A New Cement to Glue Nonconforming Grids with Robin Interface Conditions: The Finite Element Case


 Julius Clarke
 2 years ago
 Views:
Transcription
1 A New Cement to Glue Nonconforming Grids wit Robin Interface Conditions: Te Finite Element Case Martin J. Gander, Caroline Japet 2, Yvon Maday 3, and Frédéric Nataf 4 McGill University, Dept. of Matematics and Statistics, Montreal (ttp:// 2 Université Paris 3, Laboratoire d Analyse, Géométrie et Applications 3 Université Pierre et Marie Curie, Laboratoire Jacques Louis Lions (ttp:// 4 CNRS, UMR 764, CMAP, Ecole Polytecnique, 928 Palaiseau (ttp:// Summary. We present and analyze a new nonconforming domain decomposition metod based on a Scwarz metod wit Robin transmission conditions. We prove tat te metod is well posed and convergent. Our error analysis is valid in two dimensions for piecewise polynomials of low and ig order and also in tree dimensions for P elements. We furter present an efficient algoritm in two dimensions to perform te required projections between arbitrary grids. We finally illustrate te new metod wit numerical results. Introduction We propose a domain decomposition metod based on te Scwarz algoritm tat permits te use of optimized interface conditions on nonconforming grids. Suc interface conditions ave been sown to be a ey ingredient for efficient domain decomposition metods in te case of conforming approximations (see Després [99], Nataf et al. [995], Japet [998], Cevalier and Nataf [998]). Our goal is to use tese interface conditions on nonconforming grids, because tis simplifies greatly te parallel generation and adaptation of meses per subdomain. Te mortar metod, first introduced in Bernardi et al. [994], also permits te use of nonconforming grids, and it is well suited to te use of DiricletNeumann (Gastaldi et al. [996]) or NeumannNeumann metods applied to te Scur complement matrix. But te mortar metod can not be used easily wit optimized transmission conditions in te framewor of Scwarz metods. In Acdou et al. [22], te case of finite volume discretizations as been introduced and analyzed. Tis paper is a first step in te finite element case; we consider only interface conditions of order ere.
2 26 Martin J. Gander, Caroline Japet, Yvon Maday, and Frédéric Nataf 2 Definition of te metod and te iterative solver We consider te model problem (Id )u = f in Ω, u = on Ω, () were f is given in L 2 (Ω) and Ω is a C, (or convex) domain in IR d, d = 2 or 3. We assume tat it is decomposed into K nonoverlapping subdomains Ω = K Ω, were Ω, K are C, or convex polygons in two or polyedrons in tree dimensions. We also assume tat tis domain decomposition is conforming. Let n be te unit outward normal for Ω and Γ,l = Ω Ω l. Te variational statement of problem () consists of writing te problem as follows: Find u H (Ω) suc tat ( u v + uv)dx = fvdx, v H (Ω). (2) Ω Ω We introduce now te space H (Ω ) = {ϕ H (Ω ), ϕ = over Ω Ω }, and te constrained space K K V ={(v,q) ( H (Ω )) ( H /2 ( Ω )), v =v l and q = q l on Γ,l }. Problem (2) is ten equivalent to te following: Find (u,p) V suc tat Ω ( u v + u v )dx = H /2 ( Ω ) < p, v > H /2 ( Ω ) Ω f v dx, v K H (Ω ). Being equivalent wit te original problem, were p = u n over Ω, tis problem is naturally well posed. We now describe te iterative procedure in te continuous case, and ten its discrete, nonconforming analog. 2. Te continuous case We introduce for α IR, α >, te zerot order transmission condition p + αu = p l + αu l over Γ,l and te following algoritm: let (u n, pn ) H (Ω ) H /2 ( Ω ) be an approximation of (u, p) in Ω at step n. Ten, (u n+, p n+ ) is te solution in H (Ω ) H /2 ( Ω ) of
3 Nonconforming Grids wit Robin Interface Conditions 26 ( u n+ v + u n+ ) v dx H /2 ( Ω ) < p n+, v > H /2 ( Ω ) Ω = f v dx, v H (Ω ), Ω < p n+ +αu n+, v > Γ,l=< p n l +αun l, v > Γ,l, v H /2 (Γ,l ). Convergence of tis algoritm is sown in Després [99] using energy estimates and summarized in te following Teorem. Assume tat f is in L 2 (Ω) and (p ) K l H/2 (Γ,l ). Ten, algoritm (3) converges in te sense tat lim u n n ( u H (Ω ) + p n p ) H /2 ( Ω ) =, for K, were u solves (), u = u Ω, p = u n on Ω for K. 2.2 Te discrete case We introduce now te discrete spaces: eac Ω is provided wit its own mes T, K, suc tat Ω = T T T. For T T, let T be te diameter of T and te discretization parameter, = max K (max T T T ). Let ρ T be te diameter of te circle in two dimensions or spere in tree dimensions inscribed in T. We suppose tat T is uniformly regular: tere exists σ and τ independent of suc tat T T, σ T σ and τ T. We consider tat te sets belonging to te meses are of simplicial type (triangles or tetraedra), but te following analysis can be applied as well for quadrangular or exaedral meses. Let P M (T) denote