Generalized Wavelet Theory and Non-Linear, Non-Periodic Boundary Value Problems. Dr. Klaus Braun May, 2016

Transcription

1 Grlizd Wvlt Tory d No-ir, No-Priodic Boudry Vlu Problms Dr Klus Bru My, 6 Abstrct W Fourir mts Nvir i t pysicl ilbrt spc tr is globl uiqu solutio o t corrspodig wk vritio qutio o t o-lir, o-sttiory Nvir-Stoks qutios [BrK] T corrspodig umricl pproimtio mtod is t itz-glrki mtod wic is usully quippd wit iit lmts pproimtio spcs I cs o gtiv scl pysicl spcs tis is clld boudry lmt mtod rltd to t udrlyig sigulr qutio rprsttio T iit boudry lmt pproimtio proprtis c som cllgs i cs o o-lirity dor o-priodic boudry coditios T wvlt tsio mtod is usd to rprst uctios tt v discotiuitis d srp pks, d or ccurtly dcostructig d rcostructig iit, opriodic dor o-sttiory sigls [MM] T wvlt tory is stblisd i t Fourir ilbrt spc rmwork As cosquc i ordr to pply t Cldró rprducig ormul tis rquirs to t so-clld dmissibility coditio diig wvlt lyzig sigl uctio T cllg or t wvlt tsio pproimtio mtod is bout t combitio o iit lmts to pproimt o rgulr uctio domis pplid to witi t pysicl spc d wvlts trtig t sty o-lir trms socks or t o-priodic srp dgs boudry coditios pplid to witi t wvlt spc Followig t slog w Fourir grlizd wvs mts Cldró grlizd wvlts w provid Glrki-psio-wvlt mtod wic oprts o sm pysicl d wvlt spc wic is T isomtry o t grlizd wvlt trsorm is giv by ddb, T spc is motivtd by t origil Cldró ormul: lt rl, rdil wit visig vrg, i ˆ d, d : ˆ T or it olds t rproducig ormul d Tis rlcts ot blcd rltiosip btw t ilbrt spcs

2 Fourir pysicl spc mts Cldró wvlt spc Most commo umricl mtods usd or umricl solutio o prtil dirtil qutios r spctrl, iit lmt FEM or wvlt mtods Wil spctrl mtods v good ccurcy its sptil locliztio is poor Gibbs pomo t FEM or PDE wit smoot boudry coditios v good ccurcy d good sptil locliztio [BrK] I cs o sigulrity t t boudry good ccurcy o FEM c b civd, s wll owvr tis rquirs dditiol problm dpdig modiictio o t pproimtio spcs g [Bl] [NiJ] I [CF] Nitsc-bsd domi compositio mtod is did d lyzd or t solutio o 3 yprsigulr itgrl qutios govrig t plci i trior to op surc, subjct to Num boudry problm T mtod c b pplid to lir lsticity d coustics problms, but i cs t oprtor govrig t mé qutio cuss mjor diicultis A wvlt is uctio usd to divid giv uctio or cotiuous sigl ito dirt scl compots A wvlt is itgrbl uctio ulillig t dmissibility coditio ˆ c : d For t prmtrs or, b it bls t diitio o t wvlt trsormtio by t b, b : t dt, T prmtrs, b r clld diltio r-scl, zoomig rsp trsltio sit prmtr For t prmtr c b itrprtd s rciprocl o rqucy T dmissibility coditio surs t vlidity o t Cldró rcostructio ormul wrby it olds b ddb b, c ddb, Wvlt trsormtio lysis is pplid spcilly to olir PDEs vig solutios cotiig locl pom g ormtio o socks lik urrics d itrctios btw svrl scls g tmospric turbulc wr tr is motio o cotiuous rg o lgt scls Suc solutios c b wll rprstd i wvlt bss bcus o proprtis lik compct support loclity i spc d visig momt loclity i tim Most o t wvlt lgoritms c dl priodic boudry coditios owvr, dirt possibilitis o dlig grl boudry coditios v b studid d t dvtg o wvlt trsorms ovr trditiol Fourir trsorms or rprstig uctios tt v discotiuitis d srp pks, d or ccurtly dcostructig d rcostructig iit, o-priodic dor osttiory sigls r still pdig [MM] T ilbrt trsormd wvlt pproc could llow t dirct umricl simultio o scttrig d rditio pom wil voidig t limittios o boudry lmt mtods o-uiquss d t costrits o rtiicil, o-rlctiv boudry coditios T vrious proprtis o t ilbrt trsorm d its rltd ilbrt trsormd wvlts suggst tt ty migt b usul or solvig trior boudry vlu problms wit prscribd bvior t t poit t For istc, coustic rditio rom compct objct is dscribd by solutio o t wv qutio tt stisis t Sommrld rditio coditio t

