Numerische Mathematik

Transcription

1 Numer. Math. (203) : OI 0.007/s Numerische Mathematik A stochastic collocation method for the second order wave equation with a discontinuous random speed Mohammad Motamed Fabio Nobile Raúl Tempone Received: 30 October 20 / Revised: 5 July 202 / Published online: 3 August 202 Springer-Verlag 202 Abstract In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical space and depends on a finite number of random variables. The numerical scheme consists of a finite difference or finite element method in the physical space and acollocationinthezerosofsuitabletensorproductorthogonalpolynomials(gauss points) in the probability space. This approach leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. We consider both full and sparse tensor product spaces of orthogonal polynomials. We provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data. In particular, we show that, unlike in elliptic and parabolic problems, the solution to hyperbolic problems is not in general This work was supported by the King Abdullah University of Science and Technology (AEA project Bayesian earthquake source validation for ground motion simulation ), the VR project Effektiva numeriska metoder för stokastiska differentialekvationer med tillämpningar, and the PECOS center at ICES, University of Texas at Austin (Project Number , Center for Predictive Computational Science). The second author was partially supported by the Italian grant FIRB-IEAS (Project no. RBI08223Z) Advanced numerical techniques for uncertainty quantification in engineering and life science problems. M. Motamed (B) R. Tempone King Abdullah University of Science and Technology, Thuwal, Saudi Arabia mohammad.motamed@kaust.edu.sa F. Nobile EPFL, Lausanne, Switzerland F. Nobile Politecnico di Milano, Milan, Italy

2 494 M. Motamed et al. analytic with respect to the random variables. Therefore, the rate of convergence may only be algebraic. An exponential/fast rate of convergence is still possible for some quantities of interest and for the wave solution with particular types of data. We present numerical examples, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo method for this class of problems. Mathematics Subject Classification (2000) 65C20 65M70 65M60 65M06 65M5 65Z05 Introduction Partial differential equations (PEs) are important mathematical models for multidimensional physical systems. There is an increasing interest in including uncertainty in these models and quantifying its effects on the predicted solution or other quantities of physical interest. The uncertainty may be due to an intrinsic variability of the physical system. It may also reflect our ignorance or inability to accurately characterize all input data of the mathematical model. Examples include the variability of soil permeability in subsurface aquifers and heterogeneity of materials with microstructure. Probability theory offers a natural framework to describe uncertainty by parametrizing the input data either in terms of a finite number of random variables or more generally by random fields. Random fields can in turn be accurately approximated by afinitenumberofrandomvariableswhentheinputdatavaryslowlyinspace,with a correlation length comparable to the size of the physical domain. A possible way to describe such random fields is to use the truncated Karhunen-Loéve [27,28] or polynomial chaos expansion [42,45]. There are different techniques for solving PEs in probabilistic setting. The most popular one is the Monte Carlo sampling, see for instance [2]. It consists in generating independent realizations drawn from the input distribution and then computing sample statistics of the corresponding output values. This allows one to reuse available deterministic solvers. While being very flexible and easy to implement, this technique features a very slow convergence rate. In the last few years, other approaches have been proposed, which in certain situations feature a much faster convergence rate. They exploit the possible regularity that the solution might have with respect to the input parameters, which opens up the possibility to use deterministic approximations of the response function (i.e. the solution of the problem as a function of the input parameters) based on global polynomials. Such approximations are expected to yield a very fast convergence. Stochastic Galerkin [3,29,44,4,38] and Stochastic Collocation[2,32,3,43] are among these techniques. Such new techniques have been successfully applied to stochastic elliptic and parabolic PEs. In particular, we have shown in previous works [2,30] that, under particular assumptions, the solution of these problems is analytic with respect to the input random variables. The convergence results are then derived from the regularity results. For stochastic hyperbolic problems, the analysis is not well developed. In the case of linear problems, there are a few works on the one-dimensional scalar advection

3 A collocation method for the stochastic wave equation 495 equation with a time- and space-independent random wave speed [4,37,4]. Such problems also possess high regularity properties provided the data live in suitable spaces. The main difficulty arises when the coefficients vary in space or time. In this more general case, the solution of linear hyperbolic problems may have lower regularity than those of elliptic, parabolic and hyperbolic problems with constant random coefficients. There are also recent works on stochastic nonlinear conservation laws, see for instance [25,26,34,39,40]. In this paper, we consider the linear second order scalar wave equation with a piecewise smooth random wave speed. In many applications, such as seismology, acoustics, electromagnetism and general relativity, the underlying differential equations are systems of second order hyperbolic PEs. In deterministic problems, these systems are often rewritten as first order systems and then discretized. This approach has the disadvantage of introducing auxiliary variables with their associated constraints and boundary conditions. This in turn reduces computational efficiency and accuracy [2,20]. Here, we analyze the problem in the second order differential form, without reducing it to the first order form, and propose a numerical method that directly discretizes the second order PE. The analysis of the first order and other types of second order hyperbolic systems with discontinuous random coefficients will be addressed elsewhere. We propose a stochastic collocation method for solving the wave propagation problem in a medium consisting of non-overlapping sub-domains. In each sud-domain, the wave speed is smooth and is given in terms of one random variable. We assume that the interfaces of speed discontinuity are smooth. We derive a priori error estimates with respect to the number of collocation points. The main result is that unlike in elliptic and parabolic problems, the solution to hyperbolic problems is not in general analytic with respect to the random variables. Therefore, the convergence rate of error in the wave solution may only be algebraic. A fast spectral convergence is still possible for some linear quantities of interest with smooth mollifiers and for the wave solution with smooth data compactly supported within sub-domains. We also show that the semi-discrete solution is analytic with respect to the random variables with the radius of analyticity proportional to the mesh size h.wetherefore obtain anexponential rate of convergence which deteriorates as the quantity h p gets smaller, with p representing the polynomial degree in the stochastic space. The outline of the paper is as follows: in Sect. 2 we formulate the mathematical problem, prove its well-posedness, and provide regularity results on the solution and aquantityofinterest.thecollocationmethodforsolvingtheunderlyingstochastic PE is described in Sect. 3.InSect.4 we give a complete error analysis for the collocation method and obtain convergence results. In Sect. 5 we perform some numerical examples to illustrate the accuracy and efficiency of the method. Finally, we present our conclusions in Sect Mathematical setting We consider the linear second order scalar wave equation with a discontinuous random wave speed and deterministic boundary and initial conditions. We study the well-posedness of the problem and regularity of the solution and a quantity of interest with respect to the input random parameters.

4 496 M. Motamed et al. 2. Problem statement Let be a convex bounded polygonal domain in R d and (Ω, F, P) be a complete probability space. Here, Ω is the set of outcomes, F 2 Ω is the σ -algebra of events and P : F [0, ] is a probability measure. Consider the stochastic initial boundary value problem (IBVP): find a random function u :[0, T ] Ω R, suchthat P-almost everywhere in, i.e. almost surely (a.s.), the following holds u tt (t, x, ω) (a 2 (x, ω) u(t, x, ω) ) = f (t, x) u(0, x, ω) = g (x), u t (0, x, ω) = g 2 (x) u(t, x, ω) = 0 where the data in [0, T ] Ω on {t = 0} Ω on[0, T ] Ω () f L 2 ([0, T ] ), g H 0 (), g 2 L 2 (), (2) are compatible. We assume that the random wave speed a is bounded and uniformly coercive, 0 < a min a(x, ω) a max <, almost everywhere in, a.s. (3) In many wave propagation problems, the source of randomness can be described or approximated by using only a small number of uncorrelated random variables. For example, in seismic applications, a typical situation is the case of layered materials where the wave speeds in the layers are not perfectly known and therefore are described by uncorrelated random variables. The number of random variables corresponds therefore to the number of layers. In this case the randomness is described by a finite number of random variables. Another example is the approximation of the random speed by a truncated Karhunen-Loéve expansion [3]. In this case the number of random variables is the number of terms in the expansion. In this case the randomness is approximated by a finite number of random variables. This motivates us to make the following finite dimensional noise assumption on the form of the wave speed, a(x, ω) = a(x, Y (ω),..., Y N (ω)), almost everywhere in, a.s., (4) where N N + and Y =[Y,...,Y N ] R N is a random vector. We denote by Γ n Y n (Ω) the image of Y n and assume that Y n is bounded. We let Γ = N n= Γ n and assume further that the random vector Y has a bounded joint probability density function ρ : Γ R + with ρ L (Γ ).Wenotethatbyusingasimilarapproachto [2,9] wecanalsotreatunboundedrandomvariables,suchasgaussianandexponential variables. Unbounded wave speeds could theoretically arise in the case of rigid materials. In this paper we consider only bounded random variables. In this paper, in particular, we consider a heterogeneous medium consisting of N sub-domains. In each sub-domain, the wave speed is smooth and represented by one random variable. The boundaries of sub-domains, which are interfaces of speed discontinuity, are assumed to be smooth and do not overlap.