te space of all polynomials defined over T of total degree less tan or equal to M for our Lagrangian finite elements. Ten, we define over eac subdomain two conforming spaces Y and X by Y = {v, C (Ω ), v, T P M (T), T T }, X = {v, Y, v, Ω Ω = }. (4) Te space of traces over eac Γ,l of elements of Y is denoted by Y,l. In te sequel we assume for te sae of simplicity tat referring to a pair (, l) implies tat Γ,l,l is not empty. Wit eac suc interface we associate a subspace W of Y,l lie in te mortar element metod; for two dimensions, see Bernardi et al. [994], and for tree dimensions see Belgacem and Maday [997] and Braess and Damen [998]. To be more specific, we recall te situation in two dimensions: if te space X consists of continuous piecewise polynomials of degree M, ten it is readily noticed tat te restriction of X,l to Γ consists of finite element functions adapted to te (possibly curved) side Γ,l of piecewise polynomials of degree M. Tis side as two end points wic we denote by x,l and x,l n and wic belong to te set of vertices of te (3)
4 262 Martin J. Gander, Caroline Japet, Yvon Maday, and Frédéric Nataf corresponding triangulation of Γ,l : x,l, x,l,...,x,l is ten te subspace of tose elements of Y,l M over bot [x,l, x,l n, x,l,l n. Te space W tat are polynomials of degree ] and [x,l n, x,l n ]. As before, te space W,l product space of te W over eac l suc tat Γ,l. Te discrete constrained space is ten defined by K K V = {(u,p ) ( X) ( W ), is te,l ((p, + αu, ) ( p,l + αu,l ))ψ,,l =, ψ,,l W }, Γ,l and te discrete problem is te following: Find (u,p ) V suc tat v = (v,,...v,k ) K X, ( u, v, +u, v, ) dx p, v, ds= f v, dx. (5) Ω Ω Ω Te discrete algoritm is ten as follows: let (u n,, pn, ) X W be a discrete approximation of (u,p) in Ω at step n. Ten, (u n+ ) is te,, pn+, solution in X W of ( ) u n+, v,+u n+, v, dx p n+, v,ds= f v, dx, v, X, (6) Ω Ω Ω (p n+, + αun+, )ψ,,l = ( p n,l + αu n,l,l)ψ,,l, ψ,,l W. (7) Γ,l Γ,l Remar. Let π,l denote te ortogonal projection operator from L 2 (Γ,l ) onto. Ten (7) corresponds to W,l p n+, + απ,l(u n+, ) = π,l( p n,l + αun,l ) over Γ,l. (8) Remar 2. A fundamental difference between tis metod and te original mortar metod in Bernardi et al. [994] is tat te interface conditions are cosen in a symmetric way: tere is no master and no slave, see also Gander et al. [2]. Equation (8) is te transmission condition on Γ,l for Ω, and te transmission condition on Γ,l for Ω l is p n+,l + απ l, (u n+,l ) = π l,( p n, + αun, ) over Γ,l. (9) In order to analyze te convergence of tis iterative sceme, we define for any p in K L2 ( Ω ) te norm p 2, = ( K l= l p 2 ) H 2, 2 (Γ,l )
5 were. H 2 (Γ,l ) Nonconforming Grids wit Robin Interface Conditions 263 stands for te dual norm of H 2 (Γ,l ). Convergence of te algoritm (6)(7) can be sown again using an energy estimate, see Japet et al. [23]. Teorem 2. Assume tat α c for some constant c small enoug. Ten, te discrete problem (5) as a unique solution (u,p ) V. Te algoritm (6)(7) is well posed and converges in te sense tat lim n ( un, u, H (Ω ) + p n, p, ) =, for K. H 2 (Γ,l ) l 3 Best approximation properties In tis part we give best approximation results of (u,p) by elements in V. Te proofs can be found in Japet et al. [23] for te two dimensional case wit te degree of te finite element approximations M 3 and in tree dimensions for first order approximations. Teorem 3. Assume tat te solution u of () is in H 2 (Ω) H (Ω) and u = u Ω H 2+m (Ω ) wit M m, and let p,l = u n over eac Γ,l. Ten, tere exists a constant c independent of and α suc tat u u + p p 2, c(α2+m + +m ) + c( m α + +m ) u H 2+m (Ω ) p,l. H +m 2 (Γ,l ) Assuming more regularity on te normal derivatives on te interfaces, we ave Teorem 4. Assume tat te solution u of () is in H 2 (Ω) H (Ω) and u = u Ω H 2+m (Ω ) wit M m, and p,l = u n is in H 3 2 +m (Γ,l ). Ten tere exists a constant c independent of and α suc tat u u + p p 2, c(α 2+m + +m ) + c( +m α + 2+m )(log ) β(m) l u H 2+m (Ω ) p,l 3. H +m 2 (Γ,l ) Remar 3. Te Robin parameter α can depend on in te previous teorems, lie te optimal Robin parameter α opt in section 5. l
