GPU Integral Computations in Stochastic Geometry

Size: px
Start display at page:

Download "GPU Integral Computations in Stochastic Geometry"

Transcription

1 Outline GPU Integral Computations in Stochastic Geometry Department of Computer Science Western Michigan University June 25, 2013

2 Outline Outline 1 Introduction 2 3 4

3 Outline Outline 1 Introduction 2 3 4

4 Outline Outline 1 Introduction 2 3 4

5 Outline Outline 1 Introduction 2 3 4

6 Outline Introduction 1 Introduction 2 3 4

7 Problem Characteristics For given integrand f( x), over domain D, obtain integral approximation Q I = D f( x)d x It is useful to obtain an estimated error E Q I. Figure : 1 0 dx 1 0 dy 2εy (x+y 1) 2 +ε 2, ε = 0.1

8 Outline Introduction 1 Introduction 2 3 4

9 Adaptive methods Definition: Integration rule: Q = r j=1 w jf( x j ) D f( x)d x; also in composite rules Adaptive partitioning: concentrates subdivisions and function sampling in difficult areas. Competitive for: irregular function behavior, small to moderate dimensions Iterated adaptive method; e.g.: b d a c f(x, y) dx dy = [ b ] a dx d c dy f(x, y) Figure : 1 1 dx 1 1 dy εy 2 θ(1 x 2 y 2 ) with a = 0.8, ε = (x 2 +y 2 a 2 ) 2 +ε 2

10 Non-adaptive methods Sequence of rules, with increasing number of points; rules may be on a lattice. Monte Carlo (MC): Q = 1 N N j=1 f( x j) Competitive for: moderate to high dimensions, irregular function behavior, irratic integration domain Quasi-Monte Carlo (QMC): In a non-adaptive sequence, a number of randomized copies are generated for each rule, and the results over the randomized copies are averaged. Competitive: up to high dimensions, fairly smooth function behavior

11 Outline Introduction 1 Introduction 2 3 4

12 An adaptive region subdivision algorithm for distributed memory systems; layered over MPI (Message Passing Interface - see, e.g., [1]); uses task pool on each node, and load balancing among nodes to handle localized integrand difficulties such as peaks and singularities An iterated adaptive method, for shared memory/multi-core systems QMC method; load balancing is applied to deal with possible heterogeneous processors MC method: implemented using CUDA [2] on GPUs Goals: to extend ParInt to run efficiently on a hybrid architecture, with distributed nodes, each multicore, and possibly with GPUs (graphics processing units).

13 In computational geometry Example: cube tetrahedron picking yields the expected volume of a random tetrahedron in a cube. Other examples: ball tetrahedron picking, tetrahedron tetrahedron picking, also sphere tetrahedron picking Expected d-dimensional (dd) volume E[V n (K)] of the polyhedron formed by n points X 1,..., X n, uniformly distributed in the interior of a convex body K (cf., [3]): K E[V n (K)] = 1 K K Hull(X 1,...,X n ) dx 1 K K denotes the d-dimensional volume of K and dxn K Hull(X 1,...,X n ) is the volume of the convex hull generated by the n points.

14 Cube = C 3 tetrahedron picking (n = 4, d = 3): E[V 4 (C 3 )] = dx 1 dy 1 dz 1 dx 2 dy 2 dz x 1 y 1 z 1 1 with V 4 = 1 x 2 y 2 z x 3 y 3 z 3 1. x 4 y 4 z dx 3 dy 3 dz 3 dx 4 dy 4 dz 4 V

15 Tetrahedron = T 3 tetrahedron picking (cf., [4, 3, 5]) 1 1 x1 E[V 4 (T 3 )] = 6 4 dx x3 dx dy 1 1 x1 y 1 0 dy 3 1 x3 y x2 dz 1 dx x4 dz 3 dx x2 y 2 dy 2 dz x4 y 4 dy 4 dz 4 V 4 0 Ball = B 3 tetrahedron picking [6, 7]: We implemented as above but with integration over: [ x 2 ] 1 x y 1 2 dx i dy i dz i 1 1 x1 2i i=1 1 x 2 1 y 2 1

16 Sphere = S 2 tetrahedron picking: According to the interpretation of [8] (the vertices are varied over S 2, and using the parameter representation: S 2 = { s(θ, u) (sin(θ) 1 u 2, cos(θ) 1 u 2, u), the integral is: 1 du du du du 2π du du du 1 3 u [ 1, 1] and θ [0, 2π]} 1 1 du 4 2π 0 2π 2π 2π dθ 0 1 dθ 0 2 dθ 0 3 dθ 0 4 V 4 2π 2π 2π 2π dθ 0 1 dθ 0 2 dθ 0 3 dθ With some simplifications (cf., [9]: u 1 = 1 (first vertex); θ 2 = 0 (second vertex)): E[V 4 (S 2 )] = 1 du du du 2π du du 1 3 2π dθ 0 3 dθ 0 4 V 4 1 du 2π 2π 1 4 dθ 0 3 dθ 0 4 1

17 Outline Introduction 1 Introduction 2 3 4

18 Random number generation The MC estimate is obtained as the avarage of the function values at a set of N uniformly distributed random points; using the CUDA pseudo-random number generator library (curand) [10] to generate N d random floats for the coordinates of N d-dimensional points, directly on the GPU (device); and using a program section to allocate space for ndim = N d floats in array a on the device, and call the CURAND functions cudamalloc, curandcreategenerator, curandsetpseudorandomgeneratorseed and curandgenerateuniform.

