A Multigrid Tutorial part two


 Silvester Horn
 2 years ago
 Views:
Transcription
1 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 Steve McCormick Department of Applied Matematics University of Colorado at Boulder Tis work was performed, in part, under te auspices of te United States Department of Energy by University of California Lawrence Livermore National Laboratory under contract number W7405Eng48. UCRLVG3689 Pt
2 Outline Nonlinear Problems Neumann Boundary Conditions Anisotropic Problems Variable Mes Problems Variable Coefficient Problems Algebraic Multigrid of 04
3 Nonlinear Problems How sould we approac te nonlinear system A ( u) = and can we use multigrid to solve suc a system? f A fundamental relation we ve relied on, te residual equation A u Av = f Av Ae = does not old, since, if A(u) is a nonlinear operator, A ( u) A( v) A( e) r 3 of 04
4 Te Nonlinear Residual Equation We still base our development around te residual equation, now te nonlinear residual equation: A ( u) = A ( u) A( v) = f A( v) f A ( u) A( v) = r How can we use tis equation as te basis for a solution metod? 4 of 04
5 Let s consider Newton s Metod Te best known and most important metod for solving nonlinear equations! We wis to solve F(x) = 0. Expand F in a Taylor series about x : F( x + s) = F( x) + sf (x) + s F ( ξ) Dropping iger order terms, if x+s is a solution, 0 = F( s) + sf ( x) s = F( x) / F ( x) Hence, we develop an iteration F( x) x x F ( x) 5 of 04
6 6 of 04 Newton s metod for systems We wis to solve te system A(u) = 0. In vector form tis is Expanding A(v+e) in a Taylor series about v : u u u f u u u f u u u f u A = ),,, ( ),,, ( ),,, ( = ) ( N N N N e v J v A e v A + ) ( + ) ( = ) + ( iger order terms
7 Newton for systems (cont.) Were J(v) is te Jacobian system f J( v ) = f N u u f u f u N f f f u u u N If u=v+e is a solution, 0 = A(v) + J(v) e and f N u f N u N u = v e = J( v) A( v) Leading to te iteration v v J( v) A( v) 7 of 04
8 Newton s metod in terms of te residual equation Te nonlinear residual equation is A ( v + e) A( v) = r Expanding A(v+e) in a twoterm Taylor series about v : A ( v) + J( v) e A( v) = J ( v) e = r r Newton s metod is tus: r = f A( v) v v + J( v) r 8 of 04
9 How does multigrid fit in? One obvious metod is to use multigrid to solve J(v)e = r at eac iteration step. Tis metod is called Newtonmultigrid and can be very effective. However, we would like to us multigrid ideas to treat te nonlinearity directly. Hence, we need to specialize te multigrid components (relaxation, grid transfers, coarsening) for te nonlinear case. 9 of 04
10 Wat is nonlinear relaxation? Several of te common relaxation scemes ave nonlinear counterparts. For A(u)=f, we describe te nonlinear GaussSeidel iteration: For eac =,,, N Set te t component of te residual to zero and solve for v. Tat is, solve (A(v)) = f. Equivalently, For eac =,,, N Find s R suc tat ( A( v + s ε ) ) = f were ε is te canonical t unit basis vector 0 of 04
11 of 04 How is nonlinear GaussSeidel done? Eac is a nonlinear scalar equation for v. We use te scalar Newton s metod to solve! Example: may be discretized so tat is given by Newton iteration for v is given by = ) ) ( ( f v A, = ) ( + ) ( f e x u x u x u ) ( = ) ) ( ( f v A = + + f e v v v v v + N v e f e v v v v v v v v ) + (
12 How do we do coarsening for nonlinear multigrid? Recall te nonlinear residual equation A ( v + e) A( v) = In multigrid, we obtain an approximate solution v on te fine grid, ten solve te residual equation on te coarse grid. Te residual equation on appears as Ω A ( v + e ) A ( v ) = r r of 04
13 3 of 04 Look at te coarse residual equation We must evaluate te quantities on in Given v, a finegrid approximation, we restrict te residual to te coarse grid For v we restrict v by Tus, Ω r v A e v A = ) ( ) + ( v A f I r ) ) ( ( = v I v = v A f I v I A e v I A ) ) ( ( + ) ( = ) + (
14 We ve obtained a coarsegrid equation of te form A ( u ) = f. Consider te coarsegrid residual equation: A I v e ( A I v I f A v + ) = ( ) + ( ( ) ) u f coarsegrid unknown All quantities are known We solve A ( u ) = f for = I v e and obtain e u = I v We ten apply te correction: v = v + I e u + 4 of 04
15 FAS, te Full Approximation Sceme, two grid form Perform nonlinear relaxation on A ( u ) = f to obtain an approximation v. Restrict te approximation and its residual v r = I ( f A( v ) ) = I v Solve te coarsegrid residual problem A ( u) = A ( v) + r Extract te coarsegrid error = u v e Interpolate and apply te correction v + = v I e 5 of 04
16 A few observations about FAS If A is a linear operator ten FAS reduces directly to te linear twogrid correction sceme. A fixed point of FAS is an exact solution to te finegrid problem and an exact solution to te finegrid problem is a fixed point of te FAS iteration. 6 of 04
