c 2012 Society for Industrial and Applied Mathematics

Size: px
Start display at page:

Download "c 2012 Society for Industrial and Applied Mathematics"

Transcription

1 SIAM J. NUMER. ANAL. Vol. 50, No. 4, pp c 0 Socety for Industral and Appled Mathematcs A QUANTITATIVE DESCRIPTION OF MESH DEPENDENCE FOR THE DISCRETIZATION OF SINGULARLY PERTURBED NONCONVEX PROBLEMS ANDREA BRAIDES AND NUNG KWAN YIP Abstract. We nvestgate the lmtng descrpton for a fnte-dfference approxmaton of a sngularly perturbed Allen Cahn type energy functonal. The key ssue s to understand the nteracton between two small length-scales: the nterfacal thckness and the mesh sze of spatal dscretzaton. Dependng on ther relatve szes, we obtan results n the framework of Γ-convergence for the () subcrtcal ( ), () crtcal ( ), and () supercrtcal ( ) cases. The frst case leads to the same area functonal as the spatally contnuous case whle the thrd gves the same result as that comng from a ferromagnetc spn energy. The crtcal case can be regarded as an nterpolaton between the two. Key words. spatal dscretzaton, sngularly perturbed problems, nonconvex functonals, Allen Cahn functonal AMS subject classfcatons. 35B5, 49M5, 65K0 DOI. 0.37/0800. Introducton. In ths paper we descrbe the effect of dscretzaton by fnte dfferences on sngularly perturbed nonconvex varatonal problems by examnng the prototypcal case of an Allen Cahn energy ( () F (u) = W (u)+ u ) dx, Ω where Ω R n and W s a double-well energy densty wth wells n ±; e.g., W (u) = (u ). Except for the trval constant functons u ± (whch can be excluded by an ntegral constrant), a functon u that attans very small energy value F (n the sense that F (u )=O()) typcally parttons the doman Ω nto subdomans on whch u takes on the values close to or and makes a rapd transton between the subdomans (see Fgure ). The energy then concentrates on the transton regon whch s often called the nterfacal regon. Such a descrpton can be made rgorous usng the theory of Γ- convergence of the functonals F. By now t s well known that as 0 the functonals F behave as a sharp-nterface phase-transton energy, fnte only on functons takng the values n {±} and whch can be wrtten as an nterfacal energy (see, e.g., [35, 33, 37]) n the form () C W H n (Ω {u =}), where C W s a constant determned by W and the boundary of {u =} s sutably defned. In the above H n s the Hausdorff (n )-dmensonal measure whch Receved by the edtors January 4, 0; accepted for publcaton (n revsed form) May 8, 0; publshed electroncally July 0, 0. Dpartmento d Matematca, Unverstà d Roma Tor Vergata, va della rcerca scentfca, 0033 Roma, Italy (brades@mat.unroma.t). Department of Mathematcs, Purdue Unversty, West Lafayette, IN (yp@math.purdue. edu). Ths author s work was partally supported by NSF s Dvson of Mathematcal Scences. 883

2 884 ANDREA BRAIDES AND NUNG KWAN YIP Ω Interfacal regon u ~ u ~ Fg.. Parttonng of Ω nto subregons wth u and u. concdes wth the (n )-dmensonal surface measure f the surface s a smooth manfold. Dynamcal models assocated wth energy () also arse n many applcatons. A typcal equaton, derved from the negatve L -gradent flow wth respect to (), s the followng Allen Cahn equaton [5]: (3) u t = ɛ u W (u). The 0 lmt of the above equaton s also studed n many works. It s shown that the lmtng equaton s gven by the moton of a sharp nterface by ts mean curvature. See, for example, [5, 6, 30, 7]. Due to ther wde range of applcatons, t s thus of practcal mportance to consder numercal schemes assocated wth () and (3). In ths paper, we only consder statonary problems. Some dscusson on dynamcal problems wll be gven. We beleve the latter problems are mportant and yet very challengng. A formal fnte-dfference dscretzaton of F can be obtaned by ntroducng another small postve parameter whch represents the mesh sze. The followng energy defned on functons u :Ω Z n R s a typcal example: (4) E, (u) = n W (u )+,j n u u j where the frst sum s taken on Ω Z n and the second sum on all nearest neghbors (n.n.), j Ω Z n,.e., those ndces such that j =. (Each such par s thus counted twce, whch explans the factor /.) The second term s easly shown to be a dscretzaton of Ω u dx. Heurstcally, f the dscretzaton step s much smaller than, then the energy E, s an approxmaton of the sharp-nterface energy (). Ths does not happen n the more realstc case when the two scales may nteract. We are nterested n descrbng precsely ths nteracton. We wll frst show that (4) s approxmated by () for. In the case that K we wll show that the E, s are approxmately (5) ϕ K (ν)dh n, Ω {u=} where ϕ K s some ansotropc convex energy densty characterzed by a dscrete optmal profle problem. In the case, we have a dfferent scalng of the energes,

3 MESH DEPENDENCE FOR NONCONVEX PROBLEMS 885 E,, whch approxmate the crystallne nterfacal energy (6) 4 Ω {u=} ν dh n. Moreover, n ths last scalng E, are also approxmated by a ferromagnetc spn energy (7) E, ferro (u) = n u u j,j defned for u :Ω Z n {, }. Fnally, we wll show that the functon ϕ K acts as an nterpolaton between the eucldean norm ν and the crystallne norm 4 ν. The asymptotc result descrbed n the present paper has some common features wth the analyss n the contnuum settng the nteracton between phase transtons and mcroscopc oscllatons. Examples nclude the study of energes of the form (see [7, 4, ]) (8) F, (u) = Ω ( ( x ) W (u)+ a u ) dx or Ω ( ( W u, x ) + u ) dx. (See also [3], where Isng systems wth Kac potentals are analyzed.) In the above works, a lmtng energy functonal n the form of ansotropc sharp nterfacal energy s also obtaned. In our case the dscrete dmenson adds a further constrant on the dfference quotents, whch mples that ϕ K = O(/K) ask +. The man dffculty of the type of problems above s due to the combned presence of sngular perturbaton and the spatal heterogenety (whch can also come from the dscreteness of the problem). When the two scales nteract, there can be lots of local mnmzers. Ths phenomenon manfests tself even more profoundly n dynamcal problems (see, for example, [38, 6]). Even though there are works whch extend the theory of Γ-convergence to evoluton equatons [36], dynamcal versons of the problems descrbed here reman largely open. The work [7] proves the convergence of the tme dependent problem of the fnte-dfference scheme (4) to the moton by mean curvature but only n the subcrtcal case ( ). On the other hand, the works [9, 3, 3] consder from a homogenzaton pont of vew moton by mean curvature n heterogeneous meda. A frst-order Hamlton Jacob equaton s derved n the lmt. See secton 3.5 for further dscusson.. Settng of the problem. Let W : R n [0, + ) be a locally Lpschtz double-well potental wth wells at ±;.e., W s a nonnegatve functon and W (u) =0 f and only f u =oru =. Moreover, we suppose that W s coercve,.e., lm W (u) =+, u ± and that W s convex close to ±,.e., there exsts C 0 > 0 such that {u : W (u) C 0 } conssts of two ntervals on each of whch W s convex. Standard examples nclude W (u) =( u ) or W (u) =( u ) (see Fgure ). Let Ω be a bounded open subset of R n wth Lpshtz boundary Ω. We wll analyze the asymptotc behavor of E, defned by (4) on functons u :Ω Z n R by computng ther Γ-developments ([9, secton ] and [3]; see also [0, 0] for a general ntroducton to Γ-convergence). For completeness, we wll brefly state the

