Numerical approximation and simulation of the stochastic wave equation on the sphere
NNUMERICAL APPROXIMATION AND SIMULATION OF THESTOCHASTIC WAVE EQUATION ON THE SPHERE
DAVID COHEN AND ANNIKA LANG
Abstract.
Solutions to the stochastic wave equation on the unit sphere are approximatedby spectral methods. Strong, weak, and almost sure convergence rates for the proposednumerical schemes are provided and shown to depend only on the smoothness of the drivingnoise and the initial conditions. Numerical experiments confirm the theoretical rates. Thedeveloped numerical method is extended to stochastic wave equations on higher-dimensionalspheres and to the free stochastic Schr¨odinger equation on the unit sphere. Introduction
The recent years have witnessed a strong interest in the theoretical study of (regularity)properties and the simulation of random fields, especially the ones that are defined by stochas-tic partial differential equations (SPDEs) on Euclidean spaces. This increase in the interest inrandom fields is due to the huge demand from applications as diverse as models for the motionof a strand of DNA floating in a fluid [12], climate and weather forecast models [17], modelsfor the initiation and propagation of action potentials in neurons [15], random surface growmodels [21], porous media and subsurface flow [7], or modeling of fibrosis in atrial tissue [9],for instance.Yet, leaving the (by now well understood) Euclidean setting, theoretical results on randomfields on Riemannian manifolds have just started to pop up in the literature. So far, thisresearch has mostly focused on random fields on the sphere, e. g., [25, 26, 31, 29, 27, 30] andreferences therein. The interest of random fields on spheres is essentially driven by the factthat our planet Earth is approximately a sphere.One example of an SPDE on the sphere and the main subject of the numerical analysis ofthis work is the stochastic wave equation ∂ tt u ( t ) − ∆ S u ( t ) = ˙ W ( t ) , driven by an isotropic Q -Wiener process. For details on the notation, see below. Besides theintrinsic mathematical interest, one motivation to study this equation comes from [6]. Thiswork proposes and analyzes stochastic diffusion models for cosmic microwave background(CMB) radiation studies. Such models are given by damped wave equations on the spherewith random initial conditions. Since fluctuations in CMB observations may be generated Mathematics Subject Classification.
Key words and phrases.
Gaussian random fields, Karhunen–Lo`eve expansion, spherical harmonic functions,stochastic partial differential equations, stochastic wave equation, stochastic Schr¨odinger equation, sphere,spectral Galerkin methods, strong and weak convergence rates, almost sure convergence.Acknowledgment. The work of DC was partially supported by the Swedish Research Council (VR) (projectnr. 2018-04443). The work of AL was partially supported by the Swedish Research Council (VR) (project nr.2020-04170), by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knutand Alice Wallenberg Foundation, and by the Chalmers AI Research Centre (CHAIR). a r X i v : . [ m a t h . NA ] F e b D. COHEN AND A. LANG by errors in the CMB map, contamination from the galaxy or distortions in the optics of thetelescope [6], one may be interested in considering a driving noise living on the sphere.Unfortunately, to this day, available and well-analyzed algorithms for an efficient simulationof random fields on manifolds do not match the current demand from applications. To namea few results from the literature on numerics for SPDEs on manifolds: the paper [29] provesrates of convergence for a spectral discretization of the heat equation on the sphere drivenby an additive isotropic Gaussian noise; convergence rates of multilevel Monte Carlo finiteand spectral element discretizations of stationary diffusion equations on the unit sphere withisotropic lognormal diffusion coefficients are considered in [19]; [8] proposes a simulationmethod for Gaussian fields defined over spheres cross time; a numerical approximation tosolutions to random spherical hyperbolic diffusions is analyzed in [6]; rates of convergence ofapproximation schemes to solutions to fractional SPDEs on the unit sphere are shown in [2];the work [22] studies a numerical scheme for simulating stochastic heat equations on theunit sphere with multiplicative noise; in [18] multilevel algorithms for the fast simulation ofnonstationary Gaussian random fields on compact manifolds are analyzed. We are not awareof any results on numerical approximations of stochastic wave equations on manifolds.In the present publication, we derive a representation of the infinite-dimensional analyticalsolution of the stochastic wave equation on the sphere driven by an isotropic Q -Wiener noise.This needs to be numerically approximated in order to be able to efficiently generate samplepaths. The proposed algorithm is given by the truncation of a series expansion of the analyticalsolution, see (5). We prove strong and almost sure convergence rates of the fully discreteapproximation scheme in Proposition 4.1. This is then used to show weak convergence resultsin Proposition 4.3 and Proposition 4.5. It turns out that these rates depend only on the decayof the angular power spectrum of the driving noise and the smoothness of the initial conditionwhile they are independent of the chosen space and time grids. We show that depending onthe smoothness of test functions, we obtain up to twice the strong order of convergence.These results are shown for the stochastic wave equation on the unit sphere S and then,strong and almost sure convergence results are extended to higher-dimensional spheres S d − .Finally we obtain similar results for a related equation, namely the free stochastic Schr¨odingerequation on the sphere driven by an isotropic noise. Observe that the extension of our resultsto damped and nonlinear problems is not straightforward and needs further analysis. Inparticular, one would have to deal with additional errors in the space and time discretization.A peculiarity in the present approach is that we are able to obtain two equations forthe position and velocity component of the stochastic wave equation that can be simulatedseparately but with respect to two correlated driving noises. Therefore we put some focuson the properties of these correlated random fields and their simulation, see Proposition 3.1.With these in place we are able to show convergence of the position even when the seriesexpansion of the velocity does not converge.The outline of the paper is as follows: In Section 2 we recall definitions of isotropic Gaussianrandom fields on S , of the Karhunen–Lo`eve expansion in spherical harmonic functions ofthese fields from [31, 29], and of Wiener processes on the sphere. This then allows us todefine the stochastic wave equation on the sphere in Section 3 and analyze its propertiesbased on the semigroup approach. In Section 4 we approximate solutions to the SPDE withspectral methods. In addition, we provide convergence rates of these approximations in the p -th moment, in the P -almost sure sense, and in the weak sense. Details on the numericalimplementation of the studied discretizations are also presented in this section. Numericalillustrations of our theoretical findings are given in Section 5. Although the main focus of PPROXIMATION OF THE STOCHASTIC WAVE EQUATION ON THE SPHERE 3 the paper is the stochastic wave equation on the unit sphere S , we include two extensionsin the last section that can be solved with the developed theory. Namely, an extension ofthe corresponding results to higher-dimensional spheres S d − ⊂ R d and an efficient algorithmfor simulating the free stochastic Schr¨odinger equation on the sphere with its convergenceproperties.2. Isotropic Gaussian random fields and Wiener processes on the sphere
We recall some notions and results, mostly from [29], in order to be able to define SPDEson the sphere in the next section.Throughout, we denote by (Ω , A , ( F t ) t , P ) a complete filtered probability space and write S for the unit sphere in R , i. e., S = (cid:8) x ∈ R , (cid:107) x (cid:107) R = 1 (cid:9) , where (cid:107) · (cid:107) R denotes the Euclidean norm. Let ( S , d ) be the compact metric space with thegeodesic metric given by d ( x, y ) = arccos ( (cid:104) x, y (cid:105) R )for all x, y ∈ S . We denote by B ( S ) the Borel σ -algebra of S .To introduce basis expansions often also called Karhunen–Lo`eve expansions of a Q -Wienerprocess on the sphere, we first need to define spherical harmonic functions on S . We recallthat the Legendre polynomials ( P (cid:96) , (cid:96) ∈ N ) are for example given by Rodrigues’ formula (see,e. g., [33]) P (cid:96) ( µ ) = 2 − (cid:96) (cid:96) ! ∂ (cid:96) ∂µ (cid:96) ( µ − (cid:96) for all (cid:96) ∈ N and µ ∈ [ − , associated Legendre functions ( P (cid:96),m , (cid:96) ∈ N , m = 0 , . . . , (cid:96) ) by P (cid:96),m ( µ ) = ( − m (1 − µ ) m/ ∂ m ∂µ m P (cid:96) ( µ )for (cid:96) ∈ N , m = 0 , . . . , (cid:96) , and µ ∈ [ − , surface spherical harmonicfunctions Y = ( Y (cid:96),m , (cid:96) ∈ N , m = − (cid:96), . . . , (cid:96) ) as mappings Y (cid:96),m : [0 , π ] × [0 , π ) → C , which aregiven by Y (cid:96),m ( ϑ, ϕ ) = (cid:115) (cid:96) + 14 π ( (cid:96) − m )!( (cid:96) + m )! P (cid:96),m (cos ϑ )e i mϕ for (cid:96) ∈ N , m = 0 , . . . , (cid:96) , and ( ϑ, ϕ ) ∈ [0 , π ] × [0 , π ) and by Y (cid:96),m = ( − m Y (cid:96), − m for (cid:96) ∈ N and m = − (cid:96), . . . , −
1. It is well-known that the spherical harmonics form anorthonormal basis of L ( S ), the subspace of real-valued functions in L ( S ; C ). In whatfollows we set for y ∈ S Y (cid:96),m ( y ) = Y (cid:96),m ( ϑ, ϕ ) , where y = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ ), i. e., we identify (with a slight abuse of notation)Cartesian and angular coordinates of the point y ∈ S . Furthermore we denote by σ the Lebesgue measure on the sphere which admits the representation dσ ( y ) = sin ϑ dϑ dϕ for y ∈ S , y = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ ). D. COHEN AND A. LANG
The spherical Laplacian , also called
Laplace–Beltrami operator , is given in terms of sphericalcoordinates similarly to Section 3.4.3 in [31] by∆ S = (sin ϑ ) − ∂∂ϑ (cid:18) sin ϑ ∂∂ϑ (cid:19) + (sin ϑ ) − ∂ ∂ϕ . It is well-known (see, e. g., Theorem 2.13 in [32]) that the spherical harmonic functions Y arethe eigenfunctions of ∆ S with eigenvalues ( − (cid:96) ( (cid:96) + 1) , (cid:96) ∈ N ), i. e.,∆ S Y (cid:96),m = − (cid:96) ( (cid:96) + 1) Y (cid:96),m for all (cid:96) ∈ N , m = − (cid:96), . . . , (cid:96) .To characterize the regularity of solutions to SPDEs in what follows, we introduce theSobolev space on S for a smoothness index s ∈ R H s ( S ) = (Id − ∆ S ) − s/ L ( S )together with its norm (cid:107) f (cid:107) H s ( S ) = (cid:107) (Id − ∆ S ) s/ f (cid:107) L ( S ) for some f ∈ H s ( S ) with H ( S ) = L ( S ).Furthermore, we work on L p (Ω; H s ( S )) with norm (cid:107) Z (cid:107) L p (Ω; H s ( S )) = E (cid:104) (cid:107) Z (cid:107) pH s ( S ) (cid:105) /p for finite p ≥ A ⊗ B ( S )-measurable mapping Z : Ω × S → R is called a real-valued random field on theunit sphere. Such a random field is called Gaussian if for all k ∈ N and x , . . . , x k ∈ S , themultivariate random variable ( Z ( x ) , . . . , Z ( x k )) is multivariate Gaussian distributed. Finally,such a random field is called isotropic if its covariance function only depends on the distance d ( x, y ), for x, y ∈ S .We recall Theorem 2.3 and Lemma 5.1 in [29] on the series expansions of isotropic Gaussianrandom fields on the sphere. Lemma 2.1.
