Multigrid computational methods are


 Solomon Lee
 2 years ago
 Views:
Transcription
1 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, ave since been developed and applied to various problems in many disciplines. Tis introductory article provides te basic concepts and metods of analysis and outlines some of te difficulties of developing efficient multigrid algoritms /06/$ IEEE Copublised by te IEEE CS and te AIP IRAD YAVNEH Multigrid computational metods are well known for being te fastest numerical metods for solving elliptic boundaryvalue problems. Over te past 30 years, multigrid metods ave earned a reputation as an efficient and versatile approac for oter types of computational problems as well, including oter types of partial differential equations and systems and some integral equations. In addition, researcers ave successfully developed generalizations of te ideas underlying multigrid metods for problems in various disciplines. (See te Multigrid Metods Resources sidebar for more details.) Tis introductory article presents te fundamentals of multigrid metods, including explicit algoritms, and points out some of te main pitfalls using elementary model problems. Tis material is mostly intended for readers wo ave a practical interest in computational metods but little or no acquaintance wit multigrid tecniques. Te article also provides some background for tis special issue and oter, more advanced publications. Tecnion Israel Institute of Tecnology Basic Concepts Let s begin wit a simple example based on a problem studied in a different context. Global versus Local Processes Te ometown team as won te regional cup. Te players are to receive medals, so it s up to te coac to line tem up, equally spaced, along te goal line. Alas, te field is muddy. Te coac, wo abors mud, tries to accomplis te task from afar. He begins by numbering te players 0 to N and orders players 0 and N to stand by te left and rigt goal posts, respectively, wic are a distance L apart. Now, te coac could, for example, order player i, i =,, N, to move to te point on te goal line tat is at a distance il/n from te left goal post. Tis would be a global process tat would solve te problem directly. However, it would require eac player to recognize te leftand goal post, perform some fancy aritmetic, and gauge long distances accurately. Suspecting tat is players aren t up to te task, te coac reasons tat if eac player were to stand at te center point between is two neigbors (wit players 0 and N fixed at te goal posts), is task would be accomplised. From te local rule, a global order will emerge, e pilosopizes. Wit tis in mind, te coac devises te following simple iterative local process. At eac iteration, e blows is wistle, and player i, i =,, N, COMPUTING IN SCIENCE & ENGINEERING
2 MULTIGRID METHODS RESOURCES Radii Petrovic Fedorenko, and Nikolai Sergeevitc Bakvalov 3 first conceived multigrid metods in te 960s, but Aci Brandt first acieved teir practical utility and efficiency for general problems in is pioneering work in te 970s. 4 6 Classical texts on multigrid metods include an excellent collection of papers edited by Wolfgang Hackbusc and Ulric Trottenberg, 7 Brandt s guide to multigrid metods, 8 and te classical book by Hackbusc. 9 Notable recent textbooks on multigrid include te introductory tutorial by William Briggs and is colleagues 0 and a compreensive textbook by Trottenberg and is colleagues. Many oter excellent books and numerous papers are also available, mostly devoted to more specific aspects of multigrid tecniques, bot teoretical and practical. References. R.P. Fedorenko, A Relaxation Metod for Solving Elliptic Difference Equations, USSR Computational Matematics and Matematical Pysics, vol., no. 4, 96, pp R.P. Fedorenko, Te Speed of Convergence of One Iterative Process, USSR Computational Matematics and Matematical Pysics, vol. 4, no. 3, 964, pp N.S. Bakvalov, On te Convergence of a Relaxation Metod wit Natural Constraints on te Elliptic Operator, USSR Computational Matematics and Matematical Pysics, vol. 6, no. 5, 966, pp A. Brandt, A MultiLevel Adaptive Tecnique (MLAT) for Fast Numerical Solution to Boundary Value Problems, Proc. 3rd Int l Conf. Numerical Metods in Fluid Mecanics, H. Cabannes and R. Temam, eds., Lecture Notes in Pysics 8, Springer, 973, pp A. Brandt, MultiLevel Adaptive Tecniques (MLAT): Te Multigrid Metod, researc report RC 606, IBM T.J. Watson Researc Ctr., A. Brandt, MultiLevel Adaptive Solutions to BoundaryValue Problems, Matematics of Computation, vol. 3, no. 38, 977, pp W. Hackbusc and U. Trottenberg, eds., Multigrid Metods, Springer Verlag, A. Brandt, 984 Multigrid Guide wit Applications to Fluid Dynamics, GMD Studie 85, GMDFIT, Postfac 40, D505, St. Augustin, West Germany, W. Hackbusc, MultiGrid Metods and Applications, Springer, W.L. Briggs, V.E. Henson, and S.F. McCormick, A Multigrid Tutorial, nd ed., SIAM Press, U. Trottenberg, C.W. Oosterlee, and A. Scüller, Multigrid, Academic Press, 00. (a) (b) (c) (d) (e) (f) Figure. Jacobi s playeralignment iteration. Red disks sow te current position, and blue disks sow te previous position, before te last wistle blow: (a) initial position, (b) one wistle blow, (c) slow convergence, (d) fast