shishlenin
Keyword(s):  
Inverse Problem ◽  
Comparative Analysis ◽  
Parabolic Equation ◽  
Numerical Methods ◽  
Finite Difference ◽  
Difference Scheme ◽  
Gradient Method ◽  
Finite Difference Scheme ◽  
Three Dimensional ◽  
Continuation Problem

Abstract The problem of continuation of the solution of a three-dimensional parabolic equation with data given on a time-like surface is investigated. Two numerical methods for solving the continuation problem are compared: the finite-difference scheme inversion and the solution of inverse problem by gradient method. The functional gradient formula is obtained. The results of numerical calculations are presented.

scholarly journals Conclusion of discrete analog for the inverse problem of acoustics

2015 ◽  
Vol 6 (2) ◽  
pp. 56-60
Author(s):  
G.A. Tyulepberdinova ◽  
◽  
S.A. Аdilzhanovа ◽  
G.G. Gaziz ◽  
◽  
...  
Keyword(s):  
Inverse Problem ◽  
Discrete Analog

scholarly journals The Inverse Problem for Valley Glacier Flow

1981 ◽  
Vol 27 (95) ◽  
pp. 179-184 ◽  
Author(s):  
Didier Hantz ◽  
Louis Lliboutry
Keyword(s):  
Inverse Problem ◽  
Finite Difference ◽  
Analytical Solution ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Numerical Stability ◽  
Valley Glacier ◽  
Explicit Finite Difference ◽  
Glacier Flow

AbstractWe seek to infer the velocities within a cylindrical valley glacier from measured surface velocities. In the Newtonian viscous case, an explicit finite-difference scheme does not fulfil von Neumann’s condition for numerical stability. How this fact does not contradict the existence of Somigliana’s analytical solution is explained. A procedure is given which delays the onset of instability and allows velocities at shallow depths to be determined.

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