A centered, isotropic Gaussian random field Z has a converging Karhunen–Lo`eve expansion Z = ∞ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) a (cid:96),m Y (cid:96),m with a (cid:96),m = ( Z, Y (cid:96),m ) L ( S ) and A (cid:96) = E [ a (cid:96),m a (cid:96),m ] for all m = − (cid:96), . . . , (cid:96) , where ( A (cid:96) , (cid:96) ∈ N ) iscalled the angular power spectrum of Z . For (cid:96) ∈ N , m = 1 , . . . , (cid:96) , and ϑ ∈ [0 , π ] set L (cid:96),m ( ϑ ) = (cid:115) (cid:96) + 14 π ( (cid:96) − m )!( (cid:96) + m )! P (cid:96),m (cos ϑ ) . Then for y = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ ) Z ( y ) = ∞ (cid:88) (cid:96) =0 (cid:32)(cid:112) A (cid:96) X (cid:96), L (cid:96), ( ϑ ) + (cid:112) A (cid:96) (cid:96) (cid:88) m =1 L (cid:96),m ( ϑ )( X (cid:96),m cos( mϕ ) + X (cid:96),m sin( mϕ )) (cid:33) in law, where (( X (cid:96),m , X (cid:96),m ) , (cid:96) ∈ N , m = 0 , . . . , (cid:96) ) is a sequence of independent, real-valued,standard normally distributed random variables and X (cid:96), = 0 for (cid:96) ∈ N . PPROXIMATION OF THE STOCHASTIC WAVE EQUATION ON THE SPHERE 5
In order to simulate solutions to the stochastic wave equation on the sphere, we need toapproximate the driving noise which can be generated by a sequence of Gaussian randomfields. We choose to truncate the above series expansion for an index κ ∈ N and set Z κ ( y ) = κ (cid:88) (cid:96) =0 (cid:32)(cid:112) A (cid:96) X (cid:96), L (cid:96), ( ϑ ) + (cid:112) A (cid:96) (cid:96) (cid:88) m =1 L (cid:96),m ( ϑ )( X (cid:96),m cos( mϕ ) + X (cid:96),m sin( mϕ )) (cid:33) , where we recall y = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ ) and ( ϑ, ϕ ) ∈ [0 , π ] × [0 , π ).The above lemma then allows us to present the following results on L p (Ω; L ( S )) conver-gence and P -almost sure convergence of the truncated series which are proven in Theorem 5.3and Corollary 5.4 in [29]. Theorem 2.2.
Let the angular power spectrum ( A (cid:96) , (cid:96) ∈ N ) of the centered, isotropic Gauss-ian random field Z decay algebraically with order α > , i. e., there exist constants C > and (cid:96) ∈ N such that A (cid:96) ≤ C · (cid:96) − α for all (cid:96) > (cid:96) . Then the series of approximate random fields ( Z κ , κ ∈ N ) converges to the random field Z in L p (Ω; L ( S )) for any finite p ≥ , and thetruncation error is bounded by (cid:107) Z − Z κ (cid:107) L p (Ω; L ( S )) ≤ ˆ C p · κ − ( α − / for κ > (cid:96) , where ˆ C p is a constant depending on p , C , and α .In addition, ( Z κ , κ ∈ N ) converges P -almost surely and for all δ < ( α − / , the truncationerror is asymptotically bounded by (cid:107) Z − Z κ (cid:107) L ( S ) ≤ κ − δ , P -a.s. . We follow [29], where isotropic Gaussian random fields are connected to Q -Wiener pro-cesses. There it is shown that an isotropic Q -Wiener process ( W ( t ) , t ∈ T ) on some finitetime interval T = [0 , T ] with values in L ( S ) can be represented by the expansion W ( t, y ) = ∞ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) a (cid:96),m ( t ) Y (cid:96),m ( y )= ∞ (cid:88) (cid:96) =0 (cid:32)(cid:112) A (cid:96) β (cid:96), ( t ) Y (cid:96), ( y ) + (cid:112) A (cid:96) (cid:96) (cid:88) m =1 ( β (cid:96),m ( t ) Re Y (cid:96),m ( y ) + β (cid:96),m ( t ) Im Y (cid:96),m ( y )) (cid:33) = ∞ (cid:88) (cid:96) =0 (cid:32)(cid:112) A (cid:96) β (cid:96), ( t ) L (cid:96), ( ϑ ) + (cid:112) A (cid:96) (cid:96) (cid:88) m =1 L (cid:96),m ( ϑ )( β (cid:96),m ( t ) cos( mϕ ) + β (cid:96),m ( t ) sin( mϕ )) (cid:33) , (1)where (( β (cid:96),m , β (cid:96),m ) , (cid:96) ∈ N , m = 0 , . . . , (cid:96) ) is a sequence of independent, real-valued Brownianmotions with β (cid:96), = 0 for (cid:96) ∈ N and t ∈ T . The covariance operator Q is characterizedsimilarly to the introduction in [28] by QY (cid:96),m = A (cid:96) Y (cid:96),m for (cid:96) ∈ N and m = − (cid:96), . . . , (cid:96) , i. e., the eigenvalues of Q are given by the angular powerspectrum ( A (cid:96) , (cid:96) ∈ N ), and the eigenfunctions are the spherical harmonic functions.Due to the properties of Brownian motion, the above Q -Wiener process can be generated byincrements which are isotropic Gaussian random fields with angular power spectrum ( hA (cid:96) , (cid:96) ∈ N ) for a time step size h . D. COHEN AND A. LANG The stochastic wave equation on the sphere
With the preparations from the preceding section at hand, we have all necessary tools tointroduce the main subject of our study.The stochastic wave equation on the sphere is defined as(2) ∂ tt u ( t ) − ∆ S u ( t ) = ˙ W ( t )with initial conditions u (0) = v ∈ L (Ω; L ( S )) and ∂ t u (0) = v ∈ L (Ω; L ( S )), where t ∈ T = [0 , T ], T < + ∞ . For ease of presentation, we consider the case of non-random initialconditions. The case of random initial conditions follows under appropriate integrabilityassumptions. The notation ˙ W stands for the formal derivative of the Q -Wiener process withseries expansion (1) as introduced in Section 2.Denoting the velocity of the solution by u = ∂ t u = ∂ t u , one can rewrite (2) asd X ( t ) = AX ( t ) d t + G d W ( t ) X (0) = X , (3)where A = (cid:18) I ∆ S (cid:19) , G = (cid:18) I (cid:19) , X = (cid:18) u u (cid:19) , X = (cid:18) v v (cid:19) . Existence of a unique mild solution of the abstract formulation (3) of the stochastic waveequation on the sphere follows from classical results on linear SPDEs, see for instance [11],and this mild solution reads X ( t ) = e tA X + (cid:90) t e ( t − s ) A G d W ( s ) . Equivalently, the integral formulation of our problem is given by u ( t ) = v + (cid:90) t u ( s ) d su ( t ) = v + (cid:90) t ∆ S u ( s ) d s + W ( t ) . (4)Since the spherical harmonic functions Y = ( Y (cid:96),m , (cid:96) ∈ N , m = − (cid:96), . . . , (cid:96) ) form an orthonor-mal basis of L ( S ) and are eigenfunctions of ∆ S , we insert the following ansatz for a seriesexpansion of the exact solution to SPDE (2) u ( t ) = ∞ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) u (cid:96),m ( t ) Y (cid:96),m and u ( t ) = ∞ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) u (cid:96),m ( t ) Y (cid:96),m (5)into equation (4) and compare the coefficients in front of Y (cid:96),m to obtain the following system u (cid:96),m ( t ) = v (cid:96),m + (cid:90) t u (cid:96),m ( s ) d su (cid:96),m ( t ) = v (cid:96),m − (cid:96) ( (cid:96) + 1) (cid:90) t u (cid:96),m ( s ) d s + a (cid:96),m ( t ) , where v (cid:96),m , v (cid:96),m , resp. a (cid:96),m are the coefficients of the expansions of the initial values v and v , resp. weighted Brownian motions in the expansion of the noise (1). PPROXIMATION OF THE STOCHASTIC WAVE EQUATION ON THE SPHERE 7
Writing the evolution of the initial values in the above linear harmonic oscillators withrotation matrices and using the variation of constants formula, one derives the followingsystem for the coefficients of the expansions of the solution (cid:40) u (cid:96),m ( t ) = cos( t ( (cid:96) ( (cid:96) + 1)) / ) v (cid:96),m + ( (cid:96) ( (cid:96) + 1)) − / sin( t ( (cid:96) ( (cid:96) + 1)) / ) v (cid:96),m + ˆ W (cid:96),m ( t ) u (cid:96),m ( t ) = − ( (cid:96) ( (cid:96) + 1)) / sin( t ( (cid:96) ( (cid:96) + 1)) / ) v (cid:96),m + cos( t ( (cid:96) ( (cid:96) + 1)) / ) v (cid:96),m + ˆ W (cid:96),m ( t ) , (6)where ˆ W (cid:96),m ( t ) = (cid:32) ˆ W (cid:96),m ( t )ˆ W (cid:96),m ( t ) (cid:33) = (cid:90) t R (cid:96) ( t − s ) d a (cid:96),m ( s )with R (cid:96) ( t ) = (cid:18) R (cid:96) ( t ) R (cid:96) ( t ) (cid:19) = (cid:18) ( (cid:96) ( (cid:96) + 1)) − / sin( t ( (cid:96) ( (cid:96) + 1)) / )cos( t ( (cid:96) ( (cid:96) + 1)) / ) (cid:19) for (cid:96) (cid:54) = 0 and ˆ W , ( t ) = (cid:18) ˆ W , ( t )ˆ W , ( t ) (cid:19) = (cid:90) t a , ( s ) d sa , ( t ) . We now characterize the above stochastic convolutions ˆ W (cid:96),mi for i = 1 , Proposition 3.1.
