A STATIC PDE APPROACH FOR MULTI-DIMENSIONAL EXTRAPOLATION USING FAST SWEEPING METHODS

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1 A STATIC PDE APPROACH FOR MULTI-DIMENSIONAL EXTRAPOLATION USING FAST SWEEPING METHODS TARIQ ASLAM, SONGTING LUO, AND HONGKAI ZHAO Abstract. A static Partial Differential Equation (PDE) approac is presented for multidimensional etrapolation under te assumption tat a level set function eists wic separates te region of known values from te region to be etrapolated. Arbitrar orders of polnomial etrapolation can be obtained troug solutions of a series of static linear PDEs. Fast sweeping metods of first and second orders are presented to solve te PDEs for constant, linear and quadratic etrapolation. Numerical eamples are presented to demonstrate te approac. Ke words. Static PDE, Multi-dimensional etrapolation, Fast sweeping metod AMS subject classifications. 65N6, 49M5. Introduction. A great number of computational psics applications require etrapolation metods. One particularl popular metod utilized in level set metods [] can be used to etrapolate data from one region, were te function is known, to anoter region were tere is no data. A level set function demarcates te regions were te function is known and were etrapolation is needed. Te original application of tis tpe of etrapolation was te Gost Fluid Metod [7], were data is known in te real region and needs to be etrapolated into te gost region. Tis was originall done wit constant etrapolation in te normal direction to te interface. Linear and quadratic etrapolation were introduced in [] and etended to cubic etrapolation in [9]. Fast marcing and fast sweeping tecniques were discussed in [, 4] for constant etrapolation. In terms of psics applications, te PDE approac to etrapolation was utilized in [9] for solving Laplace and eat conduction equations on arbitrar domains wit applications to Stefan problems. Te tecnique as also been applied to vaporization in two-pase fluid flow [], bubble dnamics [5] and compressible reacting flow wit pase canges []. It as also been etended to work on quadtree and octree grids [6] wit applications to solving nonlinear Poisson-Boltzmann equations [7]. An ecellent summar of parabolic and elliptic problems utilizing PDE etrapolation was given in []. Te goal of tis work is to develop a static PDE approac as general as te time-dependent approac proposed in [, 9]. Te main advantage of using te static PDE approac is computation efficienc. For te time-dependent approac, were te time is artificial, etrapolation or information is advected from te known region to te unknown region. Numericall one solves a series of linear advection equations. Te time step is restricted b te Courant-Friedrics-Lew (CFL) condition for stable eplicit time scemes, wic can also be viewed as Jacobi iterations. In oter words, it takes O(d/ t) time steps or iterations to advect etrapolation into te unknown region witin a distance d from te known region, were te time step t as to be of te same order of te grid size. Using te static PDE approac, efficient iterative scemes can be applied to solve te static linear PDEs. For eample, te Los Alamos National Laborator, Los Alamos, NM (aslam@lanl.gov). Department of Matematics, Iowa State Universit, Ames, IA 5. (luos@iastate.edu). Department of Matematics, Universit of California, Irvine, CA (zao@mat.uci.edu). Researc of Zao is partiall supported b NSF grant DMS-5698.

2 T. Aslam, S. Luo and H. Zao fast sweeping metod can be easil tailored for tis task. Te number of iterations is finite, and is usuall small and almost independent of te grid size. In principle, one pass tpe of algoritms, suc as fast marcing metod [, 4] can also be used for solving tese linear static advection PDEs. Sorting is not even needed in tis case since te upwind direction is given a priori. However, depending on te geometr of te boundar of te known region, using a fied upwind stencils ma not work for te fast marcing metod. We will sow numerical eamples to demonstrate tis situation in Section 4. Tecniques from ordered upwind sceme [] ma be needed to adjust te local stencil at eac grid. On te contrar, te fast sweeping metod alwas uses fied stencils and discretization scemes at eac grid and can deal wit ig order discretization in te same eas and efficient fasion. Te outline of te paper is te following. In Section, te static PDE approac to solving constant, linear and quadratic etrapolation is presented. In Section, te fast sweeping metods of first and second orders are given. Section 4 provides numerical tests for several etrapolation metods and demonstrates efficienc and accurac of te various scemes. Finall, we draw conclusions in te last section. In te appendi, etension of te static PDE approac to triangular meses is presented.. Static PDE Approac to Multi-Dimensional Etrapolation. Assume a function u is given in a domain Ω in tat is defined b ψ, and needs to be etrapolated to te remaining portion of te space Ω\Ω in tat is defined b ψ >, were ψ is a given level set function, e.g., te signed distance function from te interface Γ = Ω in Ω\Ω in. Motivated b te time-dependent PDE approac introduced in [], we consider a static PDE approac... Constant Etrapolation. Constant etrapolation of te function u is done as a constant along a normal n to te interface Γ. n is defined wit te level set function, Te PDE used to acieve constant etrapolation is were H(ψ) is te unit Heaviside function, i.e., { if ψ >, H(ψ) = if ψ. n = ψ ψ. (.) H(ψ) n u =, (.) (.).. Linear Etrapolation. Linear etrapolation in te normal direction is done in tree steps. First, te directional derivative of u in te normal direction is defined as u n = n u. (.4) Ten u n can be etrapolated in a constant manner into te unknown region b te PDE H(ψ) n u n =. (.5) Wit u n in te wole space, we can etrapolate u into te unknown region b solving te PDE H(ψ)( n u u n ) =. (.6)

