Building More Efficient Time Parallel Solvers with Applications to Multiscale Modeling and Gyrokinetics
|
|
- Annis Wilcox
- 5 years ago
- Views:
Transcription
1 Building More Efficient Time Parallel Solvers with Applications to Multiscale Modeling and Gyrokinetics January 2015 Carl D Lederman AFTC/PA clearance No , 16 January
2 The Differential Equation The general form of the differential equation (DE) of interest is: = ff xx xx 0 = xx 0 ff: R DD R DD xx tt : R R DD where the dimension, DD, may be any size and ff may be derived from a spatial discretization of a PDE. The differential equations are assumed computationally solvable by existing serial means. The solution is represented in a compact form using propagator FF. xx tt = FF(xx 0, tt) The letter CC represents a coarse propagator, which solves the same DE with less accuracy and with more computational speed. 2
3 Serial Solvers The serial solution to the described DE is summarized as: Serial Computations xx tt3 = FF(xx tt2, ) xx tt2 = FF(xx tt1, ) xx tt1 = FF(xx tt0, ) The propagator, FF, encapsulates any necessary processes required to advance the solution such as a Newton non-linear solver or Fourier transform. This approach is standard for solving DEs of the type described and generally works well. The only drawback is that computing on modern machines works fastest when tasks can be parallelized as much as possible and this approach restricts the possible parallelization that can occur. 3
4 TP Solvers For time parallel (TP) modeling, the solution, xx vv uu, depends on both a value vv [0, VV] related to physical time as well as uu, which serves as an iteration number in the simplest case. The TP method alternates between parallel and serial steps. Serial Computations xx 3 0 = CC(xx 2 0 ) xx 2 0 = CC(xx 1 0 ) xx 1 0 = CC(xx 0 0 ) Parallel Computations 0 FF xx VV 1, 0 FF xx VV 2, FF xx 1 0, FF xx 0 0, Serial Computations xx VV = gg(xx VV 1, FF xx VV 1, ) xx 2 1 = gg(xx 1 1, FF xx 1 0, ) xx 1 1 = gg(xx 0 1, FF xx 0 0, ) As much computation as possible should be performed in parallel. A simple means of achieving this is to make FF a composition propagators with smaller time steps. 4
5 Propagator Derivatives In the TP approach, the fine propagator is run independently on each time block. Thus, in general, the fine propagator FF(xx, tt) will be starting from an xx that disagrees with the desired final solution To account for the inaccurate propagator starting location, some measure of the variation in xx, or FF is needed. xx Initial Coarse Solution Fine Propagator Difference in Fine Propagator Starting Location 5
6 TP PDE A general approach yields an equation that can be discretized in a few different ways to produce TP schemes. A generic functional for the energy of the DE is defined. φφ xx(vv) = VV ddvv ff xx ff xx ddvv ddvv The functional is minimized by gradient descent with a carefully chosen functional gradient. = DDDD(φφ xx ) The resulting PDE is: 2 xx vvvvvv xx + ff = 0 ff: R DD R DD xx uu, vv : R R R DD 6
7 Direct TP PDE Approximations The simplest approach is to simply integrate in vv using known serial propagators: dd xx uu, vv + xx uu,vv, dd xx uu, vv + xx uu, vv + FF xx uu, vv, = 0 Which can be further simplified to: mm vv+1 uu mm vv + xx vv+1 FF xx uu vv = 0 xxuu vv mm vv = xx vv uu+1 xx vv uu = xx vv uu+1 xx vv uu Additionally the propagator derivative can be approximated with the substitution: xxuu vv ddcc xx vv uu An alternative substitution yields the Parareal Method : xx vv uu mm vv CC xx vv uu + mm vv CC xx vv uu = CC xx vv uu+1 CC xx vv uu These schemes are implemented efficiently when each of the fine propagators is precomputed in parallel. 7
8 Additional TP PDE Approximations Higher Accuracy Schemes This approach is based on substituting a better derivative formula into the TP PDE. ddff ddxx 1 ddff xxuu 2 ddxx + 1 ddff vv xxuu+1 2 vv xxuu vv The discretization relies on a hybrid approach in which the required computations are made based a few solution values at uu + 1 and many at uu. Much of the computational work that relies only on uu can be done in parallel. Multi-Propagator Schemes Approximates using ddcc. Computing the derivative of the fine propagator can be the computationally expensive for some DEs. The fine propagator derivative, computed over shorter intervals, can be combined with the coarse propagator derivative, computed over longer intervals, to produce an approximation that is close to the accuracy of the pure fine propagator derivative while remaining less computation expensive. 8
