Analysis and Computation of Hyperbolic PDEs with Random Data

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1 Analysis and Computation of Hyperbolic PDEs with Random Data Mohammad Motamed 1, Fabio Nobile 2,3 and Raúl Tempone 1 1 King Abdullah University of Science and Technology, Thuwal, Saudi Arabia 2 EPFL Lausanne, Switzerland 3 Politecnico di Milano, Milan, Italy March 20, Introduction Hyperbolic partial differential equations (PDEs) are mathematical models of wave phenomena, with applications in a wide range of scientific and engineering fields such as electromagnetic radiation, geosciences, fluid and solid mechanics, aeroacoustics, and general relativity. The theory of hyperbolic problems, including Friedrichs and Kreiss theories, has been well developed based on energy estimates and the method of Fourier and Laplace transforms [8, 16]. Moreover, stable numerical methods, such as the finite difference method [14], the finite volume method [17], the finite element method [6], spectral methods [4], and the boundary element method [11], have been proposed to compute approximate solutions of hyperbolic problems. However, the development of the theory and numerics for hyperbolic PDEs has been based on the assumption that all input data, such as coefficients, initial data, boundary and force terms, and computational domain, are exactly known. 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 either an intrinsic variability of the physical system (aleatoric uncertainty), or our ignorance or inability to accurately characterize all input data (epistemic uncertainty). For example, in earthquake modeling, seismic waves propagate in a geological region where, due to soil spatial variability and the uncertainty of measured soil parameters, both kinds of uncertainties are present. Consequently, the field of uncertainty quantification (UQ) has arisen as a new scientific discipline. UQ is the science of quantitative characterization, reduction, and propagation of uncertainties and the key for achieving validated predictive computations. The numerical solution of hyperbolic problems with random inputs is a new branch of UQ, relying on a broad range of mathematics and statistics groundwork with associated algorithmic and computational development, and aiming at accurate and fast propagation of uncertainties and the quantification of uncertain simulation-based predictions. The most popular method for solving PDEs in probabilistic setting is the Monte Carlo sampling, see for instance [7]. It consists in generating independent realizations drawn from the input mohammad.motamed@kaust.edu.sa fabio.nobile@epfl.ch raul.tempone@kaust.edu.sa 1

2 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 [10, 22, 38, 3, 32] and Stochastic Collocation [2, 27, 26, 37] are among these techniques. Such new techniques have been successfully applied to stochastic elliptic and parabolic PDEs. In particular, it is shown that, under particular assumptions, the solution of these problems is analytic with respect to the input random variables [2, 25]. The convergence results are then derived from the regularity results. For stochastic hyperbolic problems, the analysis was not well developed until very recently, see [24, 23, 1]. In the case of linear problems, there are a few works on the one-dimensional scalar advection equation with a time- and space-independent random wave speed [36, 13, 31]. Such problems also possess high regularity properties provided the data live in suitable spaces. The main difficulty however arises when the coefficients vary in space or time and are possibly non-smooth. 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 [24, 23, 1]. There are also recent works on stochastic nonlinear conservation laws, see for instance [18, 19, 28, 34, 35]. In the present notes, we assume that the uncertainty in the input data is parametrized 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 a finite number of random variables when the input data vary slowly in space, 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 [20, 21] or polynomial chaos expansion [39]. We will address theoretical issues of well-posedness and stochastic regularity, as well as efficient treatments and computational error analysis for hyperbolic problems with random inputs. We refer to [24, 23, 1] for further details. 2 Problem Statement In this section, for simplicity, we consider the linear second order scalar acoustic wave equation with random wave speed and deterministic boundary and initial conditions. We motivate and describe the source of randomness and address the well-posedness of the problem. For extensions to the system of elastic wave equations see [23, 1]. Let D be a convex bounded polygonal domain in R d, d = 2, 3, 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, 1] is a probability measure. Consider the stochastic initial boundary value problem (IBVP): find a random function u : [0, T ] D Ω R, such that P -almost everywhere in Ω, 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) in [0, T ] D Ω, u(0, x, ω) = g 1 (x), u t (0, x, ω) = g 2 (x) on {t = 0} D Ω, (1) u(t, x, ω) = 0 on [0, T ] D Ω. 2