3 T sclig d wvlts uctios o compct support c b did o t rl li,, or o t circl is priod [WJ] T ilbrt trsorm o compctly-supportd wvlt is lso wvlt [WJ] Tt is, t ilbrt trsormd wvlts r ortogol to tir trslts d orm bsis or T ilbrt trsorm sclig d wvlt uctios do ot v compct support Tir support is ll o T priodic ilbrt trsorm is did by cot krl T priodic ilbrt trsorm o priodic, compct support sclig or wvlt uctio is ot priodic solutio o sclig rltio I Glrki mtod t dgrs o rdom r t psio coicits o st o bsis uctios d ts psio coicits r ot i pysicl stt i cs o wvlt Glrki mtod, ms i wvlt spc Morovr, i wvlt Glrki mtods t trtmt o oliritis is complictd w c b dld wit coupl o tciqus [MM]: - usig t coctio coicits - usig t qudrtur ormul - usig t psudo-pproc irst mp wvlt spc to pysicl spc, comput olir trm i pysicl spc d t bck to wvlt spc; tis pproc is ot vry prcticl bcus it rquirs trsormtio btw t pysicl d t wvlt spc W propos grlizd wvlts to b usd by t psudo-pproc or t trtmt o oliritis wr t wvlt spc is idticl wit t ppropritly did pysicl spc s proposd i [BrK] At t sm tim t pproc ovrcoms t limittios o boudry lmt mtods o-uiquss d t costrits o rtiicil, o-rlctiv boudry coditios o scttrig d rditio pom T pproc c b tdd to cotiuous rdil wvlt trsorm o wit scls d orittios SO big rltd to isz bsis blig diltios, rottios d trsltio o sigl uctio [] T corrspodig dmissibility coditio is giv by wit ˆ c : d ˆ : For rdil uctio F F r it olds i ˆ F r ˆ r r d wit t Fourir trsorm o t uiorm distributio o uit mss ovr t uit spr wit ctr t t origi giv by ˆ J Wit rspct to t ilbrt-court cojctur i t cott o t wv rditio problm rgrdig t istc o milis o distortio-r, progrssiv wvs d t uygs pricipl, [BrK] d t strigtorwrd grliztio o t blow or cotiuous rdil wvlt tory o []v5 w ot tt - d ˆ i t spc-tim dimsio is 4, [WG]3, 55: J - t sptil lttic is quidistt oly i t spcil css o spc-tim dimsios, 4, [] cptr 3 - t log proprtis o clssicl isz trsorms o sprs [ArN] dr

4 I tis ppr w cosidr t Poisso boudry vlu modl problm wit o-priodic boudry coditio or spc-tim dimsio d grlizd priodic uctio o t ilbrt spcs T rltd FEMBEM optiml Glrki pproimtio lysis lso i cs o loclly rducd rgulrity ssumptios to t solutio is giv i [BrK] t * wit S, i is t boudry o t uit spr t u s big priodic uctio d dots t itgrl rom to i t Cucy-ss T or : u wit : S d or rl Fourir coicits d orms r did by u i : u d : u T t Fourir coicits o t covolutio oprtor r giv by u y Au : log si u y dy : k y u y dy Au k u u T oprtor A bls crctriztio o t ilbrt spcs d i t orm A, W ot tt k I tis rmwork t dmissibility coditio or stdrd wvlts is quivlt to d t dmissibility coditio costt is giv by t orm T simpl lyr pottil itgrl qutio A g solvs t boudry vlu problm i D :, y y g o : S D T rltd itz mtod usig spli spcs is clld boudry iit lmt mtod BEM I t cs bov tis dis itz- Glrki pproimtio : S A, A, or ll S wit optiml pproimtio bvior wit rspct to t rltd rgulrity o t solutio d t trigl prmtr [BrK] t A t corrspodig pproimtio o t itz pproimtio o A g T t rror c b rprstd i t orm l, wrby l y dy iy d i, i For i wit id t uctio iy y : l is lyticl wit rspct to y From t Scwrz iqulity it tror ollows t c t t i cs S S k t