The stochastic convolution ˆ W ( t ) is Gaussian with mean zero and expansion ˆ W ( t, y ) = ∞ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) W (cid:96),m ( t ) Y (cid:96),m ( y )= ∞ (cid:88) (cid:96) =0 (cid:32)(cid:112) A (cid:96) ˆ β (cid:96), ( t ) Y (cid:96), ( y ) + (cid:112) A (cid:96) (cid:96) (cid:88) m =1 ( ˆ β (cid:96),m ( t ) Re Y (cid:96),m ( y ) + ˆ β (cid:96),m ( t ) Im Y (cid:96),m ( y )) (cid:33) = ∞ (cid:88) (cid:96) =0 (cid:32)(cid:112) A (cid:96) ˆ β (cid:96), ( t ) L (cid:96), ( ϑ ) + (cid:112) A (cid:96) (cid:96) (cid:88) m =1 L (cid:96),m ( ϑ )( ˆ β (cid:96),m ( t ) cos( mϕ ) + ˆ β (cid:96),m ( t ) sin( mϕ )) (cid:33) , where equality is in distribution.The processes ( ˆ β (cid:96),mi ( t ) , i = 1 , , (cid:96) ∈ N , m = − (cid:96), . . . , (cid:96) ) are given by ˆ β (cid:96),mi ( t ) = (cid:32) β (cid:96),mi, ( t ) β (cid:96),mi, ( t ) (cid:33) = D (cid:96) ( t ) X (cid:96),mi for a sequence ( X (cid:96),mi = ( X (cid:96),mi, , X (cid:96),mi, ) T , i = 1 , , (cid:96) ∈ N , m = − (cid:96), . . . , (cid:96) ) of independent, identi-cally distributed random variables with X (cid:96),mi,j ∼ N (0 , . The term D (cid:96) ( t ) denotes the Choleskydecomposition of the covariance matrix C (cid:96) ( t ) of ˆ W (cid:96),m ( t ) . More specifically, D (cid:96) satisfies D (cid:96) ( t ) D (cid:96) ( t ) T = C (cid:96) ( t ) with C (cid:96) ( t ) = (cid:96) ( (cid:96) +1)) / t − sin(2( (cid:96) ( (cid:96) +1)) / t )4( (cid:96) ( (cid:96) +1)) / sin(( (cid:96) ( (cid:96) +1)) / t ) (cid:96) ( (cid:96) +1))sin(( (cid:96) ( (cid:96) +1)) / t ) (cid:96) ( (cid:96) +1)) 2( (cid:96) ( (cid:96) +1)) / t +sin(2( (cid:96) ( (cid:96) +1)) / t )4( (cid:96) ( (cid:96) +1)) / for (cid:96) (cid:54) = 0 and C ( t ) = (cid:18) t / t / t / t (cid:19) . D. COHEN AND A. LANG
Proof.
We observe first that ˆ W ( t ) satisfies by (1)ˆ W ( t ) = ∞ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) ˆ W (cid:96),m ( t ) Y (cid:96),m = ˆ W , ( t ) Y , + ∞ (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) (cid:90) t R (cid:96) ( t − s ) d a (cid:96),m ( s ) Y (cid:96),m = ˆ W , ( t ) Y , + ∞ (cid:88) (cid:96) =1 (cid:112) A (cid:96) (cid:20)(cid:90) t R (cid:96) ( t − s ) d β (cid:96), ( s ) Y (cid:96), + √ (cid:96) (cid:88) m =1 (cid:18)(cid:90) t R (cid:96) ( t − s ) d β (cid:96),m ( s ) Re Y (cid:96),m + (cid:90) t R (cid:96) ( t − s ) d β (cid:96),m ( s ) Im Y (cid:96),m (cid:19)(cid:35) with independent Brownian motions ( β (cid:96),m , (cid:96) ∈ N , m = 0 , . . . , (cid:96) ) and ( β (cid:96),m , (cid:96) ∈ N , m =1 , . . . , (cid:96) ). Since all Brownian motions are centered and independent, it is sufficient to computethe following covariances which are given by C ( t ) = A − (cid:18) E [ ˆ W , ( t ) ] E [ ˆ W , ( t ) ˆ W , ( t )] E [ ˆ W , ( t ) ˆ W , ( t )] E [ ˆ W , ( t ) ] (cid:19) = A − (cid:18) E [( (cid:82) t a , ( s ) d s ) ] E [ (cid:82) t a , ( s ) a , ( t ) d s ] E [ (cid:82) t a , ( s ) a , ( t ) d s ] E [( a , ( t )) ] (cid:19) = (cid:18) t / t / t / t (cid:19) for (cid:96) = 0 and else for i = 1 , C (cid:96) ( t ) = (cid:18) E [(Int ) ] E [Int Int ] E [Int Int ] E [(Int ) ] (cid:19) = (cid:18) (cid:82) t R (cid:96) ( t − s ) d s (cid:82) t R (cid:96) ( t − s ) R (cid:96) ( t − s ) d s (cid:82) t R (cid:96) ( t − s ) R (cid:96) ( t − s ) d s (cid:82) t R (cid:96) ( t − s ) d s (cid:19) = (cid:96) ( (cid:96) +1)) / t − sin(2( (cid:96) ( (cid:96) +1)) / t )4( (cid:96) ( (cid:96) +1)) / sin(( (cid:96) ( (cid:96) +1)) / t ) (cid:96) ( (cid:96) +1))sin(( (cid:96) ( (cid:96) +1)) / t ) (cid:96) ( (cid:96) +1)) 2( (cid:96) ( (cid:96) +1)) / t +sin(2( (cid:96) ( (cid:96) +1)) / t )4( (cid:96) ( (cid:96) +1)) / , where we have set Int = (cid:82) t R (cid:96) ( t − s ) d β (cid:96),mi ( s ) and Int = (cid:82) t R (cid:96) ( t − s ) d β (cid:96),mi ( s ).Setting D (cid:96) ( t ) the Cholesky decomposition of the above covariance matrices satisfying D (cid:96) ( t ) T D (cid:96) ( t ) = C (cid:96) ( t )we obtain for (cid:96) (cid:54) = 0 and i = 1 , D (cid:96) ( t ) X (cid:96),mi = (cid:90) t R (cid:96) ( t − s ) d β (cid:96),mi ( s )in distribution and similarly for (cid:96) = 0 D (cid:96) ( t ) X (cid:96),m = ˆ W , ( t )with ( X (cid:96),mi = ( X (cid:96),mi, , X (cid:96),mi, ) T , i = 1 , , (cid:96) ∈ N , m = − (cid:96), . . . , (cid:96) ) independent and identicallydistributed standard normally distributed random variables. This concludes the proof. (cid:3) PPROXIMATION OF THE STOCHASTIC WAVE EQUATION ON THE SPHERE 9
Remark . Since we are interested in the simulation of sample paths of solutions to (3), weneed to generate increments of ˆ W (cid:96),m ( t ). Therefore it is important to observe thatˆ β (cid:96),mi ( t ) − ˆ β (cid:96),mi ( s ) = D (cid:96) ( t − s ) X (cid:96),mi in distribution for s < t . In this way we can generate sample paths of ˆ W ( t ) by sums ofindependent Gaussian increments.For completeness we also remark that the Cholesky decomposition D (cid:96) ( t ) can be computedexplicitly and is given by(7) D (cid:96) ( t ) = (cid:18) d , d , d , (cid:19) with d , = (2( (cid:96) ( (cid:96) + 1)) / t − sin(2( (cid:96) ( (cid:96) + 1)) / t )) / (cid:96) ( (cid:96) + 1)) / d , = sin(( (cid:96) ( (cid:96) + 1)) / t ) ( (cid:96) ( (cid:96) + 1)) / (2( (cid:96) ( (cid:96) + 1)) / t − sin(2( (cid:96) ( (cid:96) + 1)) / t )) / d , = (cid:32) (cid:96) ( (cid:96) + 1)) t − sin(2( (cid:96) ( (cid:96) + 1)) / t ) − (cid:96) ( (cid:96) + 1)) / t ) (cid:96) ( (cid:96) + 1)) / (2( (cid:96) ( (cid:96) + 1)) / t − sin(2( (cid:96) ( (cid:96) + 1)) / t )) (cid:33) / for (cid:96) (cid:54) = 0 and D ( t ) = t / (cid:18) t/ √ √ /
20 1 / (cid:19) . We close this section by showing regularity estimates for the solution of (2) that dependon the regularity of the initial conditions and the driving noise. These properties allow toobtain optimal weak convergence rates in Section 4.
Proposition 3.3.
Denote by X = ( u , u ) the solution to the stochastic wave equation (3) with initial value ( v , v ) . Assume that there exist (cid:96) ∈ N , α > , and a constant C > suchthat the angular power spectrum of the driving noise ( A (cid:96) , (cid:96) ∈ N ) satisfies A (cid:96) ≤ C · (cid:96) − α forall (cid:96) > (cid:96) . Then, for all t ∈ [0 , T ] , m ∈ N , and s < α/ with v ∈ H s ( S ) and v ∈ H s − ( S ) , u ( t ) ∈ L m (Ω; H s ( S )) , i. e., there exists a constant M such that (cid:107) u ( t ) (cid:107) L m (Ω; H s ( S )) ≤ M (1 + (cid:107) v (cid:107) H s ( S ) + (cid:107) v (cid:107) H s − ( S ) ) < + ∞ . And for all t ∈ [0 , T ] , m ∈ N , and s < α/ − with v ∈ H s +1 ( S ) and v ∈ H s ( S ) , u ( t ) ∈ L m (Ω; H s ( S )) , i. e., there exists a constant M such that (cid:107) u ( t ) (cid:107) L m (Ω; H s ( S )) ≤ M (1 + (cid:107) v (cid:107) H s +1 ( S ) + (cid:107) v (cid:107) H s ( S ) ) < + ∞ . Proof.