6 264 Martin J. Gander, Caroline Japet, Yvon Maday, and Frédéric Nataf 4 Efficient projection algoritm Te projection (8) between non conforming grids is not an easy tas in an algoritm, already for two dimensional problems, since one needs to find te intersections of corresponding arbitrary grid cells. A sort and efficient algoritm as been proposed in Gander et al. [2] in te finite volume case wit projections on piecewise constant functions. In our case, we denote by n te dimension of W,l, and we introduce te sape functions {ψ,l i } i n of W,l. Ten, to compute te rigt and side in (7), we need to compute te interface matrix M = ( ψ,l i Γ,l ψ l, j ) i,j n. In te same spirit as in Gander et al. [2], te following sort algoritm in Matlab computes te interface matrix M for nonmatcing grids in one pass. function M=InterfaceMatrix(ta,tb); n=lengt(tb); m=lengt(ta); ta(m)=tb(n); % must be numerically equal j=; M=zeros(n,lengt(ta)); for i=:n, tm=tb(i); wile ta(j+)<tb(i+), M(i:i+,j:j+)=M(i:i+,j:j+)+intMortar(ta(j),ta(j+),... tb(i),tb(i+),tm,ta(j+),j== j==m,i== i==n); j=j+; tm=ta(j); end; M(i:i+,j:j+)=M(i:i+,j:j+)+intMortar(ta(j),ta(j+),... tb(i),tb(i+),tm,tb(i+),j== j==m,i== i==n); end; It taes two vectors ta and tb wit ordered entries, wic represent two nonmatcing grids at te interface, wit ta()=tb(), ta(end)=tb(end), and computes te matrix M(i,j)= Γ,l b i a j, were b i is te at function for te node tb(i) and a j is te at function for te node ta(j). Te mortar condition of constant sape functions at te corners is taen into account, and from te resulting matrix M te first and last row and column needs to be removed. Tis algoritm as linear complexity; it does a single pass witout any special cases or any additional grid. It advances automatically on watever side te next cell boundary is coming and andles any possible cases of nonmatcing grids at a one dimensional interface. 5 Numerical results On te unit square Ω = (, ) (, ) we consider te problem
7 y Nonconforming Grids wit Robin Interface Conditions 265 (Id )u(x, y) = x 3 (y 2 2) 6xy 2 + ( + x 2 + y 2 )sin(xy), (x, y) Ω, u = x 3 y 2 + sin(xy), (x, y) Ω, wose exact solution is u(x, y) = x 3 y 2 + sin(xy). We decompose te unit square into four nonoverlapping subdomains wit meses generated in an independent manner, as sown in Figure on te left. Te computed solution Initial Mes Computed solution x.8.6 y x Fig.. Initial mes and computed solution after two refinements. is te solution at convergence ( of te discrete algoritm (6)(7), wit) stopping criterion max,l/γ,l ((p Γ,l, + αu, ) ( p,l + αu,l ))ψ,l < 8, and α =. On Figure on te rigt, we sow te computed solution. Figure 2 on te left corresponds to te best approximation error of Teorem 4. On te rigt, we compare in te case of two subdomains te optimal log(h discrete error) log() 7 2 subdomains Nonconforming case alpa_opt 65 6 log(error) 2 number of iterations log() alpa Fig. 2. H error versus on te left and number of iterations versus α on te rigt. numerical α to te teoretical value, wic minimizes te convergence rate at te continuous level: α opt = [(π 2 π + )(( min ) 2 + )] 4. Te nonconforming meses ave 289 and 56 nodes respectively, and te discretization parame
8 266 Martin J. Gander, Caroline Japet, Yvon Maday, and Frédéric Nataf ters are =.65 and 2 =.32. We observe tat te optimal numerical α is very close to α opt. References Y. Acdou, C. Japet, Y. Maday, and F. Nataf. A new cement to glue nonconforming grids wit Robin interface conditions: te finite volume case. Numer. Mat., 92(4):593 62, 22. F. B. Belgacem and Y. Maday. Te mortar element metod for tree dimensional finite elements. RAIRO Matematical Modelling and Numerical Analysis, 3(2):289 32, 997. C. Bernardi, Y. Maday, and A. T. Patera. A new non conforming approac to domain decomposition: Te mortar element metod. In H. Brezis and J.L. Lions, editors, Collège de France Seminar. Pitman, 994. Tis paper appeared as a tecnical report about five years earlier. D. Braess and W. Damen. Stability estimates of te mortar finite element metod for 3dimensional problems. EastWest J. Numer. Mat., 6(4): , 998. P. Cevalier and F. Nataf. Symmetrized metod wit optimized secondorder conditions for te Helmoltz equation. In Domain decomposition metods, (Boulder, CO, 997), pages Amer. Mat. Soc., Providence, RI, 998. B. Després. Domain decomposition metod and te elmoltz problem. In SIAM, editor, Matematical and Numerical aspects of wave propagation penomena, pages Piladelpia PA, 99. M. J. Gander, L. Halpern, and F. Nataf. Optimal Scwarz waveform relaxation for te one dimensional wave equation. Tecnical Report 469, CMAP, Ecole Polytecnique, September 2. F. Gastaldi, L. Gastaldi, and A. Quarteroni. Adaptive domain decomposition metods for advection dominated equations. EastWest J. Numer. Mat., 4:65 26, 996. C. Japet. Optimized KrylovVentcell metod. Application to convectiondiffusion problems. In P. E. Bjørstad, M. S. Espedal, and D. E. Keyes, editors, Proceedings of te 9t international conference on domain decomposition metods, pages ddm.org, 998. C. Japet, Y. Maday, and F. Nataf. A new cement to glue nonconforming grids wit robin interface conditions: Te finite element case. to be submitted, 23. F. Nataf, F. Rogier, and E. de Sturler. Domain decomposition metods for fluid dynamics, NavierStoes equations and related nonlinear analysis. Edited by A. Sequeira, Plenum Press Corporation, pages , 995.