convergence, (e) oversoot, and (f) damped iteration. moves to te point tat s at te center between te positions of player i and player i + just before te wistle was blown. Tese iterations are repeated until te errors in te players positions are so small tat te mayor, wo will be pinning te medals, won t notice. Tis is te convergence criterion. Undeniably, te coac deserves a name; we ll call im Mr. Jacobi. Figure a sows te players initial apazard position, and Figure b te position after one iteration (wistle blow). (For clarity, we assume tat te players aren t even on te goal line initially. Tis pessimistic view does not alter our analysis significantly. We can envision purely lateral movements by projecting te players positions onto te goal line.) As I ll sow, Jacobi s metod is con NOVEMBER/DECEMBER 006 3
3 (a) (b) (c) Figure. Multiscale player alignment. Red disks sow te current position, and blue disks sow te previous position, before te last wistle blow: (a) large, (b) medium, and (c) small scales. vergent tat is, te errors in te players positions will eventually be as small as we wis to make tem. However, tis migt take a long time. Suppose te players positions are as sown in Figure c. In tis case, te players move sort distances at eac iteration. Sadly, tis slow crawl toward convergence will persist for te duration of te process. In contrast, convergence is obtained in a single iteration in te position sown in Figure d. Wat is te essential property tat makes te convergence slow in te position of Figure c, yet fast in Figure d? In Figure c, we ave a largescale error. But Jacobi s process is local, employing only smallscale information tat is, eac player s new position is determined by is near neigborood. Tus, to eac player, it seems (from is myopic viewpoint) tat te error in position is small, wen in fact it isn t. Te point is tat we cannot correct wat we cannot detect, and a largescale error can t be detected locally. In contrast, Figure d sows a smallscale error position, wic can be effectively detected and corrected by using te local process. Finally, consider te position in Figure e. It clearly as a smallscale error, but Jacobi s process doesn t reduce it effectively; rater, it introduces an oversoot. Not every local process is suitable. However, tis problem is easily overcome witout compromising te process s local nature by introducing a constant damping factor. Tat is, eac player moves only a fraction of te way toward is designated target at eac iteration; in Figure f, for example, te players movements are damped by a factor /3. Multiscale Concept Te main idea beind multigrid metods is to use te simple local process but to do so at all scales of te problem. For convenience, assume N to be a power of two. In our simple example, te coac begins wit a largescale view in wic te problem consists only of te players numbered 0, N/, and N. A wistle blow puts player N/ alfway between is largescale neigbors (see Figure a). Next, addressing te intermediate scale, players N/4 and 3N/4 are activated, and tey move at te sound of te wistle to te midpoint between teir midscale neigbors (see Figure b). Finally, te small scale is solved by Jacobi s iteration at te remaining locations (Figure c). Tus, in just tree wistle blows generally, log (N) and only N individual moves, te problem is solved, even toug te simple local process as been used. Because te players can move simultaneously at eac iteration in parallel, so to speak te number of wistle blows is important in determining te overall time, and log (N) is encouraging. For million players, for example, Jacobi s process would require a mere 0 wistle blows. Researcers ave successfully exploited te idea of applying a local process at different scales in many fields and applications. Our simple scenario displays te concept but ides te difficulties. In general, developing a multiscale solver for a given problem involves four main tasks: coosing an appropriate local process, coosing appropriate coarse (largescale) variables, coosing appropriate metods for transferring information across scales, and developing appropriate equations (or processes) for te coarse variables. Depending on te application, eac of tese tasks can be simple, moderately difficult, or extremely callenging. Tis is partly te reason wy multigrid continues to be an active and triving researc field. We can examine tese four tasks for developing a multiscale solver by using a sligtly less trivial problem. Before doing tis, let s analyze Jacobi s playeralignment algoritm and give some matematical substance to our intuitive notions on te dependence of its efficiency on te scale of te error. 4 COMPUTING IN SCIENCE & ENGINEERING