Let us first observe that (cid:107) u ( t ) (cid:107) L m (Ω; H s ( S )) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) (cid:16) R (cid:96) ( t ) v (cid:96),m + R (cid:96) ( t ) v (cid:96),m (cid:17) Y (cid:96),m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L m (Ω; H s ( S )) + (cid:13)(cid:13)(cid:13) ˆ W ( t ) (cid:13)(cid:13)(cid:13) L m (Ω; H s ( S )) . The first term with respect to the initial conditions satisfies (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) (cid:16) R (cid:96) ( t ) v (cid:96),m + R (cid:96) ( t ) v (cid:96),m (cid:17) Y (cid:96),m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H s ( S ) ≤ ∞ (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) (cid:16) (1 + (cid:96) ( (cid:96) + 1)) s | v (cid:96),m | + (1 + (cid:96) ( (cid:96) + 1)) s ( (cid:96) ( (cid:96) + 1)) − | v (cid:96),m | (cid:17) ≤ C ( (cid:107) v (cid:107) H s ( S ) + (cid:107) v (cid:107) H s − ( S ) ) . Given the angular power spectrum of ˆ W ( t ) in Proposition 3.1, it follows for the secondmoment, i. e. m = 1, that (cid:13)(cid:13)(cid:13) ˆ W ( t ) (cid:13)(cid:13)(cid:13) L (Ω; H s ( S )) = (cid:13)(cid:13)(cid:13) (Id − ∆ S ) s/ ˆ W ( t ) (cid:13)(cid:13)(cid:13) L (Ω; L ( S )) = ∞ (cid:88) (cid:96) =0 (2 (cid:96) + 1)(1 + (cid:96) ( (cid:96) + 1)) s (cid:96) ( (cid:96) + 1)) / t − sin(2( (cid:96) ( (cid:96) + 1)) / t )4( (cid:96) ( (cid:96) + 1)) / A (cid:96) , which converges for α > s since the elements of the sum behave like (cid:96) s − α − . By Fernique’stheorem [14], this convergence implies that the norm is finite for all m and arbitrary momentbounds can for example be obtained by the Burkholder–Davis–Gundy inequality.Similar computations for u conclude the proof. (cid:3) Convergence analysis
In this section, we numerically solve the wave equation on the sphere driven by additive Q -Wiener noise with spectral methods. We approximate the solution by truncation of thederived spectral representation and show convergence rates in p -th moment, P -almost surely,and in the weak sense.An efficient simulation of numerical approximations to solutions to the stochastic waveequation on the sphere (2) is then obtained via Algorithm 1. The strong errors of thistruncation procedure are given in the following proposition. Proposition 4.1.
Let t ∈ T and t < · · · < t n = t be a discrete time partition for n ∈ N ,which yields a recursive representation of the solution X = ( u , u ) of the stochastic waveequation on the sphere (3) given by (5) . Assume that the initial values satisfy v ∈ H β ( S ) and v ∈ H γ ( S ) . Furthermore, assume that there exist (cid:96) ∈ N , α > , and a constant C > such that the angular power spectrum of the driving noise ( A (cid:96) , (cid:96) ∈ N ) satisfies A (cid:96) ≤ C · (cid:96) − α for all (cid:96) > (cid:96) . Then, the error of the approximate solution X κ = ( u κ , u κ ) , given by (8) , isbounded uniformly in time and independently of the time discretization by (cid:107) u ( t ) − u κ ( t ) (cid:107) L p (Ω; L ( S )) ≤ ˆ C p · (cid:16) κ − α/ + κ − β (cid:107) v (cid:107) H β ( S ) + κ − ( γ +1) (cid:107) v (cid:107) H γ ( S ) (cid:17) (cid:107) u ( t ) − u κ ( t ) (cid:107) L p (Ω; L ( S )) ≤ ˆ C p · (cid:16) κ − ( α/ − + κ − ( β − (cid:107) v (cid:107) H β ( S ) + κ − γ (cid:107) v (cid:107) H γ ( S ) (cid:17) for all p ≥ and κ > (cid:96) , where ˆ C p is a constant that may depend on p , C , T , and α .On top of that, the error of the approximate solution X κ is bounded uniformly in time,independently of the time discretization, and asymptotically in κ by (cid:107) u ( t ) − u κ ( t ) (cid:107) L ( S ) ≤ κ − δ , P -a.s. PPROXIMATION OF THE STOCHASTIC WAVE EQUATION ON THE SPHERE 11
Algorithm 1
Simulations of paths of the solution to (2) Fix a truncation index κ ∈ N . Compute a discrete time grid 0 = t < t < . . . < t n = T , n ∈ N , with time step h . Compute the covariance matrix C (cid:96) ( h ) of the stochastic integrals ˆ W (cid:96),m ( h ) and ˆ W (cid:96),m ( h )in Proposition 3.1. Perform a Cholesky decomposition of C (cid:96) ( h ) = D (cid:96) ( h ) T D (cid:96) ( h ) or use the explicit formula (7). Use D (cid:96) ( h ) to generate noise incrementsˆ W (cid:96),m ( h ) = (cid:32) ˆ W (cid:96),m ( h )ˆ W (cid:96),m ( h ) (cid:33) = D (cid:96) (cid:32) β (cid:96),m ( h ) β (cid:96),m ( h ) (cid:33) (cid:112) A (cid:96) . Compute u (cid:96),m ( t j + h ) and u (cid:96),m ( t j + h ) recursively using (6) u (cid:96),m ( t j +1 ) = (cid:18) R (cid:96) ( h ) − (cid:96) ( (cid:96) + 1) R (cid:96) ( h ) (cid:19) u (cid:96),m ( t j ) + R (cid:96) ( h ) u (cid:96),m ( t j ) + ˆ W (cid:96),m ( h ) . Truncate the ansatz (5) at the fixed positive integer κ to get the numerical approximations u κ ( t j ) = κ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) u (cid:96),m ( t j ) Y (cid:96),m and u κ ( t j ) = κ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) u (cid:96),m ( t j ) Y (cid:96),m . (8) (cid:107) u ( t ) − u κ ( t ) (cid:107) L ( S ) ≤ κ − ( δ − , P -a.s.for all δ < min( α/ , β, γ + 1) .Remark . We remark that it is not necessary that the angular power spectrum ( A (cid:96) , (cid:96) ∈ N )of the Q -Wiener process decays with rate (cid:96) − α for α > α > Q is a trace class operator for convergence in the first component. Proof of Proposition 4.1.
Let us first consider the convergence in p -th moment of the firstcomponent of the solution for p ≥
1. By definition of u and u κ in (5) and (8), one obtains (cid:107) u ( t ) − u κ ( t ) (cid:107) L p (Ω; L ( S )) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ (cid:88) (cid:96) = κ +1 (cid:96) (cid:88) m = − (cid:96) u (cid:96),m ( t ) Y (cid:96),m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω; L ( S )) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ (cid:88) (cid:96) = κ +1 (cid:96) (cid:88) m = − (cid:96) (cid:16) R (cid:96) ( t ) v (cid:96),m + R (cid:96) ( t ) v (cid:96),m (cid:17) Y (cid:96),m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω; L ( S )) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ (cid:88) (cid:96) = κ +1 (cid:96) (cid:88) m = − (cid:96) (cid:90) t R (cid:96) ( t − s ) d a (cid:96),m ( s ) Y (cid:96),m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω; L ( S )) ≤ (cid:107) v − v κ (cid:107) L ( S ) + (cid:32) ∞ (cid:88) (cid:96) = κ +1 (cid:96) (cid:88) m = − (cid:96) ( (cid:96) ( (cid:96) + 1)) − | v (cid:96),m | (cid:33) / + (cid:107) Z − Z κ (cid:107) L p (Ω; L ( S )) , where Z denotes an isotropic Gaussian random field with angular power spectrum (cid:101) A (cid:96) = 2( (cid:96) ( (cid:96) + 1)) / t − sin(2( (cid:96) ( (cid:96) + 1)) / t )4( (cid:96) ( (cid:96) + 1)) / A (cid:96) and Z κ its truncation. Observe that (cid:101) A (cid:96) ≤ C (cid:32) t (cid:96) ( (cid:96) + 1) + | sin(2( (cid:96) ( (cid:96) + 1)) / t ) | (cid:96) ( (cid:96) + 1)) / (cid:33) A (cid:96) ≤ CT (cid:96) − A (cid:96) ≤ CT (cid:96) − ( α +2) for large enough index (cid:96) > (cid:96) ≥ A (cid:96) . The assumptions on the initial values and the fact that the spherical harmonicfunctions are orthonormal provide us with the estimate (cid:107) v − v κ (cid:107) L ( S ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ (cid:88) (cid:96) = κ +1 (cid:96) (cid:88) m = − (cid:96) v (cid:96),m Y (cid:96),m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( S ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ (cid:88) (cid:96) = κ +1 (cid:96) (cid:88) m = − (cid:96) v (cid:96),m (Id − ∆ S ) β/ (Id − ∆ S ) − β/ Y (cid:96),m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( S ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ (cid:88) (cid:96) = κ +1 (cid:96) (cid:88) m = − (cid:96) v (cid:96),m (1 + (cid:96) ( (cid:96) + 1)) − β/ (Id − ∆ S ) β/ Y (cid:96),m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( S ) ≤ Cκ − β (cid:107) v (cid:107) H β ( S ) and similarly for the second component of the initial value.Collecting all the estimates above and using Theorem 2.2 we obtain the desired bound (cid:107) u ( t ) − u κ ( t ) (cid:107) L p (Ω; L ( S )) ≤ ˆ C p · (cid:16) κ − α/ + κ − β (cid:107) v (cid:107) H β ( S ) + κ − ( γ +1) (cid:107) v (cid:107) H γ ( S ) (cid:17) . The corresponding estimate for the second component is done in a similar way and left to thereader. Observe that the rate of convergence decays by one due to the factor ( (cid:96) ( (cid:96) + 1)) / inthe first term of (6).We continue with the rate of the almost sure convergence in the first component of thesolution. Let δ < min( α/ , β, γ + 1). The above strong L p error estimate combined withChebyshev’s inequality provide us with P (cid:16) (cid:107) u ( t ) − u κ ( t ) (cid:107) L ( S ) ≥ κ − δ (cid:17) ≤ κ δp E (cid:104) (cid:107) u ( t ) − u κ ( t ) (cid:107) pL ( S ) (cid:105) ≤ κ δp ˆ C pp (cid:16) κ − α/ + κ − β (cid:107) v (cid:107) H β ( S ) + κ − ( γ +1) (cid:107) v (cid:107) H γ ( S ) (cid:17) p . For all p > max(( α/ − δ ) − , ( β − δ ) − , ( γ + 1 − δ ) − ), the series ∞ (cid:88) κ =1 κ ( δ − min( α/ ,β,γ +1)) p < + ∞ converges which implies the claim by the Borel–Cantelli lemma. Almost sure convergence ofthe second component is shown in a similar way which concludes the proof. (cid:3) Using Proposition 4.1, we continue with bounding weak errors of the mean and secondmoment in a first step. We observe that the weak error for the mean is the error to thecorresponding deterministic wave equation on S and that the error for the second moment PPROXIMATION OF THE STOCHASTIC WAVE EQUATION ON THE SPHERE 13 satisfies the rule of thumb that the weak convergence rate is twice the strong convergencerate.
Proposition 4.3.