3 Here, the solution u is the displacement, and t and x = (x 1,..., x d ) are the time and location, respectively, and the data f L 2 ((0, T ); L 2 (D)), g 1 H 1 0 (D), g 2 L 2 (D), (2) are compatible. The only source of randomness is the wave speed a which is assumed to be bounded and uniformly coercive, 0 < a min a(x, ω) a max <, almost everywhere in D, a.s. (3) Assumption (3) guarantees that the energy is conserved and therefore the stochastic IBVP (1) is well-posed [24]. In many wave propagation problems, the source of randomness can be described or approximated by 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 is therefore the number of layers. In this case the randomness is described by a finite number of random variables. Another situation is when the wave speeds in layers are given by random fields, which in turn are approximated by a truncated Karhunen-Loéve expansion. Hence, the number of random variables corresponds to the number of layers as well as 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 1 (ω),..., Y N (ω)), almost everywhere in D, a.s, (4) where N N + and Y = [Y 1,..., 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=1 Γ n and assume further that the random vector Y has a bounded joint probability density function ρ : Γ R + with ρ L (Γ). We note that by using a similar approach to [2, 5] we can also treat unbounded random variables, such as Gaussian and exponential variables. The finite dimensional noise assumption (4) implies that the solution of the stochastic IBVP (1) can be described by only N random variables, i.e., u(t, x, ω) = u(t, x, Y 1 (ω),..., Y N (ω)). This turns the original stochastic problem into a deterministic 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 ), where t [0, T ], x D, and Y Γ. Note that the knowledge of u = u(t, x, Y ) fully determines the law of the random field u = u(t, x, ω). Here, as an example of form (4), we consider a random speed a given by a(x, ω) = a 0 (x) + N a n (x, ω) I Dn (x), a n (x, ω) = Y n (ω) α n (x), α n C (D n ), (5) n=1 where, I is the indicator function. In this case, D is a heterogeneous medium consisting of N nonoverlapping sub-domains {D n } N n=1, {Y n } N n=1 are independent random variables, and {α n } N n=1 are smooth functions defined on sub-domains. The boundaries of sub-domains, which are interfaces of speed discontinuity, are assumed to be smooth. The ultimate goal is the prediction of statistical moments of the solution u or statistics of some given quantities of physical interest. As an example, we consider the following quantity, T Q(Y ) = u(t, x, Y ) φ(x) dx dt + u(t, x, Y ) ψ(x) dx, (6) 0 D 3 D

4 where u solves (1) and the mollifiers φ and ψ are given functions of x. 3 Non-intrusive Numerical Methods There are in general three types of methods for propagating uncertainty in PDE models with random inputs; intrusive, non-intrusive, and hybrid methods. Intrusive methods, such as perturbation expansion, intrusive polynomial chaos [39], and stochastic Galerkin [10], require extensive modifications in existing deterministic solvers. On the contrary, non-intrusive methods, such as Monte Carlo and Stochastic Collocation, are sample-based approaches. They rely on a set of deterministic models corresponding to a set of realizations, and hence require no modification to the existing deterministic solvers. Finally, hybrid methods are a mixture of both intrusive and non-intrusive approaches. Non-intrusive (or sample-based) methods are attractive in the sense that they require only a deterministic solver for computing deterministic models. In addition, since the deterministic models are independent, it is possible to distribute them onto multiple processors and perform parallel computation. The most popular non-intrusive technique is the Monte Carlo method. While being very flexible and easy to implement, this technique features a very slow convergence rate. The Multi-Level Monte Carlo method has been proposed to accelerate the slow convergence of Monte Carlo sampling [12]. Quasi Monte Carlo methods can be considered as well, that aim at achieving higher rates of convergence [15]. The stochastic collocation (SC) method is another non-intrusive technique, in which, regularity features of the quantity of interest with respect to the input parameters can be exploited to obtain a much faster or possibly spectral convergence. In SC, the problem(1) is first discretized in space and time, using a deterministic numerical method. The obtained semi-discrete problem is then collocated in a set of η collocation points {Y (k) } η k=1 Γ to compute the approximate solutions u h(t, x, Y (k) ), where h represent the discretization mesh/grid size. A global polynomial approximation is then built upon those evaluations, u h,η (t, x, Y ) = η u h (t, x, Y (k) ) L k (Y ), k=1 for suitable multivariate polynomial bases {L k } η k=1 such as Lagrange polynomials. Finally, using the Gauss quadrature formula, we can easily approximate the statistical moments of the solution. For instance, E[u(., Y )] E[u h,η (., Y )] = u h,η (., Y ) ρ(y ) dy = η u h (., Y (k) ) k=1 Γ Γ L k (Y ) ρ(y ) dy η u h (., Y (k) ) θ k. A key point in SC is the choice of the set of collocation points {Y (k) }, i.e. the type of computational grid in the N-dimensional stochastic space. A full tensor grid, based on cartesian product of mono-dimensional grids, can only be used when the number of stochastic dimensions N is small, since the computational cost grows exponentially fast with N (curse of dimensionality). Alternatively, sparse grids can reduce the curse of dimensionality. They were originally introduced by Smolyak for high dimensional quadrature and interpolation computations [30]. In the following we will briefly review the sparse grid construction. k=1 4