9 Sample Results Coarsest Initial Fine The TP schemes can vary considerably in their convergence to the desired solution 9
10 Potential Applications to Multiscale Problems For multiscale (MS) applications, the goal is not just to perform fewer computations in serial, but fewer computations. To accomplish this, as much useful information as possible needs to be taken from only a small sampling of the fine scale features. One of the TP methods discussed offers an efficient means to approximate the derivative (in xx) of a fine propagator based on only a sampling of values as well a coarse propagator derivative. The fine propagator itself can be then be approximated from the following relation: (xx, tt) tt = xx, tt ff(xx) 10
11 Gyrokinetic Particle Trajectories (Preliminary) A particle rapidly orbits around a slowly varying magnetic field (cyan). A known analytical approximation is the helix trajectory (blue). Standard numerical schemes are accurate with small time steps, but numerical stability is limited based upon the gyrofrequency. Fine Coarse Multiscale B Field The coarse and multiscale solutions are implemented with a single step (but interpolated to show the full trajectory) The multiscale approximation may allow for larger time step sizes than the standard numerical schemes and better accuracy than the analytical approximation. 11
12 Extensions Solve larger scale problems and PDEs with Time Parallel methods Fully develop the multiscale gyrokinetic solver Further analyze the causes of simulation failure for multiscale and time parallel methods Thank you! 12
Using Hybrid-System Verification Tools in the Design of Simplex-Based Systems. Scott D. Stoller
Using Hybrid-System Verification Tools in the Design of Simplex-Based Systems Scott D. Stoller 2014 Annual Safe and Secure Systems and Software Symposium (S5) 1 Simplex Architecture Simplex Architecture
More informationCSE 5526: Introduction to Neural Networks Radial Basis Function (RBF) Networks
CSE 5526: Introduction to Neural Networks Radial Basis Function (RBF) Networks Part IV 1 Function approximation MLP is both a pattern classifier and a function approximator As a function approximator,
More informationSmoothers. < interactive example > Partial Differential Equations Numerical Methods for PDEs Sparse Linear Systems
Smoothers Partial Differential Equations Disappointing convergence rates observed for stationary iterative methods are asymptotic Much better progress may be made initially before eventually settling into
More informationConvolution Neural Nets meet
Convolution Neural Nets meet PDE s Eldad Haber Lars Ruthotto SIAM CS&E 2017 Convolution Neural Networks (CNN) Meet PDE s Optimization Multiscale Example Future work CNN - A quick overview Neural Networks
More informationWhat is Multigrid? They have been extended to solve a wide variety of other problems, linear and nonlinear.
AMSC 600/CMSC 760 Fall 2007 Solution of Sparse Linear Systems Multigrid, Part 1 Dianne P. O Leary c 2006, 2007 What is Multigrid? Originally, multigrid algorithms were proposed as an iterative method to
More informationODEs occur quite often in physics and astrophysics: Wave Equation in 1-D stellar structure equations hydrostatic equation in atmospheres orbits
Solving ODEs General Stuff ODEs occur quite often in physics and astrophysics: Wave Equation in 1-D stellar structure equations hydrostatic equation in atmospheres orbits need workhorse solvers to deal
More informationMultigrid Pattern. I. Problem. II. Driving Forces. III. Solution
Multigrid Pattern I. Problem Problem domain is decomposed into a set of geometric grids, where each element participates in a local computation followed by data exchanges with adjacent neighbors. The grids
More informationMass-Spring Systems. Last Time?