17 A few observations about FAS, continued Te FAS coarsegrid equation can be written as A u ( ) = f + τ were τ is te socalled tau correction. τ 0 u In general, since, te solution to te FAS coarsegrid equation is not te same as te solution to te original coarsegrid problem. A ( u Te tau correction may be viewed as a way to alter te coarsegrid equations to enance teir approximation properties. ) = f 7 of 04
18 Still more observations about FAS FAS may be viewed as an inner and outer iteration: te outer iteration is te coarsegrid correction, te inner iteration te relaxation metod. A true multilevel FAS process is recursive, using FAS to solve te nonlinear Ω problem using Ω 4. Hence, FAS is generally employed in a V or W cycling sceme. 8 of 04
19 And yet more observations about FAS! For linear problems we use FMG to obtain a good initial guess on te fine grid. Convergence of nonlinear iterations depends critically on aving a good initial guess. Wen FMG is used for nonlinear problems te interpolant I u is generally accurate enoug to be in te basin of attraction of te finegrid solver. Tus, one FMG cycle, weter FAS, Newton, or Newtonmultigrid is used on eac level, sould provide a solution accurate to te level of discretization, unless te nonlinearity is extremely strong. 9 of 04
20 Intergrid transfers for FAS Generally speaking, te standard operators (linear interpolation, full weigting) work effectively in FAS scemes. In te case of strongly nonlinear problems, te use of igerorder interpolation (e.g., cubic interpolation) may be beneficial. I For an FMG sceme, were u is te interpolation of a coarsegrid solution to become a finegrid initial guess, igerorder interpolation is always advised. 0 of 04
21 Wat is in FAS? As in te linear case, tere are two basic possibilities: A ( u ) is determined by discretizing te nonlinear operator A(u) in te same fasion as was employed to obtain A ( u ), except tat te coarser mes spacing is used. A ( u ) A ( u ) is determined from te Galerkin condition A ( u ) = I A ( u ) I were te action of te Galerkin product can be captured in an implementable formula. Te first metod is usually easier, and more common. of 04
22 Nonlinear problems: an example Consider u( x, y) + γu( x, y) e u( x, y) = f ( x, y) on te unit square [0,] x [0,] wit omogeneous Diriclet boundary conditions and a regular Cartesian grid. Suppose te exact solution is u( x, y) = (x x 3 ) sin ( 3π y) of 04
23 Discretization of nonlinear example Te operator can be written (sloppily) as u 4 u u e + γ i,, i, = i i, f A ( u ) Te relaxation is given by u ( A ( u) ) i, u i, i, 4 + i, f i, u i, γ ( + u ) e 3 of 04
24 FAS and Newton s metod on u( x, y) + γu( x, y) e u( x, y) = f ( x, y) FAS γ convergence factor number of FAS cycles 0 Newton s Metod γ convergence factor 4.00E E E04.00E04 number of Newton iterations of 04
25 Newton, NewtonMG, and FAS on u( x, y) + γu( x, y) e u( x, y) = f ( x, y) Newton uses exact solve, NewtonMG is inexact Newton wit a fixed number of inner V(,)cycles te Jacobian problem, 0 overall stopping criterion r < 0 Outer Inner Metod iterations iterations Megaflops Newton NewtonMG NewtonMG NewtonMG NewtonMG 0.3 NewtonMG FAS 7. 5 of 04
26 Comparing FMGFAS and FMGNewton u( x, y) + γu( x, y) e u( x, y) = f ( x, y) We will do one FMG cycle using a single FAS V(,)  cycle as te solver at eac new level. We ten follow tat wit sufficiently many FAS V(,)cycles as is necessary to obtain r < 00. Next, we will do one FMG cycle using a Newtonmultigrid step at eac new level (wit a single linear V(,)cycle as te Jacobian solver. ) We ten follow tat wit sufficiently many Newtonmultigrid steps as is necessary to obtain r < of 04
27 Comparing FMGFAS and FMGNewton u( x, y) + γu( x, y) e u( x, y) = f ( x, y) Cycle r e Mflops r e Mflops Cycle FMGFAS.0E0.00E E0.50E05.4 FMGNewton FAS V 6.80E04.40E E04.49E NewtonMG FAS V 5.00E05.49E E05.49E NewtonMG FAS V 3.90E06.49E E06.49E NewtonMG FAS V 3.0E07.49E E06.49E NewtonMG FAS V 3.00E08.49E E07.49E NewtonMG FAS V.90E09.49E E07.49E05.6 NewtonMG FAS V 3.00E0.49E E08.49E NewtonMG FAS V 3.0E.49E E08.49E NewtonMG 5.50E09.49E NewtonMG.80E09.49E NewtonMG 5.60E0.49E05. NewtonMG.80E0.49E05.8 NewtonMG 5.70E.49E NewtonMG 7 of 04
28 Outline Nonlinear Problems Neumann Boundary Conditions Anisotropic Problems Variable Mes Problems Variable Coefficient Problems Algebraic Multigrid 8 of 04
29 Neumann Boundary Conditions Consider te (d) problem u ( x) = f ( x) 0 < x < u ( 0) = u ( ) = 0 We discretize tis on te interval [0,] wit grid spacing x = for =,,, N+. = N + We extend te interval wit two gost points 0 x N N N+ N+ 9 of 04
30 We use central differences We approximate te derivatives wit differences, using te gost points: N N+ N+ u ( 0) u u Giving te system u ( x ) u + u u + u ( ) u N + u N u + u u + = f 0 N + u = 0 u un + u N = 0 30 of 04
31 Eliminating te gost points Use te boundary conditions to eliminate u, u N + u u = 0 u = u Eliminating te gost points in te =0 and =N+ equations gives te (N+)x(N+) system of equations: u N + u N = 0 u N + = u N u + u u + = f 0 N + u 0 u u N + un + = f = 0 f N + 3 of 04
32 We write te system in matrix form We can write, were A A = u = Note tat is (N+)x(N+), nonsymmetric, and te system involves unknowns and at te boundaries. f A u 0 u N + 3 of 04