5 A collocation method for the stochastic wave equation 497 The random speed a can for instance be given by a(x, ω) = a 0 (x) + N a n (x, ω) X n (x), a n (x, ω) = Y n (ω) α n (x), (5) n= where X n are indicator functions describing the geometry of each sub-domain, Y n are independent and identically distributed random variables, and α n are smooth functions defined on each sub-domain. Note that the representation of the coefficient a in (5) is exact, and there is no truncation error as in the Karhunen-Loéve expansion. The more general case where the wave speed in each sub-domain a n (x, ω) is represented by a Karhunen-Loéve expansion can be treated in the same way. In this case the total number of random variables is N n= M n,wherem n is the number of terms in the truncated Karhunen-Loéve expansion in each sub-domain. The case where the geometry of sub-domains is also random will be addressed elsewhere. For elliptic equations, random boundaries have been studied, e.g., in [7,5,46]. The finite dimensional noise assumption implies that the solution of the stochastic IBVP () can be described by only N random variables, i.e., u(t, x, ω) = u(t, x, Y (ω),..., Y N (ω)). Thisturnstheoriginalstochasticproblemintoadeterministic IBVP for the wave equation with an N-dimensional parameter, which allows the use of standard finite difference and finite element methods to approximate the solution of the resulting deterministic problem u = u(t, x, Y ), wheret [0, T ], x, and Y Γ.Notethattheknowledgeofu = u(t, x, Y ) fully determines the law of the random field u = u(t, x, ω). The ultimate goal isthen theprediction of statistical moments of the solution u or statistics of some given quantities of physical interest. Before studying the well-posedness and regularity in details, we start the discussion with two simple examples. Example AbasictechniqueforstudyingtheregularityofthesolutionofaPEwith respect to a parameter is based on analyzing the equation in the complex plane. In this approach, the parameter is first extended into the complex plane. Then, if the extended problem is well-posed and the first derivative of the resulting complex-valued solution with respect to the parameter satisfies the so called Cauchy Riemann conditions, the solution can analytically be extended into the complex plane. This approach has been used in [30]toprovetheanalyticityofthesolutionofparabolicPEswithstochastic parameters. As a first example, we therefore consider the Cauchy problem for the one-dimensional scalar wave equation with a complex-valued one-parameter wave speed u tt (t, x) a 2 u xx (t, x) = 0, in [0, T ] R, u(0, x) = g(x), u t (0, x) = 0, on {t = 0} R, with a constant, complex-valued coefficient a = a R + ia I, a R, a I R. Assume that g(x) is a smooth function that vanishes at infinity. We apply the Fourier transform with respect to x and get

6 498 M. Motamed et al. û tt (t, k) + a 2 k 2 û(t, k) = 0, û(0, k) =ĝ(k), û t (0, k) = 0, where û(t, k) = R u(t, x) e i kx dx and ĝ(k) = R g(x) e i kx dx are the Fourier transforms of u(t, x) and g(x) with respect to x, respectively. The solution of this linear, second order ordinary differential equation with parameter k is given by û(t, k) = ĝ(k) 2 ( e r t + e r 2 t ), r,2 =±iak. When a I = 0, then r,2 =±ia R k. Performing the inverse Fourier transform, we get the solution u(t, x) = 2 ( g(x + ar t) + g(x a R t) ), and therefore the Cauchy problem is well-posed. When a I = 0, then Re(r ) = Re(r 2 ) = a I k,and û(t, k) ĝ(k) e a I k t. Therefore, regardless of the sign of Re(a 2 ) = a 2 R a2 I,theFouriertransformofthe solution û(t, k) grows exponentially fast, i.e., e a I k t,unlessthefouriertransformof the initial solution ĝ(k) decays faster than e a I k t.thecauchyproblemistherefore well-posed only if g(x) is in a restricted class of Gevrey spaces [35]. efinition Afunctiong(x) is a Gevrey function of order q > 0, i.e., g G q (R),if g C (R) and for every compact subset R, thereexistsapositiveconstantc such that, max x n g(x) C n+ (n!) q. In particular, G (R) is the space of analytic functions [8]. For 0 < q <, the class G q (R) is a subclass of the analytic functions, while for < q < it contains the analytic functions. We now state a known result on the decay of the Fourire transform of Gevrey functions [24,35]. Lemma Afunctiong(x) belongs to the Gevrey space G q (R) if and only if there exist positive constants C and ɛ such that ĝ(k) Ce ɛ k /q. Therefore, for the Cauchy problem to be well-posed in the complex strip r ={(a R + ia I ) C : a I r}, we need g G q (R) with q <. Note that for q =, the problem is well-posed only for a finite time interval when t ɛ/r. Thisshowsthat even if the initial solution g is analytic, i.e, g G (R), thesolutionisnotanalytic for all times in r.reversingtheargument,wecansaythat,startingfromananalytic initial solution g,with ĝ(k) Ce ɛ k,thesolutionattimet will be analytic only in the strip ɛ/t, and the analyticity region becomes smaller and smaller as time increases. Example 2 An important characteristic of waves in a heterogeneous medium in which the wave speed is piecewise smooth, is scattering by discontinuity interfaces. As a

7 A collocation method for the stochastic wave equation 499 simple scattering problem, we consider the Cauchy problem for the second order scalar wave equation in a one-dimensional domain consisting of two homogeneous half-spaces separated by an interface at x = 0, u tt (t, x) ( a 2 (x) u x (t, x) ) = 0, in [0, T ] R, x with a piecewise constant wave speed { a, x < 0, a(x) = a +, x > 0. In this setting, the wave speed contains two positive parameters, a and a +.Wechoose the initial conditions such that the initial wave pulse is smooth, compactly supported, lies in the left half-space, and travels to the right. That is u(0, x) = g( x), u t (0, x) = a g ( x), g(x) C 0 (0, ). By d Alembert s formula, the solution reads { g(a t x) + Φ u(t, x) = (a t + x), x < 0, Φ 2 (a + t x), x > 0. Note that when x < a t,thesolutionispurelyright-going,u = g(a t x), and when x > a + t,thesolutioniszero,u = 0. The functions Φ and Φ 2 are obtained by the interface jump conditions at x = 0, u(t, 0 ) = u(t, 0 + ), a 2 u x(t, 0 ) = a 2 + u x(t, 0 + ). (6) After some manipulation, we get the solution { g(a t x) + a a + a u(t, x) = +a + g(a t + x), x < 0, 2 a a +a + g( a a+ (a + t x)), x > 0. The interpretation of this solution is that the initial pulse g( x) inside the left halfspace moves to the right with speed a until it reaches the interface. At the interface it is partially reflected (Φ )withspeeda and partially transmitted (Φ 2 )withspeed a +.Theinterfacebetweentwolayersgeneratesnoreflectionsifthespeedsareequal, a = a +.Fromtheclosedformofthesolution(7), we note that the solution u(t, x) is infinitely differentiable with respect to both parameters a and a + in (0, + ). Note that the smooth initial solution u(0, x),which is contained in one layer with zero value at the interface, automatically satisfies the interface conditions (6)attimezero. Otherwise, if for instance the initial solution crosses the interface without satisfying (6), a singularity is introduced in the solution, and the high regularity result does no longer hold. In the more general case of multi-dimensional heterogeneous media consisting of sub-domains, the interface jump conditions on a smooth interface ϒ between two sub-domains I and II are given by [u(t,.)] ϒ = 0, [a 2 (.) u n (t,.)] ϒ = 0. (8) (7)

8 500 M. Motamed et al. Here, the subscript n represents the normal derivative, and [v(.)] ϒ is the jump in the function v across the interface ϒ. In this general case,the high regularity with respect to parameters holds provided the smooth initial solution satisfies (8). The jump conditions are satisfied for instance when the initial data are contained within sub-domains. This result for Cauchy problems can easily be extended to IBVPs by splitting the problem to one pure Cauchy and two half-space problems. See Sect for more details. Remark Immediate results of the above two examples are the following:. For the solution of the Cauchy problem for the one-dimensional wave equation to be analytic with respect to the random wave speed at all times in a given complex strip r with r > 0, the initial datum needs to live in a space strictly contained in the space of analytic functions, which is the Gevrey space G q (R) with 0 < q <. Moreover, if the problem is well-posed and the data are analytic, the solution may be analytic with respect to the parameter in r only for a short time interval. Problems in higher physical dimensions with constant random wave speeds can be studied similarly by the Fourier transform and the generalization of univariate Gevrey functions to the case of multivariate functions. 2. In a one-dimensional heterogeneous medium with piecewise smooth wave speeds, if the data are smooth and the initial solution satisfies the interface jump conditions (8), the solution to the Cauchy problem is smooth with respect to the wave speeds. If the initial solution does not satisfy (8), the solution is not smooth with respect to the wave speeds. We refer to the proof of Theorem 3 where a more general one-dimensional problem, i.e. problem (4), with smooth data is considered. We also refer to Conjecture for problems in two dimensions. We note that the above high regularity results with respect to parameters are valid only for particular types of smooth data. In real applications, the data are not smooth. We therefore study the well-posedness and regularity properties in the more general case when the data satisfy the minimal assumptions (2). 2.2 Well-posedness We now show that the problem () with the data satisfying (2)andtheassumption(3) is well-posed. For a function of the random vector Y,weintroducethespaceofsquare integrable functions: L 2 ρ (Γ ) = v : Γ R, v(y ) 2 ρ(y ) dy <, with the inner product (v,v 2 ) L 2 ρ (Γ ) = E [v v 2 ] = v v 2 ρ(y ) dy. Γ Γ

