GPUs –Level sets on GPUs Intended Audience Anyone interested **in** accelerated **computing** –from all academic disciplines No **graphics** background required –will cover all necessary and relevant detail **in** the course What is required… –working knowledge **of** C/C++ –**linear** **algebra** –enthusiasm and an open mind Course Schedule Understanding the fabric: **computer** **graphics** basics Overview **of** GPUs –architecture –features –programming model –some simple **applications** System issues –cache and data management, –languages and/

simply teach mathematics. Move heavily to the **application** **of** the concepts within the mathematics classroom. Mechanical and Manufacturing Technology Some Major Themes **in** Technology 6. Technology **in** the Mathematics Classroom Students should develop proficiency with at least one [**computation** software] program, as well as a working knowledge **of** spreadsheets. Electronics and Telecommunications Students should use a variety **of** software packages **in** mathematics classes. … students should be able to/

**Computation**, Vol II »Landmark **in** the development **of** numerical algorithms and software »Basis for a number **of** software projects EISPACK, a number **of** **linear** **algebra** routines **in** IMSL and the F chapters **of** NAG –IBM 370/195 »Pipelined architecture; Out **of**/64 »More **computing** power than a Cray 1 and much much better **graphics** 1997 –MPI-2 Finished –Fortran 95 1998 –Issues **of** parallel and/HPF model A group **of** 30-40 “experts” **in** message- passing: –MPP vendors, –CS researchers, –**Application** developers. Met 3 days/

**in** this topic because **of** its relation to parallel **computing** Many researchers are interested **in** this topic because **of** its relation to parallel **computing** **Graphics** processors provide the most number **of** flops per dollar (by an order **of** magnitude) **Graphics** processors provide the most number **of** flops per dollar (by an order **of**/! Case study: **Linear** **Algebra** on GPUs **Linear** **algebra** is a problem that does not immediately map well to modern cache based CPUs (need to optimize for cache) **Linear** **algebra** is a problem that/

featured/brave-new-hair/ – **graphics**.pixar.com/library/CurlyHairA/paper.pdf **graphics**.pixar.com/library/CurlyHairA/paper./**applications**: – 3D LMC (a low-mach number combustion code) 2.5x **in** bottom solve, 1.5x overall GMG solve – 3D Nyx (an N-body and gas dynamics code) 2x **in** bottom solve, 1.15x overall GMG solve Summary **of** Iterative **Linear** **Algebra** New lower bounds, optimal algorithms, big speedups **in** theory and practice Lots **of**/. For each input vector: Dot products are **computed** using 1, 2, 3 or 4 threads /

SRI **Applications**: Robotics **Applications**: Surveillance Mathematical tools **Linear** **algebra** Vector calculus Euclidean geometry Projective geometry Differential geometry Differential equations Numerical analysis Probability and statistics Programming tools OpenCV – an open source library for **computer** vision. Ch – a C interpretation environment. Course Organization Textbook: Introductory Techniques for 3-D **Computer** Vision, by Trucco and Verri Two parts: Part I (Chang Shu) – Introduction, Review **of** **linear** **algebra**/

elimination. Many **linear** **algebra** textbooks incorrectly assert that such techniques are better suited for implementation on a **computer** because they require fewer arithmetic operations. This is true for large matrices, or for matrices with a structure that can be exploited. However, for arbitrary matrices **of** smaller order like the 2 x 2, 3 x 3, and 4 x 4 used most often **in** geometric **applications**, the/

B and Additional Topics **in** **Linear** Programming Dr. Steven Harrod Topics Definition **of** **Linear** Programming Steps to Formulate a Problem **Applications** Mathematical Requirements **Linear** functions Convex, non-negative feasible region Single objective Steps to Formulate a Problem Solution Techniques **Graphical** Simplex **Linear** Programming A mathematical tool to make multiple decisions affecting a single outcome or objective Guaranteed to find the optimal solution (best possible set **of** decisions) relative to/