33 We must consider a compatibility condition Te problem u ( x) = f ( x), for 0 < x < and wit u ( 0) = u ( ) = 0 is not wellposed! If u(x) is a solution, so is u(x)+c for any constant c. We cannot be certain a solution exists. If one does, it must satisfy 0 u (x) dx = 0 f ( x) dx [u () u (0)] = 0 f ( x) dx 0 = 0 f ( x) dx Tis integral compatibility condition is necessary! If f(x) doesn t satisfy it, tere is no solution! 33 of 04
34 Te wellposed system Te compatibility condition is necessary for a solution to exist. In general, it is also sufficient, wic can be proven tat is a wellbeaved x operator in te space of functions u(x) tat ave zero mean. Tus we may conclude tat if f(x) satisfies te compatibility condition, tis problem is wellposed: u ( x) = f ( x) 0 < x < u ( 0) = u ( ) = 0 0 u( x) dx = 0 Te last says tat of all solutions u(x)+c we coose te one wit zero mean. 34 of 04
35 Te discrete problem is not well posed Since all row sums of A are zero, NS( A ) Putting into rowecelon form sows tat A d im( NS( A ) ) = ence N S( A ) = span ( ) By te Fundamental Teorem of Linear Algebra, T as a solution if and only if f NS ( ( A ) ) A It is easy to sow tat T N S ( ( A ) ) = c( /,,,...,, / ) T Tus, A u = f as a solution if and only if T f c( /,,,...,, / ) Tat is, N f 0 + f + f N + = = 0 35 of 04
36 We ave two issues to consider Solvability. A solution exists iff f NS ( ( A ) T ) Uniqueness. If u solves A u = f so does u + c Note tat if A =(A ) T ten NS( A ) = NS((A ) T ) and solvability and uniqueness can be andled togeter Tis is easily done. Multiply st and last equations by /, giving A = 36 of 04
37 Te new system is symmetric We ave te symmetric system : Solvability is guaranteed by ensuring tat f is ortogonal to te constant vector : u 0 u u u N u N + = A u = f f 0 / f f f N f N + / f N +, = f = 0 = 0 37 of 04
38 Te wellposed discrete system Te (N+3)x(N+) system is: u + u u = f u 0 u f 0 = u N + u N + = N + u i = 0 i = 0 + or, more simply u f N + A u = f =, 0 0 N + (coose te zero mean solution) 38 of 04
39 Multigrid for te Neumann Problem We must ave te interval endpoints on all grids x x 0 x N/ x N x 0 x N / 4 x N / Relaxation is performed at all points, including endpoints: v0 v + f 0 v + + v + v + We add a global GramScmidt step after relaxation on eac level to enforce te zeromean condition v f v, v, v v N + N + f N + 39 of 04
40 Interpolation must include te endpoints We use linear interpolation: I = / / / / / / / / 40 of 04
41 4 of 04 Restriction also treats te endpoints For restriction, we use, yielding te values f f f = 4 f f f N N N = 4 f f f f + + = I I T ) = (
42 Te coarsegrid operator We compute te coarsegrid operator using te Galerkin condition A = I A I = 0 I A I A ε 0 I ε 0 I ε 0 ε A = 4 4 of 04
43 Coarsegrid solvability Assuming f satisfies f, = 0, te solvability condition, we can sow tat teoretically te coarsegrid problem A u = I ( f A v ) is also solvable. To be certain numerical roundoff does not perturb solvability, we incorporate a GramScmidt like step eac time a new rigtand side f is generated for te coarse grid: f f f,, 43 of 04
44 Neumann Problem: an example Consider te problem u ( x) = x, 0 < x < u ( 0) = u ( ) = 0 x x 3 (c=/ gives te zero mean solution). 3 wic as u ( x) = + c as a solution for any c grid size average number r e N conv. factor of cycles E E E E E E E E E E E E E E E E of 04
45 Outline Nonlinear Problems Neumann Boundary Conditions Anisotropic Problems Variable Mes Problems Variable Coefficient Problems Algebraic Multigrid 45 of 04
46 Anisotropic Problems All problems considered tus far ave ad as te offdiagonal entries. We consider two situations wen te matrix as two different constant on te offdiagonals. Tese situations arise wen te (d) differential equation as constant, but different, coefficients for te derivatives in te coordinate directions te discretization as constant, but different, mas spacing in te different coordinate directions 46 of 04
47 We consider two types of anisotropy Different coefficients on te derivatives u xx α discretized on a uniform grid wit spacing. u yy = f Constant, but different, mes spacings: y x = = y = x α N x 47 of 04
48 48 of 04 Bot problems lead to te same stencil + α + + u u u u u u k k k k k k +,,,, +,, u u u u u u k k k k k k α +,,,, +,, A α α + α =
49 Wy standard multigrid fails Note tat A = + α as weak connections in te α α ydirection. MG convergence factors degrade as α gets small. Poor performance at α = 0.. Consider α 0. A Tis is a collection of disconnected d problems! Point relaxation will smoot oscillatory errors in te x direction (strong connections), but wit no connections in te ydirection te errors in tat direction will generally be random, and no point relaxation will ave te smooting property in te ydirection. 0 + α 0 49 of 04
50 We analyze weigted Jacobi Te eigenvalues of te weigted Jacobi iteration matrix for tis problem are λ i, l = ω +α sin iπ N +αsin lπ N i 0 k l 0 l 5 i l 50 of 04