4 886 ANDREA BRAIDES AND NUNG KWAN YIP W(u) = ( u ) W(u) = ( u ) u u Fg.. Examples of double-well potental. necessary defntons at the end of ths secton. In ths process each u s dentfed wth ts pecewse-constant nterpolaton defned by u(x) =u on + (, )n (and equal to 0 elsewhere) so that Γ-lmts can be taken n Lebesgue spaces. Whatever the dependence of on, the Γ-lmt of E, wth respect to the weak L -convergence can be easly shown to be smply W (u) dx, Ω where W s the convex envelope of W. However, the structure of the nterface cannot be descrbed by the above lmt. It wll only be revealed by the next-order Γ-lmt whch captures energes at the frst relevant scale. Ths s descrbed by the theorem below. To prepare for ts statement and the proof, we denote by Q ν afxed n-dmensonal cube centered at 0, wth sde length and one sde parallel to ν, Q ν T = TQ ν for all T>0, and Q ν T (x) =x + TQν for all x. The lmt energes wll be defned on u BV (Ω; {±}) (or equvalently on sets of fnte permeter after dentfyng u wth A = {u =}). For such a u, thejumpsets u s defned (correspondng to the reduced boundary of A). Furthermore, on all ponts of S u, the measure-theoretcal normal ν, pontng nwards to A, s defned. (We refer to [8, 9] for precse defntons and detals.) Theorem.. Let Ω, W,andE, be as above, and let = (). We then have the followng three regmes for the Γ-lmt wth respect to the strong L -convergence. In all cases the doman of the Γ-lmt s BV (Ω; {±}) and s a surface term on the set S u (whch we wll call the nterface). () (subcrtcal case) If (lm =0), then we have (9) Γ- lm E,(u) =C W H n (Ω S u ), where C W = W (s) ds, as n the contnuous case. () (crtcal case) If lm = K wth 0 <K<+, then (0) Γ- lm E,(u) = ϕ K (ν) dh n, Ω S u where ϕ K s gven by the asymptotc formula () ϕ K (ν) = lm N + N n nf K W (v )+ K v v j,,j

5 MESH DEPENDENCE FOR NONCONVEX PROBLEMS 887 where the ndces, j are restrcted to the cube Q ν N and the nfmum s taken on all v that are equal to u ν (x) =sgn x, ν on a neghborhood of Q ν N. Furthermore, ϕ K s contnuous n the normal drecton ν. () (supercrtcal case) If ( lm =0), then we have () Γ- lm E,(u) =4 ν dh n, Ω S u where ν = ν + + ν n. (v) (nterpolaton) For all ν S n we have (3) lm K 0 ϕ K (ν) =c W, lm Kϕ K(ν) =4 ν. K + Remark.. () The exstence of the lmt n () and the contnuty of ϕ K can be proved as n the contnuous case (see [7]). Moreover, t s a posteror convex (.e., ts one-homogeneous extenson s) snce t s the ntegrand of a lower-semcontnuous nterfacal energy (see [6]). () In the one-dmensonal case formula () reduces to the computaton of an optmal-profle problem { (4) C K =nf K } W (v )+ v v : v(± ) =± ; K n partcular (takng v ± as test functons) we have (5) C K 4 K. (3) It s easly seen that for coordnate drectons mnmzers for ϕ K (e k )areonedmensonal so that whch gves the estmate ϕ K (e k )=C K, (6) ϕ K (ν) C K ν, snce the rght-hand sde s the greatest postvely one-homogeneous convex functon satsfyng ϕ K (e k )=C K for all k. (4) Note that n the supercrtcal case, the lmt nterfacal energy s degenerate, or not unformly convex. Ths s understandable as n ths case the nonlnear term W (u) domnates so that the energy concentrates on the spn functons v whch takes on only values of or. In ths case, the energy s equvalent to bond-countng, the number of bonds between and. It s lkely that ϕ K be unformly convex for 0 <K< even though ths s not mmedately clear from ts defnton. (5) For smplcty, n ths paper we do not mpose any boundary condtons for the functon space. Such effects can be consdered. However, boundary layer mght arse. In addton, the scalng assocated wth the boundary condtons mght be dfferent from that n the bulk. Hence care must be taken. See, for example, [34, ] for some works n the contnuum case whch do consder boundary energy terms. Further remarks and extensons wll be gven at the end of ths paper. Before the proof, we brefly outlne the defnton and procedure of provng Γ-convergence. Gven a sequence of functonals f : X R, t s sad to Γ-converge to f 0 : X 0 R f the followng two steps are true:

6 888 ANDREA BRAIDES AND NUNG KWAN YIP lower bound: for every x X 0 and sequence {x X } >0 such that x x 0, (7) lm nf f (x e ) f 0 (x); upper bound: gven any x 0 X 0, one can fnd a sequence x X such that (8) f 0 (x 0 ) lm sup f ( x e ). The fundamental property of Γ-convergence s that f the collecton of functonals {f } s equcoercve (every sequence wth equbounded energy has a convergent subsequence), then mnmzers of f wll have a subsequence that converges to a mnmzer of f 0. In the applcaton of ths paper, the X s and X 0 wll be taken to be subspaces of L. We are most nterested n the subspace BV (Ω; {±}) of all functons wth bounded varatons whch take values n {±} whch can also be dentfed wth sets of fnte permeter. We wll use n partcular the well-known result by Modca and Mortola (see [35]) that (9) Γ- lm sup F (u) =C W H n (Ω S u ) wth doman u BV (Ω; {±}). In ths case the sequence of energes s equcoercve n L (Ω). Besde the Γ-lmt defned above, t s useful to ntroduce the Γ-lower and upper lmts, respectvely, as (0) () f (x) := Γ- lm nf f (x) = nf f (x) := Γ- lm sup f (x) = nf { lm nf { } f (x ):x x lm sup f (x ):x x whch are always defned at all x. The desred lower bound then translates nto f (x) f 0 (x) and the upper bound nto f (x) f 0 (x). The Γ-lower and upper lmts are lower-semcontnuous functons [9, Proposton.8]. Ths s useful n the computaton of the lower nequalty, snce t allows us to restrct to classes of lower-semcontnuous f 0, whch n the present paper wll be surface energes wth a convex ntegrand of the normal. Moreover, t allows us also to restrct the verfcaton of the upper bound to a dense class of x (see [9, Remark.9]), whch n our case wll be (functons dentfed wth) polyhedral sets. 3.Proofoftheresult.In the two extreme cases () and () the proof wll be acheved by a separaton of scales argument. Frst, a lower bound s obtaned by comparng the energes wth the functonals that are formally obtaned by lettng, respectvely, 0 wth >0fxedand wth >0 fxed. Second, the bounds wll be proved to be sharp by usng sutable approxmate (or recoverng) sequences. In the ntermedate case, a new surface energy defned drectly by a famly of scaled dscrete problems has to be constructed. As s customary, the letter C wll denote a strctly postve constant whose value may vary at each of ts appearance. 3.. Subcrtcal case ( ). We wll frst show that gven {u } wth sup E,(u ) C < +, we can construct contnuous functons v such that, },

7 u u L = o() and MESH DEPENDENCE FOR NONCONVEX PROBLEMS 889 lm nf E,(u ) lm nf F (v ). By the equcoercveness of F, ths mples that we may suppose up to subsequences that v u and hence u u n L (Ω). If u u, then from the nequalty above and (9) we have lm nf E,(u ) lm nf F (v ) C W H n (Ω S u ),.e., the lower bound (7) for E,. To start the proof, upon a truncaton argument, we can frst suppose that u. Next we consder the contnuous functons v mentoned above to be the pecewseaffne nterpolatons on a trangularzaton of u.thenwehave (),j n u u j = Ω v dx + o(). Next we have to estmate the frst sum n E, (u )ntermsof Ω W (v ) dx. We wll consder the varous cases whether u together wth the values of ts nearest neghbors fall nto the same convexty regon of W ( ). The frst and smple scenaro s that when for all the neghbors j of, thevalues u and u j le n the same nterval of convexty of W as n ths case we have by Jensen s nequalty, (3) W (v ) dx T n + T W (u k) k Vertces(T ) for each smplex T of the trangulaton wth vertces on Z n,wherev s a convex combnaton of the value u k at the vertces of T and one of the vertces s as above. To contnue, let C 0 be a fxed number such that {z : W (z) <C 0 } conssts of two ntervals. Denote by J the set of ndces such that W (u ) <C 0/ and the value u j of each ts nearest neghbor j s n the same nterval of {z : W (z) <C 0} as u, and denote by J the complement of J n { : W (u ) <C 0/}. If the smplex T as above s such that one of ts vertces s n J, then (3) holds by the observaton above. As for the smplexes wth one vertex n J,notethat C n J u u j C n #(J ) so that ( ) #(J )=o n. As a consequence, f we sum over all such smplexes and smply take nto account that W s bounded on [, ] we have (4) T J T W (v ) dx C n #(J )=o ( ).