FINITE DIFFERENCE METHODS
FINITE DIFFERENCE METHODS LONG CHEN Te best known metods, finite difference, consists of replacing eac derivative by a difference quotient in te classic formulation. It is simple to code and economic to
More informationVerifying Numerical Convergence Rates
1 Order of accuracy Verifying Numerical Convergence Rates We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, suc as te grid size or time step, and
More informationOPTIMAL DISCONTINUOUS GALERKIN METHODS FOR THE ACOUSTIC WAVE EQUATION IN HIGHER DIMENSIONS
OPTIMAL DISCONTINUOUS GALERKIN METHODS FOR THE ACOUSTIC WAVE EQUATION IN HIGHER DIMENSIONS ERIC T. CHUNG AND BJÖRN ENGQUIST Abstract. In tis paper, we developed and analyzed a new class of discontinuous
More informationThe finite element immersed boundary method: model, stability, and numerical results
Te finite element immersed boundary metod: model, stability, and numerical results Lucia Gastaldi Università di Brescia ttp://dm.ing.unibs.it/gastaldi/ INdAM Worksop, Cortona, September 18, 2006 Joint
More informationProjective Geometry. Projective Geometry
Euclidean versus Euclidean geometry describes sapes as tey are Properties of objects tat are uncanged by rigid motions» Lengts» Angles» Parallelism Projective geometry describes objects as tey appear Lengts,
More informationIntroduction to the Finite Element Method
Introduction to the Finite Element Method 09.06.2009 Outline Motivation Partial Differential Equations (PDEs) Finite Difference Method (FDM) Finite Element Method (FEM) References Motivation Figure: cross
More informationPart II: Finite Difference/Volume Discretisation for CFD
Part II: Finite Difference/Volume Discretisation for CFD Finite Volume Metod of te AdvectionDiffusion Equation A Finite Difference/Volume Metod for te Incompressible NavierStokes Equations MarkerandCell
More informationThe EOQ Inventory Formula
Te EOQ Inventory Formula James M. Cargal Matematics Department Troy University Montgomery Campus A basic problem for businesses and manufacturers is, wen ordering supplies, to determine wat quantity of
More informationResearch on the Antiperspective Correction Algorithm of QR Barcode
Researc on te Antiperspective Correction Algoritm of QR Barcode Jianua Li, YiWen Wang, YiJun Wang,Yi Cen, Guoceng Wang Key Laboratory of Electronic Tin Films and Integrated Devices University of Electronic
More informationAn Additive NeumannNeumann Method for Mortar Finite Element for 4th Order Problems
An Additive eumanneumann Method for Mortar Finite Element for 4th Order Problems Leszek Marcinkowski Department of Mathematics, University of Warsaw, Banacha 2, 02097 Warszawa, Poland, Leszek.Marcinkowski@mimuw.edu.pl
More informationACT Math Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as
More informationFinite Volume Discretization of the Heat Equation
Lecture Notes 3 Finite Volume Discretization of te Heat Equation We consider finite volume discretizations of te onedimensional variable coefficient eat equation, wit Neumann boundary conditions u t x
More informationSurface Areas of Prisms and Cylinders
12.2 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.10.B G.11.C Surface Areas of Prisms and Cylinders Essential Question How can you find te surface area of a prism or a cylinder? Recall tat te surface area of
More informationON LOCAL LIKELIHOOD DENSITY ESTIMATION WHEN THE BANDWIDTH IS LARGE
ON LOCAL LIKELIHOOD DENSITY ESTIMATION WHEN THE BANDWIDTH IS LARGE Byeong U. Park 1 and Young Kyung Lee 2 Department of Statistics, Seoul National University, Seoul, Korea Tae Yoon Kim 3 and Ceolyong Park
More informationAn inquiry into the multiplier process in ISLM model
An inquiry into te multiplier process in ISLM model Autor: Li ziran Address: Li ziran, Room 409, Building 38#, Peing University, Beijing 00.87,PRC. Pone: (86) 0062763074 Internet Address: jefferson@water.pu.edu.cn
More informationComputer Science and Engineering, UCSD October 7, 1999 GoldreicLevin Teorem Autor: Bellare Te GoldreicLevin Teorem 1 Te problem We æx a an integer n for te lengt of te strings involved. If a is an nbit
More information1 The Collocation Method
CS410 Assignment 7 Due: 1/5/14 (Fri) at 6pm You must wor eiter on your own or wit one partner. You may discuss bacground issues and general solution strategies wit oters, but te solutions you submit must
More informationGeometric Stratification of Accounting Data
Stratification of Accounting Data Patricia Gunning * Jane Mary Horgan ** William Yancey *** Abstract: We suggest a new procedure for defining te boundaries of te strata in igly skewed populations, usual
More informationDomain Decomposition Methods. Partial Differential Equations
Domain Decomposition Methods for Partial Differential Equations ALFIO QUARTERONI Professor ofnumericalanalysis, Politecnico di Milano, Italy, and Ecole Polytechnique Federale de Lausanne, Switzerland ALBERTO