4 (a) (b) (c) Figure 3. Tree eigenvectors. We associate scale wit te wavelengt, wic is inversely proportional to k: (a) v (), (b) v (3), and (c) v (6). Analyzing Jacobi s PlayerAlignment Algoritm First, observe tat Jacobi s iteration acts similarly and independently in te directions parallel and perpendicular to te goal line. Terefore, it suffices to consider te convergence wit respect to just one of tese directions. Accordingly, let te column ( 0) ( 0) ( 0) T vector e = [ e,..., e N ] denote te players initial errors, eiter in te direction perpendicular or parallel to te goal line. We can define te error as te player s present coordinate, subtracted from te correct coordinate (to wic e sould eventually converge.) For convenience, we don t include players 0 and N in tis vector, but we keep in mind ( 0) ( 0) tat e 0 = e N = 0. For n =,,, te new error in te location of player i after iteration n is given by te average of te errors of is two neigbors just before te iteration namely, ( n) ( n ) ( e e e n ) i = i + + i. () We can write tis in vectormatrix form as e (n) = Re (n ), () were R is an N by N tridiagonal matrix wit onealf on te upper and lower diagonals and 0 on te main diagonal. For example, for N = 6, we ave R = (3) Te beavior of te iteration and its relation to scale is most clearly observed for an error, e, wic is an eigenvector of R, satisfying Re = e, for some (scalar) eigenvalue,. Te matrix R as N eigenvalues; for k =,, N, tey are given by ( k) kπ λ = cos. (4) Te N corresponding eigenvectors are given by kπ sin kπ ( k) sin v =. N k sin ( ) π (To verify tis, use te trigonometric identity, (5) ki ( + ) π ki sin sin + π = kπ kiπ cos sin to sow tat Rv (k) = (k) v (k).) Figure 3 sows te first, tird, and sixt eigenvectors. We associate scale wit te wavelengt, wic is inversely proportional to k. Tus, small k corresponds to largescale eigenvectors, and vice versa. Te eigenvectors v (k) are linearly independent, spanning te space R N, so we can write any initial error as a linear combination of te eigenvectors, N ( 0) ( k) e = c k v, (6) k= for some real coefficients, c k. Repeated multiplication by te iteration matrix, R, yields te error after n iterations: N ( n) n ( 0) n ( k) e = R e = R ckv k= N N n ( k) ( k) n ( k) = ckr v = ck( λ ) v k= k= N n kπ k = ck cos N v. k= (7) Because (k) = cos(k /N) < for k N, NOVEMBER/DECEMBER 006 5
5 (a) (b) (c) Figure 4. Te smooting property of te damped Jacobi relaxation. (a) Initial position, (b) after four relaxations, and (c) after eigt relaxations. we conclude tat Jacobi s metod converges because (k) n tends to zero as n tends to infinity. It s illuminating to examine te rate of convergence for different values of k. For k/n <<, we ave ( k) n kπ ( λ ) = cos n n kπ N e, (8) were we can verify te approximation by using a Taylor series expansion. We find tat te coefficients of eigenvectors corresponding to small k values cange only sligtly in eac iteration (see Figure c). Indeed, we see ere tat n = O(N ) iterations are required to reduce suc an error by an order of magnitude. For million players, for example, undreds of billions of wistle blows are required, compared to just 0 using te multiscale approac. For k = N/, (k) = cos(k /N) = 0, and convergence is immediate (see Figure d). As k approaces N, te scale of te error becomes even smaller, and cos(k /N) approaces, wic is consistent wit te oversoot we saw in Figure e tat is, at eac iteration, te error mainly canges sign, wit ardly any amplitude cange. Damped Jacobi Iteration s Smooting Factor To overcome te oversoot problem, we introduce a damping factor. If we damp te players movements by a factor /3, we obtain ( n) ( n ) ( n ) ( ei ei ei e n ) = i 3 3 and in vectormatrix form,, (9) ( n) ( n ) ( n ) e = Rde I + e, (0) R 3 3 wit R d denoting te damped Jacobi iteration matrix and I te identity matrix. Te eigenvectors clearly remain te same, but te eigenvalues of R d are given by ( k) ( k) kπ λd = + λ = + cos. () We can now quantify te claim tat Jacobi s iteration, damped by a factor /3, is efficient in reducing smallscale errors. To do tis, let s (somewat arbitrarily at tis point) split te spectrum down te middle, rougly, and declare all eigenvectors wit k < N/ smoot (large scale), and eigenvectors wit N/ k < N roug (small scale). Observe tat, as we vary k from N/ to N, spanning te roug eigenvectors, cos(k /N) ( k) varies from 0 to (nearly), and terefore λ d varies from /3 to (nearly) /3. Te fact tat ( k) λ d /3 for all te roug eigenvectors implies tat te size of te coefficient of eac roug eigenvector is reduced in eac iteration to at most /3 its size prior to te iteration independently of N. We say tat te iteration s smooting factor is /3. If we begin wit an initial error tat s some apazard combination of te eigenvectors, ten after just a few iterations, te coefficients of all te roug eigenvectors will be greatly reduced. Tus, te damped Jacobi iteration (usually called relaxation) is efficient at smooting te error, altoug, as we ve seen, it is inefficient at reducing te error once it as been smooted. Figure 4 exibits tese properties. Relaxation, one observes, irons out te wrinkles but leaves te fat. D Model Problem Te idea of using a local rule to obtain a global order is, of course, ancient. Peraps te most common example in matematics is differential equations, wic ave a global solution determined by a local rule on te derivatives (plus boundary conditions). To advance our discussion, we require an example tat s sligtly less trivial tan player alignment. Consider te D boundaryvalue problem: 6 COMPUTING IN SCIENCE & ENGINEERING
