Finite difference methods discretize the domain into a grid and approximate derivatives as algebraic equations relating variable values at neighboring grid points. First and second derivatives can be approximated using forward, backward, or central differencing schemes of varying orders of accuracy. Boundary conditions require special treatment as higher-order approximations may require values outside the domain. Solving the resulting system of algebraic equations provides an approximate numerical solution to the partial differential equation over the discretized domain.
This document provides an introduction to the basic concepts of computational fluid dynamics (CFD). It discusses the need for CFD due to the inability to analytically solve the governing equations for most engineering problems. The document then summarizes some common applications of CFD in industry, including simulating vehicle aerodynamics, mixing manifolds, and bio-medical flows. It also outlines the overall strategy of CFD in discretizing the continuous problem domain into a discrete grid before discussing specific discretization methods like the finite difference and finite volume methods.
This document provides an overview of finite difference methods for solving partial differential equations. It introduces partial differential equations and various discretization methods including finite difference methods. It covers the basics of finite difference methods including Taylor series expansions, finite difference quotients, truncation error, explicit and implicit methods like the Crank-Nicolson method. It also discusses consistency, stability, and convergence of finite difference schemes. Finally, it applies these concepts to fluid flow equations and discusses conservative and transportive properties of finite difference formulations.
This document discusses multi-dimensional finite difference formulations for derivatives used in computational fluid dynamics. It describes how one-dimensional finite difference formulas can be extended to multiple dimensions. Both first and second order derivatives in x, y, and z directions are presented. Approximations for mixed derivatives are also provided. The document outlines how boundary conditions are implemented and how the finite difference discretization results in a system of algebraic equations that must be solved to obtain the numerical solution.
The document discusses Binary Decision Diagrams (BDDs) and Ordered BDDs (OBDDs) which provide a more compact representation of Boolean functions compared to truth tables. It describes algorithms for reducing, applying logical operations, restricting variables, and checking satisfiability on BDDs/OBDDs. OBDDs ensure variables appear in the same order on all paths, allowing efficient equivalence checking. The document concludes with applications of OBDDs in symbolic model checking where sets of states are represented as OBDDs.
This document summarizes a paper that presents new algorithms for solving the cyclic order-preserving assignment problem (COPAP) and related sub-problem, the linear order-preserving assignment problem (LOPAP). It introduces a new point-assignment cost function called the Procrustean local shape distance (PLSD) and explores heuristics for using the A* search algorithm to more efficiently solve COPAP and LOPAP. Experimental results on the MPEG-7 shape dataset are presented and recommendations are made for solving COPAP/LOPAP in practice.
This document discusses the optimal synthesis of four-bar linkages to generate a desired path. It describes using the gradient descent optimization algorithm to minimize the sum of squared errors between target precision points and points reached by the coupler link. The key constraints considered are Grashof's criterion, input link angle order, and transmission angle. A computer program implements gradient descent in MATLAB to determine the optimal four-bar linkage dimensions.
This document discusses various numerical analysis methods for solving differential and partial differential equations. It begins with a brief history of numerical analysis, then discusses different interpolation methods like Lagrangian interpolation. It also covers finite difference methods, finite element methods, spectral methods, and the method of lines - explaining how each method discretizes equations. The document concludes by discussing multigrid methods, which use a hierarchy of grids to accelerate convergence in solving equations.
This document provides an introduction to the basic concepts of computational fluid dynamics (CFD). It discusses the need for CFD due to the inability to analytically solve the governing equations for most engineering problems. The document then summarizes some common applications of CFD in industry, including simulating vehicle aerodynamics, mixing manifolds, and bio-medical flows. It also outlines the overall strategy of CFD in discretizing the continuous problem domain into a discrete grid before discussing specific discretization methods like the finite difference and finite volume methods.
This document provides an overview of finite difference methods for solving partial differential equations. It introduces partial differential equations and various discretization methods including finite difference methods. It covers the basics of finite difference methods including Taylor series expansions, finite difference quotients, truncation error, explicit and implicit methods like the Crank-Nicolson method. It also discusses consistency, stability, and convergence of finite difference schemes. Finally, it applies these concepts to fluid flow equations and discusses conservative and transportive properties of finite difference formulations.
This document discusses multi-dimensional finite difference formulations for derivatives used in computational fluid dynamics. It describes how one-dimensional finite difference formulas can be extended to multiple dimensions. Both first and second order derivatives in x, y, and z directions are presented. Approximations for mixed derivatives are also provided. The document outlines how boundary conditions are implemented and how the finite difference discretization results in a system of algebraic equations that must be solved to obtain the numerical solution.
The document discusses Binary Decision Diagrams (BDDs) and Ordered BDDs (OBDDs) which provide a more compact representation of Boolean functions compared to truth tables. It describes algorithms for reducing, applying logical operations, restricting variables, and checking satisfiability on BDDs/OBDDs. OBDDs ensure variables appear in the same order on all paths, allowing efficient equivalence checking. The document concludes with applications of OBDDs in symbolic model checking where sets of states are represented as OBDDs.
This document summarizes a paper that presents new algorithms for solving the cyclic order-preserving assignment problem (COPAP) and related sub-problem, the linear order-preserving assignment problem (LOPAP). It introduces a new point-assignment cost function called the Procrustean local shape distance (PLSD) and explores heuristics for using the A* search algorithm to more efficiently solve COPAP and LOPAP. Experimental results on the MPEG-7 shape dataset are presented and recommendations are made for solving COPAP/LOPAP in practice.