9 A collocation method for the stochastic wave equation 50 We also introduce the mapping u :[0, T ] H0 () L2 ρ (Γ ),definedby [u(t)](x, Y ) := u(t, x, Y ), t [0, T ], x, Y Γ. Similarly, we introduce the function f :[0, T ] L 2 (), definedby [f(t)](x) := f (t, x), t [0, T ], x. Finally, for a real Hilbert space X with norm. X,weintroducethetime-involving space H X := L 2 (0, T ; X) L 2 ρ (Γ ) = L 2 (0, T ; X L 2 ρ (Γ )), consisting of all measurable functions v with v 2 H X = v 2 X ρ(y ) dt dy <. [0,T ] Γ Examples of X include the L 2 () space and the Sobolev space H0 () and its dual space H (). We now recall the notion of weak solutions for the IBVP (). efinition 2 The function u H H 0 () with u H L 2 () and u H H () is a weak solution to the IBVP () providedthefollowinghold: (i) u(0) = g and u (0) = g 2, (ii) for a.e. time 0 t T and v H0 () L2 ρ (Γ ): u (t)vρ dx dy + B(u(t), v) ρ dy = Γ where B(v,v 2 )(Y )= Γ Γ f(t)vρ dx dy, (9) a 2 (x, Y )( v (x, Y ) v 2 (x, Y )) dx, v,v 2 H0 () L2 ρ (Γ ). Theorem Under the assumptions (2) and (3), thereisauniqueweaksolution u H H 0 () to the IBVP ().Moreover,itsatisfiestheenergyestimate ( max u(t) H 0 t T 0 () L2 ρ (Γ ) + ) u (t) L 2 () L 2 ρ (Γ ) + u HH () ( ) C f L 2 (0,T ;L 2 ()) + g H 0 () + g 2 L 2 (). (0) Proof The proof is an easy extension of the proof for deterministic problems, see e.g. []. 2.3 Regularity In this section we study the regularity of the solution and of a quantity of interest with respect to the random input variable Y.Themainresultisthatunderthemini-

10 502 M. Motamed et al. mal assumptions (2) and(3) thesolution,whichisinl 2 ρ (Γ ), hasingeneralonlyone bounded derivative with respect to Y, while the considered quantity of interest may have many bounded derivatives. The available regularity is then used to estimate the convergence rate of the error for the stochastic collocation method Regularity of the solution We first investigate the regularity of the solution with respect to the random variable Y.Fordeterministicproblems,forinstancewhenY is a fixed constant, it is well known that in the case of x-discontinuous wave speed, with the data satisfying (2)andunder the assumption (3), the solution of () is in general only u C 0 (0, T ; H0 ()), see for instance [33,36]. In other words, in the presence of discontinuous wave speed, one should not expect higher spatial regularity than H (). To investigate the Y -regularity of the solution in the stochastic space, we differentiate the IBVP () with respect to Y and obtain ũ tt (a 2 ũ ) = (2 aa Y u ), ũ = Y u, () with homogeneous initial and boundary conditions. The force term in the above IBVP is f := (2 aa Y u ) L (0, T ; H ()) for every Y Γ.Infactif v L (0, T ; H 0 ()),then T 0 f v dx dt = (2 aa Y u ),v = 2 aa Y u, v 2 aa Y u L (0,T ;L 2 ()) v L (0,T ;L 2 ()) <. We now state an important result which is a generalization of a result given by Hörmander [6]. Lemma 2 For arbitrary f L (0, T ; H k ()), g H k+ () and g 2 H k (), with k R, foreveryy Γ, there is a unique weak solution u C 0 (0, T ; H k+ ()) C (0, T ; H k ()) of the IBVP () withthex-smooth wave speed (4) satisfying (3). Moreover,itsatisfiestheenergyestimate ( u(t) H k+ () + u (t) H ()) k max 0 t T ) C k,t ( f L (0,T ;H k ()) + g H k+ () + g 2 H k (). (2) Proof The proof is an easy extension of the proof of Lemma and Theorem in [6]. We note that Lemma 2 holds for x-smooth wave speeds. When the wave speed is non-smooth, it holds only for k = and k = 0[36]. We apply Lemma 2 to () with k = (whichisvalidalsofornon-smoothcoefficients)andobtain ũ C 0 (0, T ; L 2 ()), Y Γ.

11 A collocation method for the stochastic wave equation 503 Moreover, the solution (7) of Example 2 with g H0 (R) shows that the second and higher Y -derivatives do not exist. Therefore, under the minimal assumptions (2), the solution has at most one bounded Y -derivative in L 2 ().Wehaveprovedthefollowing result, Theorem 2 For the solution of the second order wave propagation problem () with data given by (2) and a random piecewise smooth wave speed satisfying (3) and (5), we have Y u C 0 (0, T ; L 2 ()) for every Y Γ Regularity of quantities of interest We now consider the quantity of interest Q(Y ) = T 0 u(t, x, Y ) φ(x) dx dt + u(t, x, Y ) ψ(x) dx, (3) where u solves ()andthemollifiersφ and ψ are given functions of x. As a corollary of Theorem 2,wecanwrite, Corollary With the assumptions of Theorem 2 and φ L () and ψ L (), we have dy d Q L (Γ ). We now assume that the mollifiers φ(x) C0 () and ψ(x) C 0 () are smooth functions and analytic in the interior of their supports. We further assume that their supports does not cross the speed discontinuity interfaces. We will show that the resulting quantity of interest (3) may have higher Y -regularity, without any higher regularity assumptions on the data than those in (2). For this purpose, we introduce the influence function (or dual solution) ϕ associated to the quantity of interest, Q, asthesolution of the dual problem ϕ tt (t, x, Y ) (a 2 (x, Y ) ϕ(t, x, Y ) ) = φ(x) in [0, T ] Γ ϕ(t, x, Y ) = 0, ϕ t (T, x, Y ) = ψ(x) on {t = T } Γ (4) ϕ(t, x, Y ) = 0 on [0, T ] Γ Note that this is a well-posed backward wave equation with smooth initial data at the final time T and a smooth force term. We can write Q(Y ) = T 0 = = T 0 T 0 ( u ϕ tt (a 2 ϕ )) dx dt + T u ϕ tt dx dt + 0 T u tt ϕ dx dt 0 u(t, x, Y ) ψ(x) dx a 2 u ϕ dx dt + u(t, x, Y ) ψ(x) dx ϕ (a 2 u ) dx dt + [ ϕ t u ϕ u t ] T 0

12 504 M. Motamed et al. = T 0 dx + u(t, x, Y ) ψ(x) dx ( ) ϕ(t, x, Y ) f (t, x) dx dt+ g 2 (x) ϕ(0, x, Y ) g (x) ϕ t (0, x, Y ) dx. The last equality follows from the initial condition in ()andinthedualproblem(4). This shows that the regularity of the quantity of interest depends only on Y -regularity of the dual solution. To investigate the Y -regularity of dual solution, we first note that the finite speed of wave propagation and the superposition principle due to the linearity of the dual problem (4) makesitpossibletosplittheibvpinr d into two half-space problems and a pure Cauchy problem [9]. To clarify this, consider a one-dimensional strip problem (d = ) for (4) on the physical domain =[0, ] with N = 2 layers with widths d and d 2.Letϑ j C (), j =, 2, 3, be monotone functions with and ϑ (x) = {, x d 6, 0, x d 3. ϑ 2 (x) = {, x d 2 6, 0, x d 2 3. ϑ 3 (x) = ϑ (x) ϑ 2 (x). Setϕ = ϕ + ϕ 2 + ϕ 3,whereeachϕ j solves ϕ jtt (a 2 ) ϕ j = ϑ j φ in [0, T ] Γ ϕ j = 0, ϕ jt = ϑ j ψ on {t = T } Γ ϕ j = 0 on [0, T ] Γ The finite speed of propagation implies that there is a time 0 < T T where ϕ = 0 for x [d, ] and t [T T, T ]. Therefore,wecanconsiderϕ as the solution of the right half-space problem ϕ tt (a 2 ϕ ) = ϑ φ, t [T T, T ], x 0 ϕ = 0, ϕ t = ϑ ψ, t = T, x 0 ϕ = 0, t [T T, T ], x = 0 Note that here we redefine the wave speed a by extending the speed corresponding to the left layer to the whole half space 0 x <. Similarly,ϕ 2 and ϕ 3 locally solve alefthalf-spaceandapurecauchyproblem,respectively.theseconsiderationsare valid in the time interval [T T, T ].Attimet = T T,weobtainanewfinaldual solution and can restart. The Y -regularity of the dual solution ϕ is therefore obtained by the regularity of ϕ, ϕ 2 and ϕ 3.Thefirsttwofunctions,ϕ and ϕ 2,whichsolvetwohalf-spaceproblems with smooth data and coefficients, are smooth []and have bounded Y -derivatives of any order s. The third function ϕ 3, which solves a single interface Cauchy problem with smooth data whose support does not cross the interface and with a piecewise smooth wave speed, has again bounded Y -derivatives of any order s. In one dimension (d = ), when the wave speed is piecewise constant, we can solve the Cauchy wave equation by d Alembert s formula and explicitly obtain the solution ϕ 3 which is smooth with respect to the wave speed and therefore is Y -smooth, see Example 2 as a