51 Two strategies for anisotropy Semicoarsening Because we expect MGlike convergence for te d problems along lines of constant y, we sould coarsen te grid in te x direction, but not in te ydirection. Line relaxation Because te te equations are strongly coupled in te xdirection it may be advantageous to solve simultaneously for entire lines of unknowns in te xdirection (along lines of constant y) 5 of 04
52 Semicoarsening wit point relaxation A α = Point relaxation on + α smoots in te x α direction. Coarsen by removing every oter yline. Ω Ω We do not coarsen along te remaining ylines. Semicoarsening is not as fast as full coarsening. Te number of points on Ω is about alf te number of points on Ω, instead of te usual onefourt. 5 of 04
53 Interpolation wit semicoarsening We interpolate in te dimensional way along eac line of constant y. Te coarsegrid correction equations are v = v v, k, k +, k v +, k = v +, k + v, k + v +, k 53 of 04
54 Line relaxation wit full coarsening Te oter approac to tis problem is to do full coarsening, but to relax entire lines (constant y) of variables simultaneously. Write A in block form as were c A = D ci ci D ci ci D ci ci ci D α + α = and D = + α + α 54 of 04
55 Line relaxation One sweep of line relaxation consists of solving a tridiagonal system for eac line of constant y. Te kt suc system as te form D v g k = k were vk is te kt subvector of v wit entries ( v k ) = v, k and te kt rigtand side subvector is α ( g ) = f + ( v k k + v, k,, k + Because D is tridiagonal, te kt system can be solved very efficiently. ) 55 of 04
56 Wy line relaxation works Te eigenvalues of te weigted block Jacobi iteration matrix are λ ω π π = sin i + α sin l i π N N sin + α N i, l l i i l 56 of 04
57 Semicoarsening wit line relaxation We migt not know te direction of weak coupling or it migt vary. Suppose we want a metod tat can andle eiter A = α + α α or A = α + α α We could use semicoarsening in te xdirection to andle A and line relaxation in te ydirection to take care of. A 57 of 04
58 Semicoarsening wit line relaxation Te original grid Original grid viewed as a stack of pencils. Line relaxation is used to solve problem along eac pencil. Coarsening is done by deleting every oter pencil 58 of 04
59 An anisotropic example u α u = Consider xx yy wit u=0 on te boundaries of te unit square, and stencil given by A Suppose f ( x, y) = ( y y ) + α(x x ) so te exact solution is given by u( x, y) = (y y )(x x ) f α = + α α Observe tat if α is small, te xdirection dominates wile if α is large, te ydirection dominates 59 of 04
60 Wat is smoot error? Consider α=0.00 and suppose point GaussSeidel is applied to a random initial guess. Te error after 50 sweeps appears as: Error along line of constant x y x Error along line of constant y 60 of 04
61 We experiment wit 3 metods Standard V(,)cycling, wit point GaussSeidel relaxation, full coarsening, and linear interpolation Semicoarsening in te xdirection. Coarse and fine grids ave te same number of points in te y direction. d full weigting and linear interpolation are used in te xdirection, tere is no ycoupling in te intergrid transfers Semicoarsening in te xdirection combined wit line relaxation in te ydirection. d full weigting and interpolation. 6 of 04
62 Wit semicoarsening, te operator must cange To account for unequal mes spacing, te residual and relaxation operators must use a modified stencil A = α y x x + α y y Note tat as grids become coarser, grows wile y remains constant. x x 6 of 04
63 How do te 3 metods work for various values of α? Asymptotic convergence factors: sceme E04 V(,)cycles emicoarsening in x semic / line relax α ydirection strong xdirection strong Note: semicoarsening in x works well for α <.00 but degrades noticeably even at α =. 63 of 04
64 A semicoarsening subtlety Suppose α is small, so tat semicoarsening in x is used. As we progress to coarser grids, x gets small but remains constant. y x If, on some coarse grid, becomes comparable to α y, te problem effectively becomes recoupled in te ydirection. Continued semicoarsening can produce artificial anisotropy, strong in te ydirection. Wen tis occurs, it is best to stop semicoarsening and continue wit full coarsening on any furter coarse grids. 64 of 04
65 Outline Nonlinear Problems Neumann Boundary Conditions Anisotropic Problems Variable Mes Problems Variable Coefficient Problems Algebraic Multigrid 65 of 04
66 Variable Mes Problems Nonuniform grids are commonly used to accommodate irregularities in problem domains Consider ow we migt approac te d problem u ( x) = f ( x) 0 < x < u( 0) = u( ) = 0 posed on tis grid: x = 0 x = x 0 x x x + x N 66 of 04
67 We need some notation for te mes spacing Let N be a positive integer. We define te spacing interval between and : x x + + / x + x = 0,,...,N + / x 0 x x x + x N 67 of 04
68 We define te discrete differential operator Using second order finite differences (and plugging troug a mess of algebra!) we obtain tis discrete representation for te problem: α u +(α +β ) u β u + u 0 = u N = 0 = f N were α = / ( / + + / ) β = + / ( / + + / ) 68 of 04