This document discusses the optimal synthesis of four-bar linkages to generate a desired path. It describes using the gradient descent optimization algorithm to minimize the sum of squared errors between target precision points and points reached by the coupler link. The key constraints considered are Grashof's criterion, input link angle order, and transmission angle. A computer program implements gradient descent in MATLAB to determine the optimal four-bar linkage dimensions.
This document discusses various numerical analysis methods for solving differential and partial differential equations. It begins with a brief history of numerical analysis, then discusses different interpolation methods like Lagrangian interpolation. It also covers finite difference methods, finite element methods, spectral methods, and the method of lines - explaining how each method discretizes equations. The document concludes by discussing multigrid methods, which use a hierarchy of grids to accelerate convergence in solving equations.
Pattern Recognition and Machine Learning : Graphical Modelsbutest
- Bayesian networks are directed acyclic graphs that represent conditional independence relationships between variables. They allow compact representation of high-dimensional joint distributions.
- Graphical models like Bayesian networks and Markov random fields use graphs to represent conditional independence relationships between random variables. Inference can be performed exactly using algorithms like sum-product on trees or approximately using loopy belief propagation on general graphs.
- Sum-product and max-sum algorithms allow efficient exact inference in trees by passing messages along edges until beliefs at all nodes converge. Loopy belief propagation extends this approach to general graphs but convergence is not guaranteed.
This document discusses numerical methods for solving differential and partial differential equations. It begins by providing some historical context on the development of numerical analysis. It then discusses several common numerical methods including Lagrangian interpolation, finite difference methods, finite element methods, spectral methods, and finite volume methods. For each method, it provides a brief overview of the approach and discusses aspects like discretization, accuracy, computational cost, and common applications. Overall, the document serves as an introduction to various numerical techniques for approximating solutions to differential equations.
This document summarizes an exercise on using computational methods to solve partial differential equations (PDEs). It first discusses using relaxation techniques to solve the Laplace equation for modeling electric fields around parallel plate capacitors. The Jacobi method was used to iteratively calculate potentials on a grid until convergence within a specified tolerance. Smaller tolerances and larger grids required more iterations to converge. Longer capacitor plates produced more uniform fields resembling the theoretical infinite case. The document demonstrates how computational methods can effectively solve important physical problems modeled by PDEs.
This document is a dissertation proposal by Rishideep Roy at the University of Chicago in November 2014. The proposal is to generalize results on extreme values and entropic repulsion for two-dimensional discrete Gaussian free fields to a more general class of Gaussian fields with logarithmic correlations. Specifically, the proposal plans to find the convergence in law of the maximum of these log-correlated Gaussian fields under minimal assumptions, as well as obtain finer estimates on entropic repulsion which relates to the behavior of these fields near hard boundaries. The proposal provides background on related works and outlines the key steps to be taken, including proving expectations and tightness of maxima, invariance of maximum distributions under perturbations, approximating the fields, and
This document describes a new implementation of the finite collocation method for solving time-dependent partial differential equations of parabolic type. The method discretizes the time variable using finite differences, resulting in elliptic PDEs for the spatial variables. It then uses a combination of finite collocation and local radial basis function methods for spatial discretization, dividing the domain into local regions to improve stability compared to global RBF methods. The method is computationally efficient due to using strong forms, collocation, and inverting small matrices. The document tests the method on several linear and nonlinear PDEs.
Second or fourth-order finite difference operators, which one is most effective?Premier Publishers
This paper presents higher-order finite difference (FD) formulas for the spatial approximation of the time-dependent reaction-diffusion problems with a clear justification through examples, “why fourth-order FD formula is preferred to its second-order counterpart” that has been widely used in literature. As a consequence, methods for the solution of initial and boundary value PDEs, such as the method of lines (MOL), is of broad interest in science and engineering. This procedure begins with discretizing the spatial derivatives in the PDE with algebraic approximations. The key idea of MOL is to replace the spatial derivatives in the PDE with the algebraic approximations. Once this procedure is done, the spatial derivatives are no longer stated explicitly in terms of the spatial independent variables. In other words, only one independent variable is remaining, the resulting semi-discrete problem has now become a system of coupled ordinary differential equations (ODEs) in time. Thus, we can apply any integration algorithm for the initial value ODEs to compute an approximate numerical solution to the PDE. Analysis of the basic properties of these schemes such as the order of accuracy, convergence, consistency, stability and symmetry are well examined.
This document discusses using isogeometric analysis to solve partial differential equations (PDEs) on lower dimensional manifolds, specifically surfaces. It introduces representing surfaces using non-uniform rational B-splines (NURBS) parametrization and mapping the surface to physical space. It proposes using the same NURBS basis functions for spatial discretization in isogeometric analysis to exactly represent the surface geometry. The document outlines error estimates for isogeometric analysis of second order PDEs on surfaces and highlights the accuracy and efficiency benefits of exact surface representation. Several examples of PDEs on surfaces, like the Laplace-Beltrami problem, are solved to demonstrate isogeometric analysis.
Regression models the relationship between continuous variables by fitting a line or curve to the data points. Logistic regression performs nonlinear regression by first transforming the dependent variable values to logits (log odds) and then fitting a linear regression line to the transformed data. This results in a sigmoid curve that models the probability of an output variable given continuous input variables. The sigmoid curve bounds the predicted probabilities between 0 and 1, allowing logistic regression to be used for binary classification problems.