8 890 ANDREA BRAIDES AND NUNG KWAN YIP In the above and the followng, we wll make use of the abused notaton that T J f any of the vertces of T e belongs to J. It apples to other set of ndces also. We now take nto account smplexes for whch the functons u may take values outsde the convexty doman of W. To ths end, wth fxed M>0, denote by I and I the sets of ndces { I = : W (u u ) >C 0 /, } u j M for all n.n.j, I { = : W (u ) >C u 0/, } u j > M for some n.n.j. Snce and n #(I I ) C 0 n ( C0 #(I ) + ) 4 M n W (u ) C n W (u )+ 4,j n u u j we have (5) #(I I ) C n, #(I ) C ( + M ) n. Let T be a smplex as above, and suppose that one of ts vertces s n I ;then by the Lpschtz contnuty of W vertces k of T we have W (v ) dx T W (u k) T Cn sup,j u j u C n+ M C so that T W (v ) dx n + T k W (u k) Cn sup u j u C n+ M.,j Summng over such T and takng nto account (5) we have (6) W (v ) dx T n + T W (u k) #(I )C n+ M = o(). k Next, f we sum over all smplexes wth one vertex n I, and agan smply take nto account that W s bounded on [, ], by (5) we have (7) (8) W (v ) dx C n C T #(I ) +M. Takng nto account the estmates n (3), (4), (6), and (7) we then obtan T n + W (u k)+o() + W (v ) dx Ω T k = n W (u )+o() + C +M. C +M

9 MESH DEPENDENCE FOR NONCONVEX PROBLEMS 89 Fnally, by (), (8), and the arbtrarness of M>0wehave ( ) lm nf E,(u ) lm nf Ω W (v )+ v dx (9) = lm nf F (v ). As noted at the begnnng of the secton, from ths nequalty we deduce the equcoercveness of the sequence E, wth respect to the strong L -convergence snce the functonals F are equcoercve and the constructon above mples that u v = o() as 0, as well as the desred lower bound. The upper bound (8) s obtaned by an explct constructon. It s suffcent to show that t holds for S u a planar nterface, snce the generalzaton to a general nterface s acheved by the same approxmaton usng polyhedral nterfaces as n the contnuum case (see, for example, [, secton 3.9]). We can also suppose that H n (S u Ω) = 0. Now consder u(x) = sgn x, ν. Letv be a mnmzer for the optmal profle problem gvng C W ;.e., v(± ) =± and + ( W (v)+ v ) dt = C W. The recovery sequence (u ) s then defned by u = v (, ν ), Z n ;.e., t s the dscretzaton of v( x, ν /) on the lattce Z n. After notng that f j = e k, u u j = ν ( ) k ) v, ν ( + o()), we have for fxed n n u u ±ek so that k= E,(u )= n = n k= ν k ( ) v, ν ( + o()) = n v (, ν ) ( + o()), ( ( ( )) n W v, ν + = C W H n (S u Ω) + o(). ( ) ) v, ν ( + o()) 3.. Crtcal case (0 < lm = K < ). In ths case, the proof of the equcoercveness can be acheved through reducton to the one-dmensonal case by a sectonal compactness crteron [, secton 3.7]. In dmenson one, the proof s standard and can be obtaned followng [9, secton 6.] by replacng ntegrals wth sums. Ths procedure also shows that the lmt s n BV (Ω; {±}).

10 89 ANDREA BRAIDES AND NUNG KWAN YIP ν Q (x) ρ x ν( x) S u Π ν (x) {u= } {u=} Fg. 3. Descrpton of notatons (a) (e) n the blow-up procedure. The proof of the lower bound can be acheved by a blow-up procedure as follows. Let sup E,(u ) < + and u u. By equcoercvty (as mentoned above) u BV (Ω; {±}). For each >0 we consder the measures μ = n ( W (u )+ n u u +e k k= ) (to avod confusng notaton x denotes the Drac delta n x), where the sum s performedonall Ω Z n such that + e k Ω Z n for all k =,...,n. Note that E,(u ) μ (Ω) so that the famly of measures μ s equbounded. Hence, up to further subsequences we can assume that μ converges weak- to a fnte measure μ. To contnue, we wll estmate μ on S u. For ths, we wll use a coverng argument for almost all S u wth cubes. Wth fxed h N, we consder the collecton Q h of cubes Q ν ρ (x) such that the followng condtons are satsfed: (a) x S u and ν = ν(x) s the normal to S u at x; (b) (Q ν ρ (x) {u =}) Πν (x) h ρn,where Π ν (x) ={y R n : y x, ν 0}; (c) μ(qν ρ (x)) dμ ρ n dh n S u (x) h ; (d) ρ Q ϕ K(ν(y))dH n (y) ϕ n ν K (ν(x)) ρ (x) Su h ;and (e) μ(q ν ρ (x)) = μ(qν ρ (x)). (The notaton n (a) (e) s pctured n Fgure 3.) Note that for fxed x S u and for ρ small enough, (b) s satsfed by the defnton of S u snce ts blow-up set s a hyperplane. Condton (c) follows from the Bescovtch dervaton theorem provded that dμ dh n S u (x) < + ; Condton (d) holds by the same reason, and (e) s satsfed for almost all ρ>0snce μ s a fnte measure and hence μ( Q ν ρ (x)) = 0 except for at most countably many

11 MESH DEPENDENCE FOR NONCONVEX PROBLEMS 893 ρ s. We deduce that Q h s a fne coverng of the set } Su {x μ = dμ S u : dh n (x) < +. S u By Morse s lemma [8, Theorem.47], we can extract a countable famly of dsjont closed cubes {Q νj ρ j (x j )} stll coverng S μ u.notethatwehave H n (S u \ S μ u)=0 snce μ(s u ) < +. We now fx an x Su. μ For smplcty, t s assumed to be 0. In addton, let Q ν ρ = Qν ρ (0) be a cube satsfyng (b) (e) above. Then for small enough we have (30) u u ν dy 4 h ρn Q ν ρ by (b) above, where u ν (x) =sgn x, ν. Note that n ths regme we have (3) E,(u ν ) C K Hn (Ω Π ν )+o() (wth a slght abuse of notaton we dentfy u ν wth ts restrcton to Z n )andthat the same estmate holds locally. By (30) and (3), t s not restrctve to suppose that u = u ν near the boundary of Q ν ρ. Ths can be done by usng a well-chosen cutoff functon close to Q ν ρ. Ths procedure wll ntroduce only an error n the energy functonal of order O(/h)ρ n. (Ths s a classcal argument n Γ-convergence datng back to De Gorg (see [4]). For ts formalzaton n a dscrete-to-contnuous settng we refer, e.g., to []). We then have lm nf = lm nf = lm nf = lm nf = lm nf μ (Q ν ρ(x)) ρ n ρ n Z n Q ν ρ ( ) n ρ N n N n Z n Q ν N Z n Q ν N n ( W (u )+ n k= ( W (u )+ n Z n Q ν ρ k= K W (w )+ K KW(w )+ K n u u +e k ) + O ( ) h ) ( ) u u +e k + O h w wj + O,j Z n Q ν N,j Z n Q ν N w w j + O ( h) ( ), h where N = ρ/, K = ρ/, andw = u for Zn. Snce the functons w are sutable test functons for the problem n () we obtan (3) lm nf μ (Q ν ρ (x)) ρ n ϕ K (ν(x)) + O( h ).

