Sparse tensor product spectral Galerkin BEM for elliptic problems with random input data on a spheroid

Adv Comput MathDOI 10.1007/s10444-014-9350-7

Sparse tensor product spectral Galerkin BEMfor elliptic problems with random input dataon a spheroid

Alexey Chernov ·Duong Pham

Received: 31 May 2013 / Accepted: 13 March 2014© Springer Science+Business Media New York 2014

Abstract We introduce and analyze a sparse tensor product spectral GalerkinBoundary Element Method based on spherical harmonics for elliptic problems withrandom input data on a spheroid. Problems of this type appear in geophysical appli-cations, in particular in data acquisition by satellites. Aiming at a deterministiccomputation of the k-th order statistical moments of the random solution, we estab-lish convergence theorems showing that the sparse tensor product spectral Galerkindiscretization is superior to the full tensor product spectral Galerkin discretizationin the case of mixed regularity of the data’s k-th order moments, naturally implyingmixed regularity of the k-th order moments of the random solution. We prove thatanalytic regularity of the data’s k-th order moments implies analytic regularity of thesolution’s k-th order moments. We illustrate performance of the sparse and full tensorproduct discretization schemes on several numerical examples.

Keywords Sparse spectral discretization · Dirichlet-to-Neumann operator ·Random data · Tensor product · Spherical harmonics · Spheroidal coordinates

Mathematics Subject Classifications (2010) 65N30 · 35R60 · 41A25

Dedicated to Professor Ernst P. Stephan on the occasion of his 65th anniversary.

Communicated by: Ian H. Sloan

A. Chernov (�) · D. PhamHausdorff Center for Mathematics and Institute for Numerical Simulation, University of Bonn,Endenicher Allee 64, Bonn 53115, Germanye-mail: [email protected]

Present Address:A. ChernovDepartment of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220,Reading RG6 6AX, UK

mailto:[email protected]

A. Chernov, D. Pham

1 Introduction

In geophysical applications, the Neumann problem exterior to a spheroid where theorbits of satellites are located is of interest; see e.g. Ref. [10] and [12]. Following itsorbit, a satellite collects data sets which amount to boundary conditions in scatteredpoints. Obviously, this process is prone to uncertainty for many reasons. One is theerror in measurements that cannot be avoided due to imperfect measurement devices.Furthermore, such measurements are usually done by sampling a large but finitenumber of system snapshots and therefore provide an incomplete information aboutthe system. In this paper we suggest and analyze a natural and efficient numericalapproach accounting for uncertainty in the boundary data.

As a model problem we consider the Laplace equation with Neumann boundarydata in the exterior to a prolate spheroid (the case of an oblate spheroid or a spherecan be treated analogously):

⎧⎪⎨

⎪⎩

�U = 0 in Dc,

∂νU = g on �,

U(x) = O( |x|−1 )as |x| → ∞,

(1.1)

where |·| denotes the Euclidean norm and ν denotes the unit outward normal vectoron the problate spheroid

� ={

(x1, x2, x3) ∈ R3 : x

21 + x2

2

b2+ x2

3

a2= 1, a > b > 0

}

(1.2)

and Dc is the unbounded domain outside the boundary �. An application of prob-lem (1.1) can be found in potential theory, where the solution of the problemrepresents disturbing gravity; see e.g. the survey of the related problems in [6] andreferences therein. The Dirichlet-to-Neumann map (the Steklov-Poincare operator)can be used to directly convert the boundary value problem into a pseudodifferentialequation on the spheroid. This amounts to solving the operator equation

Ku = g in H−1/2(�), (1.3)

where K is the Dirichlet-to-Neumann map, u is the unknown Dirichlet data, g is theNeumann data and � is a prolate spheroid.

Equation (1.3) with deterministic right-hand side g has been solved by usingcontinuous piecewise bilinear basis functions [13] and spherical radial basis func-tions [18]. In this paper, spaces of spherical harmonics are used to solve Eq. (1.3)with random right-hand side. These functions are important in many theoretical andpractical applications, particularly in the computation of atomic orbital electron con-figurations, representation of gravitational fields, geoids, and the magnetic fields ofplanetary bodies and stars, and characterization of the cosmic microwave backgroundradiation, see e.g. Ref. [11, 17, 24] and the references therein.