Mass-Spring Systems Last Time? Implicit Surfaces & Marching Cubes/Tetras Collision Detection & Conservative Bounding Regions Spatial Acceleration Data Structures Octree, k-d tree, BSF tree 1 Today Particle
More informationShading II. CITS3003 Graphics & Animation
Shading II CITS3003 Graphics & Animation Objectives Introduce distance terms to the shading model. More details about the Phong model (lightmaterial interaction). Introduce the Blinn lighting model (also
More informationChanging from Standard to Vertex Form Date: Per:
Math 2 Unit 11 Worksheet 1 Name: Changing from Standard to Vertex Form Date: Per: [1-9] Find the value of cc in the expression that completes the square, where cc =. Then write in factored form. 1. xx
More informationAn Introduction to Flow Visualization (1) Christoph Garth
An Introduction to Flow Visualization (1) Christoph Garth cgarth@ucdavis.edu Motivation What will I be talking about? Classical: Physical experiments to understand flow. 2 Motivation What will I be talking
More informationCOMPUTER GRAPHICS COURSE. LuxRender. Light Transport Foundations
COMPUTER GRAPHICS COURSE LuxRender Light Transport Foundations Georgios Papaioannou - 2015 Light Transport Light is emitted at the light sources and scattered around a 3D environment in a practically infinite
More informationDevelopment of a Maxwell Equation Solver for Application to Two Fluid Plasma Models. C. Aberle, A. Hakim, and U. Shumlak
Development of a Maxwell Equation Solver for Application to Two Fluid Plasma Models C. Aberle, A. Hakim, and U. Shumlak Aerospace and Astronautics University of Washington, Seattle American Physical Society
More informationA Brief Look at Optimization
A Brief Look at Optimization CSC 412/2506 Tutorial David Madras January 18, 2018 Slides adapted from last year s version Overview Introduction Classes of optimization problems Linear programming Steepest
More informationLesson 10. Homework Problem Set Sample Solutions. then Print True else Print False End if. False False True False False False
Homework Problem Set Sample Solutions 1. Perform the instructions in the following programming code as if you were a computer and your paper were the computer screen. Declare xx integer For all xx from
More informationCommunications in Applied Mathematics and Computational Science
Communications in Applied Mathematics and Computational Science A HYBRID PARAREAL SPECTRAL DEFERRED CORRECTIONS METHOD MICHAEL L. MINION vol. 5 no. 2 2010 mathematical sciences publishers COMM. APP. MATH.
More informationThe Immersed Interface Method
The Immersed Interface Method Numerical Solutions of PDEs Involving Interfaces and Irregular Domains Zhiiin Li Kazufumi Ito North Carolina State University Raleigh, North Carolina Society for Industrial
More informationIntroduction to optimization methods and line search
Introduction to optimization methods and line search Jussi Hakanen Post-doctoral researcher jussi.hakanen@jyu.fi How to find optimal solutions? Trial and error widely used in practice, not efficient and
More informationAccuracy Analysis of Charged Particle Trajectory CAE Software
www.integratedsoft.com Accuracy Analysis of Charged Particle Trajectory CAE Software Content Executive Summary... 3 Overview of Charged Particle Beam Analysis... 3 Types of Field Distribution... 4 Simulating
More informationFull waveform inversion by deconvolution gradient method
Full waveform inversion by deconvolution gradient method Fuchun Gao*, Paul Williamson, Henri Houllevigue, Total), 2012 Lei Fu Rice University November 14, 2012 Outline Introduction Method Implementation
More informationData Visualization. Fall 2017
Data Visualization Fall 2017 Vector Fields Vector field v: D R n D is typically 2D planar surface or 2D surface embedded in 3D n = 2 fields tangent to 2D surface n = 3 volumetric fields When visualizing
More informationTrajectory Optimization
C H A P T E R 12 Trajectory Optimization So far, we have discussed a number of ways to solve optimal control problems via state space search (e.g., Dijkstra s and Dynamic Programming/Value Iteration).
More informationLesson 19: Translating Functions
Student Outcomes Students recognize and use parent functions for linear, absolute value, quadratic, square root, and cube root functions to perform vertical and horizontal translations. They identify how
More informationVisual motion. Many slides adapted from S. Seitz, R. Szeliski, M. Pollefeys
Visual motion Man slides adapted from S. Seitz, R. Szeliski, M. Pollefes Motion and perceptual organization Sometimes, motion is the onl cue Motion and perceptual organization Sometimes, motion is the
More informationLecture 23: Starting to put it all together #2... More 2-Point Boundary value problems
Lecture 23: Starting to put it all together #2... More 2-Point Boundary value problems Outline 1) Our basic example again: -u'' + u = f(x); u(0)=α, u(l)=β 2) Solution of 2-point Boundary value problems
More informationCS205b/CME306. Lecture 9
CS205b/CME306 Lecture 9 1 Convection Supplementary Reading: Osher and Fedkiw, Sections 3.3 and 3.5; Leveque, Sections 6.7, 8.3, 10.2, 10.4. For a reference on Newton polynomial interpolation via divided
More informationComputer Vision Lecture 20
Computer Vision Lecture 2 Motion and Optical Flow Bastian Leibe RWTH Aachen http://www.vision.rwth-aachen.de leibe@vision.rwth-aachen.de 28.1.216 Man slides adapted from K. Grauman, S. Seitz, R. Szeliski,
More informationSpace Filling Curves and Hierarchical Basis. Klaus Speer
Space Filling Curves and Hierarchical Basis Klaus Speer Abstract Real world phenomena can be best described using differential equations. After linearisation we have to deal with huge linear systems of
More informationAdaptive-Mesh-Refinement Pattern
Adaptive-Mesh-Refinement Pattern I. Problem Data-parallelism is exposed on a geometric mesh structure (either irregular or regular), where each point iteratively communicates with nearby neighboring points
More informationRay-casting Algebraic Surfaces using the Frustum Form. Eurographics 2008 Crete, Thursday April 17.