3 Static PDE Approac for Multi-dimensional Etrapolation.. Quadratic Etrapolation. Quadratic etrapolation in te normal direction is done in four steps. First, te second directional derivative of u in te normal direction is defined as u nn = n ( n u) (.7) Ten u nn is etrapolated into te unknown region in a constant manner wit te PDE H(ψ) n u nn =. (.8) Net, wit u nn in te wole space, u n can be etrapolated into te wole space in a linear manner wit te PDE H(ψ)( n u n u nn ) =. (.9) Finall, te function u can be etrapolated to te unknown region via te PDE H(ψ)( n u u n ) =. (.).4. Higer-order Etrapolation. It is clear tat te pattern of te static PDE formulations can be etended to iger-order polnomial etrapolation. For instance in te n-t order etrapolation, te n-t order directional derivative of u is first computed in te region ψ, ten etrapolated in a constant fasion. After tat, eac successive lower-order directional derivative is integrated until u is calculated. Remark. Here te Heaviside function is equivalent to a label to indicate were etrapolation is needed.. Fast Sweeping Metods. In tis section, we present te fast sweeping metods for solving te aforementioned PDEs in te general form, a() u() = f(), Ω\Γ, u() = u Γ (), Γ, (.) were a(), f(), and u Γ () are given functions. For tis linear perbolic PDE, boundar condition sould be prescribed at te inflow boundar, i.e., te part of boundar were a() ν() <, Γ, were ν() is outward normal at te boundar. For te cases of etrapolation, a() = n(), f() =, and te inflow boundar is te boundar of te given domain Ω in or te zero level set of te level set function ψ. For simplicit, we present te fast sweeping metods on uniform meses in two-dimensional (-D) cases. Implementation on triangular meses is discussed in Appendi A. Etension to iger dimension is straigtforward. Fast sweeping metods are efficient iterative metods for solving perbolic tpe of PDEs, for eample, te Hamilton-Jacobi equations. For previous work on te development of fast sweeping metods, please refer to [8,, 4, 9,, 7] and reference terein. For te linear advection equation (.), te fast sweeping metod is even simpler since te upwind direction is give a priori b a(). Hence te discretization sceme is linear, i.e., independent of te solution. Te two ke ingredients for te success of fast sweeping metods are: () an upwind discretization, and () sstematic and alternating orderings tat can cover all directions of caracteristics. We first present a few upwind discretizations and ten two coices of orderings. Let us assume te domain is given b [ min, ma ] [ min, ma ], and is discretized wit a uniform mes of size I J. Te mes size is denoted as. At eac grid point (i, j) (or ( i, j )), for an function F, we denote F i,j = F ( i, j ). Also, assume tat a() = (a (, ), a (, )).

4 4 T. Aslam, S. Luo and H. Zao.. First-order Approimation. We present a first-order version of te fast sweeping metod. At point (i, j), witout loss of generalit, let us assume a i,j >, a i,j >. Te fast sweeping metod uses one-sided first-order finite difference to approimate te gradient u. Tat is, (u ) i,j = u i,j u i,j, (u ) i,j = u i,j u i,j. (.) B substituting (.) into (.), we obtain te discretized PDE a u i,j u i,j i,j + a u i,j u i,j i,j = f i,j, (.) wic can be reformulated to get a local updating formula for u i,j, i.e., u i,j = f i,j + a i,j u i,j/ + a i,j u i,j / a i,j / + a i,j /. (.4).. Second-order Approimation. Here we present two versions of secondorder scemes. One is a simple direction b direction upwind second-order sceme. Te oter one is an upwind second-order sceme wit compact stencils.... Direction b direction second-order upwind sceme. An eas direction b direction version is as follows. At point (i, j) wit a i,j >, a i,j >, we use one-sided second-order finite difference to approimate u, u. Tat is, (u ) i,j = /u i,j + u i,j /u i,j, (u ) i,j = /u i,j + u i,j /u i,j. B substituting (.5) into (.), we obtain te discretized PDE a i,j (.5) u i,j + u i,j u i,j + a u i,j + u i,j u i,j i,j = f i,j, (.6) from wic we can get a local updating formula for u i,j, i.e., u i,j = f i,j + a i,j (u i,j u i,j)/ + a i,j (u i,j u i,j)/ a i,j /() a i,j /(). (.7) Following te same procedure, te one-sided finite difference approimation, suc as (.) or (.5), can be easil obtained for oter cases of a(), and ence te local updating formula, suc as (.4) or (.7).... Second-order upwind sceme wit compact stencils. Here we give a second-order sceme wit compact stencils following te formulation developed in []. Te ke idea is to use te original PDE (.) to provide more relations and ence reduce te number of needed stencils to acieve certain accurac. Differentiating (.) we get a() u() + D u() a() = f(), (.8) were a denotes te Jacobian of a and D u denotes te Hessian of u. Te Talor epansion of u() to te second order is u( + δ) = u() + δ u() + δ, D u() δ + O( δ ). (.9)