Let t ∈ T and t < · · · < t n = t be a discrete time partition for n ∈ N which yields a recursive representation of the solution X = ( u , u ) of the stochastic waveequation on the sphere (3) given by (5) . Assume that the initial values satisfy v ∈ H β ( S ) and v ∈ H γ ( S ) . Furthermore, assume that there exist (cid:96) ∈ N , α > , and a constant C > such that the angular power spectrum of the driving noise ( A (cid:96) , (cid:96) ∈ N ) satisfies A (cid:96) ≤ C · (cid:96) − α for all (cid:96) > (cid:96) .Then, the errors in mean of the approximate solution X κ = ( u κ , u κ ) , given by (8) , arebounded uniformly in time and independently of the time discretization by (cid:107) E [ u ( t ) − u κ ( t )] (cid:107) L ( S ) ≤ ˆ C · (cid:16) κ − β (cid:107) v (cid:107) H β ( S ) + κ − ( γ +1) (cid:107) v (cid:107) H γ ( S ) (cid:17) (cid:107) E [ u ( t ) − u κ ( t )] (cid:107) L ( S ) ≤ ˆ C · (cid:16) κ − ( β − (cid:107) v (cid:107) H β ( S ) + κ − γ (cid:107) v (cid:107) H γ ( S ) (cid:17) for all κ > (cid:96) , where ˆ C is a constant that may depend on C , T , and α .Furthermore, the errors of the second moment are bounded by (cid:12)(cid:12)(cid:12) E (cid:104) (cid:107) u ( t ) (cid:107) L ( S ) − (cid:107) u κ ( t ) (cid:107) L ( S ) (cid:105)(cid:12)(cid:12)(cid:12) ≤ ˆ C · (cid:16) κ − α + κ − β (cid:107) v (cid:107) H β ( S ) + κ − γ +1) (cid:107) v (cid:107) H γ ( S ) (cid:17)(cid:12)(cid:12)(cid:12) E (cid:104) (cid:107) u ( t ) (cid:107) L ( S ) − (cid:107) u κ ( t ) (cid:107) L ( S ) (cid:105)(cid:12)(cid:12)(cid:12) ≤ ˆ C · (cid:16) κ − ( α − + κ − β − (cid:107) v (cid:107) H β ( S ) + κ − γ (cid:107) v (cid:107) H γ ( S ) (cid:17) for all κ > (cid:96) , where ˆ C is a constant that may depend on C , T , and α .Proof. The definition of X and its approximation X κ yield E (cid:2) u j ( t ) − u κj ( t ) (cid:3) = ∞ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) E (cid:104) u (cid:96),mj ( t ) (cid:105) Y (cid:96),m for j = 1 ,
2. Next, using (6) and the properties of ˆ W (cid:96),m ( t ) from Proposition 3.1, one obtains E (cid:104) u (cid:96),m ( t ) (cid:105) = cos( t ( (cid:96) ( (cid:96) + 1)) / ) E (cid:104) v (cid:96),m (cid:105) + ( (cid:96) ( (cid:96) + 1)) − / sin( t ( (cid:96) ( (cid:96) + 1)) / ) E (cid:104) v (cid:96),m (cid:105) and similarly for the second component. This corresponds to the errors in the initial values,see the proof of Proposition 4.1, and we thus obtain the error bound (cid:107) E [ u ( t ) − u κ ( t )] (cid:107) L ( S ) ≤ ˆ C · (cid:16) κ − β (cid:107) v (cid:107) H β ( S ) + κ − ( γ +1) (cid:107) v (cid:107) H γ ( S ) (cid:17) and correspondingly for the second component.In order to bound the second moments, we observe that E (cid:104) (cid:107) u j ( t ) (cid:107) L ( S ) − (cid:107) u κj ( t ) (cid:107) L ( S ) (cid:105) = E (cid:2) (cid:104) u j ( t ) + u κj ( t ) , u j ( t ) − u κj ( t ) (cid:105) L ( S ) (cid:3) = E (cid:42) κ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) u (cid:96),mj ( t ) Y (cid:96),m + ∞ (cid:88) (cid:96) = κ +1 (cid:96) (cid:88) m = − (cid:96) u (cid:96),mj ( t ) Y (cid:96),m , ∞ (cid:88) (cid:96) = κ +1 (cid:96) (cid:88) m = − (cid:96) u (cid:96),mj ( t ) Y (cid:96),m (cid:43) L ( S ) = 2 E (cid:34) κ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) ∞ (cid:88) (cid:96) (cid:48) = κ +1 (cid:96) (cid:48) (cid:88) m (cid:48) = − (cid:96) (cid:48) u (cid:96),mj ( t ) u (cid:96) (cid:48) ,m (cid:48) j ( t ) (cid:104) Y (cid:96),m , Y (cid:96) (cid:48) ,m (cid:48) (cid:105) L ( S ) (cid:35) + E (cid:104) (cid:107) u j ( t ) − u κj ( t ) (cid:107) L ( S ) (cid:105) , for j = 1 ,
2. Using the orthogonality of the spherical harmonics Y , the first term vanishes andthe second one is bounded by the square of the strong error in Proposition 4.1. This yields E (cid:104) (cid:107) u ( t ) (cid:107) L ( S ) − (cid:107) u κ ( t ) (cid:107) L ( S ) (cid:105) ≤ ˆ C · (cid:16) κ − α + κ − β (cid:107) v (cid:107) H β ( S ) + κ − γ +1) (cid:107) v (cid:107) H γ ( S ) (cid:17) and similarly for the second component. (cid:3) For a more general class of test functions, we obtain weak error rates that depend directlyon the regularity of the test function and indirectly on the regularity of the solution. Let usfirst state the abstract assumption on the test functions that will be required for the nextweak convergence result.
Assumption 4.4.
Consider the class of Fr´echet differentiable test functions ϕ satisfying forsome fixed s > (cid:107) (cid:90) ϕ (cid:48) ( ρu j ( t ) + (1 − ρ ) u κj ( t )) d ρ (cid:107) L (Ω; H s ( S )) ≤ (cid:101) C < + ∞ for j = 1 , H s ( S ), i. e., to take ϕ such that for all x ∈ H s ( S )(9) (cid:107) ϕ (cid:48) ( x ) (cid:107) H s ( S ) ≤ C (cid:16) (cid:107) x (cid:107) mH s ( S ) (cid:17) . Then we observe that ρu j ( t ) + (1 − ρ ) u κj ( t ) = κ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) u (cid:96),mj ( t ) Y (cid:96),m + ρ ∞ (cid:88) (cid:96) = κ +1 (cid:96) (cid:88) m = − (cid:96) u (cid:96),m ( t ) Y (cid:96),m . This would imply for ρ ∈ [0 , (cid:107) ϕ (cid:48) ( ρu j ( t ) + (1 − ρ ) u κj ( t )) (cid:107) L (Ω; H s ( S )) ≤ C E (cid:20)(cid:16) (cid:107) ρu j ( t ) + (1 − ρ ) u κj ( t ) (cid:107) mH s ( S ) (cid:17) (cid:21) ≤ C E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) κ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) u (cid:96),mj ( t ) Y (cid:96),m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H s ( S ) + ρ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ (cid:88) (cid:96) = κ +1 (cid:96) (cid:88) m = − (cid:96) u (cid:96),m ( t ) Y (cid:96),m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H s ( S ) m ≤ C (cid:16) (cid:107) u j ( t ) (cid:107) mL m (Ω; H s ( S )) (cid:17) . Therefore Assumption 4.4 is satisfied if u j ( t ) ∈ L m (Ω; H s ( S )) which is specified in Proposi-tion 3.3.Having seen that the class of test functions with derivatives of polynomial growth satisfiesAssumption 4.4, we are in place to state our general weak convergence result. Proposition 4.5.
Under the setting of Proposition 4.3 and Assumption 4.4, there exists aconstant ˆ C such that the weak errors are bounded by | E [ ϕ ( u ( t )) − ϕ ( u κ ( t ))] | ≤ ˆ C (cid:16) κ − ( α/ s ) + κ − ( β + s ) (cid:107) v (cid:107) H β ( S ) + κ − ( γ + s +1) (cid:107) v (cid:107) H γ ( S ) (cid:17) | E [ ϕ ( u ( t )) − ϕ ( u κ ( t ))] | ≤ ˆ C (cid:16) κ − ( α/ s − + κ − ( β + s − (cid:107) v (cid:107) H β ( S ) + κ − ( γ + s ) (cid:107) v (cid:107) H γ ( S ) (cid:17) for all κ > (cid:96) . PPROXIMATION OF THE STOCHASTIC WAVE EQUATION ON THE SPHERE 15
Proof.
The proof is inspired by [1]. Consider the Gelfand triple V ⊂ H ⊂ V ∗ with V = H s ( S ) , H = L ( S ) and V ∗ = H − s ( S ). The mean value theorem for Fr´echetderivatives followed by the Cauchy–Schwarz inequality yields (cid:12)(cid:12) E (cid:2) ϕ ( u j ( t )) − ϕ ( u κj ( t )) (cid:3)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) E (cid:20) V (cid:104) (cid:90) ϕ (cid:48) ( ρu j ( t ) + (1 − ρ ) u κj ( t )) d ρ, u j ( t ) − u κj ( t ) (cid:105) V ∗ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) ϕ (cid:48) ( ρu j ( t ) + (1 − ρ ) u κj ( t )) d ρ (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) (cid:107) u j ( t ) − u κj ( t ) (cid:107) L (Ω; V ∗ ) for j = 1 ,
2. The first term is bounded by Assumption 4.4 so that the convergence rate willbe obtained from the second term. Details are only given for the first component, i. e., for j = 1, and are obtained for u in a similar way.Following the proof of Proposition 4.1, we obtain (cid:107) u ( t ) − u κ ( t ) (cid:107) L (Ω; H − s ( S )) ≤ (cid:107) v − v κ (cid:107) H − s ( S ) + (cid:32) ∞ (cid:88) (cid:96) = κ +1 (cid:96) (cid:88) m = − (cid:96) (1 + (cid:96) ( (cid:96) + 1)) − s ( (cid:96) ( (cid:96) + 1)) − | v (cid:96),m | (cid:33) / + (cid:107) ˆ Z − ˆ Z κ (cid:107) L (Ω; L ( S )) with ˆ Z = (Id − ∆ S ) − s/ Z and ˆ Z κ its approximation. Therefore the angular power spectrum of the centered Gaussianrandom field ˆ Z is given by ˆ A (cid:96) = (1 + (cid:96) ( (cid:96) + 1)) − s (cid:101) A (cid:96) ≤ C(cid:96) − ( α +2+2 s ) . Applying Theorem 2.2 to ˆ Z and bounding the initial conditions as in Proposition 4.1 withthe additional weights (1 + (cid:96) ( (cid:96) + 1)) − s/ yield (cid:107) u ( t ) − u κ ( t ) (cid:107) L (Ω; H − s ( S )) ≤ C (cid:16) κ − ( α/ s ) + κ − ( β + s ) (cid:107) v (cid:107) H β ( S ) + κ − ( γ + s +1) (cid:107) v (cid:107) H γ ( S ) (cid:17) , which concludes the proof for the weak error in the first component. (cid:3) Proposition 4.3 states that the approximation of the second moment converges with twicethe strong rate of convergence obtained in Proposition 4.1. Let us now investigate whichregularity (for the noise and initial values) is required to achieve twice the strong rate inProposition 4.5. We focus on the convergence of the noise in the parameter α . Similarconsiderations hold for the initial conditions.For having the weak rate in Proposition 4.5 to be twice the strong rate from Proposition 4.1,one would need s = α/ u and s = α/ − u .The regularity result from Proposition 3.3 reads u ( t ) ∈ L m (Ω; H s ( S )) for all s < α/ u ( t ) ∈ L m (Ω; H s ( S )) for all s < α/ −
1. This together with the polynomial growthassumption (9) on the test functions would imply that Assumption 4.4 is satisfied for all s < α/ u and s < α/ − u . Therefore, in the situation of Proposition 4.5, thegeneral rule of thumb for the rate of weak convergence is also valid.We end this section by observing that several strategies for proving weak rates of conver-gence of numerical solutions to SPDEs in the literature could be extended to the present (a) Sample of solution. -3 -2 -1 Error (b) Errors in position. -1 Error (c) Errors in velocity.