19 Outline Introduction 1 Introduction 2 3 4

20 # EV. PROBLEM RESULT ABS. ERR. SEQ. TIME PAR. TIME SPEEDUP (M) (s) (s) 0.1 E[V 4 (S 2 )] e e e e e+00 E[V 4 (C 3 )] e e e e e+00 E[V 4 (T 3 )] e e e e e+00 1 E[V 4 (S 2 )] e e e e e+01 E[V 4 (C 3 )] e e e e e+01 E[V 4 (T 3 )] e e e e e E[V 4 (S 2 )] e e e e e+02 E[V 4 (C 3 )] e e e e e+01 E[V 4 (T 3 )] e e e e e E[V 4 (S 2 )] e e e e e+02 E[V 4 (C 3 )] e e e e e+02 E[V 4 (T 3 )] e e e e e E[V 4 (S 2 )] e e e e e+02 E[V 4 (C 3 )] e e e e e+02 E[V 4 (T 3 )] e e e e e E[V 4 (S 2 )] e e e e e+02 E[V 4 (C 3 )] e e e e e+02 E[V 4 (T 3 )] e e e e e+02 Table : GPU results for E[V 4 (S 2 )], E[V 4 (C 3 )] and E[V 4 (T 3 )].

21 We gave a brief overview of ParInt, some applications, and described a many-core parallelization of MC using an application to problems in stochastic geometry. Point-picking problems have applications, e.g., with tetrahedron vertices on a spherical cap, for analyzing the properties of the points in a Poisson-Vonoroi distribution [8]. With respect to the point-picking type of problem, extensions are possible as well (n > 4 points, d > 3). This work is part of efforts to port the to a hybrid parallel environment.

22 References Open-MPI: CUDA: NVIDIA Developer Zone, Zinani, A.: The expected volume of a tetrahedron whose vertices are chosen at random in the interior of a cube. Monatsh. Math. 139 (2003) DOI /s y. Buchta, C., Reitzner, M.: The convex hull of random points in a tetrahedron: Solution of Blaschke s problem and more general results.

23 J. Reine Angew. Math 536 (2001) 1 29 Philip, J.: The average volume of a random tetrahedron in a tetrahedron. TRITA MAT 06 MA 02 (2006) johanph/ev.pdf. Buchta, C., Müller, J.: Random polytopes in a ball. J. Appl. Prob. 21 (1984) Affentranger, F.: The expected volume of a random polytope in a ball. J. Microscopy 151 (1988) Heinrich, L., Körner, Mehlhorn, N., Muche, L.:

24 Numerical and analytical computation of some second-order characteristics of spacial Poisson-Vonoroi tesselations. Statistics 31 (1998) Weisstein, E.W.: Sphere tetrahedron picking. In: MathWorld A Wolfram Web Resource. (2013) NVIDIA: CUDA CURAND LIBRARY

Math 265 Exam 3 Solutions

Math 265 Exam 3 Solutions C Roettger, Fall 16 Math 265 Exam 3 Solutions Problem 1 Let D be the region inside the circle r 5 sin θ but outside the cardioid r 2 + sin θ. Find the area of D. Note that r and θ denote polar coordinates.

More information

Calculus III. Math 233 Spring In-term exam April 11th. Suggested solutions

Calculus III. Math 233 Spring In-term exam April 11th. Suggested solutions Calculus III Math Spring 7 In-term exam April th. Suggested solutions This exam contains sixteen problems numbered through 6. Problems 5 are multiple choice problems, which each count 5% of your total

More information

Math 11 Fall 2016 Section 1 Monday, October 17, 2016

Math 11 Fall 2016 Section 1 Monday, October 17, 2016 Math 11 Fall 16 Section 1 Monday, October 17, 16 First, some important points from the last class: f(x, y, z) dv, the integral (with respect to volume) of f over the three-dimensional region, is a triple

More information

MATH 2400, Analytic Geometry and Calculus 3

MATH 2400, Analytic Geometry and Calculus 3 MATH 2400, Analytic Geometry and Calculus 3 List of important Definitions and Theorems 1 Foundations Definition 1. By a function f one understands a mathematical object consisting of (i) a set X, called

More information

We can conclude that if f is differentiable in an interval containing a, then. f(x) L(x) = f(a) + f (a)(x a).

We can conclude that if f is differentiable in an interval containing a, then. f(x) L(x) = f(a) + f (a)(x a). = sin( x) = 8 Lecture :Linear Approximations and Differentials Consider a point on a smooth curve y = f(x), say P = (a, f(a)), If we draw a tangent line to the curve at the point P, we can see from the

More information

Calculus IV. Exam 2 November 13, 2003

Calculus IV. Exam 2 November 13, 2003 Name: Section: Calculus IV Math 1 Fall Professor Ben Richert Exam November 1, Please do all your work in this booklet and show all the steps. Calculators and note-cards are not allowed. Problem Possible

More information

MATH. 2153, Spring 16, MWF 12:40 p.m. QUIZ 1 January 25, 2016 PRINT NAME A. Derdzinski Show all work. No calculators. The problem is worth 10 points.