Ray-casting Algebraic Surfaces using the Frustum Form Martin Reimers Johan Seland Eurographics 2008 Crete, Thursday April 17. Algebraic Surfaces Zero set of polynomial f : R 3 R f (x, y, z) = f ijk x i
More informationOrdinary Differential Equations
Next: Partial Differential Equations Up: Numerical Analysis for Chemical Previous: Numerical Differentiation and Integration Subsections Runge-Kutta Methods Euler's Method Improvement of Euler's Method
More informationFeature Tracking and Optical Flow
Feature Tracking and Optical Flow Prof. D. Stricker Doz. G. Bleser Many slides adapted from James Hays, Derek Hoeim, Lana Lazebnik, Silvio Saverse, who 1 in turn adapted slides from Steve Seitz, Rick Szeliski,
More informationOptical flow. Cordelia Schmid
Optical flow Cordelia Schmid Motion field The motion field is the projection of the 3D scene motion into the image Optical flow Definition: optical flow is the apparent motion of brightness patterns in
More informationLesson 20: Graphing Quadratic Functions
Opening Exercise 1. The science class created a ball launcher that could accommodate a heavy ball. They moved the launcher to the roof of a 23-story building and launched an 8.8-pound shot put straight
More informationOptimization with Gradient and Hessian Information Calculated Using Hyper-Dual Numbers
Optimization with Gradient and Hessian Information Calculated Using Hyper-Dual Numbers Jeffrey A. Fike and Juan J. Alonso Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305,
More informationQ.4 Properties of Quadratic Function and Optimization Problems
384 Q.4 Properties of Quadratic Function and Optimization Problems In the previous section, we examined how to graph and read the characteristics of the graph of a quadratic function given in vertex form,
More informationAn Investigation into Iterative Methods for Solving Elliptic PDE s Andrew M Brown Computer Science/Maths Session (2000/2001)
An Investigation into Iterative Methods for Solving Elliptic PDE s Andrew M Brown Computer Science/Maths Session (000/001) Summary The objectives of this project were as follows: 1) Investigate iterative
More informationMaximizing an interpolating quadratic
Week 11: Monday, Apr 9 Maximizing an interpolating quadratic Suppose that a function f is evaluated on a reasonably fine, uniform mesh {x i } n i=0 with spacing h = x i+1 x i. How can we find any local
More informationADAPTIVE FINITE ELEMENT
Finite Element Methods In Linear Structural Mechanics Univ. Prof. Dr. Techn. G. MESCHKE SHORT PRESENTATION IN ADAPTIVE FINITE ELEMENT Abdullah ALSAHLY By Shorash MIRO Computational Engineering Ruhr Universität
More informationLesson 17: Graphing Quadratic Functions from Factored Form,
: Graphing Quadratic Functions from Factored Form, ff(xx) = aa(xx mm)(xx nn) 2 Opening Exercise 1. Solve the following equation. xx 2 + 6xx 40 = 0 0-12 -10-8 -6-4 -2-2 0 2 4 6-4 -6-8 -10 2. Consider the
More informationFeature Tracking and Optical Flow
Feature Tracking and Optical Flow Prof. D. Stricker Doz. G. Bleser Many slides adapted from James Hays, Derek Hoeim, Lana Lazebnik, Silvio Saverse, who in turn adapted slides from Steve Seitz, Rick Szeliski,
More informationIntroduction to Multigrid and its Parallelization
Introduction to Multigrid and its Parallelization! Thomas D. Economon Lecture 14a May 28, 2014 Announcements 2 HW 1 & 2 have been returned. Any questions? Final projects are due June 11, 5 pm. If you are
More informationConcept of Curve Fitting Difference with Interpolation
Curve Fitting Content Concept of Curve Fitting Difference with Interpolation Estimation of Linear Parameters by Least Squares Curve Fitting by Polynomial Least Squares Estimation of Non-linear Parameters
More informationThe Nagumo Equation with Comsol Multiphysics
The Nagumo Equation with Comsol Multiphysics Denny Otten 1 Christian Döding 2 Department of Mathematics Bielefeld University 33501 Bielefeld Germany Date: 25. April 2016 1. Traveling Front in the Nagumo
More informationLecture 2 Unstructured Mesh Generation