6 d u Lu = f( x), x in ( 0, ), dx u( 0) = ul, u( ) = u r, () were f is a given function, u and u r are te boundary conditions, and u is te solution we re seeking. We adopt te common numerical approac of finite difference discretization. A uniform mes is defined on te interval [0, ], wit mes size = /N and N + mes points given by x i + i for i = 0,,..., N. Our vector of unknowns te discrete approximation to u is denoted as u = [ u,... un ] T, were we exclude te boundary values for convenience but keep in mind tat u0 = ul, un = u. Te rigtand side vector is denoted by f r = [ f,..., fn ] T, wit fi = f( xi ). Alternatively, we could use some local averaging of f. Te standard secondorder accurate finite difference discretization of te second derivative now yields a system of equations for u, wic we can write as ui ui + ui + ( Lu ) i = fi, i =,..., N, u0 = ul, un = ur. (3) We can also write tis system more concisely as L u = f. Te system is tridiagonal and easy to solve by standard metods, but it s useful for examining te multigrid solver s aforementioned tasks. Observe te similarity to te playeralignment problem: if we set f 0, te solution u is obviously a straigt line connecting te point (0, u ) wit (, u r ). Te discrete solution is ten a set of values along tis line, eac equal to te average of its two neigbors. We can easily adapt Jacobi s algoritm to Equation 3. Let u (0) denote an initial approximation to u tat satisfies te boundary conditions. n Ten, at te nt iteration, we set u i, i =,, N, to satisfy te it equation in te system in Equation 3, wit te neigboring values taken from te n st iteration: n u n i u ( ) n i u ( ) = + + i fi, i =,..., N. (4) Te convergence beavior of Jacobi s algoritm is identical to its beavior in te playeralignment problem. Denote te error in our approximation after te nt iteration by e n u u n =, (5) n n wit e0 = en = 0 because all our approximations satisfy te boundary conditions. Now, subtract Equation 4 from te equation ui ui ui = + + fi, wic we get by isolating u i in Equation 3. Tis yields n n ( ) ei ei e n ( ) = + + i, (6) wic is exactly like Equation. Similarly, damped Jacobi relaxation, wit a damping factor of /3, is defined by n ( ) n ( ) n n ui + u + i ui = ui +, (7) 3 3 fi and its error propagation properties are identical to tose of te playeralignment problem in Equations 9 troug. Naïve Multiscale Algoritm We next naïvely extend te multiscale algoritm of te playeralignment problem to te D model problem. Te main point of tis exercise is to sow tat in order to know wat to do at te large scales, we must first propagate appropriate information from te small scales to te large scales. For convenience, we still assume N is a power of two. Similarly to te playeralignment process, we begin by performing Jacobi s iteration for te problem defined at te largest scale, wic consists of te variables [ u0, u. Te mes size at tis scale is 0.5, and u N /, un ] 0 and u N are prescribed by te boundary conditions. Te Jacobi step tus fixes u N / : un / = u0 + un 05. fn / Next, based on te values of te largest scale, we perform a Jacobi iteration at te secondlargest scale, obtaining u N /4 and u3 N / 4. We continue until we ave determined all variables, just as in te playeralignment problem. Tis algoritm is obviously computationally efficient. Unfortunately, it fails to give te correct answer (unless f is trivial) because we aren t solving te rigt equations on te coarse grids. More precisely, te rigtand sides are wrong.. NOVEMBER/DECEMBER 006 7
7 (a) (b) Figure 5. Aliasing. Red disks correspond to te coarse grid: (a) v (3) and (b) v (3). Aliasing Figure 5a sows a smoot eigenvector (k = 3) sampled on a grid wit N = 6 and also on a twicecoarser grid, N = 8, consisting of te fine grid s evenindexed mes points. Tis repeats in Figure 5b, but for a roug eigenvector (k = 3 on te finer grid), wit coefficient. Evidently, te two are indistinguisable on te coarse grid, wic demonstrates te wellknown penomenon of aliasing. Te coarse grid affords rougly alf as many eigenvectors as te fine grid, so some pairs of finegrid eigenvectors must alias. Specifically, by Equation 5, te evenindexed terms in eigenvectors v (k) and v (N k) are equal to te negative of eac oter. Tus, if f contains some rougness tat is, if we write f as a linear combination of eigenvectors, it will ave roug eigenvectors wit nonzero coefficients it won t be represented correctly on te coarse grid. As a somewat extreme example, if f is a linear combination of v (k) and v (N k) for some k, wit weigts of equal magnitude but opposite sign, it will vanis wen sampled on te coarser grid. Te definition of te smooting factor, wic was based on a partitioning of te spectrum into smoot and roug parts, foresadowed our use of a twicecoarser grid. Smoot eigenvectors are tose tat can be represented accurately on te twicecoarser grid, wereas roug eigenvectors alias wit smoot ones. We can conclude tat constructing correct equations for coarse grids requires information from te fine grid. For te D model problem, we can get around te aliasing problem by successive local averaging. However, in more general problems, we can t do tis witout spending a substantial computational effort, wic would defeat te purpose of te multiscale approac. We must come to grips wit te fact tat coarse grids are useful only for representing smoot functions. Tis means tat we must take care in coosing our