MATH. 2153, Spring 16, MWF 12:40 p.m. QUIZ 1 January 25, 2016 PRINT NAME A. Derdzinski Show all work. No calculators. The problem is worth 10 points. MATH. 2153, Spring 16, MWF 12:40 p.m. QUIZ 1 January 25, 2016 PRINT NAME A. Derdzinski Show all work. No calculators. The problem is worth 10 points. 1. Evaluate the area A of the triangle with the vertices

More information

Parallel Techniques. Embarrassingly Parallel Computations. Partitioning and Divide-and-Conquer Strategies

Parallel Techniques. Embarrassingly Parallel Computations. Partitioning and Divide-and-Conquer Strategies slides3-1 Parallel Techniques Embarrassingly Parallel Computations Partitioning and Divide-and-Conquer Strategies Pipelined Computations Synchronous Computations Asynchronous Computations Load Balancing

More information

Monte Carlo Simulations on Xeon Phi: Offload and Native Mode

Monte Carlo Simulations on Xeon Phi: Offload and Native Mode Western Michigan University ScholarWorks at WMU Master's Theses Graduate College 5-2015 Monte Carlo Simulations on Xeon Phi: Offload and Native Mode Bryar Shareef Western Michigan University, bryarmustafa@gmail.com

More information

MATH 261 EXAM III PRACTICE PROBLEMS

MATH 261 EXAM III PRACTICE PROBLEMS MATH 6 EXAM III PRACTICE PROBLEMS These practice problems are pulled from actual midterms in previous semesters. Exam 3 typically has 5 (not 6!) problems on it, with no more than one problem of any given

More information

12.5 Triple Integrals

12.5 Triple Integrals 1.5 Triple Integrals Arkansas Tech University MATH 94: Calculus III r. Marcel B Finan In Sections 1.1-1., we showed how a function of two variables can be integrated over a region in -space and how integration

More information

MULTI-DIMENSIONAL MONTE CARLO INTEGRATION

MULTI-DIMENSIONAL MONTE CARLO INTEGRATION CS580: Computer Graphics KAIST School of Computing Chapter 3 MULTI-DIMENSIONAL MONTE CARLO INTEGRATION 2 1 Monte Carlo Integration This describes a simple technique for the numerical evaluation of integrals

More information

38. Triple Integration over Rectangular Regions

38. Triple Integration over Rectangular Regions 8. Triple Integration over Rectangular Regions A rectangular solid region S in R can be defined by three compound inequalities, a 1 x a, b 1 y b, c 1 z c, where a 1, a, b 1, b, c 1 and c are constants.

More information

CS 563 Advanced Topics in Computer Graphics Monte Carlo Integration: Basic Concepts. by Emmanuel Agu

CS 563 Advanced Topics in Computer Graphics Monte Carlo Integration: Basic Concepts. by Emmanuel Agu CS 563 Advanced Topics in Computer Graphics Monte Carlo Integration: Basic Concepts by Emmanuel Agu Introduction The integral equations generally don t have analytic solutions, so we must turn to numerical

More information

Double Integrals over Polar Coordinate

Double Integrals over Polar Coordinate 1. 15.4 DOUBLE INTEGRALS OVER POLAR COORDINATE 1 15.4 Double Integrals over Polar Coordinate 1. Polar Coordinates. The polar coordinates (r, θ) of a point are related to the rectangular coordinates (x,y)

More information

Section Parametrized Surfaces and Surface Integrals. (I) Parametrizing Surfaces (II) Surface Area (III) Scalar Surface Integrals

Section Parametrized Surfaces and Surface Integrals. (I) Parametrizing Surfaces (II) Surface Area (III) Scalar Surface Integrals Section 16.4 Parametrized Surfaces and Surface Integrals (I) Parametrizing Surfaces (II) Surface Area (III) Scalar Surface Integrals MATH 127 (Section 16.4) Parametrized Surfaces and Surface Integrals

More information

Triple Integrals. MATH 311, Calculus III. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Triple Integrals

Triple Integrals. MATH 311, Calculus III. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Triple Integrals Triple Integrals MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 211 Riemann Sum Approach Suppose we wish to integrate w f (x, y, z), a continuous function, on the box-shaped region

More information

Math 241, Final Exam. 12/11/12.

Math 241, Final Exam. 12/11/12. Math, Final Exam. //. No notes, calculator, or text. There are points total. Partial credit may be given. ircle or otherwise clearly identify your final answer. Name:. (5 points): Equation of a line. Find

More information

Integration using Transformations in Polar, Cylindrical, and Spherical Coordinates

Integration using Transformations in Polar, Cylindrical, and Spherical Coordinates ections 15.4 Integration using Transformations in Polar, Cylindrical, and pherical Coordinates Cylindrical Coordinates pherical Coordinates MATH 127 (ection 15.5) Applications of Multiple Integrals The

More information

Coordinate Transformations in Advanced Calculus

Coordinate Transformations in Advanced Calculus Coordinate Transformations in Advanced Calculus by Sacha Nandlall T.A. for MATH 264, McGill University Email: sacha.nandlall@mail.mcgill.ca Website: http://www.resanova.com/teaching/calculus/ Fall 2006,

More information

First we consider how to parameterize a surface (similar to a parameterized curve for line integrals). Surfaces will need two parameters.

First we consider how to parameterize a surface (similar to a parameterized curve for line integrals). Surfaces will need two parameters. Math 55 - Vector Calculus II Notes 14.6 urface Integrals Let s develop some surface integrals. First we consider how to parameterize a surface (similar to a parameterized curve for line integrals). urfaces

More information

Homework 8. Due: Tuesday, March 31st, 2009

Homework 8. Due: Tuesday, March 31st, 2009 MATH 55 Applied Honors Calculus III Winter 9 Homework 8 Due: Tuesday, March 3st, 9 Section 6.5, pg. 54: 7, 3. Section 6.6, pg. 58:, 3. Section 6.7, pg. 66: 3, 5, 47. Section 6.8, pg. 73: 33, 38. Section

More information

Lighting affects appearance

Lighting affects appearance Lighting affects appearance 1 Image Normalization Global Histogram Equalization. Make two images have same histogram. Or, pick a standard histogram, and make adjust each image to have that histogram. Apply

More information

Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Homework 1 - Solutions 3. 2 Homework 2 - Solutions 13

Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Homework 1 - Solutions 3. 2 Homework 2 - Solutions 13 MATH 32B-2 (8) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Homework - Solutions 3 2 Homework 2 - Solutions 3 3 Homework 3 - Solutions 9 MATH 32B-2 (8) (L) G. Liu / (TA) A. Zhou Calculus

More information

Asymptotic Properties of Random Polytopes

Asymptotic Properties of Random Polytopes Asymptotic Properties of Random Polytopes SPUR inal Paper, Summer 5 Zach Izzo Project suggested and mentored by Nick Strehlke July, 5 Abstract A random polytope is the convex hull of points chosen randomly

More information

CHAPTER 5 NUMERICAL INTEGRATION METHODS OVER N- DIMENSIONAL REGIONS USING GENERALIZED GAUSSIAN QUADRATURE

CHAPTER 5 NUMERICAL INTEGRATION METHODS OVER N- DIMENSIONAL REGIONS USING GENERALIZED GAUSSIAN QUADRATURE CHAPTER 5 NUMERICAL INTEGRATION METHODS OVER N- DIMENSIONAL REGIONS USING GENERALIZED GAUSSIAN QUADRATURE 5.1 Introduction Multidimensional integration appears in many mathematical models and can seldom

More information

Lecture 7: Monte Carlo Rendering. MC Advantages

Lecture 7: Monte Carlo Rendering. MC Advantages Lecture 7: Monte Carlo Rendering CS 6620, Spring 2009 Kavita Bala Computer Science Cornell University MC Advantages Convergence rate of O( ) Simple Sampling Point evaluation Can use black boxes General

More information

Math 407 Solutions to Exam 3 April 24, Let f = u + iv, and let Γ be parmetrized by Γ = (x (t),y(t)) = x (t) +iy (t) for a t b.

Math 407 Solutions to Exam 3 April 24, Let f = u + iv, and let Γ be parmetrized by Γ = (x (t),y(t)) = x (t) +iy (t) for a t b. Math 7 Solutions to Exam 3 April, 3 1. (1 Define the following: (a f (z dz, where Γ is a smooth curve with finite length. Γ Let f u + iv, and let Γ be parmetrized by Γ (x (t,y(t x (t +iy (t for a t b.

More information

f (Pijk ) V. may form the Riemann sum: . Definition. The triple integral of f over the rectangular box B is defined to f (x, y, z) dv = lim

f (Pijk ) V. may form the Riemann sum: . Definition. The triple integral of f over the rectangular box B is defined to f (x, y, z) dv = lim Chapter 14 Multiple Integrals..1 Double Integrals, Iterated Integrals, Cross-sections.2 Double Integrals over more general regions, Definition, Evaluation of Double Integrals, Properties of Double Integrals.3

More information

Halftoning and quasi-monte Carlo

Halftoning and quasi-monte Carlo Halftoning and quasi-monte Carlo Ken Hanson CCS-2, Methods for Advanced Scientific Simulations Los Alamos National Laboratory This presentation available at http://www.lanl.gov/home/kmh/ LA-UR-04-1854

More information

18.S34 (FALL 2007) PROBLEMS ON HIDDEN INDEPENDENCE AND UNIFORMITY

18.S34 (FALL 2007) PROBLEMS ON HIDDEN INDEPENDENCE AND UNIFORMITY 18.S34 (FALL 2007) PROBLEMS ON HIDDEN INDEPENDENCE AND UNIFORMITY All the problems below (with the possible exception of the last one), when looked at the right way, can be solved by elegant arguments

More information

Laguerre Tessellations

Laguerre Tessellations Voronoi Tessellation Applications of Voronoi tessellations Laguerre tessellation Voronoi cells vs. Laguerre cells Normal tessellations are LT Claudia Lautensack 1 2 1 University of Applied Sciences, Darmstadt,

More information

Polyhedra with Spherical Faces and Quasi-Fuchsian Fractals

Polyhedra with Spherical Faces and Quasi-Fuchsian Fractals Topology and Computer 2017 Polyhedra with Spherical Faces and Quasi-Fuchsian Fractals Kento Nakamura Graduate School of Advanced Mathematical Science, Meiji University Sphairahedron sphaira- (= spherical)

More information

Integration. Volume Estimation

Integration. Volume Estimation Monte Carlo Integration Lab Objective: Many important integrals cannot be evaluated symbolically because the integrand has no antiderivative. Traditional numerical integration techniques like Newton-Cotes

More information

Chapter 1. Introduction

Chapter 1. Introduction Chapter 1 Introduction A Monte Carlo method is a compuational method that uses random numbers to compute (estimate) some quantity of interest. Very often the quantity we want to compute is the mean of

More information

Chapter 15 Notes, Stewart 7e

Chapter 15 Notes, Stewart 7e Contents 15.2 Iterated Integrals..................................... 2 15.3 Double Integrals over General Regions......................... 5 15.4 Double Integrals in Polar Coordinates..........................

More information

Efficacy of Numerically Approximating Pi with an N-sided Polygon

Efficacy of Numerically Approximating Pi with an N-sided Polygon Peter Vu Brewer MAT66 Honors Topic Efficacy of umerically Approximating Pi with an -sided Polygon The quest for precisely finding the irrational number pi has been an endeavor since early human history.

More information

MAC2313 Test 3 A E g(x, y, z) dy dx dz

MAC2313 Test 3 A E g(x, y, z) dy dx dz MAC2313 Test 3 A (5 pts) 1. If the function g(x, y, z) is integrated over the cylindrical solid bounded by x 2 + y 2 = 3, z = 1, and z = 7, the correct integral in Cartesian coordinates is given by: A.

More information

Triple Integrals. Be able to set up and evaluate triple integrals over rectangular boxes.

