Algebraic Multigrid. Multigrid

Transcription

1 Algebraic Mltigrid We re going to discss algebraic ltigrid bt irst begin b discssing ordinar ltigrid. Both o these deal with scale space eaining the iage at ltiple scales. This is iportant or segentation becase an iage segentation is reall a representation at a coarser scale than the piel. To a irst approiation think that ltigrid is like dision and algebraic ltigrid is like anisotropic dision. In both cases we bild a scale space on top o the. Mltigrid The basic idea behind ltigrid coes ro the ollowing insight: i we sooth soething we can then sbsaple it withot losing an inoration. In ltigrid we se this to sole PDEs. We start with soe rando initialization in which or error is like white noise. Then we se an iteratie soler that has the eect o soothing the error. Ater a while the error is all low reqenc. So we can sbsaple to a coarser scale withot losing inoration. We repeat bt eerthing is cheaper at the coarser scale. Eentall we reach a sall scale where we can inepensiel get an eact soltion. Then knowing the soltion at a coarse scale we interpolate this to get an approiate soltion at a ine scale. This new soltion onl has high reqenc error which we can get rid o prett eicientl. We re going to break this into two parts. First we ll jst consider the representation constrcted b ltigrid. Then we ll see how it is sed to sole PDEs. Gassian Praid The basic idea behind ltigrid is also sed in ision and called the Gassian praid. It was introdced b Brt and Adelson in the earl 8s bt sees to hae appeared in ltigrid earlier. As we discssed preiosl we can t jst shrink an iage b sapling it. I we do that we get aliasing. High reqencies hae a big rando eect on the sapled iage. So we irst sooth to get rid o high reqencies. Then when we saple we don t lose an inoration. I we sooth with a Gassian and repeat this we get the Gassian praid. This has been sggested or a bnch o ision applications: Motion: We want to ind the transoration that relates two iages taken ro dierent iewpoints. I we start at the top o the praid it s eas to ind the best soltion becase there are so ew. Then as we oe down the praid we can asse that we hae an approiate soltion ro the coarser leel. We can take a Talor series epansion arond that soltion and analticall ind the soltion at the iner scale. This kind o approach is widel sed.

2 Matching: Here we hae an iage and a teplate. We atch the at a coarse scale and then at the iner scale we search near that scale. This is like otion bt a dierent application and we a not be able to sole anthing analticall. Copression: An earl sggestion was to se this praid or copression. This was throgh the Laplacian praid. This is like the G.P. bt at each leel we store the dierence between the iage and the psapled ersion o the coarser iage. The idea is that this dierence shold be less correlated and lower energ so easier to copress. Has the nice adantage that it akes progressie encoding eas. Soe ersion o this is inclded in JPEG. The Laplacian praid isn t reall directl sed in copression bt the ltiscale idea is one o the orernners o waelets which are sed in JPEG. Mltigrid or soling PDEs We ll look at one eaple o this sing the Poisson eqation. I will be er slopp abot bondar conditions and discretization and jst tr to gie the intitions. i.e. I we write this discretel or a grid we get: h This leads to an iteratie algorith where we base one o these ales on preios ersions o the others. z h Then we can set: z Note that at least it s eas to see that the right soltion is a ied point o this iteration. We can std conergence rates and isses o local inia bt not here. This is called the Jacobi iteration. Other ethods Gass-Seidel are better bt this is sipler. It a be a good idea to slow down the pdates a bit as: w wz It trns ot that w = / is a good choice and we get:

3 conoltion indicates where h That is we are pdating b basicall soothing the old with an aeraging ilter. So let s look at what happens to the error with this iteration. We deine the error at iteration to be: Then we can show: Sbstitting the let side eqals: while the right side is: We hae h So canceling soe ters we jst need to show that: h This is jst the discrete stateent o the Poisson eqation. So the point is that this iteration sooths the error in or original estiate. Intitiel this akes sense since at eer iteration we are estiating a point b aeraging the neighbors which aerages the error. We are soothing with soething like a low pass

4 ilter so what happens is that the high reqenc coponents o the noise disappear qickl bt low reqenc noise is slow to disappear. So when high reqenc noise is ostl gone we downsaple and get a saller proble where the noise is high reqenc again. We recrsiel repeat this process and get an eact soltion to the coarser proble. Then we psaple. When we psaple a noise ree soltion we a get noise bt onl in high reqenc so soothing qickl gets rid o that. O corse we need to ill in soe details here: First we don t actall downsaple becase i has signiicant high reqenc coponents so will. Instead we orlate an iteration on the deect. I we write the Poisson eqation as L= where L is a linear operator we deine the deect d = L. The deect can be garanteed to get sooth as we iterate. To downsaple shrink the iage we jst select eer other piel in each row dierent options are possible. To psaple we interpolate the issing piels. I we are operating at an scales there are an options abot when to oe p or down in scale. Oerall this can rn in linear tie which is optial. Algebraic Mltigrid Now I jst going to gie soe qick intitions abot how these ideas are etended in algebraic ltigrid. This is a ethod that is sel when or iteratie ethod perors soe tpe o anisotropic soothing. This can occr becase o soe irreglarities or asetries in the PDE we are soling. For eaple i we hae a PDE like: This cases an asetr in the etent to which horizontal neighbors and ertical neighbors wind p inlencing the soltion at a point. Or noting that the Poisson eqation is the eqation describing electroagnetic ields o can iagine that i a sall barrier blocks part o this ield this creates an anisotrop in the soltion as well. In these cases the proper iteratie soltion ethod winds p peroring anisotropic soothing. These asetries can occr an other was and in act AMG can be applied to probles like sparse linear eqations where we a not een hae a proble on a grid bt we won t worr abot that. As an eaple or the aboe eqation we wind p with an iteration that aerages neighbors in the horizontal direction bt not the ertical one. This gies s bands in the soltion that are independentl aeraged and so sooth in one direction bt not the other. So ater iterations we hae signiicant high reqenc coponents in the error and i we sbsaple we will alias. This can be seen with an eaple where there is a sall sqare that is dierent ro the rest o the error and sbsapling can lose it. More generall we can start with or original grid and or weighted links between neighboring piels and then sooth b taking weighted aerages according to these

5 weights We can also orlate this ore generall or arbitrar weighted graphs. This is essentiall non-linear dision. Ater soothing each piel will hae a siilar intensit to the piels that it is strongl connected to or at least an intensit that can be linearl reconstrcted ro these. Then we sbsaple the graph so that eer node that is deleted is strongl connected to soe that reain so we can reconstrct it. This algorith has the sae strctre as reglar ltigrid bt the sbsapling coarsening depends on the weights. In the aboe eaple we wind p sapling in one direction bt not sbsapling in the other. This leads to a slower sapling rate abe ½ instead o ¼. Bt AMG still has the sae basic strctre o MG sooth to get rid o error at one leel scale then sbsaple and get rid o error at the net scale. In light o this we can sa that what Sharon et al. are doing is perhaps anisotropic dision with sapling or eicienc.