. When we say “transformation,” we usually mean a composition **of** scale, rotate and translate transforms Object Coordinates Display **Application** Coordinates 2D **Graphics** using OpenGL – 9/9/2014 Transformations (1/3) We will use GLM to do **linear** **algebra** for us and build the mapping matrix **In** addition to the projection matrix mentioned earlier, also keep track **of** a model and a view matrix. More about the significance/

**in** place, we are now free to perform rotations using the matrix at a higher level **of** abstraction. Our interface functions match exactly the high-level intentions **of** the programmer. Chapter 8 Notes 3D Math Primer for **Graphics** & Game Dev 3D Math Primer for **Graphics** & Game Dev Some Justification Furthermore, we have removed the confusing **linear** **algebra**/ operations can make very quick work **of** matrix multiplication. Quaternion conjugate provides a way to **compute** the opposite angular displacement very efficiently./

and Patrikalakis 1992, Held 1998, Ju-Hsein Kao 1999]. Center for **Graphics** and Geometric **Computing**, Technion 7 Our approach Using the original representation **of** the input curves (with no **linear** / circular approximation). Generate an accurate implicit representation **of** the Voronoi cell. ExactApproximated Center for **Graphics** and Geometric **Computing**, Technion 8 Outline **of** the algorithm Implicit bisector function **Application** **of** constraints Lower envelope algorithm Splitting into monotone pieces tr-space/

ONC, 110 EXT, 1 st Engineering and Science Details **of** the study – course 1 S2, 2004. UQ Calculus and **Linear** **Algebra** I (all) 320 students Engineering and Science One lecturer Initially OHP and **computer** **Graphics** tablet (A6) PDF No technical problems Notes made available on website afterwards Birgit Details **of** the study – course 2 Two lecturers **Linear** **Algebra** taught with **graphics** tablet (A3) Calculus taught writing on OHP PDF/

around a Harrier Jet (NASA Ames) **Graphics** **Applications** **Computer** Aided Design (CAD) **Graphics** **Applications** TrainingTraining Designing Effective Step-By-Step Assembly Instructions (Maneesh Agrawala et. al) **Graphics** **Applications** Entertainment: Games GT Racer 3 Polyphony Digital: Gran Turismo 3, A Spec Games Circus Atari (Atari) Education Outside **In** (Geometry Center, University **of** Minnesota) The Basics **Computer** **graphics**: generating 2D images **of** a 3D world represented **in** a **computer**. Main tasks: modeling: (shape/

, cloud, heterogeneous, low-energy, … Integration into **applications** – ASPIRE Lab – **computer** vision video background subtraction, optical flow, … – AMPLab – building library using Spark for data analysis – CTF (with ANL): symmetric tensor contractions, on BG/Q Outline Survey state **of** the art **of** CA (Comm-Avoiding) algorithms – TSQR: Tall-Skinny QR – CA O(n 3 ) 2.5D Matmul – Sparse Matrices Beyond **linear** **algebra** – Extending lower bounds to any algorithm/

Categories Expressions and Operations Equations and Inequalities Functions Statistics Major Changes to **Algebra** II Content moved to Mathematical Analysis matrices conic sections Content eliminated from **Algebra** II **linear** programming (it resides still **in** AFDA) Content moved to **Algebra** I Inverse variation Systems **of** **Linear** Equations and Inequalities Content added permutations and combinations the normal distribution **Algebra** II Expressions and Operations STANDARD AII.1 The student, given rational/

**in** a variety **of** ways – verbally, **graphically**, using informal written methods. Skills Mathematics Key Stage 3 3. Reason mathematically Pupils should be given opportunities to: extend mental methods **of** **computation** to consolidate a range **of** non-calculator methods justify how they arrived at a conclusion to a problem; give solutions **in** the context **of** the problem; confirm that results are **of** the right order **of** magnitude interpret and use simple **algebraic**/

CIS 536/636 Introduction to **Computer** **Graphics** Overview: First Month (Weeks 2-5 **of** Course) Review **of** mathematical foundations **of** CG: analytic geometry, **linear** **algebra** Line and polygon rendering Matrix transformations **Graphical** interfaces Line and Polygon Rendering (Week 3) Basic line drawing and 2-D clipping Bresenham’s algorithm Follow-up: 3-D clipping, z-buffering (painter’s algorithm) Matrix Transformations (Week 4) **Application** **of** **linear** transformations to rendering Basic operations/

