A MULTI-ORDER DISCONTINUOUS GALERKIN MONTE CARLO METHOD FOR HYPERBOLIC PROBLEMS WITH STOCHASTIC PARAMETERS

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1 A MULTI-ORDER DISCONTINUOUS GALERKIN MONTE CARLO METHOD FOR HYPERBOLIC PROBLEMS WITH STOCHASTIC PARAMETERS MOHAMMAD MOTAMED AND DANIEL APPELÖ Abstract We present a new muti-order Monte Caro agorithm for computing the statistics of stochastic quantities of interest described by inear hyperboic probems with stochastic parameters The method is a non-intrusive technique based on a recenty proposed high-order energy-based discontinuous Gaerkin method for the second-order acoustic and eastic wave equations The agorithm is buit upon a hierarchy of degrees of poynomia basis functions rather than a mesh hierarchy used in muti-eve Monte Caro Through compexity theorems and numerica experiments, we show that the proposed muti-order method is a vaid aternative to the current muti-eve Monte Caro method for hyperboic probems Moreover, in addition to the convenience of working with a fixed mesh, which is desirabe in many rea appications with compex geometries, the muti-order method is particuary beneficia in reducing errors due to numerica dispersion in ong-distance propagation of waves The numerica exampes verify that the muti-order approach is faster than the mesh-based muti-eve approach for waves that traverse ong distances Key words Hyperboic probems; Wave propagation; Stochastic parameters; Uncertainty quantification; Muti-eve Monte Caro; Discontinuous Gaerkin; Muti-order Monte Caro AMS subject cassifications 1 Introduction Wave propagation probems are mathematicay described by hyperboic partia differentia equations (PDEs) In rea appications, such as seismoogy, acoustics, and eectromagnetism, the probem is subject to uncertainty, due to the ack of knowedge (epistemic uncertainty) and/or intrinsic variabiities of the physica system (aeatoric uncertainty) For instance, in earthquake ground motion, both kinds of uncertainties are present due to the scarcity of measured soi parameters and inherent variations in the ocation of the focus and the intensity of seismic sources To account for uncertainties, PDE modes are often formuated in a probabiistic framework, where uncertain input parameters are described by stochastic fieds, which can in turn be approximated by a finite number of random variabes A major probem is then the forward propagation of uncertainty, where uncertainties in the input parameters are propagated through the mode to obtain information about uncertain output quantities of interest (QoIs) The most popuar method for propagating stochastic uncertainty in PDE modes is Monte Caro samping [8], where sampe statistics of output QoIs are computed from independent reaizations drawn from the input probabiity distributions Whie being very fexibe and easy to impement, this technique features a very sow convergence rate More recenty, spectra approaches, such as stochastic Gaerkin [9] and stochastic coocation [17, 19], have been proposed, which expoit the possibe reguarity that output QoIs might have with respect to the input parameters This opens up the possibiity to use deterministic approximations of the response function (ie the soution of the probem as a function of the input parameters) based on goba poynomias Such approximations are expected to yied a very fast convergence in Department of Mathematics and Statistics, University of New Mexico, 1 University of New Mexico, Abuquerque, NM motamed@mathunmedu Department of Mathematics and Statistics, University of New Mexico, 1 University of New Mexico, Abuquerque, NM appeo@mathunmedu Funding: Supported in part by NSF Grant DMS Any concusions or recommendations expressed in this paper are those of the author and do not necessariy refect the views of NSF 1

2 the presence of high stochastic reguarity Soutions to parametric hyperboic PDEs are in genera non-smooth with respect to the parameters, and therefore reated stochastic QoIs are often not reguar; see [15, 16, 4] Consequenty, spectra methods may not be appicabe to stochastic hyperboic probems, and Monte Caro samping needs to be empoyed Severa variants of Monte Caro samping have recenty been proposed to acceerate the sow convergence of the Monte Caro method These recent methods incude muti-eve Monte Caro(MLMC) [10, 6, 5, 7], muti-index Monte Caro [11], quasi-monte Caro [12], and muti-eve quasi Monte Caro [13] In the particuar case of hyperboic probems, muti-eve Monte Caro approaches have been deveoped [14, 18], based on finite voume and finite difference techniques In the present work, we wi deveop a new variant of Monte Caro samping, referred to as muti-order Monte Caro (MOMC) Compared to muti-eve Monte Caro, the method has two new components: 1) it is based on a recenty proposed energybased discontinuous Gaerkin method for deterministic hyperboic probems [3, 2]; and 2) it is buit upon a hierarchy of orders of discontinuous Gaerkin basis functions rather than a mesh hierarchy used in muti-eve Monte Caro The new method is particuary advantageous for deaing with wave propagation and non-smooth QoIs, because: a) the energy-based discontinuous Gaerkin method is capabe of accuratey treating discontinuities in the PDE coefficients and the PDE data; b) the construction of an order hierarchy based on high-order schemes, such as discontinuous Gaerkin, aows us to significanty reduce wave dispersion and produce smaer errors; c) the method uses a fixed mesh at a eves which is beneficia when waves propagate in compicated media where re-meshing is a cumbersome task The third advantage is of practica importance for instance when the materia parameters come from a Bayesian seismic tomography at fixed resoution Through compexity theorems and numerica experiments, we wi demonstrate that the proposed muti-order method is a vaid aternative to the current muti-eve Monte Caro method for hyperboic probems with rough parameters Moreover, in addition to the convenience of working with a fixed mesh, which is desirabe in many rea appications with compex geometries, the muti-order method is particuary beneficia in reducing errors due to numerica dispersion in ong-time propagating waves The numerica exampes verify that the muti-order approach is faster than the mesh-based muti-eve approach for waves that traverse ong distances Note that the MOMC requires that p-refinement can be efficienty used, which is the case for the exampes considered here The outine of the paper is as foows In Section 2 we formuatethe mathematica probem and briefy address the numerica treatment of the probem with reation to stochastic reguarity The energy-based discontinuous Gaerkin sover is briefy reviewed in Section 3 In Section 4, we present an adaptation of the muti-eve Monte Caro agorithm to the eastic wave equations discretized by the discontinuous Gaerkin method The new muti-order Monte Caro method is presented in Section 5 In Section 6 we perform some numerica exampes that verify our theoretica resuts Finay, we present our concusions in Section 7 2 Probem Statement In this section, we first present the mathematica formuation of the stochastic probem We then address the numerica treatment of the probem with reation to stochastic reguarity 21 Mathematica formuation Let D R d be a compact d-dimensiona spatia domain, where d = 2,3 As the prototype mode for wave propagation subject to uncertainty, we consider the foowing initia boundary vaue probem (IBVP) for 2