Triple Integrals. Be able to set up and evaluate triple integrals over rectangular boxes. SUGGESTED REFERENCE MATERIAL: Triple Integrals As you work through the problems listed below, you should reference Chapters 4.5 & 4.6 of the recommended textbook (or the equivalent chapter in your alternative

More information

Computation of Voronoi Diagrams and Delaunay Triangulation via Parametric Linear Programming

Computation of Voronoi Diagrams and Delaunay Triangulation via Parametric Linear Programming Computation of Voronoi Diagrams and Delaunay Triangulation via Parametric Linear Programming Saša V. Raković, Pascal Grieder and Colin Jones September 14, 2004 Abstract This note illustrates how Voronoi

More information

1 Double Integrals over Rectangular Regions

1 Double Integrals over Rectangular Regions Contents ouble Integrals over Rectangular Regions ouble Integrals Over General Regions 7. Introduction.................................... 7. Areas of General Regions............................. 9.3 Region

More information

Math 210, Exam 2, Spring 2010 Problem 1 Solution

Math 210, Exam 2, Spring 2010 Problem 1 Solution Math, Exam, Spring Problem Solution. Find and classify the critical points of the function f(x,y) x 3 +3xy y 3. Solution: By definition, an interior point (a,b) in the domain of f is a critical point of

More information

Math 2374 Spring 2007 Midterm 3 Solutions - Page 1 of 6 April 25, 2007

Math 2374 Spring 2007 Midterm 3 Solutions - Page 1 of 6 April 25, 2007 Math 374 Spring 7 Midterm 3 Solutions - Page of 6 April 5, 7. (3 points) Consider the surface parametrized by (x, y, z) Φ(x, y) (x, y,4 (x +y )) between the planes z and z 3. (i) (5 points) Set up the

More information

MATH 19520/51 Class 15

MATH 19520/51 Class 15 MATH 19520/51 Class 15 Minh-Tam Trinh University of Chicago 2017-11-01 1 Change of variables in two dimensions. 2 Double integrals via change of variables. Change of Variables Slogan: An n-variable substitution

More information

An Interface-fitted Mesh Generator and Polytopal Element Methods for Elliptic Interface Problems

An Interface-fitted Mesh Generator and Polytopal Element Methods for Elliptic Interface Problems An Interface-fitted Mesh Generator and Polytopal Element Methods for Elliptic Interface Problems Long Chen University of California, Irvine chenlong@math.uci.edu Joint work with: Huayi Wei (Xiangtan University),

More information

Math 113 Calculus III Final Exam Practice Problems Spring 2003

Math 113 Calculus III Final Exam Practice Problems Spring 2003 Math 113 Calculus III Final Exam Practice Problems Spring 23 1. Let g(x, y, z) = 2x 2 + y 2 + 4z 2. (a) Describe the shapes of the level surfaces of g. (b) In three different graphs, sketch the three cross

More information

1. Suppose that the equation F (x, y, z) = 0 implicitly defines each of the three variables x, y, and z as functions of the other two:

1. Suppose that the equation F (x, y, z) = 0 implicitly defines each of the three variables x, y, and z as functions of the other two: Final Solutions. Suppose that the equation F (x, y, z) implicitly defines each of the three variables x, y, and z as functions of the other two: z f(x, y), y g(x, z), x h(y, z). If F is differentiable

More information

Outline: Embarrassingly Parallel Problems. Example#1: Computation of the Mandelbrot Set. Embarrassingly Parallel Problems. The Mandelbrot Set

Outline: Embarrassingly Parallel Problems. Example#1: Computation of the Mandelbrot Set. Embarrassingly Parallel Problems. The Mandelbrot Set Outline: Embarrassingly Parallel Problems Example#1: Computation of the Mandelbrot Set what they are Mandelbrot Set computation cost considerations static parallelization dynamic parallelizations and its

More information

COMP331/557. Chapter 2: The Geometry of Linear Programming. (Bertsimas & Tsitsiklis, Chapter 2)

COMP331/557. Chapter 2: The Geometry of Linear Programming. (Bertsimas & Tsitsiklis, Chapter 2) COMP331/557 Chapter 2: The Geometry of Linear Programming (Bertsimas & Tsitsiklis, Chapter 2) 49 Polyhedra and Polytopes Definition 2.1. Let A 2 R m n and b 2 R m. a set {x 2 R n A x b} is called polyhedron

More information

Exploring Domains of Approximation in R 2 : Expository Essay

Exploring Domains of Approximation in R 2 : Expository Essay Exploring Domains of Approximation in R 2 : Expository Essay Nicolay Postarnakevich August 12, 2013 1 Introduction In this paper I explore the concept of the Domains of Best Approximations. These structures

More information

MATH 010B - Spring 2018 Worked Problems - Section 6.2. ze x2 +y 2

MATH 010B - Spring 2018 Worked Problems - Section 6.2. ze x2 +y 2 MATH B - Spring 8 orked Problems - Section 6.. Compute the following double integral x +y 9 z 3 ze x +y dv Solution: Here, we can t hope to integrate this directly in Cartesian coordinates, since the the

More information

Math 2130 Practice Problems Sec Name. Change the Cartesian integral to an equivalent polar integral, and then evaluate.

Math 2130 Practice Problems Sec Name. Change the Cartesian integral to an equivalent polar integral, and then evaluate. Math 10 Practice Problems Sec 1.-1. Name Change the Cartesian integral to an equivalent polar integral, and then evaluate. 1) 5 5 - x dy dx -5 0 A) 5 B) C) 15 D) 5 ) 0 0-8 - 6 - x (8 + ln 9) A) 1 1 + x

More information

To find the maximum and minimum values of f(x, y, z) subject to the constraints

To find the maximum and minimum values of f(x, y, z) subject to the constraints Midterm 3 review Math 265 Fall 2007 14.8. Lagrange Multipliers. Case 1: One constraint. To find the maximum and minimum values of f(x, y, z) subject to the constraint g(x, y, z) = k: Step 1: Find all values

More information

We have set up our axioms to deal with the geometry of space but have not yet developed these ideas much. Let s redress that imbalance.