13 A collocation method for the stochastic wave equation 505 simple illustration. When the wave speed is variable, we can employ the energy method to show Y -regularity, see Theorem 7 in the Appendix. Note that the same result holds also for a multiple interface Cauchy problem. Therefore the dual solution ϕ and consequently the quantity of interest Q have bounded Y -derivatives of any order s. We note that for 0 t T T,althoughthenewfinaldualsolutionϕ(T T, x, Y ) may not be contained in one layer, but since it naturally satisfies the correct jump conditions at the interface, the Y -regularity of the dual solution holds in the time interval [0, T ]. We have therefore proved the following result in a one-dimensional physical space, Theorem 3 Let R. With the assumptions of Theorem 2 and if φ C0 () and ψ C0 () and their supports do not cross the discontinuity interfaces, the quantity of interest (3) satisfies dy ds s Q L (Γ ) with any order s. In a more general case of two-dimensional physical space (d = 2), ϕ and ϕ 2 are again smooth [] andhaveboundedy -derivatives of any order s. The proof of smoothness for ϕ 3 is more complicated. However, noting that the discontinuity occurs in the normal direction to the interfaces, we can employ a localization argument and build a two-dimensional result by generalizing the one-dimensional ones. Based on this and numerical results, we therefore make the following conjecture, Conjecture Let R 2.WiththeassumptionsofTheorem2 and if φ C0 () and ψ C0 () and their supports do not cross the discontinuity interfaces, the quantity of interest (3) satisfies dy ds s Q L (Γ ) with any order s. Remark 2 For quantities of interest which are nonlinear in u the high Y -regularity property does not hold in general. In fact, the corresponding dual problems have nonsmooth forcing terms and data (assuming u is not smooth), and therefore the dual solutions are not smooth with respect to Y.InSect.5,wenumericallystudytheArias intensity [] which is a nonlinear quantity of interest and show that it is not regular with respect to Y. 3Astochasticcollocationmethod In this section, we review the stochastic collocation method for computing the statistical moments of the solution u to the problem (), see for example [2,43]. We first discretize the problem in space and time, using a deterministic numerical method, such as the finite element or the finite difference method, and obtain a semi-discrete problem. We next collocate the semi-discrete problem in a set of η collocation points {Y (k) } η k= Γ and compute the approximate solutions u h(t, x, Y (k) ). Finally, we build aglobalpolynomialapproximationu h,p upon those evaluations u h,p (t, x, Y ) = η u h (t, x, Y (k) ) L k (Y ), k= for suitable multivariate polynomials {L k } η k= such as Lagrange polynomials. Here, h and p represent the discretization mesh size and the polynomial degree, respectively. In what follows, we address in more details the choice of collocation points.

14 506 M. Motamed et al. We seek a numerical approximation to u in a finite-dimensional subspace H h,l of the space H H 0 () L2 (0, T ; H0 ()) L2 ρ (Γ ) in which the function u lives. We define the subspace based on a tensor product H h,l = H h H l,where H h ([0, T ] ) L 2 (0, T ; H0 ()) is the space of the semi-discrete solution in time and space for a constant Y.Thesubscripth denotes the spatial grid-lengths and the time step-size. stability of the numerical scheme. H l (Γ ) L 2 ρ (Γ ) is a tensor product space which is the span of the tensor product of orthogonal polynomials with degree at most p =[p (l),..., p N (l)]. The positive integer l is called the level,andp n (l) is the maximum degree of polynomials in the n-th direction, with n =,...,N,givenasafunctionofthelevell.Foreach Y n, n =,...,N, with the density ρ n,leth pn (Γ n ) be the span of ρ n -orthogonal polynomials V (n) 0, V (n),...,v p (n) n. The tensor product space is then H l (Γ ) = Nn= H pn (Γ n ).ThedimensionofH l is dim(h l ) = N n= (p n + ).Withoutloss of generality, for bounded random variables, we assume Γ =[, ] N. Having the finite-dimensional subspace H h,l constructed, we can use Lagrange interpolation to build an approximate solution u. The ultimate goal of the computations is the prediction of statistical moments of the solution u (such as the mean value and variance) or statistics of some given quantities of interest Q(Y ).ForalinearboundedoperatorΨ(u),using the Gauss quadrature formula for approximating integrals, we write E [Ψ (u(., Y ))] E [ Ψ (u h,p (., Y )) ] η = Ψ (u h,p (., Y )) ρ(y ) dy θ k Ψ (u h (., Y (k) )), where the weights are θ k = N n= Ɣ Ɣ n L kn (Y n ) ρ n (Y n ) dy n, L kn (Y n ) = k= η i=0, i =k n (i) Y n Yn Y (k n) n Y (i) n and the collocation points Y (k) =[Y (k ),...,Y (k N ) N ] Ɣ are tensorized Gauss points with Y (k n) n, k n = 0,,...,p n,beingthezerosoftheρ n -orthogonal polynomial of degree p n +. Here, for any vector of indices [k,...,k N ] with 0 k n p n the associated global index reads k = + k + (p + ) k 2 + (p + )(p 2 + ) k 3 +. Remark 3 The choice of orthogonal polynomials depends on the density function ρ. For instance, for uniform random variables Y n U(, ), Legendrepolynomials are used, V (n) k+ (Y n) = 2k + k + Y n V (n) k (Y n ) 2(k + /2) V (n) k (Y n), V (n) = 0, V (n) 0 =, k 0.,

15 A collocation method for the stochastic wave equation 507 Other well known orthogonal polynomials include Hermite polynomials for Gaussian random variables and Laguerre polynomials for exponential random variables [45]. Remark 4 There are other choices for the approximation space H l.forexample, instead of orthogonal polynomials, we can choose a piecewise constant approximation using the Haar-wavelet basis. We can also choose a piecewise polynomial approximation. The choice of the approximating space may depend on the smoothness of the function with respect to Y.Ingeneral,forsmoothfunctions,wechooseapolynomial approximation, while for non-smooth functions, we choose a low-degree piecewise polynomial or wavelet-type approximation [22,23]. We now consider two possible approaches for constructing the tensor product space H l and briefly review the Lagrange interpolation. 3. Full tensor product space and interpolation For a given multi-index j =[j,..., j N ] Z+ N,containingN non-negative integers, we define H j (Y ) = V () p( j ) (Y ) V (N) p( j N ) (Y N ). Given an index j,wecalculatethepolynomialdegreep( j) either by or by p( j) = j, (5) p( j) = 2 j for j > 0, p(0) = 0. (6) The isotropic full tensor product space is then chosen as } Hl {H T = span j, j := max j n l. n In other words, in each direction we take all polynomials of degree at most p(l),and therefore dim(h T l ) = (p(l) + )N.Sincethedimensionofthespacegrowsexponentiallyfastwith N (curse of dimensionality), the full tensor product approximation can be used only when the number of random variables N is small. The multi-dimensional Lagrange interpolation corresponding to a multi-index j is I j,n [u](., Y ) = N n= p( j ) U j n (u)(y )= k =0 p( j N ) k N =0 u h (., Y j,k,...,y j N N,k N ) N n= L j n n,k n (Y n ), where, for each value of a non-negative index j n in the multi-index j, U j n is the one-dimensional Lagrange interpolation operator, the set {Y j n n,k n } p( j n) k n =0 is a sequence of abscissas for Lagrange interpolation on Ɣ n,and{l j n n,k n (y)} p( j n) k n =0 are Lagrange polyno- (7)

16 508 M. Motamed et al. mials of degree p( j n ), L j n n,k n (y) = p( j n ) i=0, i =k n y Y j n n,i Y j n n,k n Y j n n,i The set of points where the function u h is evaluated to construct (7)isthetensorgrid H j,n ={Y k =[Y j,k,...,y j N N,k N ], 0 k n p( j n )}. The isotropic full tensor interpolation is obtained when we take j =[l, l,...,l] in (7), and the corresponding operator is denoted by I l,n,. I l,n [u](., Y ) = p(l) k =0 p(l) k N =0 u h (., Y l,k,...,y l N,k N ) N L l n,k n (Y n ). (8) n= 3.2 Sparse tensor product space and interpolation Here, we briefly describe the isotropic Smolyak formulas [5]. The sparse tensor product space is chosen as H S l,n = span{h j, j : j l}, The dimension of the sparse space is much smaller than that of the full space for large N. For example, when p( j) = j, wehavedim(hl,n S ) = j l = (N+l)! N! l!,which helps reducing the curse of dimensionality. This space corresponds to the space of polynomials of total degree at most p(l). The sparse interpolation formula can be written as a linear combination of Lagrange interpolations (7) onalltensorgridsh j,n. With U = 0, and for an index j n 0, define j n := U j n U j n. The isotropic Smolyak formula is then given by A l,n [u](., Y ) = j l( j j N ) u(., Y ). (9) Equivalently, the formula (9) canbewrittenas ( ) N A l,n [u](., Y ) = ( ) l j Ij N u(., Y ). (20) l j l N+ j l The collection of all tensor grids used in the sparse interpolation formula is called the sparse grid, Hl,N S = H j,n Ɣ. l N+ j l