3 92 the eastic wave equation with stochastic parameters: 93 (1) (x,y)u tt (t,x,y) σ(u(t,x,y)) = f(t,x,y), in [0,T] D Γ, u(0,x,y) = g 1 (x,y), u t (0,x,y) = g 2 (x,y), on {t = 0} D Γ, σ(u(t,x,y)) ˆn = 0, on [0,T] D Γ, where u = (u 1,,u d ) R d is the rea-vaued dispacement vector, t [0,T] is the time, x = (x 1,,x d ) R d is the vector of spatia variabes, y = (y 1,,y N ) Γ R N isarandomvector,representingtheuncertaintyinthe probem, and ˆndenotesthe outward unit norma to the boundary D We use the convention that represents the gradient operator with respect to the spatia variabes x The stress tensor σ for isotropic materias reads (2) σ(u) = λ(x,y) ui +µ(x,y)( u+( u) ) The materia parameters are the density and Lame s parameters λ and µ The sources of uncertainty are the materia parameters (,λ,µ), the force term f, and the initia data, g 1,g 2, characterizedbyn N + independent randomvariabesy 1,,y N with a bounded joint probabiity density ρ(y) = N n=1 ρ n(y n ) : Γ R + We take the force term and initia data as (3) f L 2 ((0,T);L 2 (D) L 2 ρ (Γ)), g 1 H 1 (D) L 2 ρ (Γ), g 2 L 2 (D) L 2 ρ (Γ), where L 2 ρ is the Hibert space of vector-vaued stochastic functions with bounded second moments, L 2 is the Hibert space of square integrabe vector-vaued functions, and H 1 is the Hibert space of vector-vaued functions whose first weak derivatives are square integrabe The notation denotes the tensor product space of Hibert spaces We further assume that the data are compatibe Moreover, we assume that the materia parameters are uniformy coercive and bounded: (4a) (4b) (4c) 0 < min (x,y) max <, x D, y Γ, 0 < λ min λ(x,y) λ max <, x D, y Γ, 0 < µ min µ(x,y) µ max <, x D, y Γ 117 We note that assumption (4) is a natura assumption for eastic materias We aso 118 note that in rea appications, the materia parameters and data are often not smooth 119 We have therefore made the minima reguarity assumptions (3)-(4) to account for 120 more genera wave propagation probems The assumptions (3)-(4) guarantee that the 121 probem (1) is we-posed: there exists a unique weak soution u C 0 ([0,T];H 1 (D) 122 L 2 ρ (Γ)) which depends continuousy on the data; see [15, 16] for more detais and 123 proofs 124 The utimate goa is the prediction of statistics of the wave soution u or some 125 physica quantities of interest (QoIs) reated to the soution, such as 126 (5) Q(y) = T 0 D Q D L(u) 2 (t,x,y)dxdt, where L(u) may be a differentia operator appied on u, and D Q D is a part of the computationa domain For instance, the cases where L(u) is u,u t, and u tt correspond to wave strength, kinetic energy, and Arias intensity, respectivey 3

4 22 Non-intrusive numerica methods and stochastic reguarity The goa of computations is to numericay approximate the statistica moments of the quantity (5) For instance, consider the first moment of the QoI and et A be its approximation: E[Q] = Q(y)ρ(y)dy A Γ Non-intrusive methods, such as Monte Caro [8] and sparse stochastic coocation [19, 17, 15], are popuar sampe-based approaches that rey on soving a set of deterministic probems corresponding to a set of reaizations In a non-intrusive method, the approximation A invoves two separate approximations: 1) the approximation of Q, denoted by Q; and 2) the approximation of the integra The former needs a deterministic sover that computes Q at a set of M quadrature points, and the atter requires a quadrature rue, such as sampe averages (in Monte Caro) or Gauss or Censhaw-Curtis quadrature (in stochastic coocation) Correspondingy, the tota error in the approximation can be spit into two parts: ε := E[Q] A E[Q] E[ }{{ Q] + E[ } Q] A }{{} ε I ε II The first error term ε I corresponds to the discretization error in the deterministic sover, and the second error term ε II is the quadrature error We note that in Monte Caro samping, ε II is a statistica error, as A is a statistica term In genera, the choice of the numerica method strongy depends on the reguarity of the mapping Q : Γ R, which in turn depends on the stochastic reguarity of the wave soution u, ie the reguarity of u with respect to y In the presence of high stochastic reguarity, sparse stochastic coocation exhibits fast convergence in the number of quadrature or coocation points and is preferabe However, if the QoI is not smooth in stochastic space, Monte Caro samping techniques must be empoyed It is known that the soutions of hyperboic probems, such as the IBVP (1) with the minima assumptions (3)-(4), are not smooth in the stochastic space; see [15, 16, 4] Consequenty, the QoI (5) does not have stochastic reguarity We therefore need to empoy MC-based samping techniques The most popuar one is the cassica Monte Caro method Whie being very fexibe and easy to impement, this technique features a very sow convergence rate More recenty, severa variants of Monte Caro are proposed to acceerate the sow convergence of the Monte Caro method, incuding muti-eve Monte Caro [10, 6, 5, 7], muti-index Monte Caro [11], quasi-monte Caro [12], and muti-eve quasi Monte Caro [13] In the particuar case of hyperboic probems, muti-eve Monte Caro approaches have been deveoped [14, 18], based on finite voume and finite difference techniques Inthe presentwork,wewideveopanewvariantofmontecarosamping, which we ca the muti-order Monte Caro method The method is based on a recenty proposed energy-based discontinuous Gaerkin method for deterministic hyperboic probems [3] and is buit upon a hierarchy of orders of basis functions rather than a mesh hierarchy used in muti-eve Monte Caro In what foows, we wi briefy review the deterministic sover in Section 3 We then present a muti-eve and the new muti-order agorithms based on the energy-based discontinuous Gaerkin method in Sections 4 and 5, respectivey 3 Deterministic sovers: energy based discontinuous Gaerkin methods In this section we briefy review the deterministic sover is the basis for our 4