Lecture 2 Unstructured Mesh Generation MIT 16.930 Advanced Topics in Numerical Methods for Partial Differential Equations Per-Olof Persson (persson@mit.edu) February 13, 2006 1 Mesh Generation Given a
More informationEuler s Methods (a family of Runge- Ku9a methods)
Euler s Methods (a family of Runge- Ku9a methods) ODE IVP An Ordinary Differential Equation (ODE) is an equation that contains a function having one independent variable: The equation is coupled with an
More informationODE IVP. An Ordinary Differential Equation (ODE) is an equation that contains a function having one independent variable:
Euler s Methods ODE IVP An Ordinary Differential Equation (ODE) is an equation that contains a function having one independent variable: The equation is coupled with an initial value/condition (i.e., value
More informationQuadratic Functions Date: Per:
Math 2 Unit 10 Worksheet 1 Name: Quadratic Functions Date: Per: [1-3] Using the equations and the graphs from section B of the NOTES, fill out the table below. Equation Min or Max? Vertex Domain Range
More informationMotion and Optical Flow. Slides from Ce Liu, Steve Seitz, Larry Zitnick, Ali Farhadi
Motion and Optical Flow Slides from Ce Liu, Steve Seitz, Larry Zitnick, Ali Farhadi We live in a moving world Perceiving, understanding and predicting motion is an important part of our daily lives Motion
More informationUsing Subspace Constraints to Improve Feature Tracking Presented by Bryan Poling. Based on work by Bryan Poling, Gilad Lerman, and Arthur Szlam
Presented by Based on work by, Gilad Lerman, and Arthur Szlam What is Tracking? Broad Definition Tracking, or Object tracking, is a general term for following some thing through multiple frames of a video
More informationIEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 14, NO. 1, JANUARY
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL 14, NO 1, JANUARY 2005 125 A General Framework for Nonlinear Multigrid Inversion Seungseok Oh, Student Member, IEEE, Adam B Milstein, Student Member, IEEE, Charles
More informationCalculus Limits Images in this handout were obtained from the My Math Lab Briggs online e-book.
Calculus Limits Images in this handout were obtained from the My Math Lab Briggs online e-book. A it is the value a function approaches as the input value gets closer to a specified quantity. Limits are
More informationarxiv: v1 [math.na] 26 Jun 2014
for spectrally accurate wave propagation Vladimir Druskin, Alexander V. Mamonov and Mikhail Zaslavsky, Schlumberger arxiv:406.6923v [math.na] 26 Jun 204 SUMMARY We develop a method for numerical time-domain
More informationNumerical Methods for (Time-Dependent) HJ PDEs
Numerical Methods for (Time-Dependent) HJ PDEs Ian Mitchell Department of Computer Science The University of British Columbia research supported by National Science and Engineering Research Council of
More informationModule 4 : Solving Linear Algebraic Equations Section 11 Appendix C: Steepest Descent / Gradient Search Method
Module 4 : Solving Linear Algebraic Equations Section 11 Appendix C: Steepest Descent / Gradient Search Method 11 Appendix C: Steepest Descent / Gradient Search Method In the module on Problem Discretization
More informationAn Efficient, Geometric Multigrid Solver for the Anisotropic Diffusion Equation in Two and Three Dimensions
1 n Efficient, Geometric Multigrid Solver for the nisotropic Diffusion Equation in Two and Three Dimensions Tolga Tasdizen, Ross Whitaker UUSCI-2004-002 Scientific Computing and Imaging Institute University
More informationModern Methods of Data Analysis - WS 07/08
Modern Methods of Data Analysis Lecture XV (04.02.08) Contents: Function Minimization (see E. Lohrmann & V. Blobel) Optimization Problem Set of n independent variables Sometimes in addition some constraints
More informationIntroduction to Deep Learning
ENEE698A : Machine Learning Seminar Introduction to Deep Learning Raviteja Vemulapalli Image credit: [LeCun 1998] Resources Unsupervised feature learning and deep learning (UFLDL) tutorial (http://ufldl.stanford.edu/wiki/index.php/ufldl_tutorial)
More informationExam 8N080 - Introduction MRI