12 894 ANDREA BRAIDES AND NUNG KWAN YIP Usng condton (d) above, we fnally deduce that lm nf μ (Ω) j ( lm nf μ (Q νj ρ j (x j ) S u )+O h) ϕ K (ν(y)) dh (y)+o( n j Q ν j ρ j (x j) S u h ) = ϕ K (ν(y)) dh (y)+o( n, Ω S u h whch gves the lmnf nequalty by the arbtrarness of h. For the upper bound, we agan treat explctly the case u(x) =sgnx n only. For ths t s not restrctve to suppose that H n (S u Ω) = 0. We fx η>0, N N, and v as a test functon for the problem n () such that (33) K N n W (v )+ v v j ϕ K (e n )+η. K Q N,j Q N We may extend u perodcally n the drectons x,...,x n nsde the strp { x n N/} and equal to u (.e., constant ±) outsde ths strp. We then set u = v / for Z n. ) For such u we have ( ) E,(u ) n # {j Z n :(Q N + j) Ω } W (u )+ u u j Q N,j Q N =(H n (S u Ω) + o()) Q N ( ) H n (S u Ω) ϕ K (e n )+η W (v )+ + o(),j Q N v v j as 0, whch proves the upper nequalty by the arbtrarness of η>0. The case of a general ν s proven lkewse wth an almost-perodc extenson of v n the drectons orthogonal to ν (see, e.g., []). As remarked above ths s suffcent to nfer the valdty of the upper bound for all u by approxmaton Supercrtcal case ( ). In ths case, formally lettng 0, we obtan the constrant u {±} for all. Ths suggests to use as a comparson functonal the spn energes { (34) G (u) =,j n u u j f u =forall, + otherwse. These energes have been studed n [4]. They are equcoercve n L, ther Γ-lmt s fnte only on BV (Ω; {±}), and (35) Γ- lm G (u) =4 ν dh n. 0 S u Ω

13 MESH DEPENDENCE FOR NONCONVEX PROBLEMS 895 Now consder a sequence u wth sup (/ )E, (u ) < + ;.e., (36) n+ W (u )+,j n u u j C. From (36) n partcular, for all η>0wehave } (37) # { : W (u ) >η C η for small enough. If we defne then condton (37) ensures that v = { f u > 0, f u 0, u v Cη + o() n+ = o() n and n partcular u v 0 by the arbtrarness of η. We can estmate (38) E,(u ) C η G (v ) wth C η asη 0sothat lm nf E,(u ) lm nf G (v ). Ths mples the coercveness of the energes (/ )E, and the desred lower bound. The proof of the upper bound s trval snce on the doman of G we have E, = G. It suffces then to take a sequence u = v,wherev u realzes the upper bound for the Γ-lmt n (35) Interpolaton. Note that n the proof of the lower nequalty, the condton <<was used only n the estmate leadng to (4). So f now / K, thenwe get nstead #(J )= O(K), n whch gves a O(K) n (4), and as a consequence lm nf n (9). Ths proves that ( ) E,(u ) C W + O(K) H n (Ω S u ). lm nf ϕ K (ν) C W. K 0 +

14 896 ANDREA BRAIDES AND NUNG KWAN YIP The opposte nequalty wth the lm sup s obtaned by estmatng ϕ K takng v as n the proof of the upper bound n the case subcrtcal case. Fnally, by (5) and (6) n Remark.() we obtan lm sup Kϕ K (ν) 4 ν. K + The opposte nequalty for the lmnf follows from the estmate K E,(u) ( + o()) n u u j and the same argument as after (38) Remarks and extensons. As mentoned n the ntroducton, due to the wde range of applcatons of sngularly perturbed problems, analyss and understandng of ther numercal schemes are of practcal mportance. However, mesh sze ntroduces another small length scale whch n essentally all practcal settngs nteracts wth the small parameter n the orgnal problem. Such an nteracton already gves nontrval descrptons for the statonary problem, as ndcated by our theorem. In ths secton, we gve some further remarks and plausble extensons of our results. It s natural to perform a smlar analyss for dynamcal problems, such as (3). The stuaton can be qute ntrcate due to the presence of a large number of crtcal ponts or local mnmzers for the dscrete functonal. Ths can lead to nterestng pnnng and de-pnnng phenomena. Such have been nvestgated n both the contnuum and dscrete cases (see, for example, [, 5]). The more recent work [] s closer n sprt wth the current paper, n partcular the supercrtcal case. A tme steppng, varatonal scheme s employed on a lattce model. Dependng on the relatve magntude of the mesh and tme step szes, the dynamcs demonstrates nterestng stck-slp phenomena. Even though our results do not drectly lead to concrete statements for dynamcal problems, t does gve a quanttatve descrpton of the lmt nterfacal energy functonal and more mportant, the energy scalng n dfferent regmes of and. They can also provde useful gudelnes f other effects are ncorporated. Here we provde some examples. Volume constrants can be mposed: n u = C. If C C, the same Γ-lmt appears as before but wth the constrant for the lmt u, u = C. Appled forces can also be consdered: E, (u) = n W (u )+ n u u j +,j If the forcng terms f, satsfy f, L f n () and () f, L f, (), Ω,j n f, u.

15 MESH DEPENDENCE FOR NONCONVEX PROBLEMS 897 then the Γ-lmt s the same wth the addton of the bulk ntegral term, fudx. Ω A complete pcture of dscrete dynamcs s not currently avalable. However, our results can shed lght n the realstc crtcal case () f K andk. For the former case, we beleve t s possble to compute asymptotcally the lmtng dynamcs and nvestgate the underlyng ansotropy front propagaton. For the latter case, the approach of [] mght stll be applcable. Ths resembles some works n the study of cell-dynamcal systems [8, 3]. Stochastc nose can certanly be used to drve the state out of local mnma. The ncorporaton of a nonunform adaptve mesh s also possble f we have some a pror knowledge about the locaton of the nterface. We wll defer quanttatve answers to these challengng questons n future works. Acknowledgments. The authors would lke to thank Robert Kohn for useful dscusson of the problem. The hosptalty of the IMA where ths work orgnated s also apprecated. REFERENCES [] G. Albert, Varatonal models for phase transtons, an approach va Γ-convergence, ncalculus of Varatons and Partal Dfferental Equatons. L. Ambroso and N. Dancer, eds., Sprnger-Verlag, Berln, 000, pp [] G. Albert, G. Bouchtt e, and P. Seppecher, Phase transton wth the lne-tenson effect, Arch. Raton. Mech. Anal., 44 (998) pp. 46. [3] G. Albert, G. Bellettn, M. Cassandro, and E. Presutt, Surface tenson n Isng system wth Kac potentals, J. Statst. Phys., 8 (996), pp [4] R. Alcandro, A. Brades, and M. Ccalese, Phase and ant-phase boundares n bnary dscrete systems: A varatonal vewpont, Netw. Heterog. Meda, (006), pp [5] S. Allen and J. W. Cahn, A mcroscopc theory for antphase boundary moton and ts applcaton to antphase doman coarsenng, Acta Metall., 7 (979), pp [6] L. Ambroso and A. Brades, Functonals defned on parttons of sets of fnte permeter, II: Semcontnuty, relaxaton and homogenzaton, J. Math. Pures Appl., 69 (990), pp [7] N. Ansn, A. Brades, and V. Chado Pat, Gradent theory of phase transtons n nhomogeneous meda, Proc. Roy. Soc. Ednburgh Sect. A, 33 (003), pp [8] A. Brades, Approxmaton of Free-Dscontnuty Problems, Lecture Notes n Math. 694, Sprnger-Verlag, Berln, 998. [9] A. Brades, Γ-convergence for Begnners, Oxford Unversty Press, Oxford, 00. [0] A. Brades, A handbook of Γ-convergence, n Handbook of Dfferental Equatons, Statonary Partal Dfferental Equatons 3, M. Chpot and P. Quttner, eds., Elsever, New York, 006. [] A. Brades, M. S. Gell, and M. Novaga, Moton and pnnng of dscrete nterfaces, Arch. Raton. Mech. Anal., 95 (00), pp [] A. Brades and A. Patntsk, Homogenzaton of surface and length energes for spn systems, J. Functonal Anal., to appear. [3] A. Brades and L. Trusknovsky, Asymptotc expansons by Gamma-convergence, Contn. Mech. Thermodyn., 0 (008), pp. 6. [4] A. Brades and C. I. Zepper, Multscale analyss of a prototypcal model for the nteracton between mcrostructure and surface energy, Interfaces Free Bound., (009), pp [5] J. W. Cahn, J. Mallet-Paret, and E. S. Van Vleck, Travelng wave solutons for systems of ODEs on a two-dmensonal spatal lattce, SIAM J. Appl. Math., 59 (999), pp [6] X. Chen, Generaton and propagaton of nterfaces for reacton-dffuson equatons, J. Dfferental Equatons, 96 (99), pp [7] X.Chen,C.M.Ellott,A.Gardner,andJ.Zhao, Convergence of numercal solutons to the Allen-Cahn equaton, Appl. Anal., 69 (998), pp [8] S. N. Chow, Lattce dynamcal systems, n Lectures from the C.I.M.E. Summer School, Lecture Notes n Math. 8, Sprnger-Verlag, Berln, 003, pp. 0.