5 muti-eve and muti-order Monte Caro methods As we aim for arbitrary order of accuracyinspaceasweasin time wecombineanewcassofspatiadg discretization with Tayor series time-stepping 31 Spatia discretization by an energy based discontinuous Gaerkin method Our spatia discretization is an direct appication of the formuation described for genera second order wave equations in [3] and for the eastic wave equation in [2] Here we outine the spatia discretization for the specia case of the scaar wave equation in one dimension and refer the reader to [3] for the genera case We note that an open source impementation of the method used for the exampe with the eastic wave equation in Section 62 is avaiabe, [1] The energy of the scaar wave equation is v 2 H(t) = 2 +G(x,u x)dx, D where G(x,u x ) = c 2 (x)u 2 x/2 is the potentia energy density, v is the veocity or the time derivative of the dispacement, v = u t The wave equation, written as a second order equation in space and first order in time then takes the form u t = v, v t = δg, where δg is the variationa derivative of the potentia energy δg = (G ux ) x = (c 2 (x)u x ) x For the continuous probem the change in energy is dh(t) (6) = vv t +u t (c 2 (x)u x ) x dx = [u t (c 2 (x)u x )] D, dt D where the ast equaity foows from integration by parts together with the wave equation Now, a variationa formuation that mimics the above energy identity can be obtained if the equation v u t = 0 is tested with the variationa derivative of the potentia energy Let Ω j be an eement and Π s (Ω j ) be the space of poynomias of degree s, then the variationa formuation on that eement is: Probem 1 Find v h Π s (Ω j ), u h Π r (Ω j ) such that for a ψ Π s (Ω j ), φ Π r (Ω j ) ( ) u c 2 h (7) φ x x Ω j t vh x dx = [c 2 φ x n ( v v h) ] Ωj, (8) Ωj ψ vh t +c2 ψ x u h x dx = [ψ(c2 u x ) ] Ωj Let [[f]] and {f} denote the jump and average of a quantity f at the interface between two eements, then, choosing the numerica fuxes as v = {v} τ 1 [[c 2 u x ]] (c 2 u x ) = {c 2 u x } τ 2 [[v]], wi yieds a contribution τ 1 ([[c 2 u x ]]) 2 τ 2 ([[v]]) 2 from each eement face to the change of the discrete energy dh h (t) dt = d (v dt j Ωj h ) 2 +G(x,u h 2 x) 5

6 Physica boundary conditions can aso be handed be appropriate specification of the numerica fuxes, see [3] for detais The above variationa formuation and choice of numerica fuxes resuts in an energy identity simiar to (6) However, as the energy is invariant to certain transformations the variationa probem does not fuy determine the time derivatives of u h on each eement and independent equations must be introduced In this case there is one invariant and an independent equation is ( ) u h Ω j t v h = 0 For the genera case and for the eastic wave equation see [3] and [2] Here we aways choose τ i > 0 (so caed upwind or Sommerfed fuxes) which typicay resut in methods that are q = r+1 order accurate in space 32 Tayor series time-stepping In order to match the order of accuracy in space and time we empoy Tayor series time-stepping Assuming that a the degrees offreedomhavebeen assembedintoavectorw wecanwrite the semi-discretemethod as w t = Aw with A being a matrix representing the spatia discretization Assuming we know the discrete soution at the time t n we can advance it to the next time step t n+1 = t n + t by the simpe formua w(t n + t) = w(t n )+ tw t (t n )+ ( t)2 w tt (t n ) 2! = w(t n )+ taw(t n )+ ( t)2 A 2 w(t n ) 2! 214 The stabiity domain of the Tayor series which truncates at time derivative number 215 N T incudes the imaginary axis if mod(n T,4) = 3 or mod(n T,4) = 0 However as we 216 use a sighty dissipative spatia discretization the spectrum of our discrete operator 217 wi be contained in the stabiity domain of a sufficienty arge choices of N T (ie the 218 N T shoud not be smaer than the spatia orderof approximation) Note aso that the 219 stabiity domain grows ineary with the number of terms We thus consider methods 220 based on the combination of the spatia dg discretization of order q combined with a 221 Tayor series with N T = 2 q 2, where is the ceiing operator This yieds methods 222 of order of accuracy min(q,n T ) We note that we wi use this choice of N T for both 223 muti-eve and muti-order based MC methods in Sections 4 and 5 We aso note 224 that beow we excusivey use the mesh size h as a discretization parameter, but that 225 it is directy proportiona to the tempora discretization size t, through the CFL 226 condition A muti-eve discontinuous Gaerkin Monte Caro method In this section, we present an adaptation of the muti-eve Monte Caro agorithm to the eastic wave equations discretized by the dg method We note that athough MLMC agorithms for hyperboic PDEs based on finite difference and finite voume methods have areadybeen introduced [14, 18], the anaysisof MLMC based on the dg method is different and resuts in new theoretica resuts It aso serves as a basis for deveoping the new MOMC method We foow [10] and buid a mesh hierarchy with a decreasing sequence of mesh sizes h 0 > h 1 > > h L For instance we take (9) h = h 0 β, = 0,1,,L, β 2 We denote by Q, the discretization of Q by the dg method on the mesh at the -th 6