5 Static PDE Approac for Multi-dimensional Etrapolation 5 B writing δ as δ = (δ â)â + (δ â )â, (.) (.9) can be written as u( + δ) =u() + δ u() + (δ â) â, D u() â + (δ â)(δ â ) â, D u() â (.) were â = + (δ â ) â, D u() â + O( δ ), a a and â =, â â. From (.8) we ave D u() â = f() a() B substituting (.) into (.), we ave u( + δ) =u() + δ u() + a() u(). (.) a() (δ â) (â f â ( a u)) a + (δ â)(δ â ) (â f â ( a u)) a + (δ â ) â, D U â + O( δ ) ( ) (δ â) =u() + δ a (â a) (δ â)(δ â ) (â a) u() a + (δ â ) â, D U â (δ â) + a â f + (δ â)(δ â ) â f a + O( δ ). (.) Notice tat te onl unknowns in te Talor epansion (.) are u() and â, D u() â. Terefore, in -D, at eac grid point (i, j), witout loss of generalit, assuming te upwind direction a i,j >, a i,j >, b appling te Talor epansion (.) to u i,j, u i,j, u i,j, we get tree linear equations of te tree unknowns, u() and â, D u() â. B solving te tree linear equations, te tree unknowns are computed in terms of u i,j, u i,j, u i,j, u i,j. Te solutions give an approimation of u() at (i, j), wic is furter substituted into te linear PDE (.) to get te discretized PDE tat can be solved to obtain a local updating formula of u i,j. For eample, let us denote u = (α, β), â, D u() â = γ and assume a is constant locall. Ten using (.), we ave te following tree equations u i,j = u i,j α + a i,j γ + F, u i,j = u i,j β + a i,j γ + F, u i.j = u i,j α β + (a i,j a i,j ) γ + F, (.4)

6 6 T. Aslam, S. Luo and H. Zao wit F = a i,j a i,j âi,j f i,j a i,j a i,j a i,j â i,j f i,j, F = a i,j f i,j + a i,j a i,j a i,j âi,j a i,j â i,j f i,j, F = (a i,j + a i,j ) a i,j We can solve (.4) to obtain α, β, γ, â i,j f i,j + (a i,j a i,j a i,j ) â i,j f i,j. α = ( a i,j a ) u i,j u i,j + a i,j u i,j u i,j i,j a i,j + a i,j F a i,j F + a i,j F + a i,j F a, i,j β = ( a i,j a ) u i,j u i,j + a i,j u i,j u i.j i,j a i,j + a i,j F a i,j F + a i,j F + a i,j F a, i,j γ = u i,j + u i,j u i,j u i,j + F F F a. i,j a i,j (.5) B rewriting α, β as α = g u i,j + g, β = g u i,j + g 4 and substituting tem into te PDE (.), we ave te discretized PDE were, g = ( a i,j a ) i,j, a i,j(g u i,j + g ) + a i,j (g u i,j + g 4 ) = f i,j, (.6) g = ( a i,j a ) u i,j + a i,j u i,j u i,j i,j a i,j g = ( a i,j a ) i,j, g 4 = ( a i,j a ) u i,j + a i,j u i,j u i.j i,j a i,j Hence we ave te local updating formula for u i,j b solving (.6) as + a i,j F a i,j F + a i,j F + a i,j F a, i,j + a i,j F a i,j F + a i,j F + a i,j F a. i,j u i,j = f i,j a i,j g a i,j g 4 g a i,j + g a. (.7) i,j From te first two equations in (.5), we can see tat te sceme is not monotone and wen a i,j is close to te grid lines, i.e., eiter a i,j or a i,j, te coefficients ma go unbounded. As discussed in [], at eac point (i, j), te stencils must be

7 Static PDE Approac for Multi-dimensional Etrapolation 7 cosen more carefull according to te upwind direction a i,j. At eac point (i, j), te local mes is divided into eigt regions wit equal inner angles (see Figure. (a)). If te upwind direction a i,j is in region (see Figure. (b)), ten we coose stencil {(i, j), (i, j), (i, j ), (i, j )}; if te upwind direction a i,j is in region (see Figure. (c)), ten we coose stencil {(i, j), (i, j ), (i, j ), (i +, j )}; and if te upwind direction a i,j is in region (see Figure. (d)), ten we coose stencil {(i, j), (i, j), (i, j ), (i, j + )}. For oter regions, te stencils can be cosen in a similar wa. (i-, j+) (i, j+) (i+, j+) (i-, j+) (i, j+) (i+, j+) (i-, j) (i, j) (i+, j) (i-, j) (i, j) (i+, j) a) (i-, j-) (i, j-) (i+, j-) (i-, j+) (i, j+) (i+, j+) b) (i-, j-) (i, j-) (i+, j-) (i-, j+) (i, j+) (i+, j+) (i-, j) (i, j) (i+, j) (i-, j) (i, j) (i+, j) c) (i-, j-) (i, j-) (i+, j-) d) (i-, j-) (i, j-) (i+, j-) Fig... Local mes and stencils for compact second-order sceme: (a) partition of te local mes into 8 regions wit equal inner angles; (b), (c) and (d) stencils (denoted as dark squares) used corresponding to te locations and directions of te caracteristics (denoted as dased arrow)... Ordering. Sstematic and alternating orderings tat can cover all directions of caracteristics effectivel is anoter crucial factor for te efficienc of te fast sweeping metod. For te first-order upwind sceme and direction b direction second-order upwind sceme on rectangular grids, alternating te following four orderings in -D during Gauss-Seidel iterations is effective since information arrives at a grid point troug one of te four quadrants. () i = : I, j = : J; () i = : I, j = J : ; () i = I :, j = : J; (4) i = I :, j = J :. (.8) On te oter and, for te second-order upwind sceme wit compact stencils, using te ordering introduced in [9] for triangular meses is more effective since information arrives at a grid point troug one of te eigt sectors illustrated in Figure.. Te idea is to coose a few reference points, e.g., te four corners points of te computational domain. All grid points are ordered b te distance to eac reference point in bot increasing and decreasing order. Guass-Seidel iterations will be carried out according to tese orderings alternatel.