69 We modify standard multigrid to accommodate variable spacing We coose every second finegrid point as a coarsegrid point x 0 x x N x 0 x x N / We use linear interpolation, modified for te spacing. If v = I, ten for v N/ v = v v + = + 3/ v + + / v + + / + + 3/ 69 of 04
70 We use te variational properties A to derive restriction and. A = I A I I = ( I ) T Tis produces a stencil on Ω tat is similar, but not identical, to te finegrid stencil. If te resulting system is scaled by ( / + + / ), ten te Galerkin product is te same as te finegrid stencil. For d problems tis approac can be generalized readily to logically rectangular grids. However, for irregular grids tat are not logically rectangular, AMG is a better coice. 70 of 04
71 Outline Nonlinear Problems Neumann Boundary Conditions Anisotropic Problems Variable Mes Problems Variable Coefficient Problems Algebraic Multigrid 7 of 04
72 Variable coefficient problems A common difficulty is te variable coefficient problem, given in d by (a( x) u ( x)) = f ( x) 0 < x < u( 0) = u( ) = 0 were a(x) is a positive function on [0,] We seek to develop a conservative, or selfadoint, metod for discretizing tis problem. Assume we ave available to us te values of a(x) at te midpoints of te grid a + / a( x + / ) x 0 x x x + x N 7 of 04
73 We discretize using central differences We can use secondorder differences to approximate te derivatives. To use a grid spacing of we evaluate a(x)u (x) at points midway between te gridpoints: ( a( x) u (x)) x ( au ) x + / (au ) x / + O( ) Points used to evaluate (au ) at x x 0 x x x + x N 73 of 04
74 We discretize using central differences To evaluate ( au ) x + / we must sample a(x) at te point and use second order differences: x + / ( au ) x + / a + / u + u were a + / a( x + / ) ( au ) x / a / u u Points used to evaluate u at x + / Points used to evaluate (au ) at x x 0 x x x + x N 74 of 04
75 Te basic stencil is given We combine te differences for u and for (au ) to obtain te operator (a( x ) u ( x )) ( x ) a + / u + u + a / u u and te problem becomes, for (a / u +(a / + a + / ) u a + / u + ) = f u 0 = u N = 0 N 75 of 04
76 Coarsening te variable coefficient problem A reasonable approac is to use a standard multigrid algoritm wit linear interpolation, full weigting, and te stencil were A = ( a ( ) / a / + a + / a + / a + / + a + 3/ a + / = ) Te same stencil can be obtained via te Galerkin relation 76 of 04
77 Standard multigrid degrades if a(x) is igly variable It can be sown tat te variable coefficient discretization is equivalent to using standard multigrid wit simple averaging on te Poisson problem on a certain variablemes spacing. (a(x)u ) = f u (x) = f But simple averaging won t accurately represent smoot components if is close to but far from. x + x x + x x + x + x x + 77 of 04
78 One remedy is to apply operator interpolation Assume tat relaxation does not cange smoot error, so te residual is approximately zero. Applying at yields x + a + / e +(a + / + a + 3/ ) e + a + 3/ e + = 0 Solving for e + e + = a + / e + a + 3/ e + a + / + a + 3/ 78 of 04
79 Tus, te operator induced interpolation is v = v v + = a + / v + a + 3/ v + a + / + a + 3/ And, as usual, te restriction and coarsegrid operators are defined by te Galerkin relations A = I A I I = c( I ) T 79 of 04
80 A Variable coefficient example We use V(,) cycle, full weigting, linear interpolation. We use a( x) = ρ sin( k π x ) and a( x) = ρ rand ( k π x) ρ a( x) = ρ sin( k π x ) a( x) = ρ rand ( k π x) k=3 k=5 k=50 k=00 k=00 k= of 04
81 Outline Nonlinear Problems Neumann Boundary Conditions Anisotropic Problems Variable Mes Problems Variable Coefficient Problems Algebraic Multigrid 8 of 04
82 Algebraic multigrid: for unstructuredgrids Automatically defines coarse grid AMG as two distinct pases: setup pase: define MG components solution pase: perform MG cycles AMG approac is opposite of geometric MG fix relaxation (point GaussSeidel) coose coarse grids and prolongation, P, so tat error not reduced by relaxation is in range(p) define oter MG components so tat coarsegrid correction eliminates error in range(p) (i.e., use Galerkin principle) (in contrast, geometric MG fixes coarse grids, ten defines suitable operators and smooters) 8 of
83 AMG as two pases: Setup Pase Select Coarse grids, Ω m +, m =,,... Define interpolation, Im m +, m =,,... Define restriction and coarsegrid operators m + m T m + = m + m Im = ( Im + ) A Im A Im + Solve Pase Standard multigrid operations, e.g., Vcycle, Wcycle, FMG, FAS, etc Note: Only te selection of coarse grids does not parallelize well using existing tecniques! m 83 of
84 AMG fundamental concept: Smoot error = small residuals Consider te iterative metod error recurrence ek + = ( I Q A ) ek Error tat is slow to converge satisfies ( I Q A) e e Q A e 0 r 0 More precisely, it can be sown tat smoot error satisfies r e D «() A 84 of