17 A collocation method for the stochastic wave equation Y 2 0 Y Y (a) Full grid Y (b) Sparse grid Fig. The full and sparse grids for a vector of two independent uniform random variables in [, ] with level l = 5 Sparse interpolation implies evaluating u h (t, x,.)in all points of the sparse grid, known as collocation points. By construction, we have A l,n [u](t, x,.) H S l,n.notethatthe number of collocation points is larger than the dimension of the approximating space H S l,n. Example 3 Let N = 2andl = 5, and consider p( j) = j.moreover,lety =[Y, Y 2 ] be a random vector with independent and uniformly distributed random variables Y n U(, ). Forafulltensorspace,thereare(5 + ) 2 = 36 collocation points in the grid, shown in Fig. a. For a sparse tensor space, there are (2+5)! 2! 5! = 2 admissible sets of indices j and 89 collocation points in the grid shown in Fig. b. Observe that the number of points in the full tensor grid grows much faster with the dimension N than the number of points in the sparse grid. 3.3 Choice of interpolation abscissas We propose two different abscissas in the construction of the Smolyak formula. Gaussian abscissas In this case, for a given index j n,thesequenceofabscissas {Y j n n,k n } p( j n) k n =0 are p( j n) + zeros of the orthogonal polynomial V p( jn )+.Asthechoice of the polynomial degree, we can use either the formula (5) or(6). Clenshaw-Courtis abscissas These abscissas are the extrema of Chebyshev polynomials and are given by ( Y j n π kn n,k n = cos p( j n ) ), k n = 0,...,p( j n ). It is recommended to use the formula (6)forthepolynomialdegree.Inthiscase,one obtains nested sets of abscissas and thereby H S l,n HS l+,n. We note that the structure of the stochastic collocation method, which involves solving η independent problems, allows us to use and perform parallel computations in a straight forward way.

18 50 M. Motamed et al. 4Convergenceanalysisforstochasticcollocation In this section, we consider a linear bounded operator Ψ (u) and give a priori estimates for the total error Ψ (u) Ψ (u h,l ) in the norm L 2 (0, T ; L 2 ()) L 2 ρ (Ɣ) when Ψ (u) = u,andinthenorml 2 ρ (Ɣ) when Ψ (u) = Q(Y ) with Q given in (3). We split the error into two parts and write ε := Ψ (u) Ψ (u h,l ) Ψ (u) Ψ (u h ) + Ψ (u h ) Ψ (u h,l ) =: ε I +ε II. (2) The first term in the right hand side ε I controls the convergence of the deterministic numerical scheme with respect to the mesh size h and is of order O(h r ),wherer is the minimum between the order of accuracy of the finite element or finite difference method used and the regularity of the solution. Notice that the constant in the term O(h r ) is uniform with respect to Y. Here, we focus on the second term ε II which is an interpolation error in the stochastic space. We first consider the case when h 0. We show that the error decays algebraically with respect to the number of collocation points η with an exponent proportional to s, provided there are s bounded Y -derivatives (i.e., Y s n Ψ < with n =,...,N) whenthefulltensorinterpolationisused,andifthemixed Y -drivatives (i.e., Y s Y s 2 Y s N Ψ < ) areboundedwhenthesmolyakinterpolation is used. We next consider the case when h β l,with β 2, is large. In this case, we show that the approximate solution u h is Y -analytic with the radius of analyticity proportional to h β.wethereforeobtaina fast exponentialrateofconvergence which deteriorates as the quantity h β l gets smaller. The effective error ε II will then be the minimum of the two errors corresponding to the case when h 0andwhen h β l is large. 4. The case when h 0 We only consider the operator Ψ (u h ) = u h and let h 0. The discrete solution u h has then a Y -regularity of order s = asthecontinuoussolutionu, i.e. Y u h C 0 (Ɣ; W ), where W := L 2 (0, T ; L 2 ()), seesec.2.thesecondtermoftheerrorε II will then be in the norm L 2 (0, T ; L 2 ()) L 2 ρ (Ɣ).WenoticethatforthecaseΨ (u) = Q(Y ), where Q is the quantity of interest in (3)withcompactlysupportedsmoothmollifiers whose supports does not cross the interfaces, the corresponding error estimates are obtained by replacing s = withs. The technique for obtaining error bounds for multivariate functions (when N > ) is based on one-dimensional results. We first quote a useful result from Erdös and Turán [0]forunivariatefunctions. Lemma 3 Let N = and Ɣ R be bounded. Let W be a Hilbert space. For every function v C 0 (Ɣ; W ) the interpolation error with Lagrange polynomials based on Gaussian abscissas satisfies v U j (v) L 2 ρ (Ɣ;W ) 2 inf v v 0 L v 0 W H (Ɣ;W ). (22) p( j)

19 A collocation method for the stochastic wave equation 5 We then recall a Jackson-type theorem on the error of the best approximation of univariate functions with bounded derivatives by algebraic polynomials, see [5] for instance. Lemma 4 Let N = and Ɣ R be bounded. Set W := L 2 (0, T ; L 2 ()). Givena function v C 0 (Ɣ; W ) with s 0 bounded derivatives in Y, there holds E p (v) := min v v 0 L v 0 W H (Ɣ;W ) C s p s p where the constant C s depends only on s. max k=0,...,s k Y v L (Ɣ;W ), (23) We consider one random variable Y n Ɣ n with density ρ n and denote by Ŷ n ˆƔ n the remainder N variableswithdensity ˆρ n = N k=,k =n ρ k.wecannowconsider := u h (., Y n, Ŷ n ) : Ɣ n W n as a univariate function of Y n with values in the Hilbert space W n = W L 2ˆρ n. We are ready to prove the following result. u (n) h Theorem 4 Consider the isotropic full tensor product interpolation formula (8),and let u h,l = I l,n [u h ].Thentheinterpolationerrorε II defined in (2) satisfies ε II Cp(l) s, where the constant C = C s Nn= max k=0,...,s k Y n u h,l L (Ɣ;W ) does not depend on l. Here, p(l) is either l or 2 l depending on the choice of formula (5) or (6) for the polynomial degree, respectively. Moreover, let η be the total number of collocation points, then ε II C 2 η s/n. (24) Proof We consider the first random variable Y and the corresponding onedimensional Lagrange interpolation operator I l, = U l : C 0 (Ɣ ; W ) L 2 ρ (Ɣ ; W ).TheglobalinterpolationI l,n can be written as the composition of two interpolations operators, I l,n = I l, Î l,,whereî l, : C 0 ( ˆƔ ; W ) L 2 ρ ( ˆƔ ; W ) is the interpolation operator in all directions Y 2,...,Y N except Y.Wehave, [ ε II = u h I l,n [u h ] L 2 ρ (Ɣ;W ) u h I l, [u h ] + I l, uh Î l, [u h ] ] }{{}}{{} ε II By (22) and(23), we can bound the first term, ε II Cp(l) s, C = 2 C s max k=0,...,s k Y u h,l L (Ɣ;W ). ε II2. To bound the second term we use the inequality (see Lemma 4.2 in [2]), I l, [v] L 2 ρ (Ɣ;W ) C v L (Ɣ;W ),withv C 0 (Ɣ; W ), for v = u h Î l, [u h ] and write ε II2 C u h Î l, [u h ] L (Ɣ;W ). The right hand side is again an interpolation error in the remainder N directions Y 2,...,Y N.Wecanproceediterativelyanddefineaninterpolationoperatorin

20 52 M. Motamed et al. direction Y 2 and so forth. Finally we arrive at u h I l,n [u h ] C s p(l) s N n= max k=0,...,s k Y n u h,l L (Ɣ;W ) =: Cp(l) s. Note that C s denotes a positive constant depending on s whose value may change from one expression to another expression. This proves the first inequality. The second inequality follows noting that the total number of collocation points is η = ( p(l)+ ) N. Remark 5 If the anisotropic full tensor interpolation [3] is used,the number of collocation points is η = N n= (p(l n ) + ),wherel n is the level in the n-th direction. In this case the error satisfies ε II C s N n= n p(l n ) s, n := max k=0,...,s k Y n u h,l L (Ɣ;W ). In order to minimize the computational work η subject to the constraint ε II TOL, we introduce the Lagrange function L = η + λ (C Nn= s n p(l n ) s TOL),with the Lagrange multiplier λ.byequatingthepartialderivativeofl with respect to p(l n ) to zero, we obtain p(l n ) n /s.notingthat n can be computed easily using just afewsamplesofy n,wecanquicklybuildafastwayonhowtochoosepolynomial degrees in different directions and build the anisotropic full tensor grid. To obtain error estimates using the isotropic Smolyak interpolation, we first recall another Jackson-type theorem on the error of the best approximation of univariate functions with bounded derivatives by algebraic polynomials, see [8] forinstance. Lemma 5 Let N = and Ɣ R be bounded. Let W be a Hilbert space. For every function v L 2 ρ (Ɣ; W ) with s square integrable Y -derivatives, the interpolation error with Lagrange polynomials based on Gauss-Legendre abscissas satisfies v U j (v) L 2 ρ (Ɣ;W ) C s ρ /2 p( j) s max k=0,...,s k Y v L 2 (Ɣ;W ), (25) where the constant C s depends only on s. We also need the following lemma, Lemma 6 In the isotropic Smolyak formula (9),withp( j) given by (6),if j n u (n) h L 2 ρn (Ɣ n;w n ) = (U j n U jn )u (n) h then N jn u h L n= 2 ρ (Ɣ;W ) 2 C s ρ n /2 2 s( j n ) max k n =0,...,s k n Y n u (n) h L 2 (Ɣ n ;W n ), (2 C s ) N ρ /2 2 s N n= ( j n ) max 0 k,...,k N s k Y... k N Y N u h L 2 (Ɣ;W ). (26)