5 T limittio o t psudo pproc [MM] i t ss tt tr is o mppig rquird rom t wvlt spc to t pysicl spc d t bck to comput olir trms i pysicl spc is bl by t ollowig isomtry proprtis o grlizd wvlts did s Torm: For vry Cldró s rproducig idtity olds tru, i Proo: ddb, ddb, or, dd c ˆ ˆ dd c ˆ ˆ d d mrk: du to corrspodig proprtis o t ilbrt trsorm i,,,, A t rltiosip btw t grlizd wvlts d t stdrd wvlt, c, ˆ is giv by wrby,,, ˆ mrk: t tsio o t bov to spc dimsios is bld by t Prdtl d t multi-dimsiol log to t ilbrt trsorm, t isz oprtors Wit rspct to t Prdtl oprtor w rcll rom [ii] Torm: T Prdtl oprtor : is boudd d corciv, t rg 3 S d t trior Num problm dmit o d oly o grlizd solutio T torm prompts to itroduc i cs o pysicl product i t orm u, v spc t corrspodig rgy ir I t bov ilbrt spc rmwork t Prdtl oprtor corrspods to t orml drivtiv o t doubl lyr pottil oprtor T [Kr] torm 8: : Tv : v y l ds y y y, wic is sldjoit wit rspct to t dul systms, d, tt is, Tv, or ll v, w, w v, Tw,

6 T Gussi uctio d its Fourir trsorm v sm dcy I cs o t wvlt trsorm t situtio is dirt dpdig rom t wvlt diitio d t prmtr g ldig to t Mic t wvlt uctio d d I cs o t proposd grlizd wvlt modl cocpt w ot tt or : d its rltd ilbrt trsorm : ; it olds ˆ,, ;, ; T rltiosip A t dis grlizd wvlt ltrtivly to t Mic t wit pproprit dcy i syc wit t dcy o t rltd Fourir trsorm o t to-b trsormd uctio

7 rcs [ArN] Arcozzi N, Xiwi, isz trsorms o sprs, Mtmticl src ttrs, [Bl] Blum, Dobrowolski M, O Fiit lmt Mtods or Elliptic Equtios o Domis wit Corrs, Computig 8, [BrK] Bru K, Itrior Error Estimts o t itz Mtod or Psudo-Dirtil Equtios, Jp J Appl Mt, 3 986, 59-7 [BrK] Bru K, Globl istc d uiquss o 3D Nvir-Stoks qutios, ttp:wwwvirstoks-qutioscom [BrK] Bru K, A uusul sit torm blig sup-orm bouddss o t itz-glrki oprtor or yprbolic problms, ttp:wwwvir-stoks-qutioscom [BrK3] Bru K, Optiml iit lmt pproimtio stimts or o-lir prbolic problms wit ot rgulr iitil vlu dt, ttp:wwwvir-stoks-qutioscom [CF] Couly F, ur N, A Nitsc-bsd domi dcompositio mtod or yprsigulr itgrl qutios, Numr Mt, [Kr] Krss, ir Itgrl Equtios, Sprigr Vrlg, Brli, idlbrg, Nw York, odo, Pris, Tokyo, og Kog, 989 [ii] iov, K Nsv A S, Grlizd uctios o ilbrt spcs, sigulr itgrl qutios, d problms o rodymics d lctrodymics, Dirtil Equtios, 7, Bd: 43, t 6, [MM] Mr M, Wvlts d Dirtil Equtios A sort rviw, Dprtmt o Mtmtics, Idi Istitut o Tcology, Nw Dli, Idi, 9 [NiJ] Nitsc J A, Dr Eiluss vo dsigulrität bim itzsc Vrr, Numr Mt, [NiJ] Nitsc J A, T rgulrity o picwis did uctios wit rspct to scls o Sobolv spcs, ot publisd [] uut, Tim-Frqucy d Wvlt Alysis o Fuctios wit Symmtry Proprtis, ogos, Brli, 5 [WG] Wtso G N, A Trtis o t Tory o Bssl Fuctios, Cmbridg Uivrsity Prss, Cmbridg, Scod Editio irst publisd 944, rpritd 996, 3, 4, 6 [WJ] Wiss J, T ilbrt Trsorm o Wvlts r Wvlts, Applid Mtmtics Group, 49 Grdviw od, Arligto, MA 74