85 AMG uses strong connection to determine MG components It is easy to sow from () tat smoot error satisfies () A e, e «De, Define i is strongly connected to by ai θ max {aik}, 0 < θ k i For Mmatrices, we ave from () a i ei e «e ii i i a e implying tat smoot error varies slowly in te direction of strong connections 85 of
86 Some useful definitions Te set of strong connections of a variable u i, tat is, te variables upon wose values te value of u i depends, is defined as S i = : ai > θ ma x ai i Te set of points strongly connected to a variable is denoted: S i T = { : Si}. Te set of coarsegrid variables is denoted C. Te set of finegrid variables is denoted F. Te set of coarsegrid variables used to interpolate te value of te finegrid variable C i is denoted. u i u i 86 of
87 Coosing te Coarse Grid Two Criteria i F (C) For eac, eac point sould eiter be in C or sould be strongly connected to at least one point in C i C S i (C) sould be a maximal subset wit te property tat no two Cpoints are strongly connected to eac oter. Satisfying bot (C) and (C) is sometimes impossible. We use (C) as a guide wile enforcing (C). 87 of
88 Selecting te coarsegrid points Cpoint selected (point wit largest value ) Neigbors of Cpoint become F points Next Cpoint selected (after updating values ) Fpoints selected, etc. 88 of
89 Examples: Laplacian Operator 5pt FD, 9pt FE (quads), and 9pt FE (stretced quads) pt FD 9pt FE (quads) 9pt FE (stretced quads) 89 of
90 Prolongation is based on smoot error, strong connections (from Mmatrices) Smoot error is given by: F C F r i = aii ei + ai e 0 i N C i F C Prolongation : ei, i C ( Pe) = i ωik ek, i F k C i 90 of
91 F Prolongation is based on smoot error, strong connections (from Mmatrices) C F Sets: C i D s i D w i Strongly connected C pts. Strongly connected Fpts. Weakly connected points. C i F a ii e i C Gives: C Te definition of smoot error, i a a i ii e e i D i s i a i a i e e w i D a i e Strong C Strong F Weak pts. 9 of
92 Finally, te prolongation weigts are defined In te smooterror relation, use e = ei for weak connections. For te strong Fpoints use : e = ak ek k C k i C i a k yielding te prolongation weigts: w i = a i a ii + D + n D a s i m C i w i ik a a k a in km 9 of
93 AMG setup costs: a bad rap Many geometric MG metods need to compute prolongation and coarsegrid operators Te only additional expense in te AMG setup pase is te coarse grid selection algoritm AMG setup pase is only 05% more expensive tan in geometric MG and may be considerably less tan tat! 93 of
94 AMG Performance: Sometimes a Success Story AMG performs extremely well on te model problem (Poisson s equation, regular grid) optimal convergence factor (e.g., 0.4) and scalability wit increasing problem size. AMG appears to be bot scalable and efficient on diffusion problems on unstructured grids (e.g., ). AMG andles anisotropic diffusion coefficients on irregular grids reasonably well. AMG andles anisotropic operators on structured and unstructured grids relatively well (e.g., 0.35). 94 of
95 So, wat could go wrong? Strong FF connections: weigts are dependent on eac oter For point i te value e is interpolated from k, k, and is needed to make te interpolation weigts for approximating e. i For point te value e i is interpolated from k, k, and is needed to make te interpolation weigts for approximating e. It s an implicit system! k k C C i i k 3 C k 4 C i k k C C i 95 of
96 Is tere a fix? A GaussSeidel like iterative approac to weigt definition is implemented. Usually two passes suffice. But does it work? Frequently, it does: Convergence factors for Laplacian, stretced quadrilaterals x = 0 y teta Standard Iterative x = 00 y of
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 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 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 informationMultigrid computational methods are
M ULTIGRID C OMPUTING Wy Multigrid Metods Are So Efficient Originally introduced as a way to numerically solve elliptic boundaryvalue problems, multigrid metods, and teir various multiscale descendants,
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 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 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 informationChapter 7 Numerical Differentiation and Integration
45 We ave a abit in writing articles publised in scientiþc journals to make te work as Þnised as possible, to cover up all te tracks, to not worry about te blind alleys or describe ow you ad te wrong idea
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 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 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 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 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 informationA.4 RATIONAL EXPRESSIONS
Appendi A.4 Rational Epressions A9 A.4 RATIONAL EXPRESSIONS Wat you sould learn Find domains of algebraic epressions. Simplify rational epressions. Add, subtract, multiply, and divide rational epressions.