The document discusses linear regression and logistic regression. Linear regression finds the best-fitting linear relationship between independent and dependent variables. Logistic regression applies a sigmoid function to the linear combination of inputs to output a probability between 0 and 1, fitting a logistic curve rather than a straight line. It works by first transforming the probabilities into log-odds (logits) and then performing linear regression on the transformed data. This allows predicting probabilities while ensuring outputs remain between 0 and 1.
Investigation on the Pattern Synthesis of Subarray Weights for Low EMI Applic...IOSRJECE
In modern radar applications, it is frequently required to produce sum and difference patterns sequentially. The sum pattern amplitude coefficients are obtained by using Dolph-Chebyshev synthesis method where as the difference pattern excitation coefficients will be optimized in this present work. For this purpose optimal group weights will be introduced to the different array elements to obtain any type of beam depending on the application. Optimization of excitation to the array elements is the main objective so in this process a subarray configuration is adopted. However, Differential Evolution Algorithm is applied for optimization method. The proposed method is reliable and accurate. It is superior to other methods in terms of convergence speed and robustness. Numerical and simulation results are presented.
This document summarizes an investigation of using a dual tree algorithm and space partitioning trees to approximate matrix multiplication more efficiently than the naive O(MDN) approach under certain conditions. It presents an algorithm that organizes the row vectors of the left matrix and column vectors of the right matrix into ball trees, then performs a dual tree comparison to estimate the product matrix entries. For this to provide better complexity than naive multiplication, the vectors must fall into clusters proportional to D^τ for some τ > 0. However, uniformly distributed vectors would result in exponentially small expected cluster sizes, limiting the practical applicability of this approach. Future work is needed to address this issue.
This document discusses using variational inference methods to perform online change point detection when the underlying model is not in the exponential family. Specifically:
- It develops variational approximations to the posterior on change point times for efficient inference when the underlying model, like a Rice distribution, does not have tractable posterior predictive distributions.
- It applies this methodology to radar tracking, using a Rice distribution to model signal-to-noise ratio features, and develops a variational method for inferring the parameters of the non-exponential Rice distribution online.
- It provides improvements to online variational inference that allow updating approximations efficiently as new data arrives, enabling the use of variational inference within the Bayesian online change point detection framework.
ON AN OPTIMIZATION TECHNIQUE USING BINARY DECISION DIAGRAMIJCSEA Journal
Two-level logic minimization is a central problem in logic synthesis, and has applications in reliability analysis and automated reasoning. This paper represents a method of minimizing Boolean sum of products function with binary decision diagram and with disjoint sum of product minimization. Due to the symbolic representation of cubes for large problem instances, the method is orders of magnitude faster than previous enumerative techniques. But the quality of the approach largely depends on the variable ordering of the underlying BDD. The application of Binary Decision Diagrams (BDDs) as an efficient approach for the minimization of Disjoint Sums-of-Products (DSOPs). DSOPs are a starting point for several applications. The use of BDDs has the advantage of an implicit representation of terms. Due to this scheme the algorithm is faster than techniques working on explicit representations and the application to large circuits that could not be handled so far becomes possible. Theoretical studies on the influence of the BDDs to the search space are carried out. In experiments the proposed technique is compared to others. The results with respect to the size of the resulting DSOP are as good or better as those of the other techniques.
IMAGE REGISTRATION USING ADVANCED TOPOLOGY PRESERVING RELAXATION LABELING cscpconf
This paper presents a relaxation labeling technique with newly defined compatibility measures
for solving a general non-rigid point matching problem. In the literature, there exists a point
matching method using relaxation labeling, however, the compatibility coefficients always take
a binary value zero or one depending on whether a point and a neighboring point have
corresponding points. Our approach generalizes this relaxation labeling approach. The
compatibility coefficients take n-discrete values which measures the correlation between edges.
We use log-polar diagram to compute correlations. Through simulations, we show that this
topology preserving relaxation method improves the matching performance significantly
compared to other state-of-the-art algorithms such as shape context, thin plate spline-robust
point matching, robust point matching by preserving local neighborhood structures and
coherent point drif
IMAGE REGISTRATION USING ADVANCED TOPOLOGY PRESERVING RELAXATION LABELING csandit
This paper presents a relaxation labeling technique with newly defined compatibility measures
for solving a general non-rigid point matching problem. In the literature, there exists a point
matching method using relaxation labeling, however, the compatibility coefficients always take
a binary value zero or one depending on whether a point and a neighboring point have
corresponding points. Our approach generalizes this relaxation labeling approach. The
compatibility coefficients take n-discrete values which measures the correlation between edges.
We use log-polar diagram to compute correlations. Through simulations, we show that this
topology preserving relaxation method improves the matching performance significantly
compared to other state-of-the-art algorithms such as shape context, thin plate spline-robust
point matching, robust point matching by preserving local neighborhood structures and
coherent point drift.
keywords; Data flow analysis, control dependency .