16 898 ANDREA BRAIDES AND NUNG KWAN YIP [9] B. Cracun and K. Bhattacharya, Effectve moton of a curvature-senstve nterface through a heterogeneous medum, Interfaces Free Bound., 6 (004), pp [0] G. Dal Maso, An Introducton to Γ-Convergence, Brkhäuser, Boston, 993. [] N. Drr, M. Luca, and M. Novaga, Γ-convergence of the Allen-Cahn energy wth an oscllatng forcng term, Interfaces Free Bound., 8 (006), pp [] N. Drr and N. K. Yp, Pnnng and de-pnnng phenomena n front propagaton n heterogeneous meda, Interfaces Free Bound., 8 (006), pp [3] N. Drr, G. Karal, and N. K. Yp, Pulsatng wave for mean curvature flow n nhomogeneous medum, European J. Appl. Math., 9 (008), pp [4] E. De Gorg, Sulla convergenza d alcune successon d ntegral del tpo dell area, Rend. Mat. Appl., 8 (975), pp [5] P. de Motton and M. Schatzman, Geometrc evoluton of developed nterfaces, Trans. Amer. Math. Soc., 347 (995), pp [6] C. M. Ellott and A. M. Stuart, The global dynamcs of dscrete semlnear parabolc equatons, SIAM J. Numer. Anal., 30 (993), pp [7] L. C. Evans, H. M. Soner, and P. E. Sougands, Phase transtons and generalzed moton by mean curvature, Comm. Pure Appl. Math., 45 (99), pp [8] I. Fonseca and G. Leon, Modern Methods n the Calculus of Varatons: L p Spaces, Sprnger, Berln, 007. [9] E. Gust, Mnmal Surfaces and Functons of Bounded Varaton, Brkhäuser, Boston, 984. [30] T. Ilmanen, Convergence of the Allen-Cahn equaton to Brakke s moton by mean curvature, J. Dfferental Geom., 38 (993), pp [3] P.-L. Lons and P. E. Sougands, Homogenzaton of degenerate second-order PDE n perodc and almost perodc envronments and applcatons, Ann. Inst. H. Poncaré Anal. Non Lnéare, (005), pp [3] J. Mallet-Paret, Travelng waves n spatally dscrete dynamcal systems of dffuson type, n Lectures from the C.I.M.E. Summer School, Lecture Notes n Math. 8, Sprnger-Verlag, Berln, 003, pp. 0. [33] L. Modca, The gradent theory of phase transtons and the mnmal nterface crteron, Arch. Raton. Mech. Anal., 98 (987), pp [34] L. Modca, The gradent theory of phase transtons wth boundary contact energy, Ann. Inst. H. Poncaré, Anal. Non Lnéare, 4 (987), pp [35] L. Modca and S. Mortola, Un esempo d Γ-convergenza, Boll. Un. Mat. Ital., 4 (977), pp [36] E. Sander and S. Serfaty, Gamma-convergence of gradent flows wth applcatons to Gnzburg-Landau, Comm. Pure Appl. Math., 57 (004), pp [37] P. Sternberg, The effect of a sngular perturbaton on nonconvex varatonal problems, Arch. Raton. Mech. Anal., 0 (988), pp [38] H.-K. Zhao, T. Chan, B. Merrman, and S. Osher, A varatonal level set approach to multphase moton, J. Comput. Phys., 7 (996), pp

More metrics on cartesian products

More metrics on cartesian products More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of

More information

Lecture 12: Discrete Laplacian

Lecture 12: Discrete Laplacian Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly

More information

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could

More information

20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The first idea is connectedness.

20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The first idea is connectedness. 20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The frst dea s connectedness. Essentally, we want to say that a space cannot be decomposed

More information

MAT 578 Functional Analysis

MAT 578 Functional Analysis MAT 578 Functonal Analyss John Qugg Fall 2008 Locally convex spaces revsed September 6, 2008 Ths secton establshes the fundamental propertes of locally convex spaces. Acknowledgment: although I wrote these

More information

Numerical Heat and Mass Transfer

Numerical Heat and Mass Transfer Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and

More information

Appendix B. Criterion of Riemann-Stieltjes Integrability

Appendix B. Criterion of Riemann-Stieltjes Integrability Appendx B. Crteron of Remann-Steltes Integrablty Ths note s complementary to [R, Ch. 6] and [T, Sec. 3.5]. The man result of ths note s Theorem B.3, whch provdes the necessary and suffcent condtons for

More information

Exercise Solutions to Real Analysis

Exercise Solutions to Real Analysis xercse Solutons to Real Analyss Note: References refer to H. L. Royden, Real Analyss xersze 1. Gven any set A any ɛ > 0, there s an open set O such that A O m O m A + ɛ. Soluton 1. If m A =, then there

More information

Curvature and isoperimetric inequality

Curvature and isoperimetric inequality urvature and sopermetrc nequalty Julà ufí, Agustí Reventós, arlos J Rodríguez Abstract We prove an nequalty nvolvng the length of a plane curve and the ntegral of ts radus of curvature, that has as a consequence

More information

The equation of motion of a dynamical system is given by a set of differential equations. That is (1)

The equation of motion of a dynamical system is given by a set of differential equations. That is (1) Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence

More information

BOUNDEDNESS OF THE RIESZ TRANSFORM WITH MATRIX A 2 WEIGHTS

BOUNDEDNESS OF THE RIESZ TRANSFORM WITH MATRIX A 2 WEIGHTS BOUNDEDNESS OF THE IESZ TANSFOM WITH MATIX A WEIGHTS Introducton Let L = L ( n, be the functon space wth norm (ˆ f L = f(x C dx d < For a d d matrx valued functon W : wth W (x postve sem-defnte for all

More information

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number

More information

The Feynman path integral

The Feynman path integral The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space

More information

MMA and GCMMA two methods for nonlinear optimization

MMA and GCMMA two methods for nonlinear optimization MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons

More information

Homogenization of reaction-diffusion processes in a two-component porous medium with a non-linear flux-condition on the interface

Homogenization of reaction-diffusion processes in a two-component porous medium with a non-linear flux-condition on the interface Homogenzaton of reacton-dffuson processes n a two-component porous medum wth a non-lnear flux-condton on the nterface Internatonal Conference on Numercal and Mathematcal Modelng of Flow and Transport n

More information

Affine transformations and convexity

Affine transformations and convexity Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/

More information

Appendix B. The Finite Difference Scheme

Appendix B. The Finite Difference Scheme 140 APPENDIXES Appendx B. The Fnte Dfference Scheme In ths appendx we present numercal technques whch are used to approxmate solutons of system 3.1 3.3. A comprehensve treatment of theoretcal and mplementaton

More information

ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM

ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look

More information

For now, let us focus on a specific model of neurons. These are simplified from reality but can achieve remarkable results.