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 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 informationMath 113 HW #5 Solutions
Mat 3 HW #5 Solutions. Exercise.5.6. Suppose f is continuous on [, 5] and te only solutions of te equation f(x) = 6 are x = and x =. If f() = 8, explain wy f(3) > 6. Answer: Suppose we ad tat f(3) 6. Ten
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 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 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 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 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 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 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 informationA New Cement to Glue Nonconforming Grids with Robin Interface Conditions: The Finite Element Case
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
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 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 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 information6.3 Multigrid Methods
6.3 Multigrid Methods The Jacobi and GaussSeidel iterations produce smooth errors. The error vector e has its high frequencies nearly removed in a few iterations. But low frequencies are reduced very
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 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 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 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 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 informationTRADING AWAY WIDE BRANDS FOR CHEAP BRANDS. Swati Dhingra London School of Economics and CEP. Online Appendix
TRADING AWAY WIDE BRANDS FOR CHEAP BRANDS Swati Dingra London Scool of Economics and CEP Online Appendix APPENDIX A. THEORETICAL & EMPIRICAL RESULTS A.1. CES and Logit Preferences: Invariance of Innovation
More informationThe differential amplifier
DiffAmp.doc 1 Te differential amplifier Te emitter coupled differential amplifier output is V o = A d V d + A c V C Were V d = V 1 V 2 and V C = (V 1 + V 2 ) / 2 In te ideal differential amplifier A c
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 informationChapter 6: Solving Large Systems of Linear Equations
! Revised December 8, 2015 12:51 PM! 1 Chapter 6: Solving Large Systems of Linear Equations Copyright 2015, David A. Randall 6.1! Introduction Systems of linear equations frequently arise in atmospheric
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 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 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 informationTESLA Report 200303
TESLA Report 233 A multigrid based 3D spacecharge routine in the tracking code GPT Gisela Pöplau, Ursula van Rienen, Marieke de Loos and Bas van der Geer Institute of General Electrical Engineering,
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 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 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 informationSolving Reservoir Simulation Equations
9th International Forum on Reservoir Simulation December 913, 2007, Abu Dhabi, United Arab Emirates Solving Reservoir Simulation Equations Fraunhofer Institute SCAI Sankt Augustin, Germany Reservoir simulation
More informationCatalogue no. 12001XIE. Survey Methodology. December 2004
Catalogue no. 1001XIE Survey Metodology December 004 How to obtain more information Specific inquiries about tis product and related statistics or services sould be directed to: Business Survey Metods
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 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 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 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 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 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 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 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 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 informationSection 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section 3.3 Differentiation of Polynomials an Rational Functions In tis section we begin te task of iscovering rules for ifferentiating various classes of
More informationWhat is Advanced Corporate Finance? What is finance? What is Corporate Finance? Deciding how to optimally manage a firm s assets and liabilities.
Wat is? Spring 2008 Note: Slides are on te web Wat is finance? Deciding ow to optimally manage a firm s assets and liabilities. Managing te costs and benefits associated wit te timing of cas in and outflows
More informationMultigrid preconditioning for nonlinear (degenerate) parabolic equations with application to monument degradation
Multigrid preconditioning for nonlinear (degenerate) parabolic equations with application to monument degradation M. Donatelli 1 M. Semplice S. SerraCapizzano 1 1 Department of Science and High Technology
More informationAn Interest Rate Model
An Interest Rate Model Concepts and Buzzwords Building Price Tree from Rate Tree Lognormal Interest Rate Model Nonnegativity Volatility and te Level Effect Readings Tuckman, capters 11 and 12. Lognormal
More informationCITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
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 informationNotes on Factoring. MA 206 Kurt Bryan
The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the yaxis We observe that
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 informationParallel Smoothers for Matrixbased Multigrid Methods on Unstructured Meshes Using Multicore CPUs and GPUs
Parallel Smooters for Matrixbased Multigrid Metods on Unstructured Meses Using Multicore CPUs and GPUs Vincent Heuveline Dimitar Lukarski Nico Trost JanPilipp Weiss No. 29 Preprint Series of te Engineering
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 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 informationAN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS
AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS Revised Edition James Epperson Mathematical Reviews BICENTENNIAL 0, 1 8 0 7 z ewiley wu 2007 r71 BICENTENNIAL WILEYINTERSCIENCE A John Wiley & Sons, Inc.,
More informationA system to monitor the quality of automated coding of textual answers to open questions
Researc in Official Statistics Number 2/2001 A system to monitor te quality of automated coding of textual answers to open questions Stefania Maccia * and Marcello D Orazio ** Italian National Statistical
More informationHOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba
HOMEWORK SOLUTIONS All questions are from Vector Calculus, by Marsden and Tromba Question :..6 Let w = f(x, y) be a function of two variables, and let x = u + v, y = u v. Show that Solution. By the chain
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 informationPretrial Settlement with Imperfect Private Monitoring
Pretrial Settlement wit Imperfect Private Monitoring Mostafa Beskar University of New Hampsire JeeHyeong Park y Seoul National University July 2011 Incomplete, Do Not Circulate Abstract We model pretrial
More informationArea of a Parallelogram
Area of a Parallelogram Focus on After tis lesson, you will be able to... φ develop te φ formula for te area of a parallelogram calculate te area of a parallelogram One of te sapes a marcing band can make
More informationNotes: Most of the material in this chapter is taken from Young and Freedman, Chap. 12.