It is known (see e.g. [13, 18]) that if � is a spheroid, K has a series representationin the basis of spherical harmonics, see Eq. (3.29) below. This representation yieldsin particular that the Galerkin matrix of K w.r.t. the basis of spherical harmonicsis diagonal. This property makes the choice of a subset of spherical harmonics as

Sparse spectral BEM for elliptic problems with random data

a Galerkin basis very natural and will be, in fact, even more important for the caseof the uncertain data g. As we will see later, numerical solution of the randomizedproblem will lead to a high-dimensional formulation, suffering from the curse ofdimensionality. Under such circumstances, diagonality of the Galerkin matrix willmake the solution of the linear system fast and simple.

As is well-known, the approximation with a high-order basis with global (or large)support is potentially dangerous. In particular, discontinuities of the approximatedfunction or its derivatives may lead to oscillations in the numerical approximation,also known as the Gibbs phenomenon. This remains true also for the model problem(1.1) and, equivalently, for its boundary reduction (1.3). We expect, however, thatthe spectral Galerkin method based on spherical harmonics will work well, if thesolution will not strongly depend on local features, as the landscape, coastline, etc.Possible examples are e.g. problems involving the measurements of the magnetic andgravity fields of the Earth. We refer to [6] and references therein for further detailson mathematical models and numerical approximations of the gravity field of theEarth and related problems in geodesy. We remark that the shape of the Earth canclosely be approximated (< 100km) by an oblate spheroid. The method described inthe present work for the prolate spheroidal geometry can be reformulated for the caseof an oblate spheroid utilizing oblate spheroidal coordinates instead of Eq. 2.7. Theprocedure is identical to the case presented here and we do not elaborate the case ofthe oblate spheroid any further.

In the present paper we assume that the input data (and therefore the solu-tion) depends on a “random event” ω ∈ � (assuming (�,�,P) is the underlyingprobability space) which allows to treat g as a random field being a measurablemapping

g :{� → H−1/2(�),

ω �→ g(·, ω). (1.4)

Continuity of K−1 obviously implies (see in particular Lemma 4.1 in Ref. [5]) that uis a random field too, i.e. a measurable mapping

u :{� → H 1/2(�),

ω �→ u(·, ω). (1.5)

Numerical solution of Eq. 1.3 is a hard task since parametrization of � requiresin general infinitely many variables and hence leads to an infinite dimensional prob-lem (see e.g. Ref. [7] for the case of a random coefficient). By a suitable truncationprocedure [29] the number of stochastic dimensions can be reduced to a finite num-ber M , which, however, can still be large. By this, Eq. 1.3 is an equation in (M + 2)dimensions and the tensor product discretization will lead to a prohibitive number ofunknowns. This effect is also known as the curse of dimensionality. Practical solu-tion of this high-dimensional problem makes mandatory the use of sparse/adaptivediscretization methods and specific solution procedures, see e.g. [14, 15, 27] and thereferences therein.

However, in many applications the solution u(x, ω) is not of interest as a func-tion of ω. Frequently the mean field Eu(x) := ∫

� u(x, ω)dP(ω), the covarianceCu(x, y) := E[(u(x, ω) − Eu(x))(u(y, ω) − Eu(y))], and higher order moments

A. Chernov, D. Pham

Mku(x1, . . . , xk) are the quantities of interest. Tensorization of Eq. 1.3 and integra-tion over the set of elementary events � provides a formulation (cf. Eq. 3.38 below)whose solution is the quantity of interest itself [26, 28, 31]:

(K⊗ · · · ⊗K)︸︷︷︸

k times

Mku = Mkg. (1.6)

Well-posedness of Eq. 1.6 and stability of its conforming discretizations follow fromcontinuity, ellipticity and symmetry of the associated bilinear form (see Lemma 3.4).The challenge in numerical approximation is in its intrinsic high dimensionality: thesolution Mku is a function on a 2k-dimensional manifold � × · · · × � (k times).This yields to an exponential in k growth of complexity for the full tensor productdiscretizations. An efficient discretization of Eq. 1.6 must be carefully adjusted tothe regularity of Mku.