5 Let j Z N + be a multi-index containing non-negative integers. For a non-negative index j n in j, we introduce a sequence of one-dimensional polynomial interpolant operators U jn : C 0 (Γ n ) P p(jn)(γ n ) of degree p(j n ) on p(j n ) + 1 suitable knots. With U 1 = 0, we define the detail operator jn := U jn U jn 1. Finally, introducing a sequence of index sets I(l) Z N +, the sparse grid approximation of u : Γ V at level l reads u η(l,n) (., Y ) = N jn [u](., Y ). (7) j I(l) n=1 Furthermore, in order for the sum (7) to have some telescopic properties, which are desirable, we impose an additional admissibility condition on the set I [9]. An index set I is said to be admissible if j I, j e n I for 1 n N, j n 1, holds. Here, e n is the n-th canonical unit vector. To fully characterize the sparse approximation operator in (7), we need to provide the following: A level l N and a function p(j) representing the relation between an index j and the number of points in the corresponding one-dimensional polynomial interpolation formula U j. A sequence of sets I(l). Typical examples include total degree and hyperbolic cross grids. The family of points to be used, such as Gauss or Clenshaw-Curtis abscissae, [33]. The rate of convergence for SC depends on the stochastic regularity of the output quantity of interest, which may be the solution or some given functional of the solution. For example, a fast spectral convergence is possible for highly regular outputs. We will discuss this issue in more details in the next section. 4 Stochastic Regularity and Convergence of Stochastic Collocation As extensively discussed in [24, 23, 1], the error in the stochastic collocation method is related to the stochastic regularity of the output quantity of interest (the solution or a given functional of the solution). It has been shown that, under particular assumptions, the solution of stochastic elliptic and parabolic PDEs is analytic with respect to the input random variables [2, 25]. In contrast, the solution of stochastic hyperbolic PDEs may possess lower regularity. In this section, first, we will convey general stochastic regularity properties of hyperbolic PDEs through simple examples. Then, we state the main regularity results followed by the convergence results for SC. 4.1 Examples: General Stochastic Regularity Properties Example 1. Consider the 1D Cauchy problem for the scalar wave equation, u tt (t, x, y) y 2 u xx (t, x, y) = 0, in [0, T ] R, u(0, x, y) = g(x), u t (0, x, y) = 0, on {t = 0} R, 5