7 eve with mesh size h We then use a teescoping sum formuation and write E[Q] = E[Q Q L ]+E[Q L ], E[Q L ] = E[Q 0 ]+ E[Q Q 1 ] = The MLMC estimator approximates the terms in the teescoping sum by sampe averages (10) A MLMC = 1 M 0 M 0 m 0=1 Q (m0) M (Q (m ) Q (m ) 1 M ) =1 m =1 Here, Q (m ) := Q (y (m) ), with m = 1,,M, are M reaizations of Q correspondingto M independent sampes{y (m) } M m =1 ofthe randomvectory The totamlmc error reads ε MLMC = E[Q] A MLMC E[Q Q L ] + E[Q L ] A MLMC }{{}}{{} ε I ε II The first error term ε I is the discretization error in the dg sover, or the weak error, which satisfies (11) ε I c 1 h q1 L, y Γ, where q 1 is reated to the order q of the dg method, and c 1 is a positive constant which may depend on Q Moreover, by the centra imit theorem, the statistica error ε II satisfies (12) ε II c α V[AMLMC ] = c V[Q 0] V[Q Q 1 ] α + =: c L V α, M 0 M M where V 0 = V[Q 0 ] and V = V[Q Q 1 ] for 1 Here, the notation is interpreted in the foowing statistica sense: ( (13) P ε II c L V ) α 2φ(c α ) 1, as M, M =0 where P is a probabiity measure, and φ(c α ) = c α 1 2π exp( τ 2 /2)dτ is the cumuative density function (CDF) of a standard norma random variabe The arger the L confidence parameter c α > 0, the higher the probabiity that ε II c α =0 V /M hods We further note that we have the strong error (14) V[Q Q ] E[(Q Q ) 2 ] c 2 h q2, y Γ =1 =0 41 Numerica agorithm An error-compexity anaysis is needed to optimay seect the computationa parameters, incuding the number of sampes at different eves {M } L =0, and the fina eve L We introduce a spitting parameter θ and write ε MLMC ε I +ε II (1 θ)ε TOL +θε TOL, θ (0,1), 7

8 where the errors ε I and ε II are given by (11) and (12), respectivey Moreover, noting that the cost of computing Q (m ) Q (m ) 1 in the MLMC estimator (10) is dominated by the cost of computing Q (m ), which is W h γ1, the tota computationa cost of MLMC reads W MLMC M W, W h γ1 =0 Here, γ 1 = d is the space-time dimension and determines the number of degrees of freedom in the deterministic sover We then take the foowing iterative strategy, consisting of two main steps: 1 Optima number of sampes We obtain the optima number of sampes at different eves by minimizing the tota computationa cost W MLMC subject to the accuracy constraint ε II θε TOL Foowing the standard approach in MLMC [10], we use the method of Lagrange mutipiers With the Lagrangian L(M,ν) := M W +ν ( L V ( θε TOL ) 2 ), M c α =0 = and the optimaity equations M L = ν L = 0, we obtain (15) M = ( θε TOL ) 2 V c α W =0 V W, W h γ1 2 Stopping criterion We start with L = 2 and iterativey add eves unti ε I (1 θ)ε TOL is achieved To find a practica stopping criterion, we start by writing (16) E[Q Q L ] = =L+1 E[Q Q 1 ] = E[Q L Q L 1 ] =L+1 E[Q Q 1 ] E[Q L Q L 1 ] Assuming E[Q Q 1 ] ch q1, we have Hence E[Q Q 1 ] E[Q L Q L 1 ] hq1 h q1 ε I = E[Q Q L ] E[Q L Q L 1 ] L = hq1 0 β q1 h = q1 β(l )q1 0 β Lq1 1 β kq1 = β q1 1 E[Q L Q L 1 ], where the ast equaity foows from the geometrica series 263 k=0 (β q1 ) k 1 = 1 β, q since β q1 < 1 Consequenty, the condition we use to add eves in the numerica 265 agorithm is k= j q1 β (17) max E[QL j Q L j 1 ] (1 θ)εtol j {0,1,2} β q1 1 This criterion wi ensure that the deterministic error approximated by an extrapoation from either of the three finest meshes is within the desired range 8

9 Now, assuming that we have made the assignment L L+1, we note that in order to compute the number of sampes by (15), we need to compute the variances {V } L =0 Setting Q(m) 1 = 0, the variances are approximated by (18) V 1 M M m=1 ( ( Q (m) ) 2 Q 1) (m) Ḡ 2, Ḡ 1 M M m=1 ( Q (m) Q (m) 1), 273 where {M } L =0 are the avaiabe number of sampes in previous iterations When a 274 newevelisadded, thevarianceatthe newevev L cannotbe approximatedby (18), 275 since the number of sampes at the new eve is not known In this case, thanks to the 276 strong error estimate (14), assuming V = V[Q Q 1 ] ch q2, we first approximate 277 V L in terms of the variance at the previous eve V L 1 by 278 (19) V L β q2 V L 1, 279 and then update the number of sampes {M } L i=0 incuding the number of sampes at 280 the new eve by (15) The expected vaues E[Q L j Q L j 1 ] in (17), with j = 0,1,2, 281 are aso approximated by ḠL j in (18) The MLMC agorithm is outined in Agorithm 1 Agorithm 1 MLMC agorithm Start with L = 2, and generate a mesh hierarchy {h } L =0 by (9) Choose an initia set {M } L =0 of sampes oop Approximate {V } L =0 by (18) Update the optima number of sampes {M } L =0 by (15) if (17) is satisfied ese Compute A MLMC by (10) and terminate the oop Set L := L+1 and h L = h 0 β L Approximate V L by (19) and compute {M } L =0 by (15) end if end oop If it is possibe to estabish bounds on the strong and weak error and the work at each eve, the compexity of Agorithm 1 is guaranteed by the foowing theorem Theorem 1 Consider a mesh hierarchy h = h 0 β, with β 2, and et Q be the q-th order accurate dg approximation of Q on a mesh with mesh size h If there are positive constants c 1,c 2,c 3,γ 1 > 0 such that (A1) E[Q Q ] c 1 h q1, q 1 γ 1 /2, (A2) V[Q Q ] E[ Q Q 2 ] c 2 h q2, q 2 > γ 1, (A3) W c 3 h γ1, 9