Extending the approach in [3, 4] we propose in this paper a sparse tensor productspectral discretization of Eq. 1.6 based on suitable combination of bases of spheri-cal harmonics on individual spheroids �. In particular, we prove (see Theorem 4.2)that if Mkg and hence Mku has a finite mixed regularity, the Galerkin discretizationby products of spherical harmonics with degrees from the hyperbolic cross achievesthe same convergence rate as the discretization by space of full tensor product spher-ical harmonics, when expressed in terms of the maximal degree L. This yields asignificant gain in the convergence rate, expressed in terms of the total number ofunknowns, since the cardinality of the sparse space grows like O(L2(logL)k−1)

compared to the cardinality of the full tensor product space growing like O(L2k).Such a significant improvement of the convergence rate vs. the total number of

unknowns by means of the hyperbolic cross construction is explained by the asymp-totic decay pattern of the Fourier coefficients of the solution Mku, cf. Eq. 4.56 andFig. 6. The hyperbolic cross construction is not beneficial any longer if the asymp-totic decay pattern of Mku’s coefficients is different. In particular, we prove thatMku is analytic if and only if the data Mkg is analytic (see Theorem 5.4) and pro-vide upper bounds on its Fourier coefficients (see also Fig. 10), indicating that not thehyperbolic cross construction, but the total degree selection (see Eq. 5.73) is optimalin this case. To the best of our knowledge, the shift theorem for analytic functions(Theorem 5.4) is a new result, not published elsewhere for operator equationsgoverned by tensor product operators.

We emphasize that our primary interest in this paper is the a priori convergenceanalysis of the spectral Galerkin schemes as well as the regularity theory for the k-thmoment equation. Other aspects related with interpolation of input data Mkg fromreal measurements, treatment of the satellite velocity, etc. are not addressed here. Theapproximation and regularity theory developed in this paper is based on the strongellipticity of the Dirichlet-to-Neumann map K which suggests that similar resultswill hold for more general strongly elliptic operators on smooth sphere-like surfaces.In this paper we restrict to the case of the spheroidal geometry for two reasons. First,although geometrically simple, this case is important in numerous practical applica-tions, see e.g. [10–12, 17] and references therein. Second, in the case of the spheroidthe Dirichlet-to-Neumann map K admits a sparse (diagonal) representation in the


basis of spherical harmonics which enables a robust solution of the resulting algebraicsystem.

The paper is organized as follows. In Section 2 we set up the functional spaceframework. In Section 3 we recapitulate some necessary results from Ref. [13] and[18], obtain an explicit series representation of the Dirichlet-to-Neumann operator,and perform the boundary reduction of the model problem (1.1). Then we derivethe deterministic k-th moment equation for the reduced problem. In Section 4 weintroduce and analyze the sparse tensor product spectral Galerkin discretizationscheme based on spherical harmonics and establish a priori convergence estimates.In Section 5 we prove that the solution of the k-th moment equation is analytic ifand only if the right-hand side is analytic and establish upper bounds for its Fouriercoefficients. In Section 6 we give numerical examples illustrating convergence of thesparse and full tensor product discretization schemes for the case of a finite mixedregularity and analytic regularity of the input data Mkg.

Throughout, we adopt the following notation: for any positive real numbers x, ywe write x � y if there exists a constant C independent of any parameters which x

and y might depend on, so that x ≤ Cy. When x and y satisfy x � y and y � x wewrite x � y.

2 Functional space framework

In this paper, we work with the prolate spheroidal coordinates (μ, θ, ϕ), in which apoint x = (x1, x2, x3) ∈ R

3 is represented by⎧⎪⎨

⎪⎩

x1 = c0 sinhμ sin θ cosϕ, μ ≥ μ0 > 0,

x2 = c0 sinhμ sin θ sin ϕ, θ ∈ [0, π],x3 = c0 coshμ cos θ, ϕ ∈ [0, 2π).

(2.7)

The prolate spheroid � defined by Eq. 1.2 is given in these coordinates as

� = {(μ, θ, ϕ) : μ = μ0 > 0, θ ∈ [0, π], ϕ ∈ [0, 2π)}where a = c0 coshμ0 and b = c0 sinhμ0 are the semi-major and semi-minor axes ofthe spheroid, so that c0 = √

a2 − b2 may be identified as the linear eccentricity andcothμ0 = a/b is the relation of the semiaxes.