6 with g C0 (R). The wave speed is constant and given by a single random variable y. We want to investigate the regularity of the solution u with respect to y. For this purpose, we extend the parameter y into the complex plane and study the extended problem in the complex plane. It is well known that if the extended problem is well-posed and the first derivative of the resulting complexvalued solution with respect to the parameter satisfies the so called Cauchy-Riemann conditions, the solution can analytically be extended into the complex plane, and hence u will be analytic with respect to y. We therefore let y = y R + i y I, where y R, y I R, and apply the Fourier transform with respect to x to obtain û(t, k, y) = ĝ(k) ( e i y k t + e i y k t), 2 where û and ĝ are the Fourier transforms of u and g with respect to x, respectively. Hence, û(t, k, y) ĝ(k) e y I k t. Therefore, the Fourier transform of the solution û(t, k, y) grows exponentially fast in time, unless the Fourier transform of the initial solution ĝ(k) decays faster than e y I k t. In order for the Cauchy problem to be well-posed, g needs to belong to the Gevrey space G q (R) of order q < 1 [29]. For ɛ k 1/q such functions, we have ĝ(k) C e for positive constants C and ɛ, and hence the problem is well-posed in the complex strip Σ r = {(y R + i y I ) C : y I r}. Note that G 1 (R) is the space of analytic functions. If g G 1 (R), the problem is well-posed only for a finite time interval when t ɛ/r. This shows that even if the initial solution g is analytic, the solution u is not analytic with respect to y for all times in Σ r. Example 2. Consider the 1D Cauchy problem for the scalar wave equation in a domain consisting of two homogeneous half-spaces separated by an interface at x = 0, u tt (t, x, Y ) ( a 2 (x, Y ) u x (t, x, Y ) ) = 0, in [0, T ] R, x u(0, x, Y ) = g( x), u t (0, x, Y ) = a g ( x), on {t = 0} R, with g C0 (0, ). The wave speed is piecewise constant and a function of a random vector of two variables Y = [y, y + ], { y, x < 0, a(x, Y ) = y +, x > 0. By d Alembert s formula and the interface jump conditions at x = 0, the solution reads u(t, 0, Y ) = u(t, 0 +, Y ), y 2 u x (t, 0, Y ) = y 2 + u x (t, 0 +, Y ), (8) u(t, x, Y ) = { g(y t x) + y y+ y +y + g(y t + x), x < 0, 2 y y +y + g( y y + (y + t x)), x > 0. (9) Clearly, the solution (9) is infinitely differentiable with respect to both parameters y and y + in (0, + ). Note that the smooth initial solution u(0, x, Y ), which is contained in one layer with zero value at the interface, automatically satisfies the interface conditions (8) at time zero. Otherwise, if for instance the initial solution crosses the interface without satisfying (8), a singularity is introduced in the solution, and the high regularity result does not hold any longer. 6

7 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 D I and D II are given by [u(t,., Y )] Υ = 0, [a 2 (., Y ) u n (t,., Y )] Υ = 0. (10) 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 (10). 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 [24] for more details. We note that the above high stochastic regularity result is valid only for particular types of smooth data. In real applications, the data are not smooth. Let us now consider a more practical datum. Example 3. Consider the 1D Cauchy problem for the scalar wave equation in a domain consisting of two homogeneous half-spaces separated by an interface at x = 0, u tt (t, x, y) ( a 2 (x, y) u x (t, x, y) ) = 0, in [0, T ] R, x u(0, x, y) = g(x), u t (0, x, y) = 0, on {t = 0} R, with g H0 1 (R) being a hat function with a narrow support supp g = [x 0 α, x 0 +α], with 0 < α 1, located at x 0 < α. The wave speed is piecewise constant and a function of a random variable y, { 1, x < 0, a(x, y) = y, x > 0. Similar to Example 2, we can obtain the closed form of the solution as 1 2 g(t + x), x < x 0 t 12 g( t + x) + 1 y 2 (1+y) u(t, x, y) = g( t x), x 0 t < x < y g( t + x y ), 0 < x < x 0 + y t 0, x > x 0 + y t Since g H 1 0, there is only one bounded derivative y u, and higher bounded y-derivatives do not exist. However, if we consider a quantity of interest Q(y) as in (6) with mollifiers φ, ψ C 0 (R) vanishing at x < 0, we can employ integration by parts and shift the derivatives on g to the mollifiers. Therefore, although the solution u has only one bounded y-derivative, the quantity of interest Q is smooth with respect to y. Remark 1. Immediate results of the above three examples are the following: 1. For the solution of the 1D Cauchy problem for the wave equation to be analytic with respect to the random wave speed at all times in a given complex strip Σ 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 < 1. Moreover, if the problem is well-posed and the data are analytic, the solution may be analytic with respect to the parameter in Σ r only for a short time interval. 2. In a 1D heterogeneous medium with piecewise smooth wave speeds, if the data are smooth and the initial solution satisfies the interface jump conditions (10), the solution to the Cauchy problem is smooth with respect to the wave speeds. If the initial solution does not satisfy (10), the solution is not smooth with respect to the wave speeds. 3. In general, the solution u has only finite stochastic regularity. The stochastic regularity of functionals of the solution (such as mollified quantities of interest) can however be considerably higher than the regularity of the solution. 7