Exam 8N080 - Introduction MRI Friday January 23 rd 2015, 13.30-16.30h For this exam you may use an ordinary calculator (not a graphical one). In total there are 6 assignments and a total of 65 points can
More informationAn Adaptive Stencil Linear Deviation Method for Wave Equations
211 An Adaptive Stencil Linear Deviation Method for Wave Equations Kelly Hasler Faculty Sponsor: Robert H. Hoar, Department of Mathematics ABSTRACT Wave Equations are partial differential equations (PDEs)
More informationMath Sections 4.4 and 4.5 Rational Functions. 1) A rational function is a quotient of polynomial functions:
1) A rational function is a quotient of polynomial functions: 2) Explain how you find the domain of a rational function: a) Write a rational function with domain x 3 b) Write a rational function with domain
More informationOutline. Level Set Methods. For Inverse Obstacle Problems 4. Introduction. Introduction. Martin Burger
For Inverse Obstacle Problems Martin Burger Outline Introduction Optimal Geometries Inverse Obstacle Problems & Shape Optimization Sensitivity Analysis based on Gradient Flows Numerical Methods University
More informationModule 2: Single Step Methods Lecture 4: The Euler Method. The Lecture Contains: The Euler Method. Euler's Method (Analytical Interpretations)
The Lecture Contains: The Euler Method Euler's Method (Analytical Interpretations) An Analytical Example file:///g /Numerical_solutions/lecture4/4_1.htm[8/26/2011 11:14:40 AM] We shall now describe methods
More informationMESHLESS METHOD FOR SIMULATION OF COMPRESSIBLE REACTING FLOW
MESHLESS METHOD FOR SIMULATION OF COMPRESSIBLE REACTING FLOW Jin Young Huh*, Kyu Hong Kim**, Suk Young Jung*** *Department of Mechanical & Aerospace Engineering, Seoul National University, **Department
More informationcomputational Fluid Dynamics - Prof. V. Esfahanian
Three boards categories: Experimental Theoretical Computational Crucial to know all three: Each has their advantages and disadvantages. Require validation and verification. School of Mechanical Engineering
More informationAutomatic Generation of Algorithms and Data Structures for Geometric Multigrid. Harald Köstler, Sebastian Kuckuk Siam Parallel Processing 02/21/2014
Automatic Generation of Algorithms and Data Structures for Geometric Multigrid Harald Köstler, Sebastian Kuckuk Siam Parallel Processing 02/21/2014 Introduction Multigrid Goal: Solve a partial differential
More informationsmooth coefficients H. Köstler, U. Rüde
A robust multigrid solver for the optical flow problem with non- smooth coefficients H. Köstler, U. Rüde Overview Optical Flow Problem Data term and various regularizers A Robust Multigrid Solver Galerkin
More informationNUMERICAL METHODS, NM (4776) AS
NUMERICAL METHODS, NM (4776) AS Objectives To provide students with an understanding that many mathematical problems cannot be solved analytically but require numerical methods. To develop a repertoire
More informationSIZE PRESERVING MESH GENERATION IN ADAPTIVITY PROCESSES
Congreso de Métodos Numéricos en Ingeniería 25-28 junio 2013, Bilbao, España c SEMNI, 2013 SIZE PRESERVING MESH GENERATION IN ADAPTIVITY PROCESSES Eloi Ruiz-Gironés 1, Xevi Roca 2 and Josep Sarrate 1 1:
More informationVisual Identity Guidelines. Abbreviated for Constituent Leagues
Visual Identity Guidelines Abbreviated for Constituent Leagues 1 Constituent League Logo The logo is available in a horizontal and vertical format. Either can be used depending on the best fit for a particular
More informationPhoton Maps. The photon map stores the lighting information on points or photons in 3D space ( on /near 2D surfaces)
Photon Mapping 1/36 Photon Maps The photon map stores the lighting information on points or photons in 3D space ( on /near 2D surfaces) As opposed to the radiosity method that stores information on surface
More information1.2 Round-off Errors and Computer Arithmetic
1.2 Round-off Errors and Computer Arithmetic 1 In a computer model, a memory storage unit word is used to store a number. A word has only a finite number of bits. These facts imply: 1. Only a small set
More informationτ-extrapolation on 3D semi-structured finite element meshes