Capter 6. Fluid Mecanics Notes: Most of te material in tis capter is taken from Young and Freedman, Cap. 12. 6.1 Fluid Statics Fluids, i.e., substances tat can flow, are te subjects of tis capter. But
More informationf(x) f(a) x a Our intuition tells us that the slope of the tangent line to the curve at the point P is m P Q =
Lecture 6 : Derivatives and Rates of Cange In tis section we return to te problem of finding te equation of a tangent line to a curve, y f(x) If P (a, f(a)) is a point on te curve y f(x) and Q(x, f(x))
More informationLecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 10
Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 10 Boundary Value Problems for Ordinary Differential Equations Copyright c 2001. Reproduction
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 informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 10 Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at UrbanaChampaign
More informationCollege Planning Using Cash Value Life Insurance
College Planning Using Cas Value Life Insurance CAUTION: Te advisor is urged to be extremely cautious of anoter college funding veicle wic provides a guaranteed return of premium immediately if funded
More informationTowards realtime image processing with Hierarchical Hybrid Grids
Towards realtime image processing with Hierarchical Hybrid Grids International Doctorate Program  Summer School Björn Gmeiner Joint work with: Harald Köstler, Ulrich Rüde August, 2011 Contents The HHG
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 informationIterative Solvers for Linear Systems
9th SimLab Course on Parallel Numerical Simulation, 4.10 8.10.2010 Iterative Solvers for Linear Systems Bernhard Gatzhammer Chair of Scientific Computing in Computer Science Technische Universität München
More informationDATA ANALYSIS II. Matrix Algorithms
DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where
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 informationPressure. Pressure. Atmospheric pressure. Conceptual example 1: Blood pressure. Pressure is force per unit area:
Pressure Pressure is force per unit area: F P = A Pressure Te direction of te force exerted on an object by a fluid is toward te object and perpendicular to its surface. At a microscopic level, te force
More informationDifferential Calculus: Differentiation (First Principles, Rules) and Sketching Graphs (Grade 12)
OpenStaxCNX moule: m39313 1 Differential Calculus: Differentiation (First Principles, Rules) an Sketcing Graps (Grae 12) Free Hig Scool Science Texts Project Tis work is prouce by OpenStaxCNX an license
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 information2.23 Gambling Rehabilitation Services. Introduction
2.23 Gambling Reabilitation Services Introduction Figure 1 Since 1995 provincial revenues from gambling activities ave increased over 56% from $69.2 million in 1995 to $108 million in 2004. Te majority
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 informationA perturbed twolevel preconditioner for the solution of threedimensional heterogeneous Helmholtz problems with applications to geophysics
Institut National Polytecnique de Toulouse(INP Toulouse) Matématiques, Informatiques et Télécommunication Xavier Pinel mardi18mai2010 A perturbed twolevel preconditioner for te solution of treedimensional
More information6. Differentiating the exponential and logarithm functions
1 6. Differentiating te exponential and logaritm functions We wis to find and use derivatives for functions of te form f(x) = a x, were a is a constant. By far te most convenient suc function for tis purpose
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 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 informationSolving partial differential equations (PDEs)
Solving partial differential equations (PDEs) Hans Fangor Engineering and te Environment University of Soutampton United Kingdom fangor@soton.ac.uk May 3, 2012 1 / 47 Outline I 1 Introduction: wat are
More information1 Review of Newton Polynomials
cs: introduction to numerical analysis 0/0/0 Lecture 8: Polynomial Interpolation: Using Newton Polynomials and Error Analysis Instructor: Professor Amos Ron Scribes: Giordano Fusco, Mark Cowlishaw, Nathanael
More informationCode Verification and Numerical Accuracy Assessment for Finite Volume CFD Codes. Subrahmanya P. Veluri
Code Verification and Numerical Accuracy Assessment for Finite Volume CFD Codes Subramanya P. Veluri Dissertation submitted to te faculty of te Virginia Polytecnic Institute and State University in partial
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 informationf(x + h) f(x) h as representing the slope of a secant line. As h goes to 0, the slope of the secant line approaches the slope of the tangent line.
Derivative of f(z) Dr. E. Jacobs Te erivative of a function is efine as a limit: f (x) 0 f(x + ) f(x) We can visualize te expression f(x+) f(x) as representing te slope of a secant line. As goes to 0,
More information