10 289 then, for any ε TOL < e 1, there exists an L N and a sequence {M } L =0 such that the 290 estimator A MLMC has an error ε MLMC ε TOL with a computationa cost proportiona to 291 ε 2 TOL Proof By writing V = V[Q Q 1 ] = V[Q Q 1 ] V[Q Q ] and using the trianguar inequaity and (A2), we get (20) V ch q2, c = c 2 (1+β q2 +2β q2/2 ) Moreover, by minimizing V[A MLMC ], or equivaenty minimizing ε II, for a fixed computationa cost W MLMC, we obtain from (A3) and (20): V (21) M h (γ1+q2)/2 W We foow [10] and seect L to be og(2c1 h q1 L = 0 ε TOL 1 ) q 1 ogβ We therefore have (22) 1 2 ε TOLβ q1 < c 1 h q1 L 1 2 ε TOL By the right inequaity in (22) and (A1), the deterministic error is bounded by (23) ε I 1 2 ε TOL Moreover, since h = β L h L, we have =0 h γ1 = h γ1 L =0 β γ1 h γ1 L =0 β γ1 = h γ1 L β γ1 β γ1 1 Now, by the eft inequaity in (22) we have h γ1 L =0 h γ1 < (2c 1 /ε TOL ) γ1/q1 β γ1, thus β2γ1 β γ1 1 (2c 1) γ1/q1 ε TOL γ 1/q 1, 302 and since ε TOL < e 1 and γ 1 /q 1 2, then ε TOL γ 1/q 1 ε TOL 2, and hence 303 (24) =0 h γ1 β2γ1 β γ1 1 (2c 1) γ1/q1 ε TOL Now, motivated by (21), we set (25) M = 4εTOL 2 cc 2 αh (q2 γ1)/2 0 1 β (q2 γ1)/2 h (γ1+q2)/2, which gives cc 2 α 1 M 4 ε 2 TOL h (q2 γ1)/2 0 (1 β (q2 γ1)/2 )h (γ1+q2)/2 10

11 Then, by (20) we have c 2 α =0 V M =0 cc 2 α h q2 1 M 4 ε 2 TOL h (q2 γ1)/2 0 (1 β (q2 γ1)/2 ) =0 h (q2 γ1)/2 306 Moreover, we have 307 (26) =0 h (q2 γ1)/2 = h (q2 γ1)/2 0 ( β (q ) 2 γ 1)/2 h (q2 γ1)/2 < =0 0 1 β (q2 γ1)/ Hence, the statistica error is bounded by (27) ε II c α L V 1 M 2 ε TOL =0 By (23) and (27), the tota error reads ε MLMC ε TOL It is eft to show that the computationa cost is proportiona to ε TOL 2 By (25), we have M < 4ε 2 TOL cc 2 α h(q2 γ1)/2 0 h (γ1+q2)/2 1 β (q2 γ1)/ Hence, by (A3), the computationa cost reads W MLMC = L M W c 3 M h γ1 =0 =0 L ( 2 4εTOL cc 2 α < c h(q2 γ1)/2 3 =0 0 1 β (q2 γ1)/2 h (q2 γ1)/2 +h γ1 ) Eventuay, by (26) and (24), we obtain W MLMC < c W ε TOL 2, c W = 4c 3 cc 2 αh q2 γ1 0 (1 β (q2 γ1)/2 ) 2 +c 3 (2c 1 ) γ1/q1 β 2γ1 β γ This competes the proof Note that for the sake of brevity we considered ony the specia case θ = 1/2 in Theorem 1 but that it can be extended to the case of a genera θ by tracking the spitting of the error in the proof 5 A muti-order discontinuous Gaerkin Monte Caro method In this section, we describe the new MOMC agorithm and present error and convergence anaysis The new agorithm wi first construct a fixed mesh, with a mesh size h, and buid an order hierarchy with an increasing sequence of degrees of poynomia basis functions, or equivaenty a sequence of dg orders q 0 < q 1 <,< q L For instance we take (28) q = q 0 +β, = 0,1,,L, q 0 1, β 1 11

12 Denote by Q the approximation of Q by a q -th order dg on the fixed mesh Using the same teescoping sum formuation as in MLMC and approximating the terms by sampe averages, we write the MOMC estimator as (29) A MOMC = 1 M 0 M 0 m 0=1 Q (m0) M (Q (m ) Q (m ) 1 M ) =1 m =1 Here, Q (m ) := Q (y (m) ), is one reaization of Q corresponding to a sampe y (m) of the random vector y The tota MOMC error reads ε MOMC = E[Q] A MOMC E[Q Q L ] + E[Q L ] A MOMC }{{}}{{} ε I ε II The first error term ε I is the discretization error in the dg sover, (30) ε I c 1 h q1l, q 1L := κ 1 q L, y Γ, where κ 1 > 0 is reated to the convergence of the dg sover in approximating E[Q] The statistica error ε II, interpreted in the statistica sense simiar to (13), satisfies (31) ε II c α V[AMOMC ] = c α V[Q 0] + M 0 The strong error aso satisfies =1 V[Q Q 1 ] M =: c α L (32) V[Q Q ] E[(Q Q ) 2 ] c 2 h q 2, q 2 = κ 2 q, y Γ, =0 V M where κ 2 > 0 is reated to the convergence of the dg sover in approximating V[Q] 51 Numerica agorithm Simiar to MLMC, an error-compexity anaysis is needed to optimay seect the computationa parameters We spit the tota error by introducing a spitting parameter θ and write ε MOMC ε I +ε II (1 θ)ε TOL +θε TOL, θ (0,1), where the errors ε I and ε II are given by (30) and (31), respectivey Moreover, noting that the cost of computing Q (m ) Q (m ) 1 in the MOMC estimator (29) is dominated by the cost of computing Q (m ), which is W q γ2, the tota computationa cost of MOMC reads W MOMC M W, W q γ2 =0 Here, γ 2 = d determines the cost of the deterministic dg sover We then take the foowing iterative strategy, consisting of two main steps: 1 Optima number of sampes We obtain the optima number of sampes in a simiar way as MLMC: (33) M = ( θε TOL ) 2 V c α W =0 12 V W, W q γ2