More informationSAT Subject Math Level 1 Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses
More informationDistances in random graphs with infinite mean degrees
Distances in random graps wit infinite mean degrees Henri van den Esker, Remco van der Hofstad, Gerard Hoogiemstra and Dmitri Znamenski April 26, 2005 Abstract We study random graps wit an i.i.d. degree
More informationOptimized Data Indexing Algorithms for OLAP Systems
Database Systems Journal vol. I, no. 2/200 7 Optimized Data Indexing Algoritms for OLAP Systems Lucian BORNAZ Faculty of Cybernetics, Statistics and Economic Informatics Academy of Economic Studies, Bucarest
More informationA Multigrid Tutorial part two
A Multigrid Tutorial part two William L. Briggs Department of Matematics University of Colorado at Denver Van Emden Henson Center for Applied Scientific Computing Lawrence Livermore National Laboratory
More informationTraining Robust Support Vector Regression via D. C. Program
Journal of Information & Computational Science 7: 12 (2010) 2385 2394 Available at ttp://www.joics.com Training Robust Support Vector Regression via D. C. Program Kuaini Wang, Ping Zong, Yaoong Zao College
More informationInstantaneous Rate of Change:
Instantaneous Rate of Cange: Last section we discovered tat te average rate of cange in F(x) can also be interpreted as te slope of a scant line. Te average rate of cange involves te cange in F(x) over
More informationArea of Trapezoids. Find the area of the trapezoid. 7 m. 11 m. 2 Use the Area of a Trapezoid. Find the value of b 2
Page 1 of. Area of Trapezoids Goal Find te area of trapezoids. Recall tat te parallel sides of a trapezoid are called te bases of te trapezoid, wit lengts denoted by and. base, eigt Key Words trapezoid
More information2.1: The Derivative and the Tangent Line Problem
.1.1.1: Te Derivative and te Tangent Line Problem Wat is te deinition o a tangent line to a curve? To answer te diiculty in writing a clear deinition o a tangent line, we can deine it as te iting position
More informationDerivatives Math 120 Calculus I D Joyce, Fall 2013
Derivatives Mat 20 Calculus I D Joyce, Fall 203 Since we ave a good understanding of its, we can develop derivatives very quickly. Recall tat we defined te derivative f x of a function f at x to be te
More informationNote nine: Linear programming CSE 101. 1 Linear constraints and objective functions. 1.1 Introductory example. Copyright c Sanjoy Dasgupta 1
Copyrigt c Sanjoy Dasgupta Figure. (a) Te feasible region for a linear program wit two variables (see tet for details). (b) Contour lines of te objective function: for different values of (profit). Te
More informationComparison between two approaches to overload control in a Real Server: local or hybrid solutions?
Comparison between two approaces to overload control in a Real Server: local or ybrid solutions? S. Montagna and M. Pignolo Researc and Development Italtel S.p.A. Settimo Milanese, ITALY Abstract Tis wor
More informationTo motivate the notion of a variogram for a covariance stationary process, { Ys ( ): s R}
4. Variograms Te covariogram and its normalized form, te correlogram, are by far te most intuitive metods for summarizing te structure of spatial dependencies in a covariance stationary process. However,
More informationSchedulability Analysis under Graph Routing in WirelessHART Networks
Scedulability Analysis under Grap Routing in WirelessHART Networks Abusayeed Saifulla, Dolvara Gunatilaka, Paras Tiwari, Mo Sa, Cenyang Lu, Bo Li Cengjie Wu, and Yixin Cen Department of Computer Science,
More informationIn other words the graph of the polynomial should pass through the points
Capter 3 Interpolation Interpolation is te problem of fitting a smoot curve troug a given set of points, generally as te grap of a function. It is useful at least in data analysis (interpolation is a form
More information1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution
1.6 Analyse Optimum Volume and Surface Area Estimation and oter informal metods of optimizing measures suc as surface area and volume often lead to reasonable solutions suc as te design of te tent in tis
More informationThe modelling of business rules for dashboard reporting using mutual information
8 t World IMACS / MODSIM Congress, Cairns, Australia 37 July 2009 ttp://mssanz.org.au/modsim09 Te modelling of business rules for dasboard reporting using mutual information Gregory Calbert Command, Control,
More informationSAMPLE DESIGN FOR THE TERRORISM RISK INSURANCE PROGRAM SURVEY
ASA Section on Survey Researc Metods SAMPLE DESIG FOR TE TERRORISM RISK ISURACE PROGRAM SURVEY G. ussain Coudry, Westat; Mats yfjäll, Statisticon; and Marianne Winglee, Westat G. ussain Coudry, Westat,
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationCan a LumpSum Transfer Make Everyone Enjoy the Gains. from Free Trade?