Denoting by S := {x ∈ R3 : |x| = 1} the unit sphere in R

3, we define atransformation F : S → � by letting

F(θ, ϕ) := (μ0, θ, ϕ). (2.8)

Notice that both � and F depend on the value μ0 which is fixed at the surface of thespheroid. Spheroidal coordinates and their properties are visualized in Figs. 1 and 2.Notice that the unit sphere is realised as the limit of � with μ0 → ∞ and e.g.c0 = 1/ sinhμ0 → 0.

For notational simplicity, we also use F to denote the transformation from Sk

into �k , namely F((θ1, ϕ1), . . . , (θk, ϕk)

) = ((μ0, θ1, ϕ1), . . . , (μ0, θk, ϕk)

). These

A. Chernov, D. Pham

Fig. 1 Spheroidal coordinatecurves in any cross-sectioncontaining x3-axis for c0 = 1.Spherical coordinate lines(dashed) are asymptotes for thecoordinate curves θ = θ0 asμ → ∞

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x1

x3 00.30.60.9µ=1.2

=0°18°

36°

54°

72°

90°

108°

126°

144°

162°180°

mappings are infinitely differentiable bijections. The space of homogeneous polyno-mials of degree � in R

3 is denoted by P� and its subspace of harmonic polynomialsis denoted by H�. We denote by P� and H� the restrictions of polynomials in P� andH�, respectively, onto the unit sphere S. The dimension of H� is 2� + 1; see e.g.Ref. [21] (page 4), one may choose for it an L2(S)-orthonormal basis {Y�,m}�m=−�.The collection of all the spherical harmonics {Y�,m : � ≥ 0, m = −�, . . . , �} formsan orthonormal basis for L2(S).

For any distribution v defined on the spheroid �, and for any point x whose prolatespheroidal coordinate is (μ0, θ, ϕ), we can write

v(x) = v(μ0, θ, ϕ) = v(F (θ, ϕ)) =∞∑

�=0

�∑

m=−�

v�,mY�,m(θ, ϕ),

where

v�,m =∫ 2π

0

(∫ π

0v(F (θ, ϕ)) Y�,m(θ, ϕ) sin θ dθ

)

dϕ (2.9)

Fig. 2 The mapping F definedin Eq. 2.8 transforms equispacedparallels on S to parallels onspheroids, shifted towards theirpoles (greater relative shift forsmaller μ0)

µ0=1.0µ0=0.7

µ0=0.4

prolate spheroids

sphere


are the Fourier coefficients of v. The Sobolev space Hs(�), for s ∈ R, is defined by

Hs(�) = {v ∈ D′(�) :∞∑

�=0

�∑

m=−�

(1 + �)2s∣∣v�,m

∣∣2

< +∞}, (2.10)

where D′(�) is the space of distributions on �. The corresponding inner product andthe norm of Hs(�) are given by

〈v, w〉Hs(�) =∞∑

�=0

�∑

m=−�

(1 + �)2s v�,mw�,m, v, w ∈ Hs(�), (2.11)

and

‖v‖Hs(�) =( ∞∑

�=0

�∑

m=−�

(1 + �)2s∣∣v�,m

∣∣2)1/2

, v ∈ Hs(�). (2.12)

We note here that the inner product (2.11) and the norm (2.12) satisfy

〈v, w〉Hs(�) = 〈v ◦ F ,w ◦ F 〉Hs(S) and ‖v‖Hs(�) = ‖v ◦ F‖Hs(S) (2.13)

for any v, w ∈ Hs(�). The set {Y�,m◦F−1 : � ∈ N, m = −�, . . . , �} is an orthogonalbasis for Hs(�). We also note that the space H 0(�) can be understood as a weightedL2-space on the spheroid.

Remark 2.1 Definition (2.10) is an equivalent definition of Sobolev spaces on a(closed) manifold, which allows to obtain explicit and dimension-independent, e.g.independent of k, constants in the convergence theorems, see e.g. Eq. 4.44. If anequivalent norm on Hs(�) is used [18, 20, 25], the estimate (4.44) will be still true,but will depend on the constant in the norm equivalence raised to the power k.