Program analysis is the method of computing properties of a program.It is useful for performing program optimiztion
CFD discretisation methods in fluid dynamicsssuser92ea91
This document summarizes Mayank Behl's presentation on discretization methods in fluid dynamics given at the Indo-German Winter Academy. The presentation covered basics of partial differential equations and their application to modeling fluid flow. It discussed the need for computational methods to solve these equations since analytical solutions are often not possible. Various discretization techniques were introduced, with a focus on the finite difference method. The finite difference method approximates partial derivatives in the governing equations with algebraic differences between node values, resulting in a system of equations that can be solved numerically.
Integration Made Easy!
The derivative of a function can be geometrically interpreted as the slope of the curve of the mathematical function f(x) plotted as a function of x. But its implications for the modeling of nature go far deeper than this simple geometric application might imply. After all, you can see yourself drawing finite triangles to discover slope, so why is the derivative so important? Its importance lies in the fact that many physical entities such as velocity, acceleration, force and so on are defined as instantaneous rates of change of some other quantity. The derivative can give you a precise intantaneous value for that rate of change and lead to precise modeling of the desired quantity.
This document introduces expander graphs and summarizes Kolmogorov and Barzdin's proof on realizing networks in three-dimensional space, which was one of the first examples of expander graphs. It defines expander graphs as sparsely populated but well-connected graphs. Kolmogorov and Barzdin constructed random graphs with properties equivalent to expander graphs and used these properties to prove that most networks can be realized in a sphere with volume proportional to the number of vertices. Their proof established lower and upper bounds on the volume needed for realization and handled both bounded and unbounded branching networks.
Embedded machine learning-based road conditions and driving behavior monitoringIJECEIAES
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
Pattern Recognition and Machine Learning : Graphical Modelsbutest
- Bayesian networks are directed acyclic graphs that represent conditional independence relationships between variables. They allow compact representation of high-dimensional joint distributions.
- Graphical models like Bayesian networks and Markov random fields use graphs to represent conditional independence relationships between random variables. Inference can be performed exactly using algorithms like sum-product on trees or approximately using loopy belief propagation on general graphs.
- Sum-product and max-sum algorithms allow efficient exact inference in trees by passing messages along edges until beliefs at all nodes converge. Loopy belief propagation extends this approach to general graphs but convergence is not guaranteed.
This document discusses numerical methods for solving differential and partial differential equations. It begins by providing some historical context on the development of numerical analysis. It then discusses several common numerical methods including Lagrangian interpolation, finite difference methods, finite element methods, spectral methods, and finite volume methods. For each method, it provides a brief overview of the approach and discusses aspects like discretization, accuracy, computational cost, and common applications. Overall, the document serves as an introduction to various numerical techniques for approximating solutions to differential equations.
This document summarizes an exercise on using computational methods to solve partial differential equations (PDEs). It first discusses using relaxation techniques to solve the Laplace equation for modeling electric fields around parallel plate capacitors. The Jacobi method was used to iteratively calculate potentials on a grid until convergence within a specified tolerance. Smaller tolerances and larger grids required more iterations to converge. Longer capacitor plates produced more uniform fields resembling the theoretical infinite case. The document demonstrates how computational methods can effectively solve important physical problems modeled by PDEs.
This document is a dissertation proposal by Rishideep Roy at the University of Chicago in November 2014. The proposal is to generalize results on extreme values and entropic repulsion for two-dimensional discrete Gaussian free fields to a more general class of Gaussian fields with logarithmic correlations. Specifically, the proposal plans to find the convergence in law of the maximum of these log-correlated Gaussian fields under minimal assumptions, as well as obtain finer estimates on entropic repulsion which relates to the behavior of these fields near hard boundaries. The proposal provides background on related works and outlines the key steps to be taken, including proving expectations and tightness of maxima, invariance of maximum distributions under perturbations, approximating the fields, and
This document describes a new implementation of the finite collocation method for solving time-dependent partial differential equations of parabolic type. The method discretizes the time variable using finite differences, resulting in elliptic PDEs for the spatial variables. It then uses a combination of finite collocation and local radial basis function methods for spatial discretization, dividing the domain into local regions to improve stability compared to global RBF methods. The method is computationally efficient due to using strong forms, collocation, and inverting small matrices. The document tests the method on several linear and nonlinear PDEs.
Second or fourth-order finite difference operators, which one is most effective?Premier Publishers
This paper presents higher-order finite difference (FD) formulas for the spatial approximation of the time-dependent reaction-diffusion problems with a clear justification through examples, “why fourth-order FD formula is preferred to its second-order counterpart” that has been widely used in literature. As a consequence, methods for the solution of initial and boundary value PDEs, such as the method of lines (MOL), is of broad interest in science and engineering. This procedure begins with discretizing the spatial derivatives in the PDE with algebraic approximations. The key idea of MOL is to replace the spatial derivatives in the PDE with the algebraic approximations. Once this procedure is done, the spatial derivatives are no longer stated explicitly in terms of the spatial independent variables. In other words, only one independent variable is remaining, the resulting semi-discrete problem has now become a system of coupled ordinary differential equations (ODEs) in time. Thus, we can apply any integration algorithm for the initial value ODEs to compute an approximate numerical solution to the PDE. Analysis of the basic properties of these schemes such as the order of accuracy, convergence, consistency, stability and symmetry are well examined.