21 A collocation method for the stochastic wave equation 53 Proof We write N 2 jn u h = n= L 2 j N 2 N jn u h ρ N dy N ρ ρ N dy dy N ρ (Ɣ;W ) Ɣ Ɣ N Ɣ n= N W (2C s ) s ( j N 2 N ) ρn max k N =0,...,s k N YN jn u h ρ ρ N dy Ɣ Ɣ N Ɣ n= N W (2C s ) s ( j N ) ρn max k N =0,...,s j N 2 2 N jn k N YN u h ρ ρ N dy. Ɣ N Ɣ Ɣ N n= W If we repeat the process, we finally arrive at (26). We can now prove the following result, Theorem 5 Consider the sparse tensor product interpolation formula (20) based on Gauss-Legendre abscissas when the formula (6) is used, and let u h,l = A l,n [u h ]. Then for the discrete solution u h with s bounded mixed derivatives in Y, the interpolation error ε II defined in (2) satisfies ε II Ĉ (l + ) 2 N 2 s (l+), with Ĉ = C 0 C0 N 2 C 0 ρ /2 max d=,...,n d (u h ),wherec 0 = 2 s+ C s and d (u h ) := max 0 k,...,k d s k Y k d Y d u h L 2 (Ɣ;W ) (27) Here, the constant Ĉdependsonthesolution,sandN,butnotonl. Moreover, let η be the total number of collocation points, then with ξ = + log 2 ( + log 2.5) 2.. ε II Ĉ ( η ) 2 N log 2 s + log 2 η ξ+log N, (28) N Proof We follow [5] and start with rewriting the isotropic Smolyak formula (9) as A l,n = N j n = ( N ) l N n= j n j n k j l n= N n= j n l n= k=0 = [( N ) ) j n (U ] l N n= j n. N n= j n l n= Let I N : Ɣ Ɣ be the identity operator on an N-dimensional space and I (n) : Ɣ n Ɣ n be a one-dimensional identity operator for n =,...,N. Wecancomputethe

22 54 M. Motamed et al. error operator recursively, E N := I N A l,n = I N N n= j n l Noting that N write E N = n= j n l N n= j n l N n= j n l [( N ) n= j n I (N) [( N j n n= ]. ) ) (U ] l N n= j n I (N) N n= j n = A l,n and that I N = I N I (N),wecan [( N j n n= If we repeat the process, we arrive at [ N N E N = R(l, d) where Then, d=2 R(l, d) = d n= j n l ) ( ) ] I (N) U l N n= j n + E N I (N). n=d+ I (n) ] [( d j n n= ) + ( I () ) N A l, n=2 I (n), ( I (d) U l d n= j n )]. (29) ( ) N N (I N A l,n )[u h ] R(l, d) I (n) [u h ] d=2 n=d+ ( ) N + (I () A l, ) I (n) [u h ], (30) where the norms are in L 2 ρ (Ɣ; W ).Wefirstbound R. By(25), we have n=2 (I (n) U j n )(u (n) h ) L 2 ρn (Ɣ n;w n ) C s ρ n /2 2 sj n and therefore, j n (u (n) h ) L 2 ρn (Ɣ n;w n ) = (U j n U jn )(u (n) h ) L 2 ρn (Ɣ n;w n ) max k n =0,...,s k n Y n u (n) h L 2 (Ɣ n ;W n ), 2 C s ρ n /2 2 s ( j n ) max k n =0,...,s k n Y n u (n) h L 2 (Ɣ n ;W n ). (3)

23 A collocation method for the stochastic wave equation 55 By Lemma 6 and (29) and(3), we then have R(l, d)[u h ] L 2 ρ (Ɣ;W ) d n= j n l (2 C s ) d 2 ( ) l + d (2 Cs ) d = l 2 with d (u h ) given by (27). Moreover, since then by (30), we get ρ /2 2 s (l d+) d (u h ) ρ /2 2 s (l d+) d (u h ), (I () A l, )[u h ] L 2 ρ (Ɣ;W ) = (I () U l )[u () h ] L 2 ρn (Ɣ n;w n ) C s ρ /2 2 s l max k =0,...,s k Y u h L 2 (Ɣ ;W ) C s ρ /2 2 s l max k =0,...,s k Y u h L 2 (Ɣ;W ), (I N A l,n )[u h ] L 2 ρ (Ɣ;W ) 2 ρ /2 N ( ) l + d (2 C s ) d 2 s (l d+) d (u h ) l d= 2 ρ /2 s (l+) 2 max d(u h ) d=,...,n N ( ) l + d (2 s+ C s ) d. l The first inequality stated in Theorem 5 follows noting that ( l+d ) l (l + ) 2 N for d =,...,N. To show the second inequality (28), we note that the number of collocation points η at level l using the Smolyak formula with Gaussian abscissas and the polynomial degree (6) satisfies(seelemma3.7in[32]) d= log η ξ + log N l + + log 2 η N, (32) with ξ = + log 2 ( + log 2.5) 2.. From the first inequality we have (I N A l,n )[u h ] L 2 ρ (Ɣ;W ) Ĉ ( η ) 2 N s + log 2 2 N This completes the proof. log η ξ+log N. Remark 6 We note that the above estimates are uniform with respect to h in the case of smooth quantity of interest. For the solution, we have one Y -derivative uniformly bounded with respect to h in L 2 (0, T ; H 0 ()). Remark 7 (algebraic rate of convergence) In full tensor interpolation, with the minimal assumptions (2)onthedata,by(24)wehaveanuppererrorboundoforderO(η s/n ) with s = whenψ (u) = u and with any order s whenψ (u) = Q(Y ).InSmolyak interpolation, with the minimal assumptions (2), when Ψ (u) = u, then(28) implies an

24 56 M. Motamed et al. upper error bound of order O(η δ s ) with s = andsome0< δ < onlywhenn = for which d (u h ) is bounded. As we showed in Sect. 2, d (u h ),whichinvolvesmixed Y -derivatives of the solution for N 2, is not bounded. This gives an algebraic error convergence for the solution when N =. When Ψ (u) = Q(Y ), withtheminimal assumptions (2), Q(Y ) has bounded mixed Y -derivatives of any order s forn, as shown in Sect. 2. We obtain an upper error bound of order O(η δ s ) with s and some 0 < δ <. This gives a faster error convergence for the quantity of interest (3). Remark 8 (full tensor versus sparse tensor) The slowdown effect that the dimension N has on the error convergence (24) when a full tensor product is employed is known as the curse of dimensionality.thisisthemainreasonfornotusingisotropicfulltensor interpolation when N is large. On the other hand, the isotropic Smolyak approximation has a larger exponent O( log N ) in (28)comparedtoO( N ) in (24). This is a clear advantage of the isotropic Smolyak method over the full tensor method when bounded mixed Y -derivatives exist. Remark 9 (computational cost versus error) In order to find the optimal choice of the mesh size h,weneedtominimizethecomputationalcomplexityofthestochasticcollocation method, η/h d+,subjecttothetotalerrorconstraintε F h r +η s/n = TOL for the isotropic full tensor interpolation and ε S h r + η s/ log N = TOL for the isotropic Smolyak interpolation. We introduce the Lagrange functions L F = η/h d+ +λ (h r +η s/n TOL) and L S = η/h d+ +λ (h r +η s/ log N TOL),with the Lagrange multiplier λ. ByequatingthepartialderivativesoftheLagrangefunc- tions with respect to η, h,andλto zero, we obtain h r TOL/( + s (d+) rn ) and hr TOL/(+ r log N s (d+) ),makingthecomputationalworksofordertol N/s (d+)/r and TOL log N/s (d+)/r for the full tensor and Smolyak interpolations, respectively. 4.2 The case when h β l is large with β 2 We consider a finite element approximation of () using a quasi-uniform triangulation of the physical domain. Let h denote the size of the largest triangle in the triangulation and u h be the semi-discrete solution. We leave t [0, T ] and Y Ɣ continuous and discretize only the spatial variables. The semi-discrete problem reads tt u h v dx + a 2 u h v dx = f v dx. (33) We differentiate the semi-discrete equation (33) with respect to the random variable Y n.wethensetũ := Yn u h and let v =ũ t to obtain (ũ tt, ũ t ) + B[ũ, ũ t ]= A [u h, ũ t ], (34) where (v,v 2 ) := v v 2 dx, B[v,v 2 ]:= a 2 v v 2 dx,

