Numerical integration in more dimensions part 2. Remo Minero
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1 Numerca ntegraton n more dmensons part Remo Mnero
2 Outne The roe of a mappng functon n mutdmensona ntegraton Gauss approach n more dmensons and quadrature rues Crtca anass of acceptabt of a gven quadrature rue 6// Semnar: Numerca Integraton n more dmensons
3 Probem defnton n! We have f : Ω R R and we want to compute I: I f d Ω! We want to mpement some numerca method, whch ought to be (as usua) accurate and cheap (e.g. sma number of operatons) 6// Semnar: Numerca Integraton n more dmensons 3
4 Coverng! Step : coverng of the doman wth repcas of a basc geometr ( ) d f ( ) f Ω e d Ω e Error : coverng error 6// Semnar: Numerca Integraton n more dmensons 4
5 Mappng functon! Step : ntroducton of a mappng functon F ŷ " Eampe : squares ê F () v () v e () v3 () v e F A [ () () () ()] () v v v v + v + b 6// Semnar: Numerca Integraton n more dmensons 5
6 6// Semnar: Numerca Integraton n more dmensons 6 Mappng functon () v e " eampe : tranges () () () () [ ] () A F b v v v v v + + F ŷ ê () v e () v
7 Propertes of the mappng functon F F + b! F s affne: ( ) A near constant F maps affne combnatons α, α to affne combnatons that s, tranges are mapped to tranges, rectanges to paraeograms, etc. ê ê F F e e 6// Semnar: Numerca Integraton n more dmensons 7
8 The roe of the mappng functon! Step : ( ) d f ( ) f Ω e d! Step : e f ( ) d f ( F ( )) det( F ) e det ( A ) f ( F ( )) e d d 6// Semnar: Numerca Integraton n more dmensons 8
9 6// Semnar: Numerca Integraton n more dmensons 9 Quadrature! Step 3: ntegraton over the basc geometr ( ) ( ) ( ) ( ) e e g w d g d F f Error : quadrature error
10 6// Semnar: Numerca Integraton n more dmensons Integraton over a surface! Suppose we have a functon defned over a surface! Thanks to the propertes of the mappng functon, we can use the same approach: () () () () [ ] () A F b v v v v v + + smp v () gven n a sutabe reference sstem
11 How can one reduce errors? Ω Coverng error Quadrature error! More accurate formuas! Smaer voumes (where necessar, dependng on f) 6// Semnar: Numerca Integraton n more dmensons
12 Open probem Gven a basc geometr, fnd the east amount of ponts and weghts such that e ( ) d w g( ) g s eact for a monomas of degree d and ower " Let s ook at some eampes 6// Semnar: Numerca Integraton n more dmensons
13 d n the square 3 equatons # pont, weght Mathematca probem - Phsca nterpretaton e e e d d w d d w d d w ŷ.5.5 w 6// Semnar: Numerca Integraton n more dmensons 3
14 6// Semnar: Numerca Integraton n more dmensons 4 d n the trange 3 equatons # pont, weght Mathematca probem - Phsca nterpretaton ŷ w 3 3 e e e w 6 d d w 6 d d w d d
15 6// Semnar: Numerca Integraton n more dmensons 5 d n the square 6 equatons # ponts, weghts Mathematca probem - Phsca nterpretaton e e e e e e w w 4 d d w w 3 d d w w 3 d d w w d d w w d d w w d d...no Mathematca souton...no Mathematca souton
16 d3 (e.g. n the trange) equatons # how man ponts? 3 ponts # too man equatons? 4 ponts # not enough 3 3 6// Semnar: Numerca Integraton n more dmensons 6
17 Bquadratc ponomas for the square! Let s choose two monomas p() and q() and et them be of degree d at most ! If we choose d3 3 3! 6 equatons # 4 ponts, 4 weghts 3...no Mathematca souton (n a reasonabe tme) 6// Semnar: Numerca Integraton n more dmensons 7
18 Cross product Gauss Gauss D n [,] Gauss D n [,] $ on for domans ke [a,b] n g(, ) dd w g(, ) d w g(, ) d w w jg(, j) j 6// Semnar: Numerca Integraton n more dmensons 8
19 Hgher degree formuas! Man of them n the terature ŷ " Eampe : degree 6 n the trange wth ponts " Eampe : degree n the trange wth 79 ponts! D.A.Dunuvant, HIGH DEGREE EFFICIENT SYMMETRICAL GAUSSIAN QUADRATURE RULES FOR THE TRIANGLE, Internatona Journa for Numerca Methods n Engneerng, vo., 9-48 (985) A.H. Stroud & D. Secrest, GAUSSIAN QUADRATURE FORMULA, Prentce-Ha, 966 6// Semnar: Numerca Integraton n more dmensons 9
20 Summar Do the number of the unknows correspond to the number of the equatons? Does the nonnear sstem have a souton? Is the found souton acceptabe? WE HAVE AN ACCEPTABLE QUADRATURE METHOD! Condton : Are a the ponts nsde the eement? Condton : Are a the weghts w postve? 6// Semnar: Numerca Integraton n more dmensons
21 Condton : e 4 3 Ω Ω F F( ) F( ) F( 4 ) F( 3) f ( ( )) ( ) What s the vaue of F f F does not beong to Ω? 4 4 6// Semnar: Numerca Integraton n more dmensons
22 Condton : w,! To awas have non-negatve ntegras for non-negatve functons f() Eact souton: b f d > a 7 ( ) < Appromaton: w f f w < a b w w w 3 w 4 w 5 w 6 w 7! Fnte Eements Methods (FEM) 6// Semnar: Numerca Integraton n more dmensons
23 Wh weghts awas n FEM! Stffness matr A s postve defnte u T Au u a u j j j u + u > f u > [ ] u u u u ϕ a ( ) j ϕ j ϕ + ϕ jϕ d,, u n! Postve defnte propert s needed for the teratve sovers of Krov tpe (a fast teratve sovers)! Negatve weghts mght cause postve defnte propert to be ost 6// Semnar: Numerca Integraton n more dmensons 3
24 Do we rea get negatve weghts?! Newton-Cotes approach n [-,]: n A A A3 A4 A5 A ! Negatve weghts aso n man formua usng wth Gauss approach 6// Semnar: Numerca Integraton n more dmensons 4
25 Coverng " Eampe : tranges # more febe Ω 6// Semnar: Numerca Integraton n more dmensons 5
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