13 2 Stopping criterion We start with L = 2 and iterativey add eves unti ε I (1 θ)ε TOL is achieved To find a practica stopping criterion, we start with (16) and assume that E[Q Q 1 ] ch κ1 ql We then have E[Q Q 1 ] E[Q L Q L 1 ] hκ1 q h κ1 ql = hκ1 (q q L) = h κ1 β( L) Hence ε I = E[Q Q L ] E[Q L Q L 1 ] k=1 h kκ1 β = hκ1 β 1 h κ1 β E[Q L Q L 1 ], where the ast equaity foows from the geometrica series k=0 ) k 1 =, (β q1 1 h κ 1 β since h κ1 β < 1 Consequenty, the condition we use to add eves in the numerica agorithm is h (j+1)κ1 β (34) max E[QL j j {0,1,2} 1 h κ1 β Q L j 1 ] (1 θ)εtol 346 This wi ensure that the deterministic error approximated by an extrapoation form 347 either of the three finest meshes is within the desired range 348 Simiar to the MLMC strategy, we approximate the variances {V } L =0 in (33) 349 by (18) When a new eve L is added, the variance at the new eve V L cannot be 350 approximated by (18), since the number of sampe at the new eve is not known In 351 this case, thanks to the strong error estimate (32), assuming V = V[Q Q 1 ] 352 ch κ2 q, we first approximate V L in terms of the variance at the previous eve V L by 354 (35) V L h κ2 β V L and then update the number of sampes {M } L i=0 incuding the number of sampes at 356 the new eve by (33) The expected vaues E[Q L j Q L j 1 ] in (34), with j = 1,2,3, 357 are aso approximated by ḠL j in (18) 358 The MOMC agorithm is outined in Agorithm Lemma 2 For every positive rea number r < 1 and every positive integer p 1, we have p (36) r (q0+β) (q 0 +β) p = c k r k f (k) (r), f(r) = rq0 1 r β =0 k=1 Hence Proof We start with the foowing geometrica series sum, thanks to r β < 1: =0 =0 (r β ) = 1 1 r β r q0+β = rq0 =: f(r) 1 rβ We differentiate the above formua with respect to r to obtain (q 0 +β)r q0+β 1 = d dr f(r) =0 13

14 Agorithm 2 MOMC agorithm Start with L = 2, and generate an order hierarchy {q } L =0 by (28) Choose an initia set {M } L =0 of sampes oop Approximate {V } L =0 by (18) Update the optima number of sampes {M } L =0 by (33) if (34) is satisfied Compute A MOMC by (29) and terminate the oop ese Set L := L+1 and q L = q 0 +βl Approximate V L by (35) and compute {M } L =0 by (33) end if end oop We then mutipy it by r to obtain (q 0 +β)r q0+β = r d dr f(r) = We obtain (36) by repeating the above process, ie differentiate with respect to r and then mutipy by r, p times Theorem 3 Consider an order hierarchy q = q 0 +β, with = 0,1,,L, where q 0 1 and β 1 are cosntants Let Q be the semi-discretization of Q by the q -th order dg method on a mesh with a fixed mesh size h < 1 If there are constants c 1,c 2,c 3,γ 1,γ 2 > 0 such that (A4) E[Q Q ] c 1 h q 1, q 1 = κ 1 q, κ 1 q 0 γ 1 /2, (A5) V[Q Q ] E[ Q Q 2 ] c 2 h q 2, q 2 = κ 2 q, κ 2 q 0 > γ 1, (A6) W c 3 h γ1 q γ2, 369 then, for any ε TOL < e 1, there exists an L N and a sequence {M } L =0 such that the 370 estimator A MOMC has an error ε MOMC ε TOL with a computationa cost proportiona to 371 ε 2 TOL Proof By writing V = V[Q Q 1 ] = V[Q Q 1 ] V[Q Q ] and using the trianguar inequaity and (A5), we get (37) V ch q 2, c = c 2 (1+h κ2β +2h κ2β/2 ) Moreover, by minimizing V[A MOMC ], or equivaenty minimizing ε II, for a fixed computationa cost W MOMC, we obtain from (A6) and (37): (38) M V /W h (γ1+q 2)/2 q γ2/2 14