Figure 1.
Sample and mean-square errors of the approximation of the sto-chastic wave equation with angular power spectrum of the Q -Wiener processwith parameter α = 3 and 100 Monte Carlo samples.setting or in the case of numerical discretizations of nonlinear stochastic wave equations onthe sphere, see for instance [13, 24, 34, 5, 20, 16, 23] and references therein. This could besubject of future research. 5. Numerical experiments
We present several numerical experiments with the aim of supporting and illustrating theabove theoretical results.In order to illustrate the rate of convergence of the mean-square error from Proposition 4.1,we consider a “reference” solution at time T = 1 with κ = 2 (since for larger κ the elementsof the angular power spectrum A (cid:96) and therefore the increments were so small that MATLABfailed to calculate the series expansion). The initial values are taken to be v = v = 0 in orderto observe the convergence rate only with respect to the regularity of the noise given by theparameter α . Afterwards we will perform a numerical example illustrating the convergencerate with respect to the regularity of the initial position given by the parameter β . We thencompute one time step of numerical solutions (since we have shown in Proposition 4.1 thatthe convergence rate is independent of the number of calculated time steps) and computethe errors for various truncation indices κ . Instead of the L ( S ) error in space, we usedthe maximum over all grid points which is a stronger error. The results and the theoreticalconvergence rates are shown for α = 3 and α = 5 in Figure 1, resp. Figure 2.In these figures, one observes that the simulation results match the theoretical results fromProposition 4.1. In addition, in order to illustrate the structure of the solution u in dependenceof the decay of the angular power spectrum, we include samples next to the convergence plots.In order to illustrate Remark 4.2 on the possibility of taking the parameter 0 < α < α = 1. The results are presented in Figure 3. There, for such non-smooth noise, one canobserve convergence in the position but not in the velocity. Similar observations were madefor time discretizations of stochastic wave equations on domains (that are not manifolds) in[10, 3], for instance.In Figure 4 we illustrate the convergence rates with respect to the regularity of the initialposition from Proposition 4.1. To ensure that the regularity of the initial position dominates PPROXIMATION OF THE STOCHASTIC WAVE EQUATION ON THE SPHERE 17 (a) Sample of solution. -6 -5 -4 -3 -2 -1 Error (b) Errors in position. -3 -2 -1 Error (c) Errors in velocity.
Figure 2.
Sample and mean-square errors of the approximation of the sto-chastic wave equation with angular power spectrum of the Q -Wiener processwith parameter α = 5 and 100 Monte Carlo samples. (a) Sample of solution. -1 Error (b) Errors in position. Error (c) Errors in velocity.
Figure 3.
Sample and mean-square errors of the approximation of the sto-chastic wave equation with angular power spectrum of the Q -Wiener processwith parameter α = 1 and 100 Monte Carlo samples.the error, we choose α = 10 and a random initial position v scaled such that it belongs to H β ( S ) with β = 2. The expected convergence rates are indeed observed in this figure.Errors of one path of the stochastic wave equation to the corresponding error plots from theprevious figures (Figure 1 and Figure 2) are presented in Figure 5. The observed convergencerates coincide with the theoretical results on P -almost sure convergence in the second part ofProposition 4.1.Let us now illustrate the weak rates of convergence from Proposition 4.3 and Proposi-tion 4.5. We consider a “reference” solution at time T = 1 with κ = 2 . The initial val-ues are taken to be v = v = 0. The test functions are given by ϕ ( u ) = (cid:107) u (cid:107) L ( S ) and ϕ ( u ) = exp( −(cid:107) u (cid:107) L ( S ) ). Observe that the second test function is of class C , bounded andwith bounded derivatives. Proposition 4.3 and Proposition 4.5 guarantee that the weak rateswill be essentially twice the strong rates in both cases. This is confirmed for α = 3 in Figure 6.6. Further extensions
In this section, we extend some of the above results first to the case of the stochastic waveequation on higher-dimensional spheres S d − , for some integer d >
3, and second to the case -4 -3 -2 -1 Error (a) Errors in position. -2 -1 Error (b) Errors in velocity.
Figure 4.
Mean-square errors of the approximation of the stochastic waveequation with angular power spectrum of the Q -Wiener process with parameter α = 10 and v ∈ H β ( S ) for β = 2 and 100 Monte Carlo samples. -3 -2 -1 Error (a) Angular power spectrum with parameter α = 3. -5 -4 -3 -2 -1 Error (b) Angular power spectrum with parameter α = 5. Figure 5.
Error of the approximation of a path of the stochastic wave equa-tion with different angular power spectra of the Q -Wiener process.of a free stochastic Schr¨odinger equation on the sphere S . We keep this section concise andfocus on strong and P -a.s. convergence.6.1. The stochastic wave equation on S d − . Let us consider the more general situation ofthe stochastic wave equation on the unit sphere S d − = { x ∈ R d , (cid:107) x (cid:107) R d = 1 } embedded into R d . The angular distance of two points x and y on S d − is given in the same way as on S ,see Section 2. Let us denote by ( S (cid:96),m , (cid:96) ∈ N , m = 1 , . . . , h ( (cid:96), d )) the spherical harmonics PPROXIMATION OF THE STOCHASTIC WAVE EQUATION ON THE SPHERE 19 -9 -8 -7 -6 -5 -4 -3 Error (a) ϕ ( u ) = (cid:107) u (cid:107) L ( S ) . -5 -4 -3 Error (b) ϕ ( u ) = (cid:107) u (cid:107) L ( S ) . -12 -10 -8 -6 -4 -2 Error (c) ϕ ( u ) = exp( −(cid:107) u (cid:107) L ( S ) ). -6 -5 -4 -3 Error (d) ϕ ( u ) = exp( −(cid:107) u (cid:107) L ( S ) ). Figure 6.
Weak errors of the approximation of the stochastic wave equationwith angular power spectrum of the Q -Wiener process with parameter α = 3and 1000 Monte Carlo samples. Left column shows position, right columnvelocity.on S d − , where h ( (cid:96), d ) = (2 (cid:96) + d −
2) ( (cid:96) + d − d − (cid:96) ! . Using the same setup as in [29] which goes back to [35], a centered isotropic Gaussianrandom field Z on S d − admits a Karhunen–Lo`eve expansion Z ( x ) = ∞ (cid:88) (cid:96) =0 h ( (cid:96),d ) (cid:88) m =1 a (cid:96),m S (cid:96),m ( x ) , where ( a (cid:96),m , (cid:96) ∈ N , m = 1 , . . . , h ( (cid:96), d )) is a sequence of independent Gaussian random vari-ables satisfying E [ a (cid:96),m ] = 0 , E [ a (cid:96),m a (cid:96) (cid:48) ,m (cid:48) ] = A (cid:96) δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) for (cid:96), (cid:96) (cid:48) ∈ N and m = 1 , . . . , h ( (cid:96), d ), m (cid:48) = 1 , . . . , h ( (cid:96) (cid:48) , d ) and ∞ (cid:88) (cid:96) =0 A (cid:96) h ( (cid:96), d ) < + ∞ . The series converges with probability one and in L p (Ω; R ) as well as in L (Ω; L p ( S d − )), p ≥ A (cid:96) , (cid:96) ∈ N ) the angular power spectrum of Z for S d − in analogy to what wasdone for S , we can rewrite Z = ∞ (cid:88) (cid:96) =0 h ( (cid:96),d ) (cid:88) m =1 a (cid:96),m S (cid:96),m = ∞ (cid:88) (cid:96) =0 (cid:112) A (cid:96) h ( (cid:96),d ) (cid:88) m =1 X (cid:96),m S (cid:96),m , where ( X (cid:96),m , (cid:96) ∈ N , m = 1 , . . . , h ( (cid:96), d )) is the sequence of independent, standard normallydistributed random variables derived by X (cid:96),m = a (cid:96),m / √ A (cid:96) . We set Z κ = κ (cid:88) (cid:96) =0 (cid:112) A (cid:96) h ( (cid:96),d ) (cid:88) m =1 X (cid:96),m S (cid:96),m for the corresponding sequence of truncated random fields ( Z κ , κ ∈ N ). It is shown in Theo-rem 5.5 in [29] that these approximations converge to the random field Z in L p (Ω; L ( S d − ))and P -almost surely with error bounds(10) (cid:107) Z − Z κ (cid:107) L p (Ω; L ( S d − )) ≤ C p · κ − ( α +1 − d ) / for κ > (cid:96) and for all δ < ( α + 1 − d ) / (cid:107) Z − Z κ (cid:107) L ( S d − ) ≤ κ − δ , P -a.s. , where A (cid:96) ≤ C · (cid:96) − α for (cid:96) ≥ (cid:96) . This generalizes Theorem 2.2 above and leads to convergencerates that depend also on the dimension of the sphere.Similarly to (1) in Section 2, we introduce a Q -Wiener process ( W ( t ) , t ∈ T ) on some finiteinterval T = [0 , T ] with values in L ( S d − ) by the expansion(12) W ( t, y ) = ∞ (cid:88) (cid:96) =0 h ( (cid:96),d ) (cid:88) m =1 a (cid:96),m ( t ) S (cid:96),m ( y ) = ∞ (cid:88) (cid:96) =0 (cid:112) A (cid:96) h ( (cid:96),d ) (cid:88) m =1 β (cid:96),m ( t ) S (cid:96),m ( y ) , where ( β (cid:96),m , (cid:96) ∈ N , m = 1 , . . . , h ( (cid:96), d )) is a sequence of independent, real-valued Brownianmotions.We next recall that the Laplace–Beltrami operator ∆ S d − on S d − has the spherical har-monics ( S (cid:96),m , (cid:96) ∈ N , m = 1 , . . . , h ( (cid:96), d )) as eigenbasis with eigenvalues given by∆ S d − S (cid:96),m = − (cid:96) ( (cid:96) + d − S (cid:96),m for (cid:96) ∈ N and m = 1 , . . . , h ( (cid:96), d ) (see, e. g., [4, Sec. 3.3]).We introduce Sobolev spaces on S d − , similarly to S , which are given for a smoothnessindex s ∈ R by H s ( S d − ) = (Id − ∆ S d − ) − s/ L ( S d − )together with the norm (cid:107) f (cid:107) H s ( S d − ) = (cid:107) (Id − ∆ S d − ) s/ f (cid:107) L ( S d − ) for some f ∈ H s ( S d − ). We also denote H ( S d − ) = L ( S d − ). PPROXIMATION OF THE STOCHASTIC WAVE EQUATION ON THE SPHERE 21