For now, let us focus on a specific model of neurons. These are simplified from reality but can achieve remarkable results. Neural Networks : Dervaton compled by Alvn Wan from Professor Jtendra Malk s lecture Ths type of computaton s called deep learnng and s the most popular method for many problems, such as computer vson

More information

PHYS 705: Classical Mechanics. Calculus of Variations II

PHYS 705: Classical Mechanics. Calculus of Variations II 1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary

More information

NUMERICAL DIFFERENTIATION

NUMERICAL DIFFERENTIATION NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the

More information

Module 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:

More information

Module 1 : The equation of continuity. Lecture 1: Equation of Continuity

Module 1 : The equation of continuity. Lecture 1: Equation of Continuity 1 Module 1 : The equaton of contnuty Lecture 1: Equaton of Contnuty 2 Advanced Heat and Mass Transfer: Modules 1. THE EQUATION OF CONTINUITY : Lectures 1-6 () () () (v) (v) Overall Mass Balance Momentum

More information

APPENDIX A Some Linear Algebra

APPENDIX A Some Linear Algebra APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,

More information

Prof. Dr. I. Nasser Phys 630, T Aug-15 One_dimensional_Ising_Model

Prof. Dr. I. Nasser Phys 630, T Aug-15 One_dimensional_Ising_Model EXACT OE-DIMESIOAL ISIG MODEL The one-dmensonal Isng model conssts of a chan of spns, each spn nteractng only wth ts two nearest neghbors. The smple Isng problem n one dmenson can be solved drectly n several

More information

REAL ANALYSIS I HOMEWORK 1

REAL ANALYSIS I HOMEWORK 1 REAL ANALYSIS I HOMEWORK CİHAN BAHRAN The questons are from Tao s text. Exercse 0.0.. If (x α ) α A s a collecton of numbers x α [0, + ] such that x α

More information

Convexity preserving interpolation by splines of arbitrary degree

Convexity preserving interpolation by splines of arbitrary degree Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete

More information

Uniqueness of Weak Solutions to the 3D Ginzburg- Landau Model for Superconductivity

Uniqueness of Weak Solutions to the 3D Ginzburg- Landau Model for Superconductivity Int. Journal of Math. Analyss, Vol. 6, 212, no. 22, 195-114 Unqueness of Weak Solutons to the 3D Gnzburg- Landau Model for Superconductvty Jshan Fan Department of Appled Mathematcs Nanjng Forestry Unversty

More information

Kernel Methods and SVMs Extension

Kernel Methods and SVMs Extension Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general

More information

Another converse of Jensen s inequality

Another converse of Jensen s inequality Another converse of Jensen s nequalty Slavko Smc Abstract. We gve the best possble global bounds for a form of dscrete Jensen s nequalty. By some examples ts frutfulness s shown. 1. Introducton Throughout

More information

Problem Set 9 Solutions

Problem Set 9 Solutions Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem

More information

Feature Selection: Part 1

Feature Selection: Part 1 CSE 546: Machne Learnng Lecture 5 Feature Selecton: Part 1 Instructor: Sham Kakade 1 Regresson n the hgh dmensonal settng How do we learn when the number of features d s greater than the sample sze n?

More information

ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION

ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215-222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EUATION

More information

THE WEIGHTED WEAK TYPE INEQUALITY FOR THE STRONG MAXIMAL FUNCTION

THE WEIGHTED WEAK TYPE INEQUALITY FOR THE STRONG MAXIMAL FUNCTION THE WEIGHTED WEAK TYPE INEQUALITY FO THE STONG MAXIMAL FUNCTION THEMIS MITSIS Abstract. We prove the natural Fefferman-Sten weak type nequalty for the strong maxmal functon n the plane, under the assumpton

More information

CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE

CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng

More information

Lecture 17 : Stochastic Processes II

Lecture 17 : Stochastic Processes II : Stochastc Processes II 1 Contnuous-tme stochastc process So far we have studed dscrete-tme stochastc processes. We studed the concept of Makov chans and martngales, tme seres analyss, and regresson analyss

More information

Report on Image warping

Report on Image warping Report on Image warpng Xuan Ne, Dec. 20, 2004 Ths document summarzed the algorthms of our mage warpng soluton for further study, and there s a detaled descrpton about the mplementaton of these algorthms.

More information

Foundations of Arithmetic

Foundations of Arithmetic Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an

More information

princeton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg

princeton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg prnceton unv. F 17 cos 521: Advanced Algorthm Desgn Lecture 7: LP Dualty Lecturer: Matt Wenberg Scrbe: LP Dualty s an extremely useful tool for analyzng structural propertes of lnear programs. Whle there

More information

Geometry of Müntz Spaces

Geometry of Müntz Spaces WDS'12 Proceedngs of Contrbuted Papers, Part I, 31 35, 212. ISBN 978-8-7378-224-5 MATFYZPRESS Geometry of Müntz Spaces P. Petráček Charles Unversty, Faculty of Mathematcs and Physcs, Prague, Czech Republc.

More information

Global Sensitivity. Tuesday 20 th February, 2018

Global Sensitivity. Tuesday 20 th February, 2018 Global Senstvty Tuesday 2 th February, 28 ) Local Senstvty Most senstvty analyses [] are based on local estmates of senstvty, typcally by expandng the response n a Taylor seres about some specfc values

More information

Difference Equations

Difference Equations Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1

More information

One-sided finite-difference approximations suitable for use with Richardson extrapolation

One-sided finite-difference approximations suitable for use with Richardson extrapolation Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,

More information

The Order Relation and Trace Inequalities for. Hermitian Operators

The Order Relation and Trace Inequalities for. Hermitian Operators Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence

More information

Random Walks on Digraphs

Random Walks on Digraphs Random Walks on Dgraphs J. J. P. Veerman October 23, 27 Introducton Let V = {, n} be a vertex set and S a non-negatve row-stochastc matrx (.e. rows sum to ). V and S defne a dgraph G = G(V, S) and a drected

More information

Errors for Linear Systems

Errors for Linear Systems Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch

More information

Edge Isoperimetric Inequalities

Edge Isoperimetric Inequalities November 7, 2005 Ross M. Rchardson Edge Isopermetrc Inequaltes 1 Four Questons Recall that n the last lecture we looked at the problem of sopermetrc nequaltes n the hypercube, Q n. Our noton of boundary

More information

Inductance Calculation for Conductors of Arbitrary Shape

Inductance Calculation for Conductors of Arbitrary Shape CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors

More information

Assortment Optimization under MNL

Assortment Optimization under MNL Assortment Optmzaton under MNL Haotan Song Aprl 30, 2017 1 Introducton The assortment optmzaton problem ams to fnd the revenue-maxmzng assortment of products to offer when the prces of products are fxed.