This document discusses using isogeometric analysis to solve partial differential equations (PDEs) on lower dimensional manifolds, specifically surfaces. It introduces representing surfaces using non-uniform rational B-splines (NURBS) parametrization and mapping the surface to physical space. It proposes using the same NURBS basis functions for spatial discretization in isogeometric analysis to exactly represent the surface geometry. The document outlines error estimates for isogeometric analysis of second order PDEs on surfaces and highlights the accuracy and efficiency benefits of exact surface representation. Several examples of PDEs on surfaces, like the Laplace-Beltrami problem, are solved to demonstrate isogeometric analysis.
Regression models the relationship between continuous variables by fitting a line or curve to the data points. Logistic regression performs nonlinear regression by first transforming the dependent variable values to logits (log odds) and then fitting a linear regression line to the transformed data. This results in a sigmoid curve that models the probability of an output variable given continuous input variables. The sigmoid curve bounds the predicted probabilities between 0 and 1, allowing logistic regression to be used for binary classification problems.
The document discusses linear regression and logistic regression. Linear regression finds the best-fitting linear relationship between independent and dependent variables. Logistic regression applies a sigmoid function to the linear combination of inputs to output a probability between 0 and 1, fitting a logistic curve rather than a straight line. It works by first transforming the probabilities into log-odds (logits) and then performing linear regression on the transformed data. This allows predicting probabilities while ensuring outputs remain between 0 and 1.
Investigation on the Pattern Synthesis of Subarray Weights for Low EMI Applic...IOSRJECE
In modern radar applications, it is frequently required to produce sum and difference patterns sequentially. The sum pattern amplitude coefficients are obtained by using Dolph-Chebyshev synthesis method where as the difference pattern excitation coefficients will be optimized in this present work. For this purpose optimal group weights will be introduced to the different array elements to obtain any type of beam depending on the application. Optimization of excitation to the array elements is the main objective so in this process a subarray configuration is adopted. However, Differential Evolution Algorithm is applied for optimization method. The proposed method is reliable and accurate. It is superior to other methods in terms of convergence speed and robustness. Numerical and simulation results are presented.
This document summarizes an investigation of using a dual tree algorithm and space partitioning trees to approximate matrix multiplication more efficiently than the naive O(MDN) approach under certain conditions. It presents an algorithm that organizes the row vectors of the left matrix and column vectors of the right matrix into ball trees, then performs a dual tree comparison to estimate the product matrix entries. For this to provide better complexity than naive multiplication, the vectors must fall into clusters proportional to D^τ for some τ > 0. However, uniformly distributed vectors would result in exponentially small expected cluster sizes, limiting the practical applicability of this approach. Future work is needed to address this issue.
This document discusses using variational inference methods to perform online change point detection when the underlying model is not in the exponential family. Specifically:
- It develops variational approximations to the posterior on change point times for efficient inference when the underlying model, like a Rice distribution, does not have tractable posterior predictive distributions.
- It applies this methodology to radar tracking, using a Rice distribution to model signal-to-noise ratio features, and develops a variational method for inferring the parameters of the non-exponential Rice distribution online.
- It provides improvements to online variational inference that allow updating approximations efficiently as new data arrives, enabling the use of variational inference within the Bayesian online change point detection framework.
ON AN OPTIMIZATION TECHNIQUE USING BINARY DECISION DIAGRAMIJCSEA Journal
Two-level logic minimization is a central problem in logic synthesis, and has applications in reliability analysis and automated reasoning. This paper represents a method of minimizing Boolean sum of products function with binary decision diagram and with disjoint sum of product minimization. Due to the symbolic representation of cubes for large problem instances, the method is orders of magnitude faster than previous enumerative techniques. But the quality of the approach largely depends on the variable ordering of the underlying BDD. The application of Binary Decision Diagrams (BDDs) as an efficient approach for the minimization of Disjoint Sums-of-Products (DSOPs). DSOPs are a starting point for several applications. The use of BDDs has the advantage of an implicit representation of terms. Due to this scheme the algorithm is faster than techniques working on explicit representations and the application to large circuits that could not be handled so far becomes possible. Theoretical studies on the influence of the BDDs to the search space are carried out. In experiments the proposed technique is compared to others. The results with respect to the size of the resulting DSOP are as good or better as those of the other techniques.
IMAGE REGISTRATION USING ADVANCED TOPOLOGY PRESERVING RELAXATION LABELING cscpconf
This paper presents a relaxation labeling technique with newly defined compatibility measures
for solving a general non-rigid point matching problem. In the literature, there exists a point
matching method using relaxation labeling, however, the compatibility coefficients always take
a binary value zero or one depending on whether a point and a neighboring point have
corresponding points. Our approach generalizes this relaxation labeling approach. The
compatibility coefficients take n-discrete values which measures the correlation between edges.
We use log-polar diagram to compute correlations. Through simulations, we show that this
topology preserving relaxation method improves the matching performance significantly
compared to other state-of-the-art algorithms such as shape context, thin plate spline-robust
point matching, robust point matching by preserving local neighborhood structures and
coherent point drif
IMAGE REGISTRATION USING ADVANCED TOPOLOGY PRESERVING RELAXATION LABELING csandit
This paper presents a relaxation labeling technique with newly defined compatibility measures
for solving a general non-rigid point matching problem. In the literature, there exists a point
matching method using relaxation labeling, however, the compatibility coefficients always take
a binary value zero or one depending on whether a point and a neighboring point have
corresponding points. Our approach generalizes this relaxation labeling approach. The
compatibility coefficients take n-discrete values which measures the correlation between edges.