τ-extrapolation on 3D semi-structured finite element meshes European Multi-Grid Conference EMG 2010 Björn Gmeiner Joint work with: Tobias Gradl, Ulrich Rüde September, 2010 Contents The HHG Framework τ-extrapolation
More informationPost Processing, Visualization, and Sample Output
Chapter 7 Post Processing, Visualization, and Sample Output Upon successful execution of an ADCIRC run, a number of output files will be created. Specifically which files are created depends upon how the
More informationPartial Differential Equations
Simulation in Computer Graphics Partial Differential Equations Matthias Teschner Computer Science Department University of Freiburg Motivation various dynamic effects and physical processes are described
More informationNumerical Simulation of Dynamic Systems XXIV
Numerical Simulation of Dynamic Systems XXIV Prof. Dr. François E. Cellier Department of Computer Science ETH Zurich May 14, 2013 Introduction Introduction A number of important simulation applications
More informationUpgraded Swimmer for Computationally Efficient Particle Tracking for Jefferson Lab s CLAS12 Spectrometer
Upgraded Swimmer for Computationally Efficient Particle Tracking for Jefferson Lab s CLAS12 Spectrometer Lydia Lorenti Advisor: David Heddle April 29, 2018 Abstract The CLAS12 spectrometer at Jefferson
More informationThe Use of Projection Operators with the Parareal Algorithm to Solve the Heat and the KdVB Equation
MATIMYÁS MATEMATIKA Journal of the Mathematical Society of the Philippines ISSN 0115-6926 Vol. 40 Nos. 1-2 (2017) pp. 41-56 The Use of Projection Operators with the Parareal Algorithm to Solve the Heat
More informationSeven Techniques For Finding FEA Errors
Seven Techniques For Finding FEA Errors by Hanson Chang, Engineering Manager, MSC.Software Corporation Design engineers today routinely perform preliminary first-pass finite element analysis (FEA) on new
More informationThe WENO Method in the Context of Earlier Methods To approximate, in a physically correct way, [3] the solution to a conservation law of the form u t
An implicit WENO scheme for steady-state computation of scalar hyperbolic equations Sigal Gottlieb Mathematics Department University of Massachusetts at Dartmouth 85 Old Westport Road North Dartmouth,
More informationover The idea is to construct an algorithm to solve the IVP ODE (9.1)
Runge- Ku(a Methods Review of Heun s Method (Deriva:on from Integra:on) The idea is to construct an algorithm to solve the IVP ODE (9.1) over To obtain the solution point we can use the fundamental theorem
More informationSTATISTICS 579 R Tutorial: More on Writing Functions
Fall 2005 1. Iterative Methods: STATISTICS 579 R Tutorial: More on Writing Functions Three kinds of looping constructs in R: the for loop, the while loop, and the repeat loop were discussed previously.
More informationHomogenization and numerical Upscaling. Unsaturated flow and two-phase flow
Homogenization and numerical Upscaling Unsaturated flow and two-phase flow Insa Neuweiler Institute of Hydromechanics, University of Stuttgart Outline Block 1: Introduction and Repetition Homogenization
More informationMultidimensional scaling
Multidimensional scaling Lecture 5 Alexander & Michael Bronstein tosca.cs.technion.ac.il/book Numerical geometry of non-rigid shapes Stanford University, Winter 2009 Cinderella 2.0 2 If it doesn t fit,
More informationMath 5BI: Problem Set 2 The Chain Rule
Math 5BI: Problem Set 2 The Chain Rule April 5, 2010 A Functions of two variables Suppose that γ(t) = (x(t), y(t), z(t)) is a differentiable parametrized curve in R 3 which lies on the surface S defined
More informationJournal of Engineering Research and Studies E-ISSN
Journal of Engineering Research and Studies E-ISS 0976-79 Research Article SPECTRAL SOLUTIO OF STEADY STATE CODUCTIO I ARBITRARY QUADRILATERAL DOMAIS Alavani Chitra R 1*, Joshi Pallavi A 1, S Pavitran
More information9. Lecture Neural Networks
Soft Control (AT 3, RMA) 9. Lecture Neural Networks Application in Automation Engineering Outline of the lecture 1. Introduction to Soft Control: definition and limitations, basics of "smart" systems 2.