25 A collocation method for the stochastic wave equation 57 A [v,v 2 ]:= 2 aa Yn v v 2 dx. n We observe that (ũ tt, ũ t ) = 2 dt d ũ t 2.Moreover,sincea is time-independent, L 2 () then B[ũ, ũ t ]= 2 dt d B[ũ, ũ] = 2 dt d a ũ 2 L 2 ().Furthermore,byHölder,inverse and Cauchy inequalities [], we have A [u h, ũ t ] C n u h L 2 ( n ) ũ t L 2 ( n ) C n C inv h u h L 2 ( n ) ũ t L 2 ( n ) T 2 C2 n C2 inv h 2 u h 2 L 2 () + 2 T ũ t 2 L 2 (), where C n := 2 aa Yn L ( n Ɣ n ),andc inv is the constant in the inverse inequality. From (34) wethereforeget d dt ũ t 2 L 2 () + d dt a ũ 2 L 2 () T ũ t 2 L 2 () + TC2 n C2 inv h 2 u h 2 L 2 (). (35) Now write y := ũ t 2 L 2 () + a ũ 2 L 2 (), y 2 := TC 2 n C2 inv h 2 u h 2 L 2 (). From the inequality (35), we have y (t) T y (t)+ y 2 (t).bythegronwall sinequality []andnotingthaty (0) = 0, we obtain ũ t 2 L 2 () + a ũ 2 L 2 () etc2 n C2 inv h 2 We now define the energy norm u h 2 E := ( t u h (t,.) 2 L 2 () + a u h(t,.) 2 ) L 2 (), and the Sobolev norm u h 2 S := sup t (0,T ) Y Ɣ sup ( t u h (t,.) 2 L t (0,T ) 2 () + u h(t,.) 2 ) L 2 (). Y Ɣ T 0 u h 2 L 2 dt. (36) () We consider two different cases. One case is when the uniform coercivity assumption (3)holds.Theothercaseiswhenthewavespeeda(x, Y ) may be zero or negative due to possible negative values in the random vector Y,andtherefore(3) doesnothold The case of uniformly coercive wave speed Under the uniform coercivity assumption (3), we have u h S ã min u h E, ã min := min{a min, } > 0.

26 58 M. Motamed et al. Moreover, by (36), we obtain Therefore, ũ 2 E et2 C 2 n C2 inv h 2 u h 2 S. Yn u h S e/2 TC n C inv h ã min u h S. (37) We now obtain the estimate on the growth of all mixed Y -derivatives of u h.let k Z+ N be a multi-index and k Y u h := k u h.inordertofindanupperboundforthe k Y k N YN k -th order mixed Y -derivative Y ku h,wefollow[6] andintroduceasetk of indices with cardinality n K such that Y Ku h := n K u h k K Yk = Y ku h.asanexample,letn = 5 and consider the set K ={,, 2, 3, 5, 5, 5} with n K = 7. Then the corresponding multi-index is k =[203] with k =7, and we have K Y u h = 7 u h 7 u h = Y Y Y2 Y3 Y5 Y5 Y5 Y 2 Y2 Y3 Y 3 = Y k u h. 5 Before deriving the estimates, we need the following two lemmas. Lemma 7 (generalized Leibniz rule) Given a set of indices K with cardinality n K and two functions f, g C K (Ɣ), Y K ( fg) = Y S f K\S Y g, S P(K) where P(K) represents the power set of K. Lemma 8 Let C R + and n Z +.Thenwehave i=0 C n j=0 i=0 (C + ) i (n i)! (C + ) n. (38) Proof The left hand side of (38) canbewrittenas n i ( ) ( i n n i j) C j+ = C j+ (n i)! j (n i)!. The right hand side of (38) canbewrittenas We now show that + n i= j n j= ( ) n C j = + j ( i j) (n i)! n j=0 j=0 i= j ( ) n C j+. j + ( ) n, 0 j n, (39) j +

27 A collocation method for the stochastic wave equation 59 from which the inequality (38) follows.weprove(39) byinductiononn j +. Case n = j +. In this case (39) reads whichistrue. General case. We assume that (39) holdsforn j + andshowthat n i= j ( i j) (n + i)! ( ) n +. j + We can use the induction hypothesis (39) andwrite ( n i ( j) (n + i)! = n i ( n j) j) (n i)! (n + i) +! n + j i= j i= j ( ) ( ) ( ) n n n + + n + j j + j j + n i= j ( n j ( i ( ) j) n (n i)! + j ) ( ) n + =, j + where the last equality is the Pascal s rule. Therefore, by induction the proof is complete. We are now ready to prove the following result, Theorem 6 The Y -derivatives of the semi-discrete solution u h which solves (33) can be bounded as k Y u h S k! (C + ) k u h S, C = ĈT h ã min, (40) where k Z+ N is a multi-index, and Ĉisindependentofh. Proof Let K be the index set corresponding to the multi-index k. Then, according to Lemma 7, the Y K derivative of the semi-discrete equation (33) is Y K ttu h v dx + Y S u h K\S Y a 2 v dx = 0. S P(K) Noting that P(K) = K ( P(K) \ K ),wewrite Y K ttu h v dx + a 2 Y K u h v dx = Now let v = K Y tu h and obtain where 2 d dt K Y tu h 2 L 2 () + 2 A K [v,v 2 ]:= S P(K)\K Y S u h K\S Y a 2 v dx. d dt a K Y u h 2 L 2 () = A K[u h, K Y tu h ], (4) S P(K)\K Y S v K\S Y a 2 v 2 dx.

28 520 M. Motamed et al. As before, by Hölder, inverse and Cauchy inequalities [], we have A K [u h, Y K t u h ] K\S Y a 2 L () Y S u h L 2 () Y K t u h L 2 () S P(K)\K CC inv h K Y t u h L 2 () T 2 C 2 C 2 inv h 2 ( S P(K)\K S P(K)\K S Y u h L 2 () ) 2 Y S u h L 2 () + 2 T K Y t u h 2 L 2 (), where C := max S P(K) S Y a2 L ( Ɣ), andc inv is the constant in the inverse inequality. From (4) wethereforeget d dt K Y tu h 2 L 2 () + d dt a K Y u h 2 L 2 () T C 2 Cinv 2 h 2 Y S u h L 2 () Now we write S P(K)\K 2 + T K Y tu h 2 L 2 () (42) y := K Y tu h 2 L 2 () + a K Y u h 2 L 2 (), y 2 := T C 2 Cinv 2 h 2 S P(K)\K Y S u h L 2 (). 2 From the inequality (42), we have y (t) T y (t)+ y 2 (t).bythegronwall sinequality []andnotingthaty (0) = 0, we obtain and therefore, Y K tu h 2 L 2 () + a K Y u h 2 L 2 () T et C 2 Cinv 2 h 2 Y S u h L 2 () K Y u h 2 E et2 C 2 C 2 inv h 2 sup t,y We finally obtain the formula K Y u h S C We now by induction show that 0 S P(K)\K S P(K)\K S Y u h S, S P(K)\K 2 dt, Y S u h L 2 (). C = e/2 T CC inv h ã min. (43) K Y u h S n K! (C + ) n K u h S, (44) 2

29 A collocation method for the stochastic wave equation 52 which is equivalent to the corresponding multi-index formulation (40). Case n K = 0. In this case the set K is empty, and (44) reads u h S u h S, which is true. Case n K =. In this case K ={k}, k N, and(44) reads Yk u h S (C + ) u h S, which follows from (37). General case. We now assume that (44) holds for all sets S with cardinality 0 n S n K. We have then the induction hypothesis, From (43) wehave S Y u h S n S! (C + ) n S u h S, 0 n S n K. (45) K Y u h S C C S P(K)\K n K i=0 S P(K) n S =i S Y u h S = C n K i=0 S P(K) n S =i n S! (C + ) n S u h S. S Y u h S Note that the number of subsets S of P(K) with cardinality i is ( n Ki ).Then K Y u h S C n K i=0 i! (C + ) i ( nk i ) u h S n K! (C + ) n K u h S, where the last inequality follows from Lemma 8. Thiscompletestheproof. Remark 0 We note that the optimal choice of the mesh size h in Remark 9 in Sect. 4. is obtained by assuming that the Y -derivatives of the solution up to order s,which appear in the coefficients C and Ĉ in the error estimates (24) and(28), are uniformly bounded with respect to h.intheabsenceofsuchassumption,wecanemploytheesti- mate (40)andfindthecoefficientsintheerrorbounds.Forinstance,forthefulltensor interpolation, the coefficient C in the interpolation error (24) is C Nh s. The total error is then ε F h r + Nh s η s/n = TOL.ByintroducingtheLagrangefunction L F = η/h d+ + λ (h r + Nh s η s/n TOL) and equating its partial derivatives with respect to η, h, andλ to zero, we obtain h r TOL/( + the computational work of order TOL N/s (N+d+)/r. rn s (N+d+) ),making We now define for every Y Ɣ the power series u h : C N L (0, T ; H0 ()) as (Z Y ) k u h (t, x, Z) = Y k k! u h(t, x, Y ), (46) k=0 k =k