15 We seect L to be Hence og(2c 1 h κ1 q0 ε TOL 1 ) og(h κ1 β ) og(2c1 h κ1 q0 ε 1 TOL ) L = og(h κ1 β ) L < og(2c 1h κ1 q0 ε TOL 1 ) og(h κ1 β ) +1 By the rues of ogarithms and simpe agebraic manipuations, we get (39) 1 2 ε TOLh κ1 β < c 1 h κ1 ql 1 2 ε TOL By the right inequaity in (39) and (A4), the deterministic error is bounded by (40) ε I 1 2 ε TOL By the eft inequaity in (39), we have h κ1 ql < 2c 1 h κ1 β ε TOL 1, and hence by taking ogarithm and rearranging we obtain q L < og(2c 1h κ1 β ) og(h κ1 ) + 1 og(h κ1 ) og(ε 1 TOL ) =: ĉ 1 +ĉ 2 og(ε 1 TOL ), where the constants ĉ 1 and ĉ 2 are independent of ε TOL and L The right hand side of the above inequaity is a ogarithmic growth and can be bounded by an agebraic growth In particuar, there exists a constant ĉ such that ĉ 1 +ĉ 2 og(ε TOL 1 ) < ĉε TOL 2/(γ 2+1), and hence We therefore have (41) =0 q L < ĉε TOL 2/(γ 2+1) q γ2 < (L+1)q γ2 L < qγ2+1 L < ĉ γ2+1 ε TOL Now, motivated by (38), we set (42) M = 4ε 2 TOL cc 2 αsh (γ1+κ2 q)/2 q γ2/2, where γ 2/2 S = h γ1/2 k=1 c k h kκ2/2 f (k) (r), f(r) = rq0 1 r β, r = hκ2/2 < 1 From (42) we get cc 2 α 1 M 4 ε 2 TOL S 1 h (γ1+κ2 q)/2 q γ2/2 15

16 Then, by (37) we have c 2 α =0 V M =0 cc 2 α M h κ2 q 1 4 ε TOL 2 S 1 L =0 h (κ2 q γ 1)/2 q γ2/2 386 Moreover, by Lemma 2 with r := h κ2/2 < 1 and p := γ 2 /2 > 1, we have 387 (43) =0 h (κ2 q γ 1)/2 q γ2/2 < h γ1/2 h κ2 (q0+β)/2 (q 0 +β) γ2/2 < S = Hence, the statistica error is bounded by (44) ε II c L V α 1 M 2 ε TOL =0 By (40) and (44), the tota error reads ε MOMC ε TOL It is eft to show that the computationa cost is proportiona to ε TOL 2 By (42), we have M < 4ε TOL 2 cc 2 α Sh(γ1+κ2 q )/2 q γ2/ Hence, by (A6), the computationa cost reads W MOMC = By (43) we obtain L M W c 3 M h γ1 q γ2 =0 < 4c 3 ε TOL 2 cc 2 αs =0 =0 h (κ2 q γ 1)/2 q γ2/2 +c 3 h γ1 =0 q γ2 W MOMC < c W ε TOL 2, c W = 4c 3 cc 2 αs 2 +c 3 ĉ γ2+1 h γ This competes the proof Remark 1 Theorems 1 and 3 show that, under the same accuracy constraint (ε MLMC ε TOL and ε MOMC ε TOL ), the tota computationa cost of both MLMC and MOMC is proportiona to ε 2 TOL This verifies that MOMC is a vaid aternative to MLMC for hyperboic probems In addition, we wi present numerica evidence (see Figures 1 and 4) that the muti-order approach is faster than the mesh-based mutieve approach for waves that traverse ong distances This superiority of MOMC over MLMC is due to the advantage of high-order schemes in controing dispersive errors in wave propagation probems In principe, this caim can be made rigorous by carefuy tracking the the effects of dispersive errors on the constant c W, which appears in the computationa cost Such anaysis is the topic of future work 6 Numerica exampes In this section we present numerica resuts from probems in (1+1) and (2+1) dimensions demonstrating the performance of the two methods described above The first exampe considers the scaar wave equation, and the second exampe considers the eastic wave equation For the detais of the dg deterministic sover, we refer to [2, 3] 16

17 Exampe 1 We first consider the scaar wave equation in(1+1) dimensions on the domain D = [0,10] and with a potentia energy density G(x) = (c(x)u x ) 2 /2 We take the initia data to be u(x,0) = e (x 5)2, v(x,0) = 0, and impose homogenous Dirichet conditions on both boundaries Here the square of the wave speed is assumed to be uncertain in the x direction and modeed by 10 independent and uniformy distributed random variabes y i = U[0,1] Precisey the piecewise constant wave speed is c 2 (x,y) = 1+ y i, x [i 1,i], i = 1,, We perform the simuations by starting both methods on a uniform grid conforming with the wave speed In MLMC we choose the base eement size is h 0 = 1 with the fixed order q = 4, and in MOMC we choose the fixed mesh size h = 1 and start with the order q 0 = 4, corresponding to a fourth order space-time accurate method in the dispacement u The quantity of interest is ( Q(y) = u(t,x,y) 2 dx D ) 1 2, where T is the fina time and the integra in x is approximated by sufficienty accurate Gauss-Legendre-Lobatto quadrature The parameters in MOMC are (γ 2,κ 1,κ 2 ) = (2,1,2) and the parameters in MLMC are (γ 1,q 1,q 2 ) = (2,4,8) For MLMC we set β = 2 and for MOMC we present resuts for β = 1 and 2 For both methods we set θ = 1/2 and c α = 196 CPU ε 2 TOL 10-3 CPU ε 2 TOL ε -4 TOL ε -1 TOL Fig 1 Product of CPU time and the square of the toerance The red curves with trianges represents MLMC and the back curves with squares represents MOMC with β = 2 and the dashed ine MOMC with β = 1 The eft figure is for T = 20 and the right is for T = 200 For this probem MOMC uses abut 18 and 32 times ess CPU-time to reach a given toerance To iustrate the advantage of using MOMC we consider two fina simuation times First we set T = 20 and perform simuations with toerances ε TOL = 2 8 s, s = 0,, 10 These simuations are performed three times with different random seeds In a second set of simuations we increase the fina time to T = 200 and perform simuations with toerances ε TOL = 2 7 s, s = 0,,5 For the second set of simuations we ony present resuts for β = 2 for MOMC 17