More informationHSC Mathematics - Extension 1. Workshop E2
HSC Mathematics - Extension Workshop E Presented by Richard D. Kenderdine BSc, GradDipAppSc(IndMaths), SurvCert, MAppStat, GStat School of Mathematics and Applied Statistics University of Wollongong Moss
More informationIMPROVING THE NUMERICAL ACCURACY OF HYDROTHERMAL RESERVOIR SIMULATIONS USING THE CIP SCHEME WITH THIRD-ORDER ACCURACY
PROCEEDINGS, Thirty-Seventh Workshop on Geothermal Reservoir Engineering Stanford University, Stanford, California, January 30 - February 1, 2012 SGP-TR-194 IMPROVING THE NUMERICAL ACCURACY OF HYDROTHERMAL
More informationScientific Visualization. CSC 7443: Scientific Information Visualization
Scientific Visualization Scientific Datasets Gaining insight into scientific data by representing the data by computer graphics Scientific data sources Computation Real material simulation/modeling (e.g.,
More informationOptical flow. Cordelia Schmid
Optical flow Cordelia Schmid Motion field The motion field is the projection of the 3D scene motion into the image Optical flow Definition: optical flow is the apparent motion of brightness patterns in
More informationIntroduction to C omputational F luid Dynamics. D. Murrin
Introduction to C omputational F luid Dynamics D. Murrin Computational fluid dynamics (CFD) is the science of predicting fluid flow, heat transfer, mass transfer, chemical reactions, and related phenomena
More informationA Low Level Introduction to High Dimensional Sparse Grids
A Low Level Introduction to High Dimensional Sparse Grids http://people.sc.fsu.edu/ jburkardt/presentations/sandia 2007.pdf... John 1 Clayton Webster 2 1 Virginia Tech 2 Sandia National Laboratory. 21
More informationInteractive Graphics. Lecture 9: Introduction to Spline Curves. Interactive Graphics Lecture 9: Slide 1
Interactive Graphics Lecture 9: Introduction to Spline Curves Interactive Graphics Lecture 9: Slide 1 Interactive Graphics Lecture 13: Slide 2 Splines The word spline comes from the ship building trade
More information3D Ball Skinning using PDEs for Generation of Smooth Tubular Surfaces
3D Ball Skinning using PDEs for Generation of Smooth Tubular Surfaces Greg Slabaugh a,, Brian Whited b, Jarek Rossignac b, Tong Fang c, Gozde Unal d a Research and Development Department, Medicsight, London
More informationNeuro-Fuzzy Inverse Forward Models
CS9 Autumn Neuro-Fuzzy Inverse Forward Models Brian Highfill Stanford University Department of Computer Science Abstract- Internal cognitive models are useful methods for the implementation of motor control
More informationMathematical Methods and Modeling Laboratory class. Numerical Integration of Ordinary Differential Equations
Mathematical Methods and Modeling Laboratory class Numerical Integration of Ordinary Differential Equations Exact Solutions of ODEs Cauchy s Initial Value Problem in normal form: Recall: if f is locally
More informationEllipsoid Algorithm :Algorithms in the Real World. Ellipsoid Algorithm. Reduction from general case
Ellipsoid Algorithm 15-853:Algorithms in the Real World Linear and Integer Programming II Ellipsoid algorithm Interior point methods First polynomial-time algorithm for linear programming (Khachian 79)
More informationAdaptive Node Selection in Periodic Radial Basis Function Interpolations
Adaptive Node Selection in Periodic Radial Basis Function Interpolations Muhammad Shams Dept. of Mathematics UMass Dartmouth Dartmouth MA 02747 Email: mshams@umassd.edu December 19, 2011 Abstract In RBFs,
More informationMultilevel Optimization for Multi-Modal X-ray Data Analysis
Multilevel Optimization for Multi-Modal X-ray Data Analysis Zichao (Wendy) Di Mathematics & Computer Science Division Argonne National Laboratory May 25, 2016 2 / 35 Outline Multi-Modality Imaging Example:
More information