30 522 M. Motamed et al. where k! = N n= (k n!) and Y k = N n= Y k n n.by(40) weget (Z Y ) k u h (Z) S Y k k! u h(y ) S k=0 k =k k=0 k =k k! k! (C + ) k (Z Y ) k u h (Y ) S. We exploit the generalized Newton binomial formula for v =[v,...,v N ] R N + and k Z +, and obtain k =k k! k! ( N ) k v k = v n, n= u h (Z) S ( N k (C + ) Z n Y n ) u h (Y ) S. k=0 n= Therefore, the series (46) convergesforallz C N such that Z n Y n τ < N (C + ) = O(h). By a continuation argument, the function u h can analytically be extended on the whole region (Ɣ, τ) = {Z C N, dist(ɣ n, Z n ) τ, n =,...,N}. Wenotethattheradiusofanalyticityisproportionaltoh. We now build an approximate solution u h,l to u h based on Lagrange interpolation in Y.WeinvestigateonlythecaseofatensorproductinterpolationonGauss-Legendre points as described in Sect. 3.Werecallaresultontheerrorofthebestapproximation of univariate analytic functions by polynomials [2]. Lemma 9 Let N = and Ɣ R be bounded. Set W := L 2 (0, T ; L 2 ()). Then, given a function v(y ) L (Ɣ; W ) which admits an analytic extension in the region of the complex plane (Ɣ, τ) ={Z C, dist(ɣ, Z) τ},forsomeτ > 0,thereholds E p (v) := 2 min v v 0 L v 0 W H (Ɣ;W ) p e σ e σ p max v(z) W, (47) Z where 0 < σ = log ( 2τ Ɣ + + 4τ 2 Ɣ 2 ). In the above lemma, τ is smaller than the distance between Ɣ and the closest singularity of the extended function v(z) : C W in the complex plane. In the multidimensional case when N 2, we note that σ n = log ( 2τ Ɣ n + + 4τ 2 ) Ɣ n 2 depends on the direction n. Wethereforeset σ = min n N min σ n, Ŷ n ˆƔ n M (v) = max n N max Ŷ n ˆƔ n max v(z) W. Z (Ɣ,τ) Similar to the proof of Theorem 4, using(22) and(47), we can show that for the isotropic full tensor product interpolation formula (8), with u h,l = I l,n [u h ],the

31 A collocation method for the stochastic wave equation 523 interpolation error ε II defined in (2) satisfies ε II = u h u h,l L 2 ρ (Ɣ;W ) 2 NM (u h ) e σ p(l). (48) We now consider the Smolyak interpolation formula (20) based on Gaussian abscissas when the formula (6) isusedandletu h,l = A l,n [u h ].Similartotheproofof Lemma 3.6 in [32], we can show that the interpolation error ε II defined in (2) satisfies { e σ l e log 2, 0 l log N 2 ε II = u h u h,l L 2 ρ (Ɣ;W ) Ĉg(l), g(l) =, e σ N 2 l/n, otherwise, with Ĉ = C 0 C0 N 2 C 0 and C 0 = 6 M (u h ) e 4 σ e 2 σ ( + log π 2 2 σ ). From (48) and(49), we note that for both full tensor and Smolyak interpolations, since σ = O(h),wewillhaveafastexponentialdecayintheerrorwhentheproduct h l is large. As a result, with a fixed h,theerrorconvergenceisslow(algebraic)fora small l and fast (exponential) for a large l.moreover,therateofconvergencedeteriorates as h gets smaller. These results are precisely what we observe in the numerical experiments presented in Sect. 5. (49) The case of non-coercive wave speed We now relax the uniform coercivity assumption (3)andinsteadassumethat 0 a min a(x, ω) a max <, x, ω Ω. We apply the inverse inequality to (36) andwrite ũ 2 E etc2 n C4 inv h 4 et 2 C 2 n C4 inv h 4 T et 2 C 2 n C4 inv h 4 (T et 4 C 2 n C4 inv h 4 u h 2 E. 0 u h 2 L 2 () dt sup u h (t) 2 L t (0,T ) 2 () sup t u h (t) L 2 () + u h (0) L 2 ()) 2 t (0,T ) In the last inequality, we assume for simplicity that u h (0) = 0. Similar to Sect. 4.2., we obtain k Y u h E k! (C 0 + ) k u h E, C 0 := e/2 T 2 CC 2 inv h 2. (50) Therefore, the series (46) convergesforallz C N such that Z n Y n τ < N (C 0 + ) = O(h 2 ).Comparingthiswiththecasewhenthecoercivityassump-

32 524 M. Motamed et al. Fig. 2 Two layered computational domain layer Layer2 tion (3) holds,we observe that as h decreases the radius of analyticity shrinks faster (proportional to h 2 ) for the non-coercive case than for the coercive case (proportional to h). We obtain the same estimates as (48) and(49) withσ = O(h 2 ).Wewill therefore have a fast exponential decay in the error when the product h 2 p(l) is large. Note that these estimates may not be sharp, as the numerical test 2 in Sect. 5 suggests that σ O(h.2 ).Thismayberelatedtotheuseofinverseinequality(whichisnot sharp) twice while obtaining (50). 5Numericalexamples In this section, we consider the IBVP () inatwodimensionallayeredmedium.we numerically simulate the problem by the stochastic collocation method and study the convergence of the statistical moments of the solution u,thelinear quantity of interest (3) andanonlinearquantityofinterestcalledtheariasintensity[] I A (Y ) = T 0 S u tt (t, x, Y ) 2 dx dt, (5) where, S is a sub-domain of the physical domain, andt is a positive final time. We show that the computational results are in accordance with the convergence rates predicted by the theory. We consider a rectangular physical domain =[ L x, L x ] [ L z, 0] and a random wave speed a of form (5) for a two-layered medium (N = 2). The computational domain containing two layers with widths d and d 2 is shown in Fig. 2. The deterministic solver employs a finite difference scheme based on second-order central difference approximation. Let x = 2 L x N x and z = L z N z denote the spatial grid-lengths, where N x and N z are natural numbers. For i = 0,,...,N x and j = 0,,...,N z,let(x i, z j ) = ( L x + i x, L z + j z) and u i, j (t) denote the corresponding grid point and the grid function approximating u(t, x i, z j ),respectively. On this spatial grid, we discretize the PE in ()andobtainthesemi-discretization,

33 A collocation method for the stochastic wave equation 525 d 2 u i, j (t) dt 2 = ( ) a 2 x i+ 2, j +iu i, j (t) a 2 i 2, j iu i, j (t) + ( ) a 2 z i, j+ + j u i, j (t) a 2 2 i, j 2 j u i, j (t) + f i, j (t). Here, + and are forward and backward first-order difference operators, respectively. We then use the second-order central difference approximation in time to obtain the fully discrete deterministic scheme. In the stochastic space, we use the isotropic Smolyak formula (20) basedongaussianabscissas,describedinsect.3. We perform four numerical tests. In the first test, we consider a zero force term and smooth initial data and study the mean and standard deviation of the solution u. In the second test, we consider the same data as in the first test and select random variables so that the uniform coercivity assumption (3) is not satisfied, and we have amin 2 = 0. We study the expected value of the solution u in this case and compare it with the case when amin 2 > 0. In the third test, we consider zero initial data and a discontinuous time-independent forcing term and study the quantity of interest (3). Finally, in the fourth test, we study the Arias intensity (5) onthefreesurfacedueto arickerwavelet[7]. In all computations, we use a time step-size t = x/5which guarantees the stability of the deterministic numerical solver. We use homogeneous Neumann boundary conditions in all tests. 5. Numerical test In the first test, we choose a computational domain =[ 2, 2] [ 3.5, 0] with two layers with widths d = 0.5 andd 2 = 3. We consider a wave speed of form (5) with a 0 = 0, α = 2andα 2 = 3, and let Y n U(0., 0.5), n =, 2, be two independent and uniformly distributed random variables. We set f = g 2 = 0andconsideran initial Gaussian wave pulse, g (x, z) = e (x xc)2 2 σx 2 (z zc)2 2 σz 2. For computing the convergence rate of error, we consider a set of spatial grid-lengths x = z = 0., 0.05, 0.025, For each grid-length x = h,weconsiderdifferent levels l and compute the L 2 -norm of error in the expected value of the solution at a fixed time t = T by ε h (l) = E [ /2 ] u h,l (T, x) E [uref ] (T, x) 2 dx. Here, the reference solution u ref is computed with a high level l ref for a fixed x = h. 5.. An irregular solution We first put the center of the initial pulse at (x c, z c ) = (0, ) and let σ x = σ z = 0.2. The initial solution is then in both layers and does not vanishes on the inter-

34 526 M. Motamed et al. Fig. 3 Test. The initial solution (top), the expected value of the solution (middle) and the standard deviation of the solution (bottom)at t = face. In this case, since the smooth initial solution does not satisfy the interface jump conditions (8), the solution is not highly regular in Y. In fact, we only have u Y L (Ɣ; C 0 (0, T ; L 2 ())), andtherefore,thesolutionhasonlyonebounded Y -derivative and no bounded mixed derivatives in Y.Figure3shows the initial solution and the expected value and standard deviation of the solution at time t =, computed with level l = 5and x = z = Figure 4 shows the L 2 -norm of error in the expected value of the solution at T = versus the number of collocation points η(l). We observe a slow convergence of order O(η δ ) with 0 < δ <, as expected due to low Y -regularity of the solution. We also note that for large values of h η, we