We use log-polar diagram to compute correlations. Through simulations, we show that this
topology preserving relaxation method improves the matching performance significantly
compared to other state-of-the-art algorithms such as shape context, thin plate spline-robust
point matching, robust point matching by preserving local neighborhood structures and
coherent point drift.
keywords; Data flow analysis, control dependency .
Program analysis is the method of computing properties of a program.It is useful for performing program optimiztion
CFD discretisation methods in fluid dynamicsssuser92ea91
This document summarizes Mayank Behl's presentation on discretization methods in fluid dynamics given at the Indo-German Winter Academy. The presentation covered basics of partial differential equations and their application to modeling fluid flow. It discussed the need for computational methods to solve these equations since analytical solutions are often not possible. Various discretization techniques were introduced, with a focus on the finite difference method. The finite difference method approximates partial derivatives in the governing equations with algebraic differences between node values, resulting in a system of equations that can be solved numerically.
Integration Made Easy!
The derivative of a function can be geometrically interpreted as the slope of the curve of the mathematical function f(x) plotted as a function of x. But its implications for the modeling of nature go far deeper than this simple geometric application might imply. After all, you can see yourself drawing finite triangles to discover slope, so why is the derivative so important? Its importance lies in the fact that many physical entities such as velocity, acceleration, force and so on are defined as instantaneous rates of change of some other quantity. The derivative can give you a precise intantaneous value for that rate of change and lead to precise modeling of the desired quantity.
This document introduces expander graphs and summarizes Kolmogorov and Barzdin's proof on realizing networks in three-dimensional space, which was one of the first examples of expander graphs. It defines expander graphs as sparsely populated but well-connected graphs. Kolmogorov and Barzdin constructed random graphs with properties equivalent to expander graphs and used these properties to prove that most networks can be realized in a sphere with volume proportional to the number of vertices. Their proof established lower and upper bounds on the volume needed for realization and handled both bounded and unbounded branching networks.
Embedded machine learning-based road conditions and driving behavior monitoringIJECEIAES
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
Harnessing WebAssembly for Real-time Stateless Streaming PipelinesChristina Lin
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
A review on techniques and modelling methodologies used for checking electrom...nooriasukmaningtyas
The proper function of the integrated circuit (IC) in an inhibiting electromagnetic environment has always been a serious concern throughout the decades of revolution in the world of electronics, from disjunct devices to today’s integrated circuit technology, where billions of transistors are combined on a single chip. The automotive industry and smart vehicles in particular, are confronting design issues such as being prone to electromagnetic interference (EMI). Electronic control devices calculate incorrect outputs because of EMI and sensors give misleading values which can prove fatal in case of automotives. In this paper, the authors have non exhaustively tried to review research work concerned with the investigation of EMI in ICs and prediction of this EMI using various modelling methodologies and measurement setups.
Redefining brain tumor segmentation: a cutting-edge convolutional neural netw...IJECEIAES
Medical image analysis has witnessed significant advancements with deep learning techniques. In the domain of brain tumor segmentation, the ability to
precisely delineate tumor boundaries from magnetic resonance imaging (MRI)
scans holds profound implications for diagnosis. This study presents an ensemble convolutional neural network (CNN) with transfer learning, integrating
the state-of-the-art Deeplabv3+ architecture with the ResNet18 backbone. The
model is rigorously trained and evaluated, exhibiting remarkable performance
metrics, including an impressive global accuracy of 99.286%, a high-class accuracy of 82.191%, a mean intersection over union (IoU) of 79.900%, a weighted
IoU of 98.620%, and a Boundary F1 (BF) score of 83.303%. Notably, a detailed comparative analysis with existing methods showcases the superiority of
our proposed model. These findings underscore the model’s competence in precise brain tumor localization, underscoring its potential to revolutionize medical
image analysis and enhance healthcare outcomes. This research paves the way
for future exploration and optimization of advanced CNN models in medical
imaging, emphasizing addressing false positives and resource efficiency.
ACEP Magazine edition 4th launched on 05.06.2024Rahul
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1. 1
Finite Difference Methods
Introduction
„ All conservation equations have similar structure ->
regarded as special cases of a generic transport equation
„ Equation we shall deal with is:
„ Equation we shall deal with is:
Treat φ as the only unknown.
Convection Diffusion Sources
2. 2
Basic Concept
„ First step in numerical solution -> discretization of
geometric domain -> grid generation
‰ In FDM grid is locally structure i e each grid node is
‰ In FDM, grid is locally structure, i.e., each grid node is
considered the origin of a local coordinate system, whose axes
coincide with grid lines.
‰ In 3D, 3 grid lines intersect at each node; none of these lines
intersect each other at any other point.
‰ Each node is uniquely identified by a set of indices: (i, j) in 2D;
(i j k) in 3D
(i, j, k) in 3D.
Basic Concept – Cont.
3. 3
Basic Concept – Cont.
„ Eq. (3.1) is linear in φ, it will be approximated by a
system of linear algebraic equations, in which the
variable values at the grid nodes are unknowns. -> The
solution of the system approximates the solution to the
PDE.
„ Each node has 1 unknown variable value and provides 1
algebraic equation.
„ This algebraic equation is a relation between the
variable value at that node and those at neighbors It is
variable value at that node and those at neighbors. It is
obtained by replacing each term of the PDE at the
particular node by a FD approximation.