We have set up our axioms to deal with the geometry of space but have not yet developed these ideas much. Let s redress that imbalance. Solid geometry We have set up our axioms to deal with the geometry of space but have not yet developed these ideas much. Let s redress that imbalance. First, note that everything we have proven for the

More information

7. The Gauss-Bonnet theorem

7. The Gauss-Bonnet theorem 7. The Gauss-Bonnet theorem 7.1 Hyperbolic polygons In Euclidean geometry, an n-sided polygon is a subset of the Euclidean plane bounded by n straight lines. Thus the edges of a Euclidean polygon are formed

More information

Contents. 3 Multiple Integration. 3.1 Double Integrals in Rectangular Coordinates

Contents. 3 Multiple Integration. 3.1 Double Integrals in Rectangular Coordinates Calculus III (part 3): Multiple Integration (by Evan Dummit, 8, v. 3.) Contents 3 Multiple Integration 3. Double Integrals in Rectangular Coordinates............................... 3.. Double Integrals

More information

MAC2313 Final A. a. The vector r u r v lies in the tangent plane of S at a given point. b. S f(x, y, z) ds = R f(r(u, v)) r u r v du dv.

MAC2313 Final A. a. The vector r u r v lies in the tangent plane of S at a given point. b. S f(x, y, z) ds = R f(r(u, v)) r u r v du dv. MAC2313 Final A (5 pts) 1. Let f(x, y, z) be a function continuous in R 3 and let S be a surface parameterized by r(u, v) with the domain of the parameterization given by R; how many of the following are

More information

Math 241 Spring 2015 Final Exam Solutions

Math 241 Spring 2015 Final Exam Solutions Math 4 Spring 5 Final Exam Solutions. Find the equation of the plane containing the line x y z+ and the point (,,). Write [ pts] your final answer in the form ax+by +cz d. Solution: A vector parallel to

More information

University of California, Berkeley

University of California, Berkeley University of California, Berkeley FINAL EXAMINATION, Fall 2012 DURATION: 3 hours Department of Mathematics MATH 53 Multivariable Calculus Examiner: Sean Fitzpatrick Total: 100 points Family Name: Given

More information

MATH 251 Fall 2016 EXAM III - VERSION A

MATH 251 Fall 2016 EXAM III - VERSION A MATH 51 Fall 16 EXAM III - VERSION A LAST NAME: FIRST NAME: SECTION NUMBER: UIN: DIRECTIONS: 1. You may use a calculator on this exam.. TURN OFF cell phones and put them away. If a cell phone is seen during

More information

Examples from Section 7.1: Integration by Parts Page 1

Examples from Section 7.1: Integration by Parts Page 1 Examples from Section 7.: Integration by Parts Page Questions Example Determine x cos x dx. Example e θ cos θ dθ Example You may wonder why we do not add a constant at the point where we integrate for

More information

of Nebraska - Lincoln

of Nebraska - Lincoln University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2008 Archimedean Solids Anna Anderson University of

More information

MIDTERM. Section: Signature:

MIDTERM. Section: Signature: MIDTERM Math 32B 8/8/2 Name: Section: Signature: Read all of the following information before starting the exam: Check your exam to make sure all pages are present. NO CALCULATORS! Show all work, clearly

More information

Applications of MPI I

Applications of MPI I Applications of MPI I N-Dimensional Integration Using Monte Carlo Techniques Gürsan ÇOBAN 4.06.202 Outline Numerical Integration Techniques Monte Carlo Techniques for Numerical Integration Some MPI Examples

More information

POLYHEDRAL GEOMETRY. Convex functions and sets. Mathematical Programming Niels Lauritzen Recall that a subset C R n is convex if

POLYHEDRAL GEOMETRY. Convex functions and sets. Mathematical Programming Niels Lauritzen Recall that a subset C R n is convex if POLYHEDRAL GEOMETRY Mathematical Programming Niels Lauritzen 7.9.2007 Convex functions and sets Recall that a subset C R n is convex if {λx + (1 λ)y 0 λ 1} C for every x, y C and 0 λ 1. A function f :

More information

2 Geometry Solutions

2 Geometry Solutions 2 Geometry Solutions jacques@ucsd.edu Here is give problems and solutions in increasing order of difficulty. 2.1 Easier problems Problem 1. What is the minimum number of hyperplanar slices to make a d-dimensional

More information

CS 372: Computational Geometry Lecture 10 Linear Programming in Fixed Dimension

CS 372: Computational Geometry Lecture 10 Linear Programming in Fixed Dimension CS 372: Computational Geometry Lecture 10 Linear Programming in Fixed Dimension Antoine Vigneron King Abdullah University of Science and Technology November 7, 2012 Antoine Vigneron (KAUST) CS 372 Lecture

More information

Math 241, Exam 3 Information.

Math 241, Exam 3 Information. Math 241, xam 3 Information. 11/28/12, LC 310, 11:15-12:05. xam 3 will be based on: Sections 15.2-15.4, 15.6-15.8. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

More information

Linear Programming in Small Dimensions

Linear Programming in Small Dimensions Linear Programming in Small Dimensions Lekcija 7 sergio.cabello@fmf.uni-lj.si FMF Univerza v Ljubljani Edited from slides by Antoine Vigneron Outline linear programming, motivation and definition one dimensional

More information

Convex Hulls in Three Dimensions. Polyhedra

Convex Hulls in Three Dimensions. Polyhedra Convex Hulls in Three Dimensions Polyhedra Polyhedron 1.A polyhedron is the generalization of a 2- D polygon to 3-D A finite number of flat polygonal faces The boundary or surface of a polyhedron - Zero-dimensional

More information

8(x 2) + 21(y 1) + 6(z 3) = 0 8x + 21y + 6z = 55.