coarsegrid variables. Taxonomy of Errors We ve seen tat damped Jacobi relaxation smootes te error. In tis context, it is important to distinguis between two distinct types of error. Te discretization error, given at mes point i by ux ( i) ui, is te difference between te solution of te differential equation (sampled on te grid) and tat of te discrete system. Tis error is determined by te problem, te discretization, and te mes size. Iteration as no effect on tis error, of course, because it only affects our approximation to u. Only te algebraic error, wic is te difference between te discrete solution and our current approximation to it (Equation 5), is smooted. Our plan, terefore, is to first apply damped Jacobi relaxation, wic smootes te algebraic error, and ten construct a linear system for tis error vector and approximate it on a coarser grid. Tis can continue recursively, suc tat, on eac coarser grid, we solve approximate equations for te nextfiner grid s algebraic error. Te motivation for using te algebraic error as our coarsegrid variable is twofold. Because we re limited in our ability to compute an accurate approximation to u on te coarse grid due to aliasing, we need to resort to iterative improvement. We sould terefore use te coarse grid to approximately correct te finegrid algebraic error, rater tan to dictate te finegrid solution. Moreover, relaxation as smooted te algebraic error, and tus we can approximate it accurately on te coarse grid. Let ũ denote our current approximation to u after performing several damped Jacobi relaxations, and let e = u u denote te corres ponding algebraic error, wic is known to be smoot (see Figure 4). Subtract from Equation 3 te trivial relation = = Lu Lu, i,..., N, i i u 0 = ul, u N = ur, to obtain a system for te algebraic error, 8 COMPUTING IN SCIENCE & ENGINEERING
8 = = Le fi Lu ri, i,... N, i i e0 = 0, en = 0, were r i is te it residual. Tis is te problem we must approximate on te coarser grid. We can write it concisely as L e = r. Once we compute a good approximation to e, we ten add it to ũ, and tus improve our approximation to u. Finest grid Relaxation Multigrid Algoritm To transfer information between grids, we need suitable operators. Consider just two grids: te fine grid, wit mes size and wit N mes points excluding boundary points, and te coarse grid, wit mes size and N/ mes points. We use a restriction operator, denoted I, for finetocoarse data transfer. Common coices are injection, defined for a finegrid vector, g, by = =, N I g gi, i,..., i tat is, te coarsegrid value is simply given by te finegrid value at te corresponding location, or full weigting, wic is defined by = ( + + ) I g gi gi i +, i 4 N i =,...,, were we apply a local averaging. For coarsetofine data transfer, we use a prolongation operator, denoted by I. A common coice is te usual linear interpolation, defined for a coarsegrid vector g by gi /, if i is even, ( I g ) = i g i ( ( + )/ + g( i )/ ), if i is odd, i =,..., N. Additionally, we must define an operator on te coarse grid, denoted L, tat approximates te fine grid s L. A common coice is to simply discretize te differential equation on te coarse grid using a sceme similar to te one applied on te fine grid. Armed wit tese operators, and using recursion to solve te coarsegrid problem approximately, we can now define te multigrid VCycle (see Figure 6). We require two parameters, commonly denoted by and, tat designate, respectively, te number of damped Jacobi relaxations performed before and after we apply te coarsegrid correction. Te multigrid algoritm is defined as Coarsest grid Algoritm VCycle: u = VCycle(u,, N, f,, ). If N ==, solve L u = f and return u ;. u = Relax(u,, N, f, ); relax times. 3. r = f L u ; compute residual vector. 4. f = I r ; restrict residual. 5. u = 0; set u to zero, including boundary conditions. 6. u = VCycle(u,, N/, f,, ); treat coarsegrid problem recursively. 7. u = u + I u ; interpolate and add correction. 8. u = Relax(u,, N, f, ); relax times; 9. Return u. Let s examine te algoritm line by line. In line, we ceck weter we ve reaced te coarsest grid, N =. If so, we solve te problem. Because tere s only a single variable tere, one Jacobi iteration witout damping does te job. In line, damped Jacobi iterations are applied. We compute te residual in line 3, and restrict it to te coarse grid in line 4. We now need to tackle te coarsegrid problem and obtain te coarsegrid approximation to te finegrid error. Noting tat te coarsegrid problem is similar to te finegrid one, we apply a recursive call in line 6. Note tat te boundary conditions on te coarse grid are zero (line 5) because tere s no error in te finegrid boundary values, wic are prescribed. Also, te Restriction Prolongation Figure 6. A scematic description of te VCycle. Te algoritm proceeds from left to rigt and top (finest grid) to bottom (coarsest grid) and back up again. On eac grid but te coarsest, we relax times before transferring (restricting) to te nextcoarser grid and times after interpolating and adding te correction. NOVEMBER/DECEMBER 006 9