8 4.2 General Results: Stochastic Regularity We now state some regularity results in the more general case when the data satisfy the minimal assumptions (2). See [24, 23] for proofs. Theorem 1. For the solution of the stochastic IBVP (1) 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 (D)), Y Γ. Theorem 2. Consider the quantity of interest Q(Y ) in (6) and let the smooth mollifiers φ, ψ C 0 (D) vanish at the discontinuity interfaces. Then with the assumptions of Theorem 1, we have Q C (Γ). In practical applications, quantities of interest, such as Arias intensity, spectral acceleration, and von Mises stress, are usually nonlinear in u. In such cases, the high stochastic regularity property might not hold. However, one can perform a low-pass filtering (LPF) on the low regular solutions or quantities of interest by convolving them with smooth kernels such as Gaussian functions, K δ (x) = 1 ( 2 π δ) d exp ( x 2 2 δ 2 ), x R d. (11) Here, the standard deviation δ is inversely proportional to the maximum frequency that is allowed to pass. For instance, the filtered solution is given by the convolution Q δ (u)(t, x, Y ) = (u K δ )(t, x, Y ) = K δ (x x) u(t, x, Y ) d x. (12) The filtered solution (12) is of a type similar to the quantity of interest (6) with smooth mollifiers. However, the main difference here is the boundary effects introduced by the convolution. Therefore, in the presence of a compactly supported smooth kernel K C 0 (D) as mollifier, the quantity (12) will have high Stochastic regularity on a smaller domain x D δ D with dist( D δ, D) d > 0. We choose d so that K δ (x 0 ) with x 0 = d is essentially zero. This implies that for any x D δ, the support of K δ (x x) is essentially vanishing at D. We note that by performing filtering, we introduce an error u Q δ (u) which is proportional to O(δ 2 ). This error needs to be taken into account. We refer to [1] for a more rigorous error analysis of filtered quantities. 4.3 Convergence Results for Stochastic Collocation In order to obtain a priori estimates for the total error u u h,η W L 2 ρ (Γ), with W := L 2 (0, T ; L 2 (D)), we split it into two parts D ε := u u h,η u u h + u h u h,η =: ε I + ε II. (13) The first term ε I controls the convergence of the deterministic numerical scheme with respect to the mesh size h and is of order O(h r ), where r 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. The second term ε II is derived as follows from the stochastic regularity results. See [24] for proofs. 8

9 Theorem 3. Consider the isotropic full tensor product interpolation. Then ε II C η s/n. (14) where the constant C = C s N n=1 max k=0,...,s D k Y n u h,l L (Γ;W ) does not depend on l. Theorem 4. Consider the Smolyak sparse tensor product interpolation based on Gauss-Legendre abscissas, and let u h,η be given by (7). Then for the solution u h with s 1 bounded mixed Y - derivatives, where the constant ε II C ( 1 + log 2 1 Cs N C = C s ρ 1/2 1 C s depends on u h, s, and N, but not on l. η ) 2 N log 2 η s ξ+log N, ξ = 1 + log 2 (1 + log2 1.5) 2.1, (15) N max d=1,...,n max 0 k 1,...,k d s Dk1 Y 1... D k d Y d u h L 2 (Γ;W ), Remark 2. It is possible to show that the semi-discrete solution u h can analytically be extended on the whole region Σ(Γ, τ) = {Z C N, dist(γ n, Z n ) τ, n = 1,..., N}, with the radius of analyticity τ = O(h) [24]. This can be used to show that for both full tensor and Smolyak interpolation, we will have a fast exponential decay in the error when the product h p(l) is large. As a result, with a fixed h, the error convergence is slow (algebraic) for a small l and fast (exponential) for a large l. Moreover, the rate of convergence deteriorates as h gets smaller. Remark 3. The main parameters in the computations include h, η(l), and possibly δ if filtered quantities are used. In order to find the optimal choice of the parameters, we need to minimize the computational complexity of the SC method, subject to the total error constraint ε = TOL. We refer to [1] for more details. References [1] I. Babuška, M. Motamed, and R. Tempone. A stochastic multiscale method for the elastodynamic wave equations arising from fiber composites Preprint. [2] I. Babuška, F. Nobile, and R. Tempone. A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal., 45: , [3] I. Babuška, R. Tempone, and G. E. Zouraris. Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation. Computer Methods in Applied Mechanics and Engineering, 194: , [4] J. P. Boyd. Chebyshev and Fourier Spectral Methods. Springer, [5] J. Charrier. Strong and weak error estimates for elliptic partial differential equations with random coefficients. IMA J. Numer. Anal., 50: , [6] K. Eriksson, D. Estep, P. Hansbo, and C. Johnson. Computational differential equations. Studentlitteratur,

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