Can a LumpSum Transfer Make Everyone Enjoy te Gains from Free Trade? Yasukazu Icino Department of Economics, Konan University June 30, 2010 Abstract I examine lumpsum transfer rules to redistribute te
More informationM(0) = 1 M(1) = 2 M(h) = M(h 1) + M(h 2) + 1 (h > 1)
Insertion and Deletion in VL Trees Submitted in Partial Fulfillment of te Requirements for Dr. Eric Kaltofen s 66621: nalysis of lgoritms by Robert McCloskey December 14, 1984 1 ackground ccording to Knut
More informationLecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function
Lecture 10: Wat is a Function, definition, piecewise defined functions, difference quotient, domain of a function A function arises wen one quantity depends on anoter. Many everyday relationsips between
More informationImproved dynamic programs for some batcing problems involving te maximum lateness criterion A P M Wagelmans Econometric Institute Erasmus University Rotterdam PO Box 1738, 3000 DR Rotterdam Te Neterlands
More information2 Limits and Derivatives
2 Limits and Derivatives 2.7 Tangent Lines, Velocity, and Derivatives A tangent line to a circle is a line tat intersects te circle at exactly one point. We would like to take tis idea of tangent line
More informationInner product. Definition of inner product
Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product
More information 1  Handout #22 May 23, 2012 Huffman Encoding and Data Compression. CS106B Spring 2012. Handout by Julie Zelenski with minor edits by Keith Schwarz
CS106B Spring 01 Handout # May 3, 01 Huffman Encoding and Data Compression Handout by Julie Zelenski wit minor edits by Keit Scwarz In te early 1980s, personal computers ad ard disks tat were no larger
More informationSAT Math MustKnow Facts & Formulas
SAT Mat MustKnow Facts & Formuas Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Rationas: fractions, tat is, anyting expressabe as a ratio of integers Reas: integers pus rationas
More informationf(a + h) f(a) f (a) = lim
Lecture 7 : Derivative AS a Function In te previous section we defined te derivative of a function f at a number a (wen te function f is defined in an open interval containing a) to be f (a) 0 f(a + )
More informationTangent Lines and Rates of Change
Tangent Lines and Rates of Cange 922005 Given a function y = f(x), ow do you find te slope of te tangent line to te grap at te point P(a, f(a))? (I m tinking of te tangent line as a line tat just skims
More informationFORCED AND NATURAL CONVECTION HEAT TRANSFER IN A LIDDRIVEN CAVITY
FORCED AND NATURAL CONVECTION HEAT TRANSFER IN A LIDDRIVEN CAVITY Guillermo E. Ovando Cacón UDIM, Instituto Tecnológico de Veracruz, Calzada Miguel Angel de Quevedo 2779, CP. 9860, Veracruz, Veracruz,
More informationA Domain Embedding/Boundary Control Method to Solve Elliptic Problems in Arbitrary Domains
A Domain Embedding/Boundary Control Method to Solve Elliptic Problems in Arbitrary Domains L. Badea 1 P. Daripa 2 Abstract This paper briefly describes some essential theory of a numerical method based
More informationAbstract. Introduction
Fast solution of te Sallow Water Equations using GPU tecnology A Crossley, R Lamb, S Waller JBA Consulting, Sout Barn, Brougton Hall, Skipton, Nort Yorksire, BD23 3AE. amanda.crossley@baconsulting.co.uk
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationDuality of linear conic problems
Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least
More informationis called the geometric transformation. In computers the world is represented by numbers; thus geometrical properties and transformations must
Capter 5 TRANSFORMATIONS, CLIPPING AND PROJECTION 5.1 Geometric transformations Treedimensional grapics aims at producing an image of 3D objects. Tis means tat te geometrical representation of te image
More information2.28 EDGE Program. Introduction
Introduction Te Economic Diversification and Growt Enterprises Act became effective on 1 January 1995. Te creation of tis Act was to encourage new businesses to start or expand in Newfoundland and Labrador.
More informationMath Test Sections. The College Board: Expanding College Opportunity
Taking te SAT I: Reasoning Test Mat Test Sections Te materials in tese files are intended for individual use by students getting ready to take an SAT Program test; permission for any oter use must be sougt
More informationLecture 10. Limits (cont d) Onesided limits. (Relevant section from Stewart, Seventh Edition: Section 2.4, pp. 113.)
Lecture 10 Limits (cont d) Onesided its (Relevant section from Stewart, Sevent Edition: Section 2.4, pp. 113.) As you may recall from your earlier course in Calculus, we may define onesided its, were
More informationWe call this set an ndimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.
Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationHøgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver
Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin email: chechkin@mech.math.msu.su Narvik 6 PART I Task. Consider twopoint
More informationPATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION
PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION Introduction In the previous chapter, we explored a class of regression models having particularly simple analytical
More information4.1 Rightangled Triangles 2. 4.2 Trigonometric Functions 19. 4.3 Trigonometric Identities 36. 4.4 Applications of Trigonometry to Triangles 53
ontents 4 Trigonometry 4.1 Rigtangled Triangles 4. Trigonometric Functions 19 4.3 Trigonometric Identities 36 4.4 pplications of Trigonometry to Triangles 53 4.5 pplications of Trigonometry to Waves 65
More informationDefinition of a Linear Program
Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1
More informationPerimeter, Area and Volume of Regular Shapes
Perimeter, Area and Volume of Regular Sapes Perimeter of Regular Polygons Perimeter means te total lengt of all sides, or distance around te edge of a polygon. For a polygon wit straigt sides tis is te
More information7.6 Complex Fractions
Section 7.6 Comple Fractions 695 7.6 Comple Fractions In tis section we learn ow to simplify wat are called comple fractions, an eample of wic follows. 2 + 3 Note tat bot te numerator and denominator are
More informationMultivariate time series analysis: Some essential notions
Capter 2 Multivariate time series analysis: Some essential notions An overview of a modeling and learning framework for multivariate time series was presented in Capter 1. In tis capter, some notions on
More informationEquilibria in sequential bargaining games as solutions to systems of equations
Economics Letters 84 (2004) 407 411 www.elsevier.com/locate/econbase Equilibria in sequential bargaining games as solutions to systems of equations Tasos Kalandrakis* Department of Political Science, Yale
More informationA network flow algorithm for reconstructing. binary images from discrete Xrays
A network flow algorithm for reconstructing binary images from discrete Xrays Kees Joost Batenburg Leiden University and CWI, The Netherlands kbatenbu@math.leidenuniv.nl Abstract We present a new algorithm
More information1 Density functions, cummulative density functions, measures of central tendency, and measures of dispersion
Density functions, cummulative density functions, measures of central tendency, and measures of dispersion densityfunctionsintro.tex October, 9 Note tat tis section of notes is limitied to te consideration
More information22.1 Finding the area of plane figures
. Finding te area of plane figures a cm a cm rea of a square = Lengt of a side Lengt of a side = (Lengt of a side) b cm a cm rea of a rectangle = Lengt readt b cm a cm rea of a triangle = a cm b cm = ab
More informationAn Intuitive Framework for RealTime Freeform Modeling
An Intuitive Framework for RealTime Freeform Modeling Mario Botsc Leif Kobbelt Computer Grapics Group RWTH Aacen University Abstract We present a freeform modeling framework for unstructured triangle
More informationNew Vocabulary volume
. Plan Objectives To find te volume of a prism To find te volume of a cylinder Examples Finding Volume of a Rectangular Prism Finding Volume of a Triangular Prism 3 Finding Volume of a Cylinder Finding
More informationLinear Threshold Units
Linear Threshold Units w x hx (... w n x n w We assume that each feature x j and each weight w j is a real number (we will relax this later) We will study three different algorithms for learning linear
More informationMATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1  BASIC DIFFERENTIATION
MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1  BASIC DIFFERENTIATION Tis tutorial is essential prerequisite material for anyone stuing mecanical engineering. Tis tutorial uses te principle of
More informationInverse Optimization by James Orlin
Inverse Optimization by James Orlin based on research that is joint with Ravi Ahuja Jeopardy 000  the Math Programming Edition The category is linear objective functions The answer: When you maximize
More informationOPTIMAL FLEET SELECTION FOR EARTHMOVING OPERATIONS
New Developments in Structural Engineering and Construction Yazdani, S. and Sing, A. (eds.) ISEC7, Honolulu, June 1823, 2013 OPTIMAL FLEET SELECTION FOR EARTHMOVING OPERATIONS JIALI FU 1, ERIK JENELIUS
More informationStrategic trading in a dynamic noisy market. Dimitri Vayanos
LSE Researc Online Article (refereed) Strategic trading in a dynamic noisy market Dimitri Vayanos LSE as developed LSE Researc Online so tat users may access researc output of te Scool. Copyrigt and Moral
More informationIntegrating Benders decomposition within Constraint Programming
Integrating Benders decomposition within Constraint Programming Hadrien Cambazard, Narendra Jussien email: {hcambaza,jussien}@emn.fr École des Mines de Nantes, LINA CNRS FRE 2729 4 rue Alfred Kastler BP
More informationChannel Allocation in NonCooperative MultiRadio MultiChannel Wireless Networks
Cannel Allocation in NonCooperative MultiRadio MultiCannel Wireless Networks Dejun Yang, Xi Fang, Guoliang Xue Arizona State University Abstract Wile tremendous efforts ave been made on cannel allocation
More information1 Derivatives of Piecewise Defined Functions
MATH 1010E University Matematics Lecture Notes (week 4) Martin Li 1 Derivatives of Piecewise Define Functions For piecewise efine functions, we often ave to be very careful in computing te erivatives.