8 8 T. Aslam, S. Luo and H. Zao.4. Scematic sketc of te fast sweeping metod. Wit bot upwind scemes and sstematic orderings available, We summarize te fast sweeping metods below.. Initialization: impose te eact value of u at grid points on or near Γ, wic are fied during iterations; assign arbitrar values at all oter points.. Gauss-Seidal iterations: sweep te wole mes following alternating orderings: At eac point (i, j), update u i,j wit (.4) or (.7) or (.7), One iteration corresponds to sweeping all te grid points once.. Termination: stop te iterations if two successive iterations ave small canges, i.e., u k u k δ for some stopping criterion δ >. Narrow band implementation. In applications, te etrapolation is often needed onl in a narrow band adjacent to te known region. Te narrow band is tpicall defined b te level set function, e.g., a signed distance function. An eas modification of te fast sweeping metod described above can serve te purpose. First we collect all grid points in te narrow band and create a list b a simple one pass region growing algoritm: starting wit an point in te narrow band, we add its neigbors tat are in te narrow band to te list, ten we furter add te neigbors of te newl added points tat are in te narrow band but not in te current list et to te updated list. We continue tis process until no more points are added to te list. Since tere is no regular data structure for points in te list, distance to a few references points is used to provide alternating orderings. Te fast sweeping metods will be applied to te grid points on te list onl, i.e., te PDEs will be solved numericall on te narrow band onl..5. Convergence Analsis. Here we provide a convergence stud for te firstorder sceme defined in Section.. In Section, te etrapolation problem is formulated as a series of static linear perbolic PDEs in te form of (.), were te inflow boundar is te zero level set of te level set function ψ() and a() = n() = ψ() ψ(). Assume ψ() is continuous and < c < ψ < C < for ψ wit c and C some constants. Te computation domain Ω is discretized b a Cartesian grid wit grid size. For simplicit we present te analsis for D. Etension to iger dimensions is straigtforward. Define Ω + = {( i, j ) Ω ψ i,j > }, Ω = {( i, j ) Ω ψ i,j }, and Γ ± to be te set of boundar grid points, i.e., Γ = {( i, j ) Ω and at least one of its four neigbors ( i±, j ), ( i, j± ) belongs to Ω + } and Γ+ = {( i, j ) Ω + and at least one of its four neigbors ( i±, j ), ( i, j± ) belongs to Ω }. Te value of u i,j is prescribed at Γ. Let A denote te matri of te linear sstem for {u i,j, ( i, j ) Ω + } corresponding to te first-order upwind sceme defined b (.) wit given inflow boundar condition for u i,j, ( i, j ) Γ. We prove te following statements about te linear sstem. Teorem.. Te discretized linear sstem as te following properties: () Maimum principle olds, i.e., if f i,j =, te maimum or minimum value of u i,j is attained at te boundar Γ. () A is a non-singular M-matri. () Te fast sweeping metod for te linear sstem converges. Proof. () We first sow ma (i, j) Ω + u i,j = ma Γ (i, j) Γ u i,j. Suppose te maimum value of u i,j is attained at ( m, n ) Ω +. Witout loss of generalit