9 initial approximation to u must be zero. After returning from te recursion, we interpolate and add te correction to te finegrid approximation (line 7). In line 8, we relax te finegrid equations times, and te resulting approximation to te solution is returned in line 9. Generally, te VCycle doesn t solve te problem exactly, and it needs to be applied iteratively. However, as I ll demonstrate later, eac suc iteration typically reduces te algebraic error dramatically. D Model Problem Te multiscale algoritm isn t restricted to one dimension, of course. Consider te D boundaryvalue problem: u u Lu + = x y u = b( x, y), ( x, y) on Ω, f( x, y), ( x, y) inω, (8) were f and b are given functions and u is te solution we re seeking. For simplicity, our domain is te unit square, = (0, ), wit denoting te boundary of. Tis is te wellknown Poisson problem wit Diriclet boundary conditions. To discretize te problem, we define a uniform grid wit mes size and mes points given by ( xi, yj ) = ( i, j), i, j = 0,,..., N, were = /N. Our vector of unknowns te discrete approximation to u is again denoted u, b denotes te discrete boundary conditions and f is te discrete rigtand side function. Te standard secondorderaccurate finite difference discretization now yields a system of equations for u, wic we can write as ui, j + ui+, j + ui, j + u i, j+ 4ui, j ( Lu ) i, j = fi, j, i, j =,... N, ui, j= bi, j, i = 0 or i = N or j = 0 or j = N, (9) or more concisely as L u = f. We can easily adapt Jacobi s algoritm to te D problem: at eac iteration, we cange our approximation to u i, j, i, j =,, N, to satisfy te (i, j)t equation in te system in Equation 9, wit te neigboring values taken from te previous iteration. We can generalize our smooting analysis of damped Jacobi relaxation to tis case. By minimizing te smooting factor wit respect to te damping, we ten find tat te best acievable smooting factor is 0.6, obtained wit a damping factor of 0.8. To apply te multigrid VCycle to te D model problem, we still need to define our coarse grids, along wit appropriate intergrid transfer operators. We coarsen naturally by eliminating all te mes points wit eiter i or j odd, suc tat rougly onefourt of te mes points remain. For restriction, we can once again use simple injection, defined for a finegrid vector, g, by = = N I g gi, j, i, j,...,, i, j tat is, te coarsegrid value is simply given by te finegrid value at te corresponding location, or full weigting, defined by = ( + I g gi, j gi, j+ i, j 6 + gi +, j + gi+, j+ gi, j + g i+, j + + gi, j + gi, j+, + 4g i, j N i, j =,...,, were a local averaging is applied. For coarsetofine data transfer, a common coice is bilinear interpolation, defined for a coarsegrid vector g by I g = i, j g i /, j /, if i is even and j is even, g ( (i +)/, j / + g (i )/, j / ), if i is odd and j is even, g ( i /,( j +)/ + g i /,( j )/ ), if i is even and j is odd, 4 g ( (i +)/,( j +)/ + g (i )/, (j+)/ + g (i +)/,( j )/ + g (i )/,( j )/ ) if i is odd and j is odd, for i =,, N. 0 COMPUTING IN SCIENCE & ENGINEERING
10 Numerical Tests To demonstrate te multigrid algoritm s effectiveness, we test it on te D and D model problems wit N = 8 and compare its convergence beavior to tat of simple Jacobi relaxation. We apply VCycles wit =, = denoted V(, ) to a problem wit a known solution, beginning wit a random initial guess for te solution. (It s easy to sow tat te asymptotic convergence beavior is independent of te particular solution.) We use (bi)linear interpolation and fullweigting restriction and plot te finegrid algebraic error s root mean square (RMS) versus te number of iterations. Figure 7 sows te results for te D and D model problems. Te V Cycle s computational cost is similar to tat of a few Jacobi iterations, as I will sow in te next section. Hence, for fairness, we consider every 0 Jacobi relaxations as a single iteration in te comparison, so tat 00 iterations marked in te plot actually correspond to,000 Jacobi relaxations. Tus, a VCycle in tese plots is no more costly tan a single Jacobi iteration. We can see tat eac VCycle reduces te error by rougly an order of magnitude, wereas Jacobi relaxation converges slowly. Te smooting analysis can approximately predict te fast multigrid convergence: on te fine grid, te reduction factor of roug errors per eac relaxation sweep is bounded from above by te smooting factor, wereas te smoot errors are nearly eliminated altogeter by te coarsegrid correction. Computational Complexity Te number of operations performed in te V Cycle on eac grid is clearly proportional to te number of mes points because tis olds for eac subprocess relaxation, residual computation, restriction, and prolongation. Because te number of variables at eac grid is approximately a constant fraction of tat of te next finer grid (onealf in D and onefourt in D), te total number of operations per VCycle is greater tan te number of operations performed on just te finest grid only by a constant factor. In D, tis factor is bounded by + / + /4 + =, and in D by + /4 + /6 + = 4/3. Te amount of computation required for a single Jacobi relaxation on te finest grid (or a calculation of te residual, wic is approximately te same) is called a work unit (WU). Our VCycle requires tree WUs for relaxation on te finest grid and one more for computing te residual. Te prolongation plus te restriction operations togeter require no more tan one WU. Tus, te