More informationUnemployment insurance/severance payments and informality in developing countries
Unemployment insurance/severance payments and informality in developing countries David Bardey y and Fernando Jaramillo z First version: September 2011. Tis version: November 2011. Abstract We analyze
More informationMAT1A01: Differentiation of Polynomials & Exponential Functions + the Product & Quotient Rules
MAT1A01: Differentiation of Polynomials & Exponential Functions + te Prouct & Quotient Rules Dr Craig 17 April 2013 Reminer Mats Learning Centre: CRing 512 My office: CRing 533A (Stats Dept corrior)
More information1. Use calculus to derive the formula for the area of a parallelogram of base b and height. y f(x)=mx+b
Area and Volume Problems. Use calculus to derive te formula for te area of a parallelogram of base b and eigt. y f(x)=mxb b g(x)=mx Te area of te parallelogram is given by te integral of te dierence of
More informationCHAPTER 7. Di erentiation
CHAPTER 7 Di erentiation 1. Te Derivative at a Point Definition 7.1. Let f be a function defined on a neigborood of x 0. f is di erentiable at x 0, if te following it exists: f 0 fx 0 + ) fx 0 ) x 0 )=.
More informationSolutions by: KARATUĞ OZAN BiRCAN. PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set in
KOÇ UNIVERSITY, SPRING 2014 MATH 401, MIDTERM1, MARCH 3 Instructor: BURAK OZBAGCI TIME: 75 Minutes Solutions by: KARATUĞ OZAN BiRCAN PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set
More informationSurface bundles over S 1, the Thurston norm, and the Whitehead link
Surface bundles over S 1, the Thurston norm, and the Whitehead link Michael Landry August 16, 2014 The Thurston norm is a powerful tool for studying the ways a 3manifold can fiber over the circle. In
More informationOn Distributed Key Distribution Centers and Unconditionally Secure Proactive Verifiable Secret Sharing Schemes Based on General Access Structure
On Distributed Key Distribution Centers and Unconditionally Secure Proactive Verifiable Secret Saring Scemes Based on General Access Structure (Corrected Version) Ventzislav Nikov 1, Svetla Nikova 2, Bart
More informationOrthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
More informationCyber Epidemic Models with Dependences
Cyber Epidemic Models wit Dependences Maocao Xu 1, Gaofeng Da 2 and Souuai Xu 3 1 Department of Matematics, Illinois State University mxu2@ilstu.edu 2 Institute for Cyber Security, University of Texas
More informationMean Value Coordinates
Mean Value Coordinates Michael S. Floater Abstract: We derive a generalization of barycentric coordinates which allows a vertex in a planar triangulation to be expressed as a convex combination of its
More informationCompute the derivative by definition: The four step procedure
Compute te derivative by definition: Te four step procedure Given a function f(x), te definition of f (x), te derivative of f(x), is lim 0 f(x + ) f(x), provided te limit exists Te derivative function
More informationMinimize subject to. x S R
Chapter 12 Lagrangian Relaxation This chapter is mostly inspired by Chapter 16 of [1]. In the previous chapters, we have succeeded to find efficient algorithms to solve several important problems such
More informationThe Derivative as a Function
Section 2.2 Te Derivative as a Function 200 Kiryl Tsiscanka Te Derivative as a Function DEFINITION: Te derivative of a function f at a number a, denoted by f (a), is if tis limit exists. f (a) f(a+) f(a)
More informationDiscussion on the paper Hypotheses testing by convex optimization by A. Goldenschluger, A. Juditsky and A. Nemirovski.
Discussion on the paper Hypotheses testing by convex optimization by A. Goldenschluger, A. Juditsky and A. Nemirovski. Fabienne Comte, Celine Duval, Valentine GenonCatalot To cite this version: Fabienne
More informationDetermine the perimeter of a triangle using algebra Find the area of a triangle using the formula
Student Name: Date: Contact Person Name: Pone Number: Lesson 0 Perimeter, Area, and Similarity of Triangles Objectives Determine te perimeter of a triangle using algebra Find te area of a triangle using
More informationFactoring Synchronous Grammars By Sorting
Factoring Syncronous Grammars By Sorting Daniel Gildea Computer Science Dept. Uniersity of Rocester Rocester, NY Giorgio Satta Dept. of Information Eng g Uniersity of Padua I Padua, Italy Hao Zang Computer
More informationWe shall turn our attention to solving linear systems of equations. Ax = b
59 Linear Algebra We shall turn our attention to solving linear systems of equations Ax = b where A R m n, x R n, and b R m. We already saw examples of methods that required the solution of a linear system
More information100 Austrian Journal of Statistics, Vol. 32 (2003), No. 1&2, 99129
AUSTRIAN JOURNAL OF STATISTICS Volume 3 003, Number 1&, 99 19 Adaptive Regression on te Real Line in Classes of Smoot Functions L.M. Artiles and B.Y. Levit Eurandom, Eindoven, te Neterlands Queen s University,
More information