9 Static PDE Approac for Multi-dimensional Etrapolation 9 assume a m,n, a m,n. From te upwind sceme we ave u m,n = a m,nu m,n / + a m,nu m,n / a m,n/ + a. m,n/ So u m,n = u m,n = u m,n. If eiter ( m, n ) Γ or ( m, n ) Γ, te result is proved. Oterwise, te argument can be continued all te wa to reac a point on te boundar Γ. Similarl, one can sow min ( i, j) Ω + u i,j = min Γ (i, j) Γ u i,j. () Tere are several equivalent definitions of an M-matri [, 8]. Denote A = (a i,j ), due to te upwind sceme, a i,j, i j. We sow tat A is nonsingular and A []: (a) denote u to be te vector of values u i,j at ( i, j ) Ω + corresponding to f i,j =, ( i, j ) Ω + and u i,j =, ( i, j ) Γ, i.e., A u =. Witout loss of generalit, assume u m,n = ma i,j u i,j >, from te te maimum principle proved above, it contradicts to te fact tat u i,j =, ( i, j ) Γ. So u =. Hence A is non-singular; and (b) we sow tat ever element in an column of A is non-negative. Te j-t column of A corresponds to a solution u = {u i,j ( i, j ) Ω + } wit u i,j =, ( i, j ) Γ and A u = e j, were e j is a column vector wit j-t element equal to and all oter elements equal to. In anoter word, u corresponds to a discrete Green s function wit zero Diriclet boundar condition and a source at some grid point ( m, n ) Ω +, i.e., f i,j = if (i, j) (m, n) and f m,n =, were (m, n) is indeed as j. Suppose u p,q = min i,j u i,j <, ten it is eas to see (p, q) (m, n). Ten appl te argument for maimum principle one can easil sow min (i, j) Γ u i,j = u p,q <, wic is a contradiction. () Since te fast sweeping metod is a Gauss-Seidel iteration wit alternating orderings in eac iteration, it converges for an M-matri [, 5]. Remark. Altoug te fast sweeping algoritm is a tpe of Gauss-Seidel iteration, te use of different orderings during te iterations can greatl accelerate te convergence for linear or nonlinear sstems arising from upwind discretization for perbolic problems. Once correct causalit is enforced in te discretization and during te update at eac grid point, alternating orderings allow efficient propagation of information along caracteristics in all directions. Remark. For te linear perbolic PDEs for te etrapolation problem, te main complication comes from te geometr of te inflow boundar Γ. Near te concave part of te boundar, te caracteristics following ψ ψ converge. Near te conve part, te caracteristics diverge. Tese scenarios are analogous to socks and rarefactions for te perbolic conservation law. For te first-order upwind monotone sceme, tere is an eplicit numerical viscosit wit coefficient of grid size (b Talor epansion of te finite difference sceme). So it can andle bot cases. However, te divergence of caracteristics near te conve part of te boundar can make one-pass algoritm fail as sown in numerical eample 4 and Figure 4.4 in te net Section. Tis penomenon can be eplained from linear algebra point of view, te discretized sstem can not be transformed to a triangular sstem no matter wat ordering is used for te grid points, wic is also equivalent to te fact tat te discretized problem becomes locall elliptic near te conve part of te boundar due to te numerical viscosit. Since te fast sweeping metod is an iterative metod, it works efficientl in all scenarios. Te following teorem sows error estimate for te first-order upwind sceme. For simplicit, we state te result in D and etension to iger dimension is straigtforward.

10 T. Aslam, S. Luo and H. Zao Teorem.. Assume ψ is te signed distance function to a smoot interface Γ. Let u() be te solution to (.) wit a() = ψ()/ ψ(), f() = and smoot boundar condition, and u i,j be te corresponding numerical solution to (.). For an point ( i, j ) in te region were a() is smoot, u i,j u( i, j ) C as, (.9) were < C < is some constant wic depends on Γ and ψ. Proof. Denote e i,j = u i,j u( i, j ). We order all grid points in Ω + b laers suc tat te k-t laer contains all te points wit (k ) < ψ( i, j ) k. We prove te error estimate (.9) b induction. Assume E k = ma (k )<ψ( m, n) k e m,n is attained at point ( i, j ), i.e., E k = e i,j. And let us assume a i,j, a i,j witout loss of generalit. We ave a i,j(e i,j e i,j ) + a i,j (e i,j e i,j ) = T i,j, (.) were T i,j = O( ) is te local truncation error and te constant in O( ) depends on te second derivatives of u() wic are bounded b te smootness assumption. Ten wic implies a i,j a i,j T i,j e i,j = a i,j + e i,j + a i,j a i,j + e i,j + a i,j a i,j +, a i,j a i,j a i,j e i,j a i,j + e i,j + a i,j a i,j + a i,j e i,j + T i,j a i,j +. a i,j Since a i,j + a i,j =, a i,j + a i,j. We ave te following cases: (a) If bot ( i, j ) and ( i, j ) are in te (k )-t laer, ten E k E k + O( ). (b) If onl one of ( i, j ), ( i, j ) is in te (k )-t laer (assume it is ( i, j )), ten a i,j a i,j a i,j e i,j a i,j + e i,j + a i,j a i,j + a i,j a i,j a i,j + e i,j a i,j a i,j + a i,j wic implies a i,j a i,j + e i,j + a i,j a i,j + a i,j a i,j e i,j + T i,j a i,j +, a i,j e i,j + T i,j a i,j + a i,j e i,j + T i,j a i,j + a i,j E k e i,j + T i,j /a i,j E k + T i,j /a i,j.