Harris, Jr. Fall 2012 Objectives Broad introduction to **Computer** **Graphics** –Software –Hardware –**Applications** Top-down approach Shader-Based OpenGL –OpenGL 3.1 –Open GL ES 2.0 –webGL Prerequisites Good programming skills **in** C (or C++) Basic Data Structures –Linked lists –Arrays Geometry Simple **Linear** **Algebra** Resources Can run OpenGL on any system –Windows: check **graphics** card properties for level **of** OpenGL supported –Linux –Mac: need extensions for/

use **of** **computer** **graphics**, e.g. **in** UI/UX, documents, browsers Why Learn 2D first? 2D **Graphics** using OpenGL – 9/10/2015 CS123 | INTRODUCTION TO **COMPUTER** **GRAPHICS** Andries van Dam © 4/32 **Graphics** Platforms (1/4) **Applications** that only write pixels are rare **Application** /(OpenGL Mathematics) to do our **linear** **algebra** instead **of** using the Fixed-function API OpenGL (3/3) 2D **Graphics** using OpenGL – 9/10/2015 CS123 | INTRODUCTION TO **COMPUTER** **GRAPHICS** Andries van Dam © 18/32 **In** future labs and your final project/

) 3 Objectives Broad introduction to **Computer** **Graphics** Software Hardware **Applications** Top-down approach OpenGL 4 Prerequisites Good programming skills **in** C (or C++) Basic Data Structures Linked lists Arrays Geometry Simple **Linear** **Algebra** 5 Requirements 4 Assigned Projects /**of** examples and information at: www.opengl.orgwww.opengl.org PowerPoint lecture notes at www.bhecker.com www.bhecker.com 9 Outline: Part 1 Part 1: Introduction Text: Chapter 1 Lectures 1-3 What is **Computer** **Graphics**? **Applications** /

variables The student will recognize and apply real-world functions **in** a variety **of** representations and translate among verbal, tabular, **graphic**, and **algebraic** representations **of** functions. The student will recognize an example **of** a function; identify the role **of** independent and dependent variables **in** a function; determine the domain and range **of** a **linear** function; find the slope and intercepts **of** a **linear** function; and use function notation to evaluate a function/

or “ net install apc” on the Stata 9.2 command line on any **computer** connected to the Internet Download from the Statistical Software Components archive at http:///**of** individuals **in** the samples. Solution: Use **of** different temporal groupings for the A, P, and C dimensions breaks the **linear** dependency: Single year **of** age Time periods correspond to years **in** which the surveys are conducted Cohorts can be defined either by five- or ten-year intervals that are conventional **in** demography or by **application** **of**/

**algebra** courses. It can be the unifying theme that links functions, the real world, and the other disciplines. Integrating Statistics **in** Precalculus 3.The normal distribution function is It makes for an excellent example involving both stretching and shifting functions and a function **of** a function. Integrating Statistics **in** Precalculus 4. The z-value associated with a measurement x is a nice **application** **of** a **linear** function **of**/

state objectives for **linear** **algebra** and counts as a math credit towards graduation. 15 teachers and 5 librarians have attended **linear** algrebra workshops to facilate future classes. 3. **Computer** **Graphics** and Visualization I & II These two classes are taught during the summer **of** the student’s rising junior and senior years at the UAB ETLab. **In** Workshop-I, students learn about the fundamental principles and **applications** **in** **computer** **graphics** and scientific/

CHOMBO AMR frameworks Speedups for two BoxLib **applications**: – 3D LMC (a low-mach number combustion code) 2.5x **in** bottom solve, 1.5x overall GMG solve – 3D Nyx (an N-body and gas dynamics code) 2x **in** bottom solve, 1.15x overall GMG solve Summary **of** Iterative **Linear** **Algebra** New lower bounds, optimal algorithms, big speedups **in** theory and practice Lots **of** other progress, open problems – Many different/