„ # of equations = # of unknowns
Basic Concept – Cont.
„ The idea behind FD is from the definition of a derivative
„ Geometrical interpretation
‰ at a point is the slope of the tangent to the curve at that point.
4. 4
Basic Concept – Cont.
„ Some approximations are better than others.
„ Approximation quality improves when the additional
points are close to x In other words as the grid is
points are close to xi. In other words, as the grid is
refined, the approximation improves.
Approximation of the First Derivative
‰ For discretization of convective term
„ Taylor series expansion
‰ Any continuous differentiable function can be expressed as
‰ Any continuous differentiable function can be expressed as
a Taylor series, in the vicinity of xi,
‰ By replacing x by xi+1 or xi-1, we obtain expressions for the
variable values at these points in terms of the variable and its
derivatives at xi.
5. 5
First Derivative – Cont.
‰ Using these expressions, we can obtain approximate
expressions for 1st and higher derivatives at point xi in terms of
the function values at neighboring points.
‰ For example, (HW)
First Derivative – Cont.
‰ If the distance between grid points is small, HOTs will be
small.
‰ FDS
‰ BDS
‰ CDS
6. 6
First Derivative – Cont.
‰ Truncation error
„ Terms deleted from RHS
„ Sum of products of a power of the spacing and a higher derivative at
i HOT f E (3 4)
point compare HOTs of Eq. (3.4)
„ For example,
Note: α’s are higher-order derivatives multiplied by constant factors
Note: α s are higher-order derivatives multiplied by constant factors.
„ The order of an approximation indicates how fast the error is reduced
when the grid is refined; it does not indicate the absolute magnitude of
the error.
First Derivative – Cont.
„ Polynomial fitting
‰ Fit the function, φ, to an interpolation curve and differentiate
the resulting curve.
g
‰ Fitting a parabola to the data at xi-1, xi, and xi+1, and computing
the first derivative at xi from the interpolant, (HW)
‰ Second order truncation error; identical to CDS with uniform
i
spacing.
‰ In general, approximation of the first derivative possesses a
truncation error of the same order as the degree of the
polynomial used to approximate the function.
7. 7
First Derivative – Cont.
3rd order
by a cubic
polynomial
at 4 points
‰ In FDS and BDS, the major contribution to the approximation
comes from one side -> Upwind schemes (UDS).
4th order by
a 4-degree
polynomial
at 5 points
‰ 1st order UDS are very inaccurate, because of false diffusion.
‰ CDS can be easily implemented, since it is not necessary to
check the flow direction.
First Derivative – Cont.
„ Compact schemes
‰ Compact schemes can be derived through the use of
polynomial fitting.
p y g
‰ However, instead of using only the variable values at
computational nodes to derive the coefficients of the
polynomial, derivatives at some points are also used.
‰ 4th order Pade scheme
„ Use variable values at nodes, i, i+1, and i-1, and first derivatives at i+1
and i-1, to obtain approximation for the 1st derivative at i.
, pp
„ A polynomial of degree 4 in the vicinity of i:
8. 8
First Derivative – Cont.
„ Since we are interested in the first derivative at i, we only need to
compute a1.
„ Differentiating Eq. (3.15),
so that
„ By writing Eq. (3.15) for and Eq. (3.16)
for we obtain
First Derivative – Cont.
„ A family of compact centered approximations of up to 6 order
„ Obviously, for the same order of approximation, Pade schemes use
fewer computational nodes and thus have more compact computational
molecules than CDS.
9. 9
First Derivative – Cont.
„ Non-uniform grids
‰ Since the truncation error depends not only on the grid spacing
but also on the derivatives of the variable, we cannot achieve a
,
uniform distribution of discretization error on a uniform grid.
Æ We need to use a non-uniform grid.
‰ Use a smaller Δx in regions where derivatives of the function
are large and a larger Δx in regions where the function is
smooth. Æ Spread the error nearly uniformly over the domain.
‰ Even though different approximations are formally of the
‰ Even though different approximations are formally of the
same order for non-uniform spacing, they do not have the
same truncation error.
First Derivative – Cont.
‰ Leading truncation errors in CDS and FDS/BDS with grid
expansion ratio of re
‰ Grid refinement
„ Halving the spacing between 2 coarse grid points -> Grid becomes
uniform everywhere except near the coarsest grid points
„ Inserting new points so that the fine grid also has a constant ratio of
g p g
spacings
10. 10
First Derivative – Cont.
„ In the second case, the expansion factor of the fine grid is smaller than
on the coarse grid
„ At a common node the ratio of the leading truncation error
„ At a common node, the ratio of the leading truncation error
Then
„ rr is 4 when the grid is uniform,
„ rr > 4 when expanding or contracting
rr 4 when expanding or contracting
„ Generation of effective grids remains one of the most difficult problems
in CFD
Approximation of the Second Derivative
‰ For discretization of diffusion term
‰ Geometrically, slope of the line tangent to the curve
representing the first derivative
‰ All such approximations involve data from at least 3 points.
‰ FDS for outer derivative and BDS for inner derivative (HW)
11. 11
Second Derivative – Cont.
‰ A better choice is to evaluate at halfway points.
‰ The resulting expression (HW)
‰ For equidistant spacing,
Second Derivative – Cont.
‰ Yet another approach is using Taylor series expansion (HW)
‰ Increase the accuracy of approximations to the first derivatives,
using the second derivatives. Æ Keeping 2 RHS terms in Eq.