The stochastic wave equation on S d − is defined as(13) ∂ tt u ( t ) − ∆ S d − u ( t ) = ˙ W ( t ) , with initial conditions u (0) = v ∈ L (Ω; L ( S d − )) and ∂ t u (0) = v ∈ L (Ω; L ( S d − )), where t ∈ T = [0 , T ], T < + ∞ . The notation ˙ W stands for the formal derivative of the Q -Wienerprocess.Denoting as before the velocity of the solution by u = ∂ t u = ∂ t u , one can rewrite (13) asd X ( t ) = AX ( t ) d t + G d W ( t ) X (0) = X , (14)where A = (cid:18) I ∆ S d − (cid:19) , G = (cid:18) I (cid:19) , X = (cid:18) u u (cid:19) , X = (cid:18) v v (cid:19) . Existence of a unique mild solution follows as before.Using the same ansatz as in Section 3 with respect to the spherical harmonics on S d − u ( t ) = ∞ (cid:88) (cid:96) =0 h ( (cid:96),d ) (cid:88) m =1 u (cid:96),m ( t ) S (cid:96),m and u ( t ) = ∞ (cid:88) (cid:96) =0 h ( (cid:96),d ) (cid:88) m =1 u (cid:96),m ( t ) S (cid:96),m (15)we obtain u (cid:96),m ( t ) = v (cid:96),m + (cid:90) t u (cid:96),m ( s ) d su (cid:96),m ( t ) = v (cid:96),m − (cid:96) ( (cid:96) + d − (cid:90) t u (cid:96),m ( s ) d s + a (cid:96),m ( t ) , where v (cid:96),m , v (cid:96),m , resp. a (cid:96),m are the coefficients of the expansions of the initial values v and v , resp. weighted Brownian motion in the expansion of the noise (12).Similarly to (6), the variation of constants formula yields (cid:40) u (cid:96),m ( t ) = R (cid:96) ( t ) v (cid:96),m + R (cid:96) ( t ) v (cid:96),m + ˆ W (cid:96),m ( t ) u (cid:96),m ( t ) = − ( (cid:96) ( (cid:96) + d − R (cid:96) ( t ) v (cid:96),m + R (cid:96) ( t ) v (cid:96),m + ˆ W (cid:96),m ( t ) , where ˆ W (cid:96),m ( t ) = (cid:32) ˆ W (cid:96),m ( t )ˆ W (cid:96),m ( t ) (cid:33) = (cid:90) t R (cid:96) ( t − s ) d a (cid:96),m ( s )with R (cid:96) ( t ) = (cid:18) R (cid:96) ( t ) R (cid:96) ( t ) (cid:19) = (cid:18) ( (cid:96) ( (cid:96) + d − − / sin( t ( (cid:96) ( (cid:96) + d − / )cos( t ( (cid:96) ( (cid:96) + d − / ) (cid:19) for (cid:96) (cid:54) = 0 and ˆ W , ( t ) = (cid:18) ˆ W , ( t )ˆ W , ( t ) (cid:19) = (cid:90) t a , ( s ) d sa , ( t ) . Note that the only change compared to Section 3 is the value of the coefficients given by theeigenvalues ∆ S d − and the renaming of the spherical harmonics. As in Section 4, we approximate the solution to the stochastic wave equation (14) bytruncation of the series expansion at some finite index κ > u κ ( t j ) = κ (cid:88) (cid:96) =0 h ( (cid:96),d ) (cid:88) m =1 u (cid:96),m ( t j ) S (cid:96),m and u κ ( t j ) = κ (cid:88) (cid:96) =0 h ( (cid:96),d ) (cid:88) m =1 u (cid:96),m ( t j ) S (cid:96),m . (16)Then replacing the eigenvalues − (cid:96) ( (cid:96) + 1) with − (cid:96) ( (cid:96) + d − (cid:96) + 1 with h ( (cid:96), d ) and applying (10) and (11) instead of Theorem 2.2 in the proof ofProposition 4.1 yields directly the following extension of Proposition 4.1. Proposition 6.1.
Let t ∈ T and t < · · · < t n = t be a discrete time partition for n ∈ N ,which yields a recursive representation of the solution X = ( u , u ) of the stochastic waveequation (14) on S d − given by (15) . Assume that the initial values satisfy v ∈ H β ( S d − ) and v ∈ H γ ( S d − ) . Furthermore, assume that there exist (cid:96) ∈ N , α > , and a constant C > such that the angular power spectrum of the driving noise ( A (cid:96) , (cid:96) ∈ N ) satisfies A (cid:96) ≤ C · (cid:96) − α for all (cid:96) > (cid:96) . Then, the error of the approximate solution X κ = ( u κ , u κ ) , given by (16) , isbounded uniformly on any finite time interval and independently of the time discretization by (cid:107) u ( t ) − u κ ( t ) (cid:107) L p (Ω; L ( S d − )) ≤ ˆ C p · (cid:16) κ − ( α +3 − d ) / + κ − β (cid:107) v (cid:107) H β ( S d − ) + κ − ( γ +1) (cid:107) v (cid:107) H γ ( S d − ) (cid:17) (cid:107) u ( t ) − u κ ( t ) (cid:107) L p (Ω; L ( S d − )) ≤ ˆ C p · (cid:16) κ − ( α +1 − d ) / + κ − ( β − (cid:107) v (cid:107) H β ( S d − ) + κ − γ (cid:107) v (cid:107) H γ ( S d − ) (cid:17) for all p ≥ and κ > (cid:96) , where ˆ C p is a constant that may depend on p , C , T , and α .Additionally, the error is bounded uniformly in time, independently of the time discretiza-tion, and asymptotically in κ by (cid:107) u ( t ) − u κ ( t ) (cid:107) L ( S d − ) ≤ κ − δ , P -a.s. (cid:107) u ( t ) − u κ ( t ) (cid:107) L ( S d − ) ≤ κ − ( δ − , P -a.s.for all δ < min(( α + 3 − d ) / , β, γ + 1) . The free stochastic Schr¨odinger equation on S . We consider efficient simulationsof paths of solutions to the free stochastic Schr¨odinger equation on the sphere S (17) i ∂ t u ( t ) = ∆ S u ( t ) + ˙ W ( t ) , with initial condition (possibly complex-valued) u (0) ∈ L (Ω; L ( S )). Here, the unknown u ( t ) = u R ( t ) + i u I ( t ), with t ∈ [0 , T ] for some T < + ∞ , is a complex valued stochasticprocess. Furthermore, the notation ˙ W stands for the formal derivative of the (real-valued) Q -Wiener process with series expansion (1).Considering the real and imaginary parts of the above SPDE, one can rewrite (17) asd X ( t ) = AX ( t ) d t + G d W ( t ) X (0) = X , (18)where A = (cid:18) S − ∆ S (cid:19) , G = (cid:18) − I (cid:19) , X = (cid:18) u R u I (cid:19) , X = (cid:18) u R (0) u I (0) (cid:19) . The existence of a mild form of the abstract formulation (18) of the stochastic Schr¨odingerequation on the sphere follows like for the above stochastic wave equation. The mild form
PPROXIMATION OF THE STOCHASTIC WAVE EQUATION ON THE SPHERE 23 reads X ( t ) = e tA X + (cid:90) t e ( t − s ) A G d W ( s )(19)with the semigroup e tA = (cid:18) cos( t ∆ S ) sin( t ∆ S ) − sin( t ∆ S ) cos( t ∆ S ) (cid:19) . Finally, one obtains the integral formulation of the above problem as u R ( t ) = u R (0) + (cid:90) t ∆ S u I ( s ) d su I ( t ) = u I (0) − (cid:90) t ∆ S u R ( s ) d s − W ( t ) . As it was done for the stochastic wave equation in Section 3, one can make the followingansatz for the real and imaginary part of solutions to (19) u R ( t ) = ∞ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) u (cid:96),mR Y (cid:96),m and u I ( t ) = ∞ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) u (cid:96),mI Y (cid:96),m and find the following system of equations defining the coefficients of these expansions: (cid:40) u (cid:96),mR ( t ) = cos( t ( (cid:96) ( (cid:96) + 1)) / ) v (cid:96),mR + sin( t ( (cid:96) ( (cid:96) + 1)) / ) v (cid:96),mI + ˆ W (cid:96),mR ( t ) u (cid:96),mI ( t ) = − sin( t ( (cid:96) ( (cid:96) + 1)) / ) v (cid:96),mR + cos( t ( (cid:96) ( (cid:96) + 1)) / ) v (cid:96),mI + ˆ W (cid:96),mI ( t ) , where ˆ W (cid:96),m ( t ) = (cid:32) ˆ W (cid:96),mR ( t )ˆ W (cid:96),mI ( t ) (cid:33) = (cid:90) t sin(( t − s )( (cid:96) ( (cid:96) + 1)) / ) d a (cid:96),m ( s ) (cid:90) t cos(( t − s )( (cid:96) ( (cid:96) + 1)) / ) d a (cid:96),m ( s ) and v (cid:96),mR , resp. v (cid:96),mI , are the coefficients of the real, resp. imaginary, part of the initial value u (0).It is clear that the analysis from Section 4 can directly be extend to the case of the stochasticSchr¨odinger equation on the sphere (17). The errors in the truncation procedure, denotedby u κR and u κI , of the above ansatz are given by the following proposition (presented for zeroinitial data for simplicity). Proposition 6.2.