finegrid work is rougly Error norm (a) Error norm (b) Iterations Iterations five WUs per VCycle. Terefore, te entire work is approximately 0 WUs per VCycle in D and seven in D. V(,) cycles Ten Jacobi relaxations V(,) cycles Ten Jacobi relaxations Figure 7. Convergence of te algebraic error. Multigrid V(, ) cycles versus Jacobi iterations comprised of 0 relaxations eac, wit N = 8: (a) D and (b) D. Finest grid Coarsest grid Relaxation No relaxation Restriction Prolongation Figure 8. A scematic description of te full multigrid algoritm for =. Te algoritm proceeds from left to rigt and top (finest grid) to bottom (coarsest grid.) Initially, we transfer te problem down to te coarsest grid. We solve te problem on te coarsest grid and ten interpolate tis solution to te secondcoarsest grid and perform a VCycle. Tese two steps are repeated recursively to finer and finer grids, ending wit te finest grid. NOVEMBER/DECEMBER 006
Verifying 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 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 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 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 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 informationFINITE 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 information2.28 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationExercises for numerical integration. Øyvind Ryan
Exercises for numerical integration Øyvind Ryan February 25, 21 1. Vi ar r(t) = (t cos t, t sin t, t) Solution: Koden blir % Oppgave.1.11 b) quad(@(x)sqrt(2+t.^2),,2*pi) a. Finn astigeten, farten og akselerasjonen.
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 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 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 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 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 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 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 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 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 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 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 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 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 informationStrategic trading and welfare in a dynamic market. Dimitri Vayanos
LSE Researc Online Article (refereed) Strategic trading and welfare in a dynamic market Dimitri Vayanos LSE as developed LSE Researc Online so tat users may access researc output of te Scool. Copyrigt
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 informationPredicting the behavior of interacting humans by fusing data from multiple sources
Predicting te beavior of interacting umans by fusing data from multiple sources Erik J. Sclict 1, Ritcie Lee 2, David H. Wolpert 3,4, Mykel J. Kocenderfer 1, and Brendan Tracey 5 1 Lincoln Laboratory,
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 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 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 informationFor Sale By Owner Program. We can help with our for sale by owner kit that includes:
Dawn Coen Broker/Owner For Sale By Owner Program If you want to sell your ome By Owner wy not:: For Sale Dawn Coen Broker/Owner YOUR NAME YOUR PHONE # Look as professional as possible Be totally prepared
More informationDesign and Analysis of a FaultTolerant Mechanism for a ServerLess VideoOnDemand System
Design and Analysis of a Faultolerant Mecanism for a ServerLess VideoOnDemand System Jack Y. B. Lee Department of Information Engineering e Cinese University of Hong Kong Satin, N.., Hong Kong Email:
More informationTheoretical calculation of the heat capacity
eoretical calculation of te eat capacity Principle of equipartition of energy Heat capacity of ideal and real gases Heat capacity of solids: DulongPetit, Einstein, Debye models Heat capacity of metals
More informationStaffing and routing in a twotier call centre. Sameer Hasija*, Edieal J. Pinker and Robert A. Shumsky
8 Int. J. Operational Researc, Vol. 1, Nos. 1/, 005 Staffing and routing in a twotier call centre Sameer Hasija*, Edieal J. Pinker and Robert A. Sumsky Simon Scool, University of Rocester, Rocester 1467,
More informationBackground Facts on Economic Statistics
Background Facts on Economic Statistics 2003:3 SAMU Te system for coordination of frame populations and samples from te Business Register at Statistics Sweden epartment of Economic Statistics Te series
More informationBonferroniBased SizeCorrection for Nonstandard Testing Problems
BonferroniBased SizeCorrection for Nonstandard Testing Problems Adam McCloskey Brown University October 2011; Tis Version: October 2012 Abstract We develop powerful new sizecorrection procedures for
More informationWelfare, financial innovation and self insurance in dynamic incomplete markets models
Welfare, financial innovation and self insurance in dynamic incomplete markets models Paul Willen Department of Economics Princeton University First version: April 998 Tis version: July 999 Abstract We
More informationAreaSpecific Recreation Use Estimation Using the National Visitor Use Monitoring Program Data