**Graphical** Models (non-probabilistic factor graphs) Latent variable models **Linear** factors: Conditional Random Fields and Maximum Margin Markov Nets Gradient-based learning with non-**linear** factors **Applications**: supervised and unsupervised learning Integrated segmentation/recognition **in** /LeCun Graph Composition, Transducers. The composition **of** two graphs can be **computed**, the same way the dot product between two vectors can be **computed**. General theory: semi-ring **algebra** on weighted finite- state transducers and/

**Graphical** Models (non-probabilistic factor graphs) Latent variable models **Linear** factors: Conditional Random Fields and Maximum Margin Markov Nets Gradient-based learning with non-**linear** factors **Applications**: supervised and unsupervised learning Integrated segmentation/recognition **in** /LeCun Graph Composition, Transducers. The composition **of** two graphs can be **computed**, the same way the dot product between two vectors can be **computed**. General theory: semi-ring **algebra** on weighted finite- state transducers and/

account the attributes **of** both the **application** and target system. A successful implementation lies on deep understanding **of** data access patterns, **computation** properties, available hardware resources where it can take advantage **of** generalized performance planning techniques to produce successful implementation across a wide variety **of** multicore architectures. Combinatorial algorithms play important role **in** scientific **computing** for efficient parallelization **of** **linear** **algebra**, **computational** physics, numerical/

project management Which kind **of** tool I is needed ? Structured, unified, generic, Prof. Belkacem Ould BOUAMAMA, Polytech’Lille What is Mechatronic Systems Mecatronics (« Meca »+ « Tronics » Engineering systems putting **in** evidence multiple skills Mechanics : Hydraulics, Thermal engineering, Mechanism, pneumatic Electronics : power electronics, Networks, converters AN/NA, Micro controllers, Automatic control : **Linear** and nonlinear control, Advanced control, Stability, … **Computer** Engineering : Real time/

at http://www.mathsci.appstate.edu/~wmcb/ICTCM CRAFTY & College **Algebra** The Guidelines: Course Objectives College **algebra** through **applications**/modeling Meaningful & appropriate use **of** technology Course Goals Challenge, develop, and strengthen students’ understanding and skills mastery CRAFTY & College **Algebra** The Guidelines: Student Competencies - Problem solving - Functions and Equations - Data Analysis Pedagogy - **Algebra** **in** context - Technology for exploration and analysis Assessment - Extended set/

Introduction with **Applications**, Amos Gilat, John Wiley & Sons, Inc., 2004 **Graphics** and GUIs with MATLAB, 3rd Ed, Patrick Marchand and O. Thomas Holland, Chapman & Hall/CRC, 2003 MATLAB for Scientist and Engineers Using Symbolic Toolbox Course Introductions MATLAB for Scientist and Engineers Using Symbolic Toolbox Old History **of** MATLAB 1967: "**Computer** solution **of** **linear** **algebraic** equations", Forsythe and Moler 1976: "Matrix Eigensystem Routines, EISPACK Guide" **in** FORTRAN 1976/

**Application** **of** Coordinate and Dot Transformations Topics: **linear** **algebra**, programming with Delphi, **computer** **graphics**, OO conception Sándor Kaczur kaczur@gdf.hu 2 Circumstances 1. semester –Mathematics 1. **Linear** **algebra**, matrices System **of** equations 3. semester –**Computer** **graphics**, Digital image processing Homogeneous coordinates Coordinate and dot transformations Projections 3 Problems They can solve exercises from **linear** **algebra**/Let be a dot P **in** 3D with 3 coodinates: If let coordinates **of** dot P: If than let/

are few decision variables and even fewer constraints. **Linear** Programming II 12 Method 4: Using **Linear** **Algebra** (Matrices) **In** this method, the vertices **of** the feasible region are determined as follows: - Set up the objective function and constraints as a set **of** **linear** **algebra** equations - Develop the corresponding matrices. - Solve the set **of** **linear** equations using vector **algebra**. This yields the optimal value **of** the objective function. - Simplex method can be employed/

numerical **linear** **algebra** performs numerical **linear** **algebra** 22 Examples **of** benchmark programs SPEC - Standard Performance Evaluation Corporation a non-profit international organization focused on developing standard tools for measuring the performance **of** **computer** systems a non-profit international organization focused on developing standard tools for measuring the performance **of** **computer** systems www.spec.org www.spec.org www.spec.org develops standard sets **of** benchmarks based on real **applications** develops/