(3.4)
‰ Higher order approximations always involve more nodes,
yielding more complex equations to solve and more
complicated treatment of BCs, so a trade-off has to be made.
12. 12
Approximation of Mixed Derivatives
‰ Mixed derivatives occur only when the transport equations are
expressed in non-orthogonal coordinate systems.
‰ It may be treated by combining 1D approximations as for the
second derivative.
‰ The mixed second derivative at (xi, yi) can be estimated using
CDS by first evaluating the first derivative w.r.t. y at (xi+1, yj)
and (xi 1 yj)
and (xi-1, yj).
Implementation of Boundary Conditions
„ Continuous problem requires information about the solution at
the domain boundaries.
‰ Dirichlet: Variable value at the boundary
‰ Neumann: Variable’s gradient in a particular direction
‰ Combination of the above
„ Problem when higher order approximations of the derivatives are
used; since they require data at more than 3 points,
approximations at interior nodes may demand data at points
beyond the boundary.
„ It may then be necessary to use different approximations for the
derivatives at points close to boundary; usually these are of lower
order than the approximations used deeper in the interior and
may be one-sided differences.
13. 13
Boundary Conditions – Cont.
„ Examples
‰ Cubic polynomial fitting & 1st derivative at i=2
‰ 4th-order polynomial & 1st derivative at i=2
‰ Same polynomial & 2nd derivative
p y
Boundary Conditions – Cont.
„ If the gradient is prescribed at the boundary, a suitable
FD approximation for it (one-sided approximation) can
be used to compute the boundary value of the variable.
be used to compute the boundary value of the variable.
‰ Zero gradient in the normal direction, FDS leads to:
‰ Parabolic fit and 2 inner points, 2nd order approximation for 1st
derivative at the boundary
14. 14
Algebraic Equation System
„ FD approximation provides an algebraic equation at
each grid node.
„ It contains the variable value at that node as well as
„ It contains the variable value at that node as well as
values at neighboring nodes.
„ If the differential equation is non-linear, the
approximation will contain some non-linear terms. Æ
Linearization is required (Chap. 5)
„ For now consider only the linear case
„ For now, consider only the linear case.
Algebraic Equation System – Cont.
„ P and its neighbors form computational molecule.
„ A depend on geometrical quantities fluid properties
„ Al depend on geometrical quantities, fluid properties,
and variable values (for non-linear equations).
„ QP contain all the terms which do not contain unknown
variable values.
15. 15
Algebraic Equation System – Cont.
„ # equations must be equal to # of unknowns. In other
words, there has to be one equation for each grid node.
Æ Large set of linear algebraic equations, which must
Æ Large set of linear algebraic equations, which must
be solved numerically.
„ This system is sparse, meaning that each equation
contains only a few unknowns.
„ In matrix notation,
‰ A: square sparse coefficient matrix
‰ φ: vector containing variable values at grid nodes
‰ Q: vector containing RHS terms
Algebraic Equation System – Cont.
„ The structure of A depends on the ordering of variables in φ. For
structured grids, if the variables are labeled starting at a corner
and traversing line after line in a regular manner, the matrix has a
poly-diagonal structure.
„ For the case of 5-point molecule, all the non-zero coefficients lie
on the main diagonal, the two neighboring diagonals, and two
other diagonals removed by N positions from the main diagonal.
„ This system is sparse, meaning that each equation contains only a
few unknowns.
„ The variables are normally stored in computers in 1D arrays.
„
16. 16
Algebraic Equation System – Cont.
„ The linearized algebraic equations in 2D can be written
Di l k t i t d i h
„ Diagonals are kept in separate arrays and give each
diagonal a separate name.
Algebraic Equation System – Cont.
„ In this notation, Eq. (3.44) can be written
F t t d id th ffi i t t i i
„ For unstructured grids, the coefficient matrix remains
sparse, but it no longer has banded structure.
17. 17
Discretiztion Errors
„ Since the discretized equations represent approximations to the
differential equation, the exact solution of the latter, Φ, does not
satisfy the difference equation. Æ This imbalance is called
i
truncation error.
„ For a grid with a reference spacing h,
„ The exact solution of the discretized equations on grid h, φh,
satisfies
„ It differs from the exact solution of the PDE by the discretization
error, i.e.,
Discretiztion Errors – Cont.
„ From Eqs. (3.46) and (3.47),
‰ This equation states that the truncation error acts as a source of the
di i i hi h i d d diff d b h L
discretization error, which is convected and diffused by the operator Lh.
‰ Because the exact solution Φ is not known, the truncation error cannot be
calculated exactly.
„ For sufficiently fine grids, the truncation error (and discretization
error as well) is proportional to the leading term in the Taylor
series:
‰ H: HOT
‰ α: depends on the derivatives at the given point but is independent of h
‰ Note
18. 18
Discretiztion Errors – Cont.
„ The discretization error can be estimated from the difference
between solutions obtained on systematically refined grids.
„ Since the exact solution may be expressed as
„ p, which is the order of the scheme, may be estimated as (HW)
„ The discretization error on grid h can be approximated by (HW)
„ The order of convergence estimated using Eq. (3.52) is valid only
when the convergence is monotonic. Monotonic convergence is
expected only on sufficiently fine grids.
Examples
„ Read through and try yourself!