8(x 2) + 21(y 1) + 6(z 3) = 0 8x + 21y + 6z = 55. MATH 24 -Review for Final Exam. Let f(x, y, z) x 2 yz + y 3 z x 2 + z, and a (2,, 3). Note: f (2xyz 2x, x 2 z + 3y 2 z, x 2 y + y 3 + ) f(a) (8, 2, 6) (a) Find all stationary points (if any) of f. et f.

More information

Math Boot Camp: Coordinate Systems

Math Boot Camp: Coordinate Systems Math Boot Camp: Coordinate Systems You can skip this boot camp if you can answer the following question: Staying on a sphere of radius R, what is the shortest distance between the point (0, 0, R) on the

More information

Voronoi Diagram. Xiao-Ming Fu

Voronoi Diagram. Xiao-Ming Fu Voronoi Diagram Xiao-Ming Fu Outlines Introduction Post Office Problem Voronoi Diagram Duality: Delaunay triangulation Centroidal Voronoi tessellations (CVT) Definition Applications Algorithms Outlines

More information

Outline: Embarrassingly Parallel Problems

Outline: Embarrassingly Parallel Problems Outline: Embarrassingly Parallel Problems what they are Mandelbrot Set computation cost considerations static parallelization dynamic parallelizations and its analysis Monte Carlo Methods parallel random

More information

Explore 3D Figures. Dr. Jing Wang (517) , Lansing Community College, Michigan, USA

Explore 3D Figures. Dr. Jing Wang (517) , Lansing Community College, Michigan, USA Explore 3D Figures Dr. Jing Wang (517)2675965, wangj@lcc.edu Lansing Community College, Michigan, USA Part I. 3D Modeling In this part, we create 3D models using Mathematica for various solids in 3D space,

More information

) in the k-th subbox. The mass of the k-th subbox is M k δ(x k, y k, z k ) V k. Thus,

) in the k-th subbox. The mass of the k-th subbox is M k δ(x k, y k, z k ) V k. Thus, 1 Triple Integrals Mass problem. Find the mass M of a solid whose density (the mass per unit volume) is a continuous nonnegative function δ(x, y, z). 1. Divide the box enclosing into subboxes, and exclude

More information

Monte Carlo Methods and Statistical Computing: My Personal E

Monte Carlo Methods and Statistical Computing: My Personal E Monte Carlo Methods and Statistical Computing: My Personal Experience Department of Mathematics & Statistics Indian Institute of Technology Kanpur November 29, 2014 Outline Preface 1 Preface 2 3 4 5 6

More information

MATH 52 MIDTERM I APRIL 22, 2009

MATH 52 MIDTERM I APRIL 22, 2009 MATH 52 MIDTERM I APRIL 22, 2009 THIS IS A CLOSED BOOK, CLOSED NOTES EXAM. NO CALCULATORS OR OTHER ELECTRONIC DEVICES ARE PERMITTED. YOU DO NOT NEED TO EVALUATE ANY INTEGRALS IN ANY PROBLEM. THERE ARE

More information

Contents. Preface... VII. Part I Classical Topics Revisited

Contents. Preface... VII. Part I Classical Topics Revisited Contents Preface........................................................ VII Part I Classical Topics Revisited 1 Sphere Packings........................................... 3 1.1 Kissing Numbers of Spheres..............................

More information

ORIE 6300 Mathematical Programming I September 2, Lecture 3

ORIE 6300 Mathematical Programming I September 2, Lecture 3 ORIE 6300 Mathematical Programming I September 2, 2014 Lecturer: David P. Williamson Lecture 3 Scribe: Divya Singhvi Last time we discussed how to take dual of an LP in two different ways. Today we will

More information

Background for Surface Integration

Background for Surface Integration Background for urface Integration 1 urface Integrals We have seen in previous work how to define and compute line integrals in R 2. You should remember the basic surface integrals that we will need to

More information

Curves: We always parameterize a curve with a single variable, for example r(t) =

Curves: We always parameterize a curve with a single variable, for example r(t) = Final Exam Topics hapters 16 and 17 In a very broad sense, the two major topics of this exam will be line and surface integrals. Both of these have versions for scalar functions and vector fields, and

More information

be a polytope. has such a representation iff it contains the origin in its interior. For a generic, sort the inequalities so that

be a polytope. has such a representation iff it contains the origin in its interior. For a generic, sort the inequalities so that ( Shelling (Bruggesser-Mani 1971) and Ranking Let be a polytope. has such a representation iff it contains the origin in its interior. For a generic, sort the inequalities so that. a ranking of vertices

More information

A small review, Second Midterm, Calculus 3, Prof. Montero 3450: , Fall 2008

A small review, Second Midterm, Calculus 3, Prof. Montero 3450: , Fall 2008 A small review, Second Midterm, Calculus, Prof. Montero 45:-4, Fall 8 Maxima and minima Let us recall first, that for a function f(x, y), the gradient is the vector ( f)(x, y) = ( ) f f (x, y); (x, y).