18 To confirm the compexity resuts of Theorem 1 and 3 we pot the product of the square of the toerance and tota CPU time as a function of toerance The resuts for both fina times can be found in Figure 1 As can be seen to the eft in Figure 1, for the short time T = 20, MOMC is about 18 times faster than MLMC for both choices of β in MOMC For the onger simuation time we find that MOMC is about 32 times faster than MLMC For the short time simuation we aso report the distributions of the number of sampes per eve for the three different methods and for the different toerances As can be seen in Figure 2, at higher eves, where the deterministic sover per sampe is computationay costy, the number of sampes (or the number of the deterministic soves) are much smaer than the number of sampes at ower eves, where the deterministic sover per sampe is computationay cheap This intuitivey show why MLMC and MOMC work 10 8 Sampes M Leve Leve Leve Fig 2 Number of sampes per eve for different toerances ε TOL = 2 8 s, s = 0,,10 In each figure, the strictest toerance corresponds to the curve with the highest number of sampes The number of sampes are monotonicay decreasing as the toerance is reaxed On top MLMC, bottom eft MOMC with β = 1, and bottom right MOMC with β = Exampe 2 In this exampe we consider the eastic wave equation with 450 traction free boundary conditions on the domain D = (x 1,x 2 ) [ 1,1] [ 2,2] In 451 MLMC, the domain D is discretized with square eements with sides h = h 0 β, with 452 β = 2 and h 0 = 1 2, and we use dg with the fixed order q = 4 In MOMC we choose 453 the fixed mesh size h = 1/2 and start with the order q 0 = 4 The materia properties 454 are taken to be piecewise constant but different above and beow x 2 = 0, precisey 18

19 Error Error ε -4 TOL ε -1 TOL Fig 3 Verification that the toerance is met Potted is the error as a function of the toerance for MLMC (soid red with trianges), MOMC and β = 1 (dashed back with pentagrams) and MOMC and β = 2 (soid back with squares) The eft figure is for T = 20 and the right is for T = they are (45) (ρ,λ,µ) = { (1,4,1+y1 ) x 2 > 0, (1,4,1+y 2 ) x 2 < 0, where y 1 and y 2 are uniform random variabes on [0,1] Note that the grid coincides with the materia interface at x 2 = 0, so that the order of the dg method is not affected by the jump discontinuity in µ The initia data is chosen to be u 1 = e 6((x1 015)2 +(x 2 01) 2), u 2 = e 6((x1 012)2 +(x 2 014) 2), v 1 = v 2 = 0 Here u 1 and u 2 are the dispacements and v 1 and v 2 are the veocities The quantity of interest is ( Q(y) = u 1 (T,x,y) 2 +u 2 (T,x,y) 2 dx D ) 1 2, where T is the fina time of the simuation We present resuts for three different fina times T = 1,10 and 100 In Figure 4 we pot the product of the square of the toerance and tota CPU time as a function of toerance for the three different fina times The resuts are for MOMC with (γ 2,κ 1,κ 2 ) = (4,1,2) and β = 1 and for MLMC with (γ 1,q 1,q 2 ) = (3,4,8) and β = 2 For both methods we set θ = 1/2 and c α = CPU ε 2 TOL 10-3 CPU ε 2 TOL 10 0 CPU ε 2 TOL 10-4 ε TOL ε TOL ε TOL Fig 4 From eft to right T = 1,10,100 MLMC (red), MOMC with β = 1 (back) As in the previous exampe, as the fina time becomes onger the advantage of the MOMC method becomes more pronounced In this case the computationa time 19

20 is about the same when T = 1, about 19 times faster when T = 10 and 24 times faster when T = 100 Again we find that the toerance is met for both the methods and for a the fina times, see Figure Error Error Error ε TOL ε TOL ε TOL Fig 5 Verification that the toerance is met From eft to right T = 1,10,100 MLMC (red), MOMC with β = 1 (back) Concusion We presented the muti-order Monte Caro method for forward propagation of uncertainties in hyperboic probems Here the method used an arbitrary order energy-based discontinuous Gaerkin discretization to buid order hierarchies but we note that the MOMC method coud of course aso use any suitabe discretization capabe of discretization at arbitrary order In fact, the compexity theorems we have presented do not rey on the dg method but are genera in that they accept any discretization We found that the optima compexity of the origina MLMC method aso carries over to our muti-order Monte Caro method and that the new method is faster when the probem at hand requires propagation of waves over ong distances Another feature of the MOMC is that a singe mesh can be generated, something that may be of practica importance both for increased set-up time as we as for the abiity to oad baance parae computations once and for a in the beginning of a simuation Our method iustrates the power of the muti-eve Monte Caro framework and how easiy it can be adopted and extended We are currenty exporing extensions of the MOMC method to other hyperboic probems and more reaistic appications and we are aso working on extending it to the hp-refinement regime REFERENCES [1] D Appeö and T Hagstrom, dg-dath: Discontinuous Gaerkin Sover by Danie Appeo & Thomas Hagstrom for the Eastic Wave Equation, 2015, dath eastic v10 [2] D Appeö and T Hagstrom, An energy-based discontinuous Gaerkin discretization of the eastic wave equation in second order form, Computer Methods in Appied Mechanics and Engineering, xxx (2016), p xxx [3] D Appeö and T Hagstrom, A new discontinuous Gaerkin formuation for wave equations in second order form, Siam Journa On Numerica Anaysis, 53 (2016), pp [4] I Babuška, M Motamed, and R Tempone, A stochastic mutiscae method for the eastodynamic wave equations arising from fiber composites, Computer Methods in Appied Mechanics and Engineering, 276 (2014), pp [5] A Barth, C Schwab, and N Zoinger, Mutieve Monte Caro finite eement method for eiptic PDEs with stochastic coefficients, Numer Math, 199 (2011), pp [6] K A Ciffe, M B Gies, R Scheich, and A L Teckentrup, Mutieve Monte Caro methods and appications to eiptic PDEs with random coefficients, Comput Visua Sci, 14 (2011), pp 3 15 [7] N Coier, A-L Haji-Ai, F Nobie, E von Schwerin, and R Tempone, A continuation mutieve Monte Caro agorithm, BIT Numer Math, (2014) (in press) 20

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