Let t ∈ T = [0 , T ] and t < . . . < t n = t be a discrete time partition for n ∈ N , which yields a recursive representation of the solution X = ( u R , u I ) of the stochasticSchr¨odinger equation on the sphere (18) with initial data u (0) = 0 . Assume that there exist (cid:96) ∈ N , α > , and a constant C > such that the angular power spectrum of the driving noise ( A (cid:96) , (cid:96) ∈ N ) decays with A (cid:96) ≤ C · (cid:96) − α for all (cid:96) > (cid:96) . Then, the error of the approximate solution X κ = ( u κR , u κI ) is bounded uniformly in time and independently of the time discretization by (cid:107) u R ( t ) − u κR ( t ) (cid:107) L p (Ω; L ( S )) ≤ ˆ C p · κ − ( α/ − (cid:107) u I ( t ) − u κI ( t ) (cid:107) L p (Ω; L ( S )) ≤ ˆ C p · κ − ( α/ − for all p ≥ and κ > (cid:96) , where ˆ C p is a constant that may depend on p , C , T , and α . (a) Sample of solution. -1 Error (b) Strong errors (real part). -1 Error (c) Error in approximation of onepath (imaginary part).
Figure 7.
Sample, mean-square errors, and error of one path of the approx-imation of the stochastic Schr¨odinger equation with angular power spectrumof the Q -Wiener process with parameter α = 4 and 100 Monte Carlo samples(for the mean-square errors). On top of that, the error is bounded uniformly in time, independently of the time discretiza-tion, and asymptotically in κ by (cid:107) u R ( t ) − u κR ( t ) (cid:107) L ( S ) ≤ κ − ( δ − , P -a.s. (cid:107) u I ( t ) − u κI ( t ) (cid:107) L ( S ) ≤ κ − ( δ − , P -a.s.for all δ < α/ . Since the proof of this proposition follows the lines of the proof of Proposition 4.1, we omitit and instead present some numerical experiments illustrating these theoretical results.We compute the errors when approximating solutions to (17) for various truncation in-dices κ for a Q -Wiener process with parameter α = 4. All other parameters are the same asin Section 5. In Figure 7, we display a sample at time T = 1 and a strong convergence plot ofthe real part of the numerical approximation, as well as errors in the approximation of a path(imaginary part) to the stochastic Schr¨odinger equation on the sphere. These illustrationsare in agreement with the results from Proposition 6.2. References [1] Adam Andersson, Raphael Kruse, and Stig Larsson. Duality in refined Sobolev–Malliavin spaces and weakapproximations of SPDE.
Stoch. PDE: Anal. Comp. , 4(1):113–149, 2016.[2] Vo V. Anh, Philip Broadbridge, Andriy Olenko, and Yu Guang Wang. On approximation for fractionalstochastic partial differential equations on the sphere.
Stoch. Environ. Res. Risk Assess , 32(9):2585–2603,2018.[3] Rikard Anton, David Cohen, Stig Larsson, and Xiaojie Wang. Full discretization of semilinear stochasticwave equations driven by multiplicative noise.
SIAM J. Numer. Anal. , 54(2):1093–1119, 2016.[4] Kendall Atkinson and Weimin Han.
Spherical Harmonics and Approximations on the Unit Sphere: AnIntroduction , volume 2044 of
Lecture Notes in Mathematics . Springer-Verlag, 2012.[5] Charles-Edouard Br´ehier, Martin Hairer, and Andrew M. Stuart. Weak error estimates for trajectories ofSPDEs under spectral Galerkin discretization.
J. Comp. Math. , 36(2):159–182, 2018.[6] Phil Broadbridge, Alexander D. Kolesnik, Nikolai Leonenko, and Andriy Olenko. Random spherical hy-perbolic diffusion.
J. Stat. Phys. , 177(5):889–916, 2019.[7] Julia Charrier. Strong and weak error estimates for elliptic partial differential equations with randomcoefficients.
SIAM J. Numer. Anal. , 50(1):216–246, 2012.
PPROXIMATION OF THE STOCHASTIC WAVE EQUATION ON THE SPHERE 25 [8] Jorge Clarke De la Cerda, Alfredo Alegr´ıa, and Emilio Porcu. Regularity properties and simulations ofGaussian random fields on the sphere cross time.
Electron. J. Stat. , 12(1):399–426, 2018.[9] Richard H. Clayton. Dispersion of recovery and vulnerability to re-entry in a model of human atrial tissuewith simulated diffuse and focal patterns of fibrosis.
Frontiers in Physiology , 9:1052, 2018.[10] David Cohen, Stig Larsson, and Magdalena Sigg. A trigonometric method for the linear stochastic waveequation.
SIAM J. Numer. Anal. , 51(1):204–222, 2013.[11] Giuseppe Da Prato and Jerzy Zabczyk.
Stochastic Equations in Infinite Dimensions , volume 152 of
En-cyclopedia of Mathematics and its Applications . Cambridge University Press, second edition, 2014.[12] Robert Dalang, Davar Khoshnevisan, Carl Mueller, David Nualart, and Yimin Xiao.
A Minicourse onStochastic Partial Differential Equations , volume 1962 of
Lecture Notes in Mathematics . Springer-Verlag,2009. Held at the University of Utah, Salt Lake City, UT, May 8–19, 2006, Edited by Khoshnevisan andFiras Rassoul-Agha.[13] Arnaud Debussche and Jacques Printems. Weak order for the discretization of the stochastic heat equation.
Math. Comput. , 78(266):845–863, 2009.[14] Xavier Fernique. Int´egrabilit´e des vecteurs gaussiens.
C. R. Acad. Sci., Paris, S´er. A , 270:1698–1699,1970.[15] Joshua H. Goldwyn and Eric Shea-Brown. The what and where of adding channel noise to the Hodgkin–Huxley equations.
PLOS Comp. Biol. , 7(11):1–9, 11 2011.[16] Philipp Harms and Marvin S. M¨uller. Weak convergence rates for stochastic evolution equations and appli-cations to nonlinear stochastic wave, HJMM, stochastic Schr¨odinger and linearized stochastic Korteweg–deVries equations.
Z. Angew. Math. Phys. , 70(1):Paper No. 16, 28, 2019.[17] Klaus Hasselmann. Stochastic climate models part I. Theory.
Tellus , 28(6):473–485, 1976.[18] Lukas Herrmann, Kristin Kirchner, and Christoph Schwab. Multilevel approximation of Gaussian randomfields: fast simulation.
Math. Models Methods Appl. Sci. , 30(1):181–223, 2020.[19] Lukas Herrmann, Annika Lang, and Christoph Schwab. Numerical analysis of lognormal diffusions on thesphere.
Stoch. PDE: Anal. Comp. , 6(1):1–44, 2018.[20] Ladislas Jacobe de Naurois, Arnulf Jentzen, and Timo Welti. Lower bounds for weak approximation errorsfor spatial spectral galerkin approximations of stochastic wave equations. In Andreas Eberle, MartinGrothaus, Walter Hoh, Moritz Kassmann, Wilhelm Stannat, and Gerald Trutnau, editors,
StochasticPartial Differential Equations and Related Fields , pages 237–248, Cham, 2018. Springer InternationalPublishing.[21] Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang. Dynamic scaling of growing interfaces.
Phys. Rev.Lett. , 56:889–892, 1986.[22] Yoshihito Kazashi and Quoc T. Le Gia. A non-uniform discretization of stochastic heat equations withmultiplicative noise on the unit sphere.
J. Complexity , 50:43–65, 2019.[23] Mih´aly Kov´acs, Annika Lang, and Andreas Petersson. Weak convergence of fully discrete finite elementapproximations of semilinear hyperbolic SPDEs with additive noise.
ESAIM:M2AN , 54(6):2199–2227,2020.[24] Mih´aly Kov´acs, Stig Larsson, and Fredrik Lindgren. Weak convergence of finite element approximationsof linear stochastic evolution equations with additive noise.
BIT Num. Math , 52(1):85–108, 2012.[25] Yuriy V. Kozachenko and L. F. Kozachenko. Modeling Gaussian isotropic random fields on a sphere.
J.Math. Sci. , 107(2):3751–3757, 2001.[26] Xiaohong Lan and Domenico Marinucci. On the dependence structure of wavelet coefficients for sphericalrandom fields.
Stochastic Process. Appl. , 119(10):3749–3766, 2009.[27] Xiaohong Lan, Domenico Marinucci, and Yimin Xiao. Strong local nondeterminism and exact modulusof continuity for spherical Gaussian fields.
Stochastic Process. Appl. , 128(4):1294–1315, 2018.[28] Annika Lang, Stig Larsson, and Christoph Schwab. Covariance structure of parabolic stochastic partialdifferential equations.
Stoch. PDE: Anal. Comp. , 1(2):351–364, 2013.[29] Annika Lang and Christoph Schwab. Isotropic Gaussian random fields on the sphere: regularity, fastsimulation and stochastic partial differential equations.
Ann. Appl. Probab. , 25(6):3047–3094, 2015.[30] Quoc Thong Le Gia, Ian H. Sloan, Robert S. Womersley, and Yu Guang Wang. Isotropic sparse regular-ization for spherical harmonic representations of random fields on the sphere.
Appl. Comput. Harmon.Anal. , 49(1):257–278, 2020.[31] Domenico Marinucci and Giovanni Peccati.
Random Fields on the Sphere. Representation, Limit Theoremsand Cosmological Applications . Cambridge University Press, 2011. [32] Mitsuo Morimoto.
Analytic Functionals on the Sphere , volume 178 of
Translations of Mathematical Mono-graphs . American Mathematical Society, 1998.[33] G´abor Szeg˝o.
Orthogonal Polynomials , volume XXIII of
Colloquium Publications . American MathematicalSociety, fourth edition, 1975.[34] Xiaojie Wang. An exponential integrator scheme for time discretization of nonlinear stochastic waveequation.
J. Sci. Comput. , 64(1):234–263, 2015.[35] Myhailo I. Yadrenko.
Spectral Theory of Random Fields . Translation Series in Mathematics and Engineer-ing. Optimization Software, Inc., Publications Division; Springer-Verlag, 1983. Transl. from the Russian.(David Cohen)
Department of Mathematical SciencesChalmers University of Technology & University of GothenburgS–412 96 G¨oteborg, Sweden.
Email address : [email protected] (Annika Lang) Department of Mathematical SciencesChalmers University of Technology & University of GothenburgS–412 96 G¨oteborg, Sweden.
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