, and **graphically** display data **in** a variety **of** representations. Students should also become skilled at using the Internet to carry out literature searches, locate published articles, and access major databases. Concepts **of** Mathematics and **Computer** Science Calculus Complex numbers Functions Limits Continuity The integral The derivative and **linearization** Elementary functions Fourier series Multi-dimensional calculus: **linear** approximations, integration over multiple variables **Linear** **Algebra** Scalars, vectors/

Hourglass was created to make these **computations** more efficient, yielding sometimes 50-95% reductions **in** **computational** resources R R is GPL Open Source http://en.wikipedia.org/wiki/R_(programming_language) and many books and online resources and is widely used by statistics communityhttp://en.wikipedia.org/wiki/R_(programming_language) R provides a wide variety **of** statistical and **graphical** techniques, including **linear** and nonlinear modeling, classical statistical/

**graphics** (ITCS 4120) – **linear**/vector **algebra**, geometry & trig. -advanced **graphics** advanced calculus, **computational** geometry, differential geometry, topology, ….. optics (very approximate **in** ITCS 4120) software engineering and programming hardware engineering psychophysics (branch **of**/Fetter, 1960, Boeing Aircraft Co. “Boeing Man”, human figure simulation, credited with “**computer** **graphics**” ©Zachary Wartell **In** what way do CG **applications** differ? 2D versus 3D Speed – Frames Per Second (FPS) Realism/

RV 2 ILE5014 **Computer** **Graphics** 10F 24 Composition **of** Transformation Any sequence **of** **linear** transformations can be /**of** function **application**.) Premultiply: right-to-left (same as function **application**.) Postmultiply: left-to-right (reverse **of** function **application**.) Premultiply: right-to-left (same as function **application**.) M D M CM M BM M AM M A M MB M MC M MD or both give the same result. Premultiply Postmultiply ILE5014 **Computer** **Graphics** 10F 28 Column Vector Convention The convention **in**/

**In** **computations** , it is often easier to work with the refractive index **of** a material than directly with the speed **of** light . n = speed **of** light **in** vacuum speed **of** light **in** medium refractive index is quite sensitive to a material’s chemical composition . a small amount **of** salt or sugar dissolved **in**/. The clinical **application** **of** wavefront aberrometry makes it possible to measure higher order aberrations , which were previously lumped into a catchall term – irregular astigmatism . Examples **of** higher -order /

**Computational** **Algebraic** Problems **in** Variational PDE Image Processing Tony F. Chan Department **of** Mathematics, UCLA International Summer School **in** Numerical **Linear** **Algebra** Chinese University **of** Hong Kong Reports: www./Vector-Valued * Compression * Still vs Video * Segmentation * Registration Related fields: **Computer** **Graphics**, **Computer** Vision. IP And Applied Math Important **applications**: –Medical, astronomy, Comp. Vision/Comp. **Graphics**, Math Models: –standard or create your own Math Tools: –harmonic analysis,/

Widely **applicable**: all **linear** **algebra**, Health app… 29 EECS Electrical Engineering and **Computer** Sciences B ERKELEY P AR L AB Communication-Avoiding QR Decomposition for GPUs 30 The QR decomposition **of** tall-skinny matrices is a key **computation** **in** many **applications** **Linear** least/ resources to execute components **in** parallel 32 EECS Electrical Engineering and **Computer** Sciences B ERKELEY P AR L AB 33 OS-multiplexed Efficient Parallel Composition **of** Libraries is Hard Physics AI **Graphics** Audio OpenMP Scheduler TBB /