11 Static PDE Approac for Multi-dimensional Etrapolation Since ψ i,j < ψ i,j and ψ i,j = ψ i,j a i,j + O( ), ψ i,j = ψ i,j a i,j + O( ), we ave a i,j > a i,j O(). So a i,j > O() > if is small enoug. So E k E k + T i,j. (c) If bot ( i, j ) and ( i, j ) are not in te (k )-t laer, let us assume a i,j a i,j witout loss of generalit. From (.), we ave wic implies Also we ave a i,j e i,j e i,j + a i,j e i,j e i,j = O( ), e i,j = e i,j + O( ). (.) ψ i,j = ψ i,j a i,j + O( ) ψ i,j + O( ) < ψ i,j (.) for small enoug. For e i,j at point ( i, j ), it satisfies a relation as (.) wit two neigbors coming from te upwind discretization determined b (a i,j, a i,j ), for eample, a i,j(e i,j e i,j ) + a i,j (e i,j e i,j ) = T i,j, wic implies from (.) a i,j(e i,j e i,j ) + a i,j (e i,j e i,j ) = O( ). (.) If eiter bot ( i, j ) and ( i, j ) are or one of tem is in (k )-t laer, ten it is reduced to case (a) or (b) above and te proof is completed. Oterwise we repeat te previous step: if a i,j a i,j, we consider e i,j at point ( i, j ) and ave e i,j = e i,j + O( ) and ψ i,j < ψ i,j < ψ i,j < (k ); oterwise we consider e i,j at point ( i, j ) and ave e i,j = e i,j + O( ) and ψ i,j < ψ i,j < (k ) for small enoug. Hence we get E k = e i,j E k + O( ). Terefore, for an laer k, E k = ko( ). Since ψ is te signed distance function, for a domain of O() size, tere are at most O(/) laers in total. Hence te global maimum error is O(). Remark 4. If Γ is conve, ten a() = ψ() is smoot everwere. So te above error bound is true everwere. If Γ is concave, ten a() is smoot up to te socks of te distance function were ψ() is discontinuous. Also te above result can be etended to a general level set function ψ() as long as ψ() > c > for some constant c. Te onl difference is te number of laers in a bounded domain and te case (c) in te above proof ma ave to go troug a few more but a finite number (depending on c) of recursive relations.

12 T. Aslam, S. Luo and H. Zao 4. Numerical Tests. We present more numerical implementation details and a few eamples to demonstrate bot efficienc and accurac of our approac. In all te eamples, level set function, ψ, is te signed distance to te boundar between known and unknown region. Level set function is negative inside te known region. 4.. Numerical Implementations. For constant etrapolation in te normal direction, we solve (.) wit te first-order fast sweeping metod. Te normal components n are computed wit te tird-order Weigted Essentiall Non-Oscillator (WENO) finite difference approimations (e.g., see [6]). For linear etrapolation in te normal direction, we solve (.5) and (.6) wit te first-order fast sweeping metod. u n = n u is first computed wit te tirdorder WENO approimations in te region ψ band. Ten it is etrapolated in a constant manner to te wole domain b solving te modified equation (.5), i.e., H(ψ + band ) n u n =. (4.) After tat, we solve (.6) to etrapolate u into te wole domain. Here band =. For quadratic etrapolation in te normal direction, we solve (.8), (.9) and (.) wit te second-order fast sweeping metods. u nn = n ( n u) is first computed in te region ψ band. Ten it is etrapolated in a constant manner to te wole domain b solving te modified equation (.8), i.e., H(ψ + band ) n u nn =. (4.) After tat, u n is computed b solving te modified equation (4.). Finall u is computed b solving (.). Here band =. Note tat te Heaviside function serves as te indicator of unknown and known regions in te PDEs. In te numerical implementation, te PDEs onl need to be solved in te unknown region were H. Remark 5. For te computation of te derivatives in n, u n, u nn, besides te tird-order WENO approimations, we also implement te centered finite differences. For constant and linear etrapolation, second-order centered finite difference is used (band = ). For quadratic etrapolation, fourt-order centered finite difference is used (band =, band = ). And te results sow desired accurac of etrapolation as in te following eamples. 4.. Numerical Eamples. We coose δ = 9 as te stopping criterion for te fast sweeping metods. In computation of n, u n, u nn, we implement bot te tird-order WENO approimations and te centered finite difference approimations. Numerical results including te accurac, te order of convergence and te number of iterations are recorded. One iteration corresponds to sweeping te wole mes once. Eample : Te computational domain is [ π, π] [ π, π], te level set function is ψ = +. Te function u is initiall given as u = {, if ψ >, cos() sin(), if ψ. (4.) Table 4. sows te accurac of constant, linear and quadratic etrapolation. Figure 4. sows te contour plots of te numerical solutions. From Table 4., we observe first, second and tird order convergence for constant, linear and quadratic etrapolation inside a narrow band of size net to te interface Γ, respectivel. And te number of iterations is almost constant as te mes is refined.

