CS475 Parallel Programming

Transcription

1 CS475 Parallel Programmig Dieretiatio ad Itegratio Wim Bohm Colorado State Uiversity Ecept as otherwise oted, the cotet o this presetatio is licesed uder the Creative Commos Attributio.5 licese.

2 Pheomea Physics: heat, low, space, time Mathematics: cotiuous uctios, partial dieretial equatios Computer sciece: Discrete simulatio o physical pheomea through Fiite Dierece Methods

3 Dieretials Physical pheomea like the low o heat are modeled with dieretials: d Δ = lim d Δ Δ o A dieretial describes rate o chage, e.g. velocity is the rate o chage o positio, v = d/d, ad acceleratio is the rate o chage o velocity, a = dv/d, which is the secod derivative the derivative o the derivative o positio

4 Partial Dieretial Equatios Partial dieretial equatios are dieretial equatios i higher dimesios epressed i a coordiate system, e.g i D: u ad u y describe the chage o u i the ad y directio.

5 Laplace Laplace described physical pheomea i ad 3D, e.g. heat i D V Δy Δ Vy ΔVy Vy V ΔV I X directio: cell receives heat VΔy, loses heat VΔV Δy, hece ΔV Δy heat removed Similarly, i Y directio: ΔVy Δ heat removed

6 trick ΔVΔy = ΔV Δ ΔVyΔ = ΔVy Δy ΔΔy V ΔΔy ΔΔy Vy y ΔΔy Combied loss : V Vy y ΔΔy

7 More tricks Heat coservatio law: Feyma: heat lows at a rate proportioal to the temperature u gradiet These two combied: V u Vy y V Vy u y = k = k = = 0 0 u u y

8 heat Heat at boudary kow What is the heat iside? Discretize it w c e s? u c = u,y, u = u,yh, u s = u,y-h, u e = uh,y, u w = u-h,y

9 Taylor series: uctio approimatio We ca epress a uctio i terms o its derivatives, The more derivatives the closer at least that was the wisdom util chaos got discovered Poitcare. h = k i = 1 1 i! i

10 Taylor approimatio u e = uh,y = u c u w = u-h,y = u c - u e u w = u c u e u w u s u = 4u c 1 u h u h 1 u h u h u h u h y u h

11 Taylor Heat coservatio u h Taylor: u e u w u s u = 4u c Heat coservatio: u u y = 0 u h y thereore: u c = u u s u e u w / 4 Thermal equilibrium: temperature at,y is average o surroudig temperatures

12 Solvig the heat equatio grid: we could have a direct solutio equatios with ukows Too Comple! iterative solutio: relaatio Keep doig c s e at every poit util equilibrium reached Jacobi versio: pig pog with two arrays Nice parallelism, slow covergece Gauss-Seidel: oe array, use latest versio u = u u u u / w More comple data depedece, aster covergece 4

13 CS view Nearest eighbor computatio, checkerboard or block row partitioig Echage o data alog borders Trick: overlappig areas see e.g. Qui Ch. 13 Re-computatio Reduced commuicatio requecy Potetially more complicated commuicatio patter

14 Itegratio Dieretiatio: idig rate o chage i y y y y = = = z, w, z. w, = u / v, dy 1 = d dy dz dw = d d d dy dz dw = w z d d d dy du dv = v u / v d d d Itegratio: idig surace uder

15 Itegratio = = ʹ = a b d F where a F b F d b a b a

16 Numerical itegratio Approimate ad derive simple ormula or itegral Liear: two poits, quadratic: three, etc. Two approaches: ope vs, closed: ope: poits do t iclude a ad b closed: poits iclude a ad b dieret math Approimate i a umber o itervals Applyig ay orm o above approimatio methods

17 Trapezoidal rule I ~ b-a.ab/ Itervals: 0, 1 = 0 h, = 0 h,. X, h = b-a/ I ~ h 0 1 / -1 / = a b i i 1 0 =1

18 Better approimatios Either: more poits icrease or higher order polyomials E.g. Simpsos rule uses quadratic approimatio over 3 poits I = Itervals: I= h ,3,5.. 1,4,6.. 0 i i i i a b = =

19 Iterative / adaptive approach Iterate with smaller ad smaller segmets util I i ~ I i1 h 1 =b-a/ Error: use relative error h etc. = h 1 / etc. ε = r preset appro preset previous appro appro.100% 0.5*10 % : umber o sigiicat digits

20 Recursive approach: adaptive quadrature traplet,right = { retur right-let*letright/;} tol = 0.5*ep10,-; arealet,right,est ={ mid=letright/; } a1=traplet,mid; a=trapmid,right; ewest = a1a; iabsewest-est/ewest<tol retur ewest; else retur arealet,mid,a1 areamid,right,a