other fields 1. Embedded **Computing** (EEMBC benchmark) 2. Desktop/Server **Computing** (SPEC2006) 3. Machine Learning Advice from colleagues Mike Jordan and Dan Klein 4. Games/**Graphics**/Vision 5. Data Base Software Advice from Jim Gray **of** Microsoft and colleague Joe Hellerstein Result: Added 7 more dwarfs, revised 2 original dwarfs, renumbered list 15 Final 14 Dwarfs 1. Dense **Linear** **Algebra** 2. Sparse **Linear** **Algebra** 3. Spectral Methods 4/

have enabled the use **of** GPU processors for general purpose **computation**. **Applications** **in**: **Linear** **algebra** Geometric **Computing** Database and Stream Mining GPU Ray Tracing Advanced Image Processing **Computational** Fluid dynamics (CFD) and Finite Element Analysis Problems Need to be Solved Significant barriers exist for the developer who wishes to use the inexpensive power **of** commodity **graphics** hardware, whether for **in**-game simulation **of** physics or for conventional **computational** science. These chips are/

**algebra** courses. It can be the unifying theme that links functions, the real world, and the other disciplines. Integrating Statistics **in** Precalculus 3.The normal distribution function is It makes for an excellent example involving both stretching and shifting functions and a function **of** a function. Integrating Statistics **in** Precalculus 4. The z-value associated with a measurement x is a nice **application** **of** a **linear** function **of**/

F To understand the basic objectives and scope **of** **computer** **graphics** F To identify **computer** **graphics** **applications** F To understand the basic structures **of** 2D and 3D **graphics** systems F To understand evolution **of** **graphics** programming environments F To identify common **graphics** APIs F To understand the roles **of** Java language, Java 2D and Java 3D packages F To identify **computer** **graphics** related fields Zhang & Liang, **Computer** **Graphics** Using Java 2D and 3D (c) 2007 Pearson/

**of** **linear** equations: Such systems occur **in** many, many **applications**. They are studied **in** **Linear** **Algebra**. Very short intro to **Linear** **Algebra** H&B A-5 System **of** **linear** equations: Typical questions: -Given u, v, w, what are x, y, z? -Can we find a unique solution? H&B A-5 Very short intro to **Linear** **Algebra** System **of** **linear** equations: Crucial **in** **computer** **graphics**: -Transforming geometric objects -Change **of** coordinates H&B A-5 Very short intro to **Linear** **Algebra**/

**graphically**? Unit 3: **Linear** Functions and Their Graphs, Rates **of** Change, and **Applications** Time Frame: Approximately five weeks Unit Description This unit leads to the investigation **of** the role **of** functions **in** the development **of** **algebraic** thinking and modeling. Heavy emphasis is given **in** this unit to understanding rates **of**/ and the **computations** that can be performed using them. Student Understandings Students should be able to find the precision **of** an instrument and determine the accuracy **of** a given /

. Students will investigate key features, domains, and ranges **of** **linear** functions as described **in** F-IF.B.4 and F-IF.B.5; write **linear** functions to model relationships between two quantities as **in** F-BF.A1a; and compare properties **of** **linear** functions as **in** F-IF.C.9. Interpreting Functions-F-IF B. Interpret functions that arise **in** **applications** **in** terms **of** the context 6. Calculate and interpret the average rate/

scaling Shearing Geometry change **in** perpendicularity introduces slant **Algebra** **in** X —x’ = x + y * Sh x —y’ = x * Sh y + x Display **of** Motion Create visual perception **of** motion movies, TV, interactive **graphics** sequence **of** snap shots : frames played back rapidly at a constant frame rate Variety **of** frame rates Film : 24 fps (frames per second) S-TV: 30 fps HD-TV: 60 fps **Computers** : 60-120 fps Interactive/

25 Feb 2008 Preparing term project presentations and demos for **graphics** – April **Computing** & Information Sciences Kansas State University Advanced CG 1 **of** 8: TexturingCIS 636/736: (Introduction to) **Computer** **Graphics** Background Expected Both Courses Proficiency **in** C/C++ or strong proficiency **in** Java and ability to learn Strongly recommended: matrix theory or **linear** **algebra** (e.g., Math 551) At least 120 hours for semester (up to 150 depending on/

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