13 Static PDE Approac for Multi-dimensional Etrapolation Table 4. Numerical accurac for etrapolation in a neigborood of size near te interface Γ. Te numbers of iterations correspond to solving te different PDEs: constant etrapolation: equation (.); linear etrapolation: (4.) and (.6); and quadratic etrapolation: (4.), (4.) and (.). Mes Tird-Order WENO constant etrapolation L error.59e E- 4.4E-.9E- Conv. Order # Iter linear etrapolation L error 4.9E-.79E-.88E E-5 Conv. Order # Iter 9, 9 9, 9 9, 9 9, 9 quadratic etrapolation L error.6e-4.76e-5 4.8E E-7 Conv. Order # Iter 7, 7, 7 7, 7, 7 7, 7, 7 7, 7, 7 quadratic etrapolation (wit compact upwind second-order sceme) L error 9.779E-4.45E-4.6E-5.4E-6 Conv. Order # Iter 8, 8, 8 8, 8, 8 8, 8, 8 8, 8, 8 Centered finite difference constant etrapolation L error.59e E- 4.4E-.9E- Conv. Order # Iter linear etrapolation L error 5.7E-.4E-.89E E-5 Conv. Order # Iter 5, 5 7, 5 5, 5 5, 5 quadratic etrapolation L error.7e-.86e-4.89e-5.74e-6 Conv. Order # Iter 8, 8, 7 8, 8, 7 8, 8, 7 8, 8, 7 quadratic etrapolation (wit compact upwind second-order sceme) L error.977e-.54e-4.4e-5 4.4E-6 Conv. Order # Iter 8, 8, 8 8, 8, 8 8, 8, 8 8, 8, 8 Table 4. sows te comparison of CPUtime between te time-dependent approac [] and te static approac for constant and linear etrapolation. For te time-dependent approac, we use second-order upwind finite difference sceme and second-order Runge-Kutta metod as in []. Te PDE must be solved to te stead state. In te numerical implementation, te computation stops wen te cange between two successive time steps is less tan δ. For te static approac, we use second-order upwind sceme (.7) for te purpose of comparison. It is clear tat te time-dependent approac requires muc more CPUtime tan te static approac.

14 4 T. Aslam, S. Luo and H. Zao a) c) b) d) Fig. 4.. Contour plots of te numerical solutions: (a) initial u, (b) constant etrapolation, (c) linear etrapolation, and (d) quadratic etrapolation. Black dots indicate te interface Γ. Mes: 4 4. Table 4. CPUtime (seconds) comparison between te time-dependent approac (TD) in [] and te static approac. Same order of accurac is obtained in te band near te interface for bot approaces. Mes Centered finite difference, constant etrapolation CPUtime(Static)....5 CPUtime(TD) Centered finite difference, linear etrapolation CPUtime(Static) CPUtime(TD) Bot approaces are implemented wit C codes on a Desktop. Eample : Te setup is te same as in Eample, ecept tat te level set function is given as ψ = min{ (.8) +, ( +.8) + }. Table 4. sows te performance of te fast sweeping metods for constant, linear and quadratic etrapolation. Figure 4. sows te contour plots of te numerical solutions. From Table 4., we observe again tat te number of iterations canges ver little as te mes is refined. Eample : Tis eample demonstrates narrow band implementation described in Section.4. Four corner points of te computational domain are cosen as te reference points for alternating orderings. Figure 4. sows te results of etrapolation into a narrow band of size near te interface Γ. Eample 4: Here we use a simple eample to demonstrate tat one pass algo-

15 Static PDE Approac for Multi-dimensional Etrapolation 5 Table 4. Te numbers of iterations correspond to solving te different PDEs: constant etrapolation: equation (.); linear etrapolation: (4.) and (.6); and quadratic etrapolation: (4.), (4.) and (.). Mes Tird-order WENO constant etrapolation # Iter 7 9 linear etrapolation # Iter, 7, 7 9, 9, quadratic etrapolation # Iter 5, 5, 5,, 5, 5, 5 5, 5, 5 quadratic etrapolation (wit compact upwind second-order sceme) # Iter,, 4, 4, 4 4, 4, 4,, Centered finite difference constant etrapolation # Iter 7 9 linear etrapolation # Iter, 9, 7, 9, quadratic etrapolation # Iter 4, 4, 5 9, 7, 4, 9, 5 45, 4, 5 quadratic etrapolation (wit compact upwind second-order sceme) # Iter,,,,,,,, ritm wit fied upwind stencils does not work. For simplicit we use te first-order upwind sceme (.4) as an eample. At an given point, at most two of its four immediate neigbors in te upwind direction are used to determine its value. So te value at tis point can get te correct value onl if its upwind neigbors ave correct values alread. If a one pass algoritm wit fied stencil as to work, tere as to be an ordering of all grid points suc tat ever grid point can onl depend on grid points tat eiter ave assigned value (from initialization) or ave been updated correctl. Figure 4.4(a) sows a scenario were two grid points, A and B, net to te interface depend on eac oter. According to te upwind direction, sown b arrows in te figure, value at point B and C are required to determine te value at A. Similarl te value at point A and D are required to determine te value at B. In anoter word, te values at A and B are coupled in te upwind sceme and can not be solved one b one as in one pass algoritms. Figure 4.4(b) eactl verifies tis scenario b enforcing a one pass condition in te sweeping algoritm for constant etrapolation. We update values onl at tose grid points were teir upwind neigbors eiter ave assigned value (from initialization) or ave been updated correctl. We see tat etrapolation get stuck immediatel at te four points were te interface is tangent to te grid lines wic is eactl demonstrated in Figure 4.4(a). Terefore te stencils must be modified on te fl, suc as tose used in te ordered upwind sceme []. On te oter and, te fast sweeping metod is an iterative metod wic andles tis situation naturall and effectivel as investigated in [5]. Te fast sweeping metod does not onl allow te fast propagation of information along caracteristics b using upwind sceme, causalit and alternating orderings during te iterations but also ave an effective built in relaation mecanism tat can andle more complicated