More information

Fluctuation Results For Quadratic Continuous-State Branching Process

Fluctuation Results For Quadratic Continuous-State Branching Process IOSR Journal of Mathematcs (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 13, Issue 3 Ver. III (May - June 2017), PP 54-61 www.osrjournals.org Fluctuaton Results For Quadratc Contnuous-State Branchng

More information

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,

More information

Salmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2

Salmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2 Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to

More information

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons

More information

Application of B-Spline to Numerical Solution of a System of Singularly Perturbed Problems

Application of B-Spline to Numerical Solution of a System of Singularly Perturbed Problems Mathematca Aeterna, Vol. 1, 011, no. 06, 405 415 Applcaton of B-Splne to Numercal Soluton of a System of Sngularly Perturbed Problems Yogesh Gupta Department of Mathematcs Unted College of Engneerng &

More information

A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS

A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS Journal of Mathematcal Scences: Advances and Applcatons Volume 25, 2014, Pages 1-12 A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS JIA JI, WEN ZHANG and XIAOFEI QI Department of Mathematcs

More information

Math 702 Midterm Exam Solutions

Math 702 Midterm Exam Solutions Math 702 Mdterm xam Solutons The terms measurable, measure, ntegrable, and almost everywhere (a.e.) n a ucldean space always refer to Lebesgue measure m. Problem. [6 pts] In each case, prove the statement

More information

STAT 309: MATHEMATICAL COMPUTATIONS I FALL 2018 LECTURE 16

STAT 309: MATHEMATICAL COMPUTATIONS I FALL 2018 LECTURE 16 STAT 39: MATHEMATICAL COMPUTATIONS I FALL 218 LECTURE 16 1 why teratve methods f we have a lnear system Ax = b where A s very, very large but s ether sparse or structured (eg, banded, Toepltz, banded plus

More information

Randić Energy and Randić Estrada Index of a Graph

Randić Energy and Randić Estrada Index of a Graph EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No., 202, 88-96 ISSN 307-5543 www.ejpam.com SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA 29 JUNE -02JULY 20, ISTANBUL

More information

Psychology 282 Lecture #24 Outline Regression Diagnostics: Outliers

Psychology 282 Lecture #24 Outline Regression Diagnostics: Outliers Psychology 282 Lecture #24 Outlne Regresson Dagnostcs: Outlers In an earler lecture we studed the statstcal assumptons underlyng the regresson model, ncludng the followng ponts: Formal statement of assumptons.

More information

The Finite Element Method: A Short Introduction

The Finite Element Method: A Short Introduction Te Fnte Element Metod: A Sort ntroducton Wat s FEM? Te Fnte Element Metod (FEM) ntroduced by engneers n late 50 s and 60 s s a numercal tecnque for solvng problems wc are descrbed by Ordnary Dfferental

More information

DECOUPLING THEORY HW2

DECOUPLING THEORY HW2 8.8 DECOUPLIG THEORY HW2 DOGHAO WAG DATE:OCT. 3 207 Problem We shall start by reformulatng the problem. Denote by δ S n the delta functon that s evenly dstrbuted at the n ) dmensonal unt sphere. As a temporal

More information

Perron Vectors of an Irreducible Nonnegative Interval Matrix

Perron Vectors of an Irreducible Nonnegative Interval Matrix Perron Vectors of an Irreducble Nonnegatve Interval Matrx Jr Rohn August 4 2005 Abstract As s well known an rreducble nonnegatve matrx possesses a unquely determned Perron vector. As the man result of

More information

College of Computer & Information Science Fall 2009 Northeastern University 20 October 2009

College of Computer & Information Science Fall 2009 Northeastern University 20 October 2009 College of Computer & Informaton Scence Fall 2009 Northeastern Unversty 20 October 2009 CS7880: Algorthmc Power Tools Scrbe: Jan Wen and Laura Poplawsk Lecture Outlne: Prmal-dual schema Network Desgn:

More information

U.C. Berkeley CS294: Spectral Methods and Expanders Handout 8 Luca Trevisan February 17, 2016

U.C. Berkeley CS294: Spectral Methods and Expanders Handout 8 Luca Trevisan February 17, 2016 U.C. Berkeley CS94: Spectral Methods and Expanders Handout 8 Luca Trevsan February 7, 06 Lecture 8: Spectral Algorthms Wrap-up In whch we talk about even more generalzatons of Cheeger s nequaltes, and

More information

NON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS

NON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS IJRRAS 8 (3 September 011 www.arpapress.com/volumes/vol8issue3/ijrras_8_3_08.pdf NON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS H.O. Bakodah Dept. of Mathematc

More information

Lecture 10 Support Vector Machines II

Lecture 10 Support Vector Machines II Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed

More information

Boundary Layer to a System of Viscous Hyperbolic Conservation Laws

Boundary Layer to a System of Viscous Hyperbolic Conservation Laws Acta Mathematcae Applcatae Snca, Englsh Seres Vol. 24, No. 3 (28) 523 528 DOI: 1.17/s1255-8-861-6 www.applmath.com.cn Acta Mathema ca Applcatae Snca, Englsh Seres The Edtoral Offce of AMAS & Sprnger-Verlag

More information

DIFFERENTIAL FORMS BRIAN OSSERMAN

DIFFERENTIAL FORMS BRIAN OSSERMAN DIFFERENTIAL FORMS BRIAN OSSERMAN Dfferentals are an mportant topc n algebrac geometry, allowng the use of some classcal geometrc arguments n the context of varetes over any feld. We wll use them to defne

More information

REGULAR POSITIVE TERNARY QUADRATIC FORMS. 1. Introduction

REGULAR POSITIVE TERNARY QUADRATIC FORMS. 1. Introduction REGULAR POSITIVE TERNARY QUADRATIC FORMS BYEONG-KWEON OH Abstract. A postve defnte quadratc form f s sad to be regular f t globally represents all ntegers that are represented by the genus of f. In 997

More information

CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 13

CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 13 CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 13 GENE H GOLUB 1 Iteratve Methods Very large problems (naturally sparse, from applcatons): teratve methods Structured matrces (even sometmes dense,

More information

Asymptotics of the Solution of a Boundary Value. Problem for One-Characteristic Differential. Equation Degenerating into a Parabolic Equation

Asymptotics of the Solution of a Boundary Value. Problem for One-Characteristic Differential. Equation Degenerating into a Parabolic Equation Nonl. Analyss and Dfferental Equatons, ol., 4, no., 5 - HIKARI Ltd, www.m-har.com http://dx.do.org/.988/nade.4.456 Asymptotcs of the Soluton of a Boundary alue Problem for One-Characterstc Dfferental Equaton

More information

2.3 Nilpotent endomorphisms

2.3 Nilpotent endomorphisms s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms

More information

SUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION

SUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION talan journal of pure appled mathematcs n. 33 2014 (63 70) 63 SUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION M.R. Farhangdoost Department of Mathematcs College of Scences Shraz Unversty Shraz, 71457-44776

More information

Bézier curves. Michael S. Floater. September 10, These notes provide an introduction to Bézier curves. i=0

Bézier curves. Michael S. Floater. September 10, These notes provide an introduction to Bézier curves. i=0 Bézer curves Mchael S. Floater September 1, 215 These notes provde an ntroducton to Bézer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of

More information

Complete subgraphs in multipartite graphs

Complete subgraphs in multipartite graphs Complete subgraphs n multpartte graphs FLORIAN PFENDER Unverstät Rostock, Insttut für Mathematk D-18057 Rostock, Germany Floran.Pfender@un-rostock.de Abstract Turán s Theorem states that every graph G

More information

High resolution entropy stable scheme for shallow water equations

High resolution entropy stable scheme for shallow water equations Internatonal Symposum on Computers & Informatcs (ISCI 05) Hgh resoluton entropy stable scheme for shallow water equatons Xaohan Cheng,a, Yufeng Ne,b, Department of Appled Mathematcs, Northwestern Polytechncal

More information

A note on almost sure behavior of randomly weighted sums of φ-mixing random variables with φ-mixing weights