United States Department of Agriculture Forest Service Pacific Nortwest Researc Station Researc Note PNWRN557 July 2007 AreaSpecific Recreation Use Estimation Using te National Visitor Use Monitoring
More informationA FLOW NETWORK ANALYSIS OF A LIQUID COOLING SYSTEM THAT INCORPORATES MICROCHANNEL HEAT SINKS
A FLOW NETWORK ANALYSIS OF A LIQUID COOLING SYSTEM THAT INCORPORATES MICROCHANNEL HEAT SINKS Amir Radmer and Suas V. Patankar Innovative Researc, Inc. 3025 Harbor Lane Nort, Suite 300 Plymout, MN 55447
More informationWorking Capital 2013 UK plc s unproductive 69 billion
2013 Executive summary 2. Te level of excess working capital increased 3. UK sectors acieve a mixed performance 4. Size matters in te supply cain 6. Not all companies are overflowing wit cas 8. Excess
More informationA strong credit score can help you score a lower rate on a mortgage
NET GAIN Scoring points for your financial future AS SEEN IN USA TODAY S MONEY SECTION, JULY 3, 2007 A strong credit score can elp you score a lower rate on a mortgage By Sandra Block Sales of existing
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 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 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 informationReferendumled Immigration Policy in the Welfare State
Referendumled Immigration Policy in te Welfare State YUJI TAMURA Department of Economics, University of Warwick, UK First version: 12 December 2003 Updated: 16 Marc 2004 Abstract Preferences of eterogeneous
More informationNonlinear Iterative Partial Least Squares Method
Numerical Methods for Determining Principal Component Analysis Abstract Factors Béchu, S., RichardPlouet, M., Fernandez, V., Walton, J., and Fairley, N. (2016) Developments in numerical treatments for
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 informationSWITCH T F T F SELECT. (b) local schedule of two branches. (a) ifthenelse construct A & B MUX. one iteration cycle
768 IEEE RANSACIONS ON COMPUERS, VOL. 46, NO. 7, JULY 997 Compileime Sceduling of Dynamic Constructs in Dataæow Program Graps Soonoi Ha, Member, IEEE and Edward A. Lee, Fellow, IEEE Abstract Sceduling
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 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 informationChapter 11. Limits and an Introduction to Calculus. Selected Applications
Capter Limits and an Introduction to Calculus. Introduction to Limits. Tecniques for Evaluating Limits. Te Tangent Line Problem. Limits at Infinit and Limits of Sequences.5 Te Area Problem Selected Applications
More informationRecall from last time: Events are recorded by local observers with synchronized clocks. Event 1 (firecracker explodes) occurs at x=x =0 and t=t =0
1/27 Day 5: Questions? Time Dilation engt Contraction PH3 Modern Pysics P11 I sometimes ask myself ow it came about tat I was te one to deelop te teory of relatiity. Te reason, I tink, is tat a normal
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 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 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 informationSimultaneous Location of Trauma Centers and Helicopters for Emergency Medical Service Planning
Simultaneous Location of Trauma Centers and Helicopters for Emergency Medical Service Planning SooHaeng Co Hoon Jang Taesik Lee Jon Turner Tepper Scool of Business, Carnegie Mellon University, Pittsburg,
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationWe consider the problem of determining (for a short lifecycle) retail product initial and
Optimizing Inventory Replenisment of Retail Fasion Products Marsall Fiser Kumar Rajaram Anant Raman Te Warton Scool, University of Pennsylvania, 3620 Locust Walk, 3207 SHDH, Piladelpia, Pennsylvania 191046366
More informationCHAPTER TWO. f(x) Slope = f (3) = Rate of change of f at 3. x 3. f(1.001) f(1) Average velocity = 1.1 1 1.01 1. s(0.8) s(0) 0.8 0
CHAPTER TWO 2.1 SOLUTIONS 99 Solutions for Section 2.1 1. (a) Te average rate of cange is te slope of te secant line in Figure 2.1, wic sows tat tis slope is positive. (b) Te instantaneous rate of cange
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 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 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 informationHaptic Manipulation of Virtual Materials for Medical Application
Haptic Manipulation of Virtual Materials for Medical Application HIDETOSHI WAKAMATSU, SATORU HONMA Graduate Scool of Healt Care Sciences Tokyo Medical and Dental University, JAPAN wakamatsu.bse@tmd.ac.jp
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 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 informationDivide and Conquer Approach
Divide and Conquer Approac Deiverabes Divide and Conquer Paradigm nteger Mutipication Strassen Matrix Mutipication Cosest Pair of points nfinite Wa Probem 6/7/01 8:58 PM Copyrigt @ gdeepak.com Divide and
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 informationFree Shipping and Repeat Buying on the Internet: Theory and Evidence
Free Sipping and Repeat Buying on te Internet: eory and Evidence Yingui Yang, Skander Essegaier and David R. Bell 1 June 13, 2005 1 Graduate Scool of Management, University of California at Davis (yiyang@ucdavis.edu)
More information