16 6 T. Aslam, S. Luo and H. Zao 4 4 a) 4 b) c) 4 d) Fig. 4.. Contour plots of te numerical solutions: (a) initial u, (b) constant etrapolation, (c) linear etrapolation, and (d) quadratic etrapolation. Black dots indicate te interface Γ. Mes: situation wen one pass algoritms fail. Eample 5: Te setup is te same as in Eample, ecept tat te level set function is te signed distance function to a triangle (see Figure 4.5). In tis case, te interface Γ is not smoot. Te derivatives are computed wit tird-order WENO approimations. Figure 4.5 sows te contour plots of te numerical solutions. 5. Conclusion. We developed te fast sweeping metod for te static PDE based etrapolation wic can be applied to a variet of psics problems often solved in conjunction wit level set tecniques. Te proposed metod is efficient, in tat te scale linearl wit te size of te grid, and eliminate te need to integrate in time to stead state as was done in te original approac []. Altoug parallelization tecniques were not discussed ere, approaces in [8, 6] can easil be applied to gain furter efficienc. Acknowledgment Te autors would like to tank te anonmous referees for elpful suggestions on tis work. Appendi A. Implementations on Triangular Meses. In tis section, we present numerical implementations of te constant and linear etrapolation wit firstorder fast sweeping metod on a triangular mes. Assume tat te computational domain is discretized b a triangular mes. Given an point C = ( C, C ) wit its two neigbors A = ( A, A ) and B = ( B, B ) on te same simple CAB (see Figure A.), and assume tat te caracteristic passing troug C falls in CAB. B appling linear Talor epansion witin CAB, we ave

17 Static PDE Approac for Multi-dimensional Etrapolation Fig. 4.. Contour plots of te numerical solutions wit etrapolation to a narrow band of size near Γ: from top to bottom: constant, linear and quadratic etrapolation, respectivel. Left: setup as in Eample ; Rigt: setup as in Eample. Black dots indicate te interface Γ. Mes size: = π/ ( ) u(c) u(a) ( ) u(c) P l (C b u(c) u(b), wit P = A )/l b, ( C A )/l b, ( C B )/l a, ( C B )/l a l a ( ) (A.) were l a = C B, and l b = C A. If denoting P p p =, we ave p p ( g u(c) + g u(c) g u(c) + g 4 ), g = p /l b + p /l a, g = (u(a)p /l b + u(b)p /l a ), g = (p /l b + p /l a ), g 4 = (u(a)p /l b + u(b)p /l a ). (A.)

18 8 T. Aslam, S. Luo and H. Zao B C D A Known Region a) b) Fig Numerical test for one pass algoritm. (a) demonstration of upwind stencils for two special points net to te interface (te part inside te rectangle of (b)). Black dots indicate te interface. Arrows indicate upwind directions. (b) numerical test for one pass algoritm for constant etrapolation. 4 a) b) c) d) 8 Fig Contour plots of te numerical solutions: (a) initial u, (b) constant etrapolation, (c) linear etrapolation, and (d) quadratic etrapolation. Black dots indicate te interface Γ. Mes: 4 4. We substitute u(c) into te PDE (.) to ave te discretized PDE on CAB, a (C, C )(g u(c) + g ) + a (C, C )(g u(c) + g4 ) = f (C), (A.) wic provides a local updating formula for u(c), u(c) = f (C) a (C, C )g a (C, C )g4. a (C, C ) + g + a (C, C )g (A.4) Wit te local solver (A.4), we can simpl use te fast sweeping algoritm summarized in Section.4. B coosing a few reference points, triangular meses are ordered according to te distance to tese reference points as in [9]. We appl te fast

19 Static PDE Approac for Multi-dimensional Etrapolation 9 Caracteristic C C A A B B Fig. A.. Illustrations of local triangular mes: left: local mes; rigt: a simple were te caracteristic passes troug c falls into. Table A. Numerical accurac for etrapolation in a neigborood of size near te interface Γ. (Nodes, Elements) (6, 66) (45, 774) (589, 66) (7, 45946) constant etrapolation L error.9e-.7e- 5.E-.47E- Conv. Order # Iter 8 4 linear etrapolation L error.76e-.88e-.e-.55e- Conv. Order # Iter 8, 8,, 4, 4 sweeping metod for Eamples and on triangular meses. Te computation domain is now given as a disk centered at te origin wit radius π. And te domain is discretized wit triangular mes, e.g., see Figure A.. Fig. A.. A Triangular mes. Table A. sows te accurac for constant and linear etrapolation for Eample. Te order of convergence is epected as in Eample. For linear etrapolation, for simplicit, we compute u n directl wit u initiall given as in Eample. Figure A. (Eample ) and Figure A.4 (Eample ) sow plots of numerical solutions b constant and linear etrapolation. REFERENCES [] D. Adalsteinsson and J. Setian. Te fast construction of etension velocities in level set metods. J. Comput. Ps., 48():, 999.

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