A note on almost sure behavior of randomly weighted sums of φ-mixing random variables with φ-mixing weights ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 7, Number 2, December 203 Avalable onlne at http://acutm.math.ut.ee A note on almost sure behavor of randomly weghted sums of φ-mxng

More information

Additional Codes using Finite Difference Method. 1 HJB Equation for Consumption-Saving Problem Without Uncertainty

Additional Codes using Finite Difference Method. 1 HJB Equation for Consumption-Saving Problem Without Uncertainty Addtonal Codes usng Fnte Dfference Method Benamn Moll 1 HJB Equaton for Consumpton-Savng Problem Wthout Uncertanty Before consderng the case wth stochastc ncome n http://www.prnceton.edu/~moll/ HACTproect/HACT_Numercal_Appendx.pdf,

More information

Lecture 4. Instructor: Haipeng Luo

Lecture 4. Instructor: Haipeng Luo Lecture 4 Instructor: Hapeng Luo In the followng lectures, we focus on the expert problem and study more adaptve algorthms. Although Hedge s proven to be worst-case optmal, one may wonder how well t would

More information

Indeterminate pin-jointed frames (trusses)

Indeterminate pin-jointed frames (trusses) Indetermnate pn-jonted frames (trusses) Calculaton of member forces usng force method I. Statcal determnacy. The degree of freedom of any truss can be derved as: w= k d a =, where k s the number of all

More information

Stanford University CS359G: Graph Partitioning and Expanders Handout 4 Luca Trevisan January 13, 2011

Stanford University CS359G: Graph Partitioning and Expanders Handout 4 Luca Trevisan January 13, 2011 Stanford Unversty CS359G: Graph Parttonng and Expanders Handout 4 Luca Trevsan January 3, 0 Lecture 4 In whch we prove the dffcult drecton of Cheeger s nequalty. As n the past lectures, consder an undrected

More information

CHAPTER 4. Vector Spaces

CHAPTER 4. Vector Spaces man 2007/2/16 page 234 CHAPTER 4 Vector Spaces To crtcze mathematcs for ts abstracton s to mss the pont entrel. Abstracton s what makes mathematcs work. Ian Stewart The man am of ths tet s to stud lnear

More information

(Online First)A Lattice Boltzmann Scheme for Diffusion Equation in Spherical Coordinate

(Online First)A Lattice Boltzmann Scheme for Diffusion Equation in Spherical Coordinate Internatonal Journal of Mathematcs and Systems Scence (018) Volume 1 do:10.494/jmss.v1.815 (Onlne Frst)A Lattce Boltzmann Scheme for Dffuson Equaton n Sphercal Coordnate Debabrata Datta 1 *, T K Pal 1

More information

Lab session: numerical simulations of sponateous polarization

Lab session: numerical simulations of sponateous polarization Lab sesson: numercal smulatons of sponateous polarzaton Emerc Boun & Vncent Calvez CNRS, ENS Lyon, France CIMPA, Hammamet, March 2012 Spontaneous cell polarzaton: the 1D case The Hawkns-Voturez model for

More information

ON THE BURGERS EQUATION WITH A STOCHASTIC STEPPING STONE NOISY TERM

ON THE BURGERS EQUATION WITH A STOCHASTIC STEPPING STONE NOISY TERM O THE BURGERS EQUATIO WITH A STOCHASTIC STEPPIG STOE OISY TERM Eaterna T. Kolovsa Comuncacón Técnca o I-2-14/11-7-22 PE/CIMAT On the Burgers Equaton wth a stochastc steppng-stone nosy term Eaterna T. Kolovsa

More information

A new construction of 3-separable matrices via an improved decoding of Macula s construction

A new construction of 3-separable matrices via an improved decoding of Macula s construction Dscrete Optmzaton 5 008 700 704 Contents lsts avalable at ScenceDrect Dscrete Optmzaton journal homepage: wwwelsevercom/locate/dsopt A new constructon of 3-separable matrces va an mproved decodng of Macula

More information

arxiv: v1 [math.co] 1 Mar 2014

arxiv: v1 [math.co] 1 Mar 2014 Unon-ntersectng set systems Gyula O.H. Katona and Dánel T. Nagy March 4, 014 arxv:1403.0088v1 [math.co] 1 Mar 014 Abstract Three ntersecton theorems are proved. Frst, we determne the sze of the largest

More information

SIMULATION OF WAVE PROPAGATION IN AN HETEROGENEOUS ELASTIC ROD

SIMULATION OF WAVE PROPAGATION IN AN HETEROGENEOUS ELASTIC ROD SIMUATION OF WAVE POPAGATION IN AN HETEOGENEOUS EASTIC OD ogéro M Saldanha da Gama Unversdade do Estado do o de Janero ua Sào Francsco Xaver 54, sala 5 A 559-9, o de Janero, Brasl e-mal: rsgama@domancombr

More information

Every planar graph is 4-colourable a proof without computer

Every planar graph is 4-colourable a proof without computer Peter Dörre Department of Informatcs and Natural Scences Fachhochschule Südwestfalen (Unversty of Appled Scences) Frauenstuhlweg 31, D-58644 Iserlohn, Germany Emal: doerre(at)fh-swf.de Mathematcs Subject

More information

CSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography

CSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography CSc 6974 and ECSE 6966 Math. Tech. for Vson, Graphcs and Robotcs Lecture 21, Aprl 17, 2006 Estmatng A Plane Homography Overvew We contnue wth a dscusson of the major ssues, usng estmaton of plane projectve

More information

The Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction

The Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also

More information

Lecture 7: Boltzmann distribution & Thermodynamics of mixing

Lecture 7: Boltzmann distribution & Thermodynamics of mixing Prof. Tbbtt Lecture 7 etworks & Gels Lecture 7: Boltzmann dstrbuton & Thermodynamcs of mxng 1 Suggested readng Prof. Mark W. Tbbtt ETH Zürch 13 März 018 Molecular Drvng Forces Dll and Bromberg: Chapters

More information

Physics 5153 Classical Mechanics. Principle of Virtual Work-1

Physics 5153 Classical Mechanics. Principle of Virtual Work-1 P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal

More information

U.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017

U.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017 U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that

More information

Yong Joon Ryang. 1. Introduction Consider the multicommodity transportation problem with convex quadratic cost function. 1 2 (x x0 ) T Q(x x 0 )

Yong Joon Ryang. 1. Introduction Consider the multicommodity transportation problem with convex quadratic cost function. 1 2 (x x0 ) T Q(x x 0 ) Kangweon-Kyungk Math. Jour. 4 1996), No. 1, pp. 7 16 AN ITERATIVE ROW-ACTION METHOD FOR MULTICOMMODITY TRANSPORTATION PROBLEMS Yong Joon Ryang Abstract. The optmzaton problems wth quadratc constrants often

More information

Maximizing the number of nonnegative subsets

Maximizing the number of nonnegative subsets Maxmzng the number of nonnegatve subsets Noga Alon Hao Huang December 1, 213 Abstract Gven a set of n real numbers, f the sum of elements of every subset of sze larger than k s negatve, what s the maxmum

More information

CHAPTER 14 GENERAL PERTURBATION THEORY

CHAPTER 14 GENERAL PERTURBATION THEORY CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves

More information

FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP

FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class

More information

Vapnik-Chervonenkis theory

Vapnik-Chervonenkis theory Vapnk-Chervonenks theory Rs Kondor June 13, 2008 For the purposes of ths lecture, we restrct ourselves to the bnary supervsed batch learnng settng. We assume that we have an nput space X, and an unknown

More information

Supplement: Proofs and Technical Details for The Solution Path of the Generalized Lasso

Supplement: Proofs and Technical Details for The Solution Path of the Generalized Lasso Supplement: Proofs and Techncal Detals for The Soluton Path of the Generalzed Lasso Ryan J. Tbshran Jonathan Taylor In ths document we gve supplementary detals to the paper The Soluton Path of the Generalzed

More information