More information

Chapter 10 Homework: Parametric Equations and Polar Coordinates

Chapter 10 Homework: Parametric Equations and Polar Coordinates Chapter 1 Homework: Parametric Equations and Polar Coordinates Name Homework 1.2 1. Consider the parametric equations x = t and y = 3 t. a. Construct a table of values for t =, 1, 2, 3, and 4 b. Plot the

More information

implicit surfaces, approximate implicitization, B-splines, A- patches, surface fitting

implicit surfaces, approximate implicitization, B-splines, A- patches, surface fitting 24. KONFERENCE O GEOMETRII A POČÍTAČOVÉ GRAFICE ZBYNĚK ŠÍR FITTING OF PIECEWISE POLYNOMIAL IMPLICIT SURFACES Abstrakt In our contribution we discuss the possibility of an efficient fitting of piecewise

More information

MONTE CARLO EVALUATION OF DEFINITE INTEGRALS

MONTE CARLO EVALUATION OF DEFINITE INTEGRALS MISN-0-355 MONTE CARLO EVALUATION OF DEFINITE INTEGRALS by Robert Ehrlich MONTE CARLO EVALUATION OF DEFINITE INTEGRALS y 1. Integrals, 1-2 Dimensions a. Introduction.............................................1

More information

Outline. CGAL par l exemplel. Current Partners. The CGAL Project.

Outline. CGAL par l exemplel. Current Partners. The CGAL Project. CGAL par l exemplel Computational Geometry Algorithms Library Raphaëlle Chaine Journées Informatique et GéomG ométrie 1 er Juin 2006 - LIRIS Lyon Outline Overview Strengths Design Structure Kernel Convex

More information

The Volume of a Platonic Solid

The Volume of a Platonic Solid University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-007 The Volume of a Platonic Solid Cindy Steinkruger

More information

MATH 200 WEEK 9 - WEDNESDAY TRIPLE INTEGRALS

MATH 200 WEEK 9 - WEDNESDAY TRIPLE INTEGRALS MATH WEEK 9 - WEDNESDAY TRIPLE INTEGRALS MATH GOALS Be able to set up and evaluate triple integrals using rectangular, cylindrical, and spherical coordinates MATH TRIPLE INTEGRALS We integrate functions

More information

Outline of the presentation

Outline of the presentation Surface Reconstruction Petra Surynková Charles University in Prague Faculty of Mathematics and Physics petra.surynkova@mff.cuni.cz Outline of the presentation My work up to now Surfaces of Building Practice

More information

Math 209 (Fall 2007) Calculus III. Solution #5. 1. Find the minimum and maximum values of the following functions f under the given constraints:

Math 209 (Fall 2007) Calculus III. Solution #5. 1. Find the minimum and maximum values of the following functions f under the given constraints: Math 9 (Fall 7) Calculus III Solution #5. Find the minimum and maximum values of the following functions f under the given constraints: (a) f(x, y) 4x + 6y, x + y ; (b) f(x, y) x y, x + y 6. Solution:

More information

Lesson 10. Transforming 3D Integrals

Lesson 10. Transforming 3D Integrals Lesson 10 Transforming D Integrals Example 1: Triple Integrals to Compute Volume ecall that in previous chapters we could find the length of an interval I by computing dx or the area of a region by computing

More information

Worksheet 3.4: Triple Integrals in Cylindrical Coordinates. Warm-Up: Cylindrical Volume Element d V

Worksheet 3.4: Triple Integrals in Cylindrical Coordinates. Warm-Up: Cylindrical Volume Element d V Boise State Math 275 (Ultman) Worksheet 3.4: Triple Integrals in Cylindrical Coordinates From the Toolbox (what you need from previous classes) Know what the volume element dv represents. Be able to find

More information

R f da (where da denotes the differential of area dxdy (or dydx)

R f da (where da denotes the differential of area dxdy (or dydx) Math 28H Topics for the second exam (Technically, everything covered on the first exam, plus) Constrained Optimization: Lagrange Multipliers Most optimization problems that arise naturally are not unconstrained;

More information

UNIVERSITI TEKNOLOGI MALAYSIA SSCE 1993 ENGINEERING MATHEMATICS II TUTORIAL 2. 1 x cos dy dx x y dy dx. y cosxdy dx

UNIVERSITI TEKNOLOGI MALAYSIA SSCE 1993 ENGINEERING MATHEMATICS II TUTORIAL 2. 1 x cos dy dx x y dy dx. y cosxdy dx UNIVESITI TEKNOLOI MALAYSIA SSCE 99 ENINEEIN MATHEMATICS II TUTOIAL. Evaluate the following iterated integrals. (e) (g) (i) x x x sinx x e x y dy dx x dy dx y y cosxdy dx xy x + dxdy (f) (h) (y + x)dy

More information

Multiple Integrals. max x i 0

Multiple Integrals. max x i 0 Multiple Integrals 1 Double Integrals Definite integrals appear when one solves Area problem. Find the area A of the region bounded above by the curve y = f(x), below by the x-axis, and on the sides by

More information

Week 7 Convex Hulls in 3D

Week 7 Convex Hulls in 3D 1 Week 7 Convex Hulls in 3D 2 Polyhedra A polyhedron is the natural generalization of a 2D polygon to 3D 3 Closed Polyhedral Surface A closed polyhedral surface is a finite set of interior disjoint polygons

More information

Q-Gorenstein deformation families of Fano varieties or The combinatorics of Mirror Symmetry. Alexander Kasprzyk

Q-Gorenstein deformation families of Fano varieties or The combinatorics of Mirror Symmetry. Alexander Kasprzyk Q-Gorenstein deformation families of Fano varieties or The combinatorics of Mirror Symmetry Alexander Kasprzyk Fano manifolds Smooth varieties, called manifolds, come with a natural notion of curvature,

More information

Chapter 4 Concepts from Geometry

Chapter 4 Concepts from Geometry Chapter 4 Concepts from Geometry An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Line Segments The line segment between two points and in R n is the set of points on the straight line joining

More information