In the sequel, we will utilize the Cauchy–Schwarz inequality

〈v, w〉Hs(�) ≤ ‖v‖Hs(�) ‖w‖Hs(�) ∀v, w ∈ Hs(�), (2.14)

and the following identity which can be easily derived from Eqs. 2.12 and 2.14

‖v‖Hs(�) = supw∈Ht (�)

w �=0

〈v, w〉H

s+t2 (�)

‖w‖Ht (�)

∀v ∈ Hs(�), for all s, t ∈ R. (2.15)

We now introduce tensor products of Sobolev spaces. Given s ∈ R, the Sobolevspace Hs

mix(�2) is defined to be the space of all distributions v on �2 := � × �

satisfying

∞∑

�1=0

∞∑

�2=0

�1∑

m1=−�1

�2∑

m2=−�2

(1 + �1)2s (1 + �2)

2s∣∣v(�1,�2),(m1,m2)

∣∣2 < ∞, (2.16)

with the Fourier coefficients

v(�1,�2),(m1,m2) :=∫

S

∫

S

v(F (x), F (y)) Y�1,m1(x)Y�2,m2(y) dσx dσy (2.17)

A. Chernov, D. Pham

The inner product on Hsmix(�

2) is given by

〈v,w〉Hsmix (�

2) :=∞∑

�1=0

∞∑

�2=0

�1∑

m1=−�1

�2∑

m2=−�2

(1 + �1)2s(1 + �2)

2s v(�1,�2),(m1,m2)w(�1,�2),(m1,m2)

(2.18)

for any v, w ∈ Hsmix(�

2) and ‖v‖Hsmix(�

2) := 〈v, v〉1/2Hsmix (�

2).

We denote by⊗k

i=1 Xi the tensor product of separable Hilbert spaces Xi , for i =1, . . . , k; see e.g. Ref. [2] (Definition 12.3.2). For the corresponding inner productsthere holds (Ref. [2], page 298)

⟨k⊗

i=1

vi,

k⊗

i=1

wi

⟩

⊗ki=1 Xi

=k∏

i=1

〈vi, wi〉Xi∀vi, wi ∈ Xi, i = 1, . . . , k. (2.19)

In the case Xi = X for i = 1, . . . , k, we denote X(k) := ⊗ki=1 X. It has the property

that‖v1 ⊗ · · · ⊗ vk‖X(k) = ‖v1‖X · · · ‖vk‖X (2.20)

holds for every v1, . . . , vk ∈ X. Relation (2.19) and definition (2.18) provide theisometry Hs(�) ⊗ Hs(�) = Hs

mix(�2). In what follows, we identify the space

Hsmix(�

2) and the tensor product Hs(�)⊗Hs(�). By this agreement, we define forany s ∈ R and k ≥ 2 the tensor product space on �k := � × . . .× � by

Hsmix(�

k) :=k⊗

i=1

Hs(�).

Inequality (2.14) and the identity (2.15) combined with above definitions imply

〈v, w〉Hsmix(�

k) ≤ ‖v‖Hsmix (�

k) ‖w‖Hsmix (�

k) ∀v, w ∈ Hsmix(�

k), ∀s ∈ R, (2.21)

and

‖v‖Hsmix (�

k) = supw∈Ht

mix(�k)

w �=0

〈v, w〉H

s+t2

mix (�k)

‖w‖Htmix (�

k)

∀v ∈ Hsmix(�

k), ∀s, t ∈ R. (2.22)

3 Variational formulation

In this section we obtain the Dirichlet-to-Neumann Eq. 1.3 and the deterministic k-thmoment equation for the case of random data.

3.1 Boundary reduction for the model problem

In this subsection we will briefly represent the boundary reduction which convertsthe exterior Neumann problem (1.1) outside the boundary � into the Dirichlet-to-Neumann Eq. 1.3 on the spheroid. In our presentation, we follow the lines of Ref. [13]and [18].


In the prolate spheroidal coordinates (2.7), the Laplace operator is expressed as

�U = 1

c20

√cosh2 μ− cos2 θ

·(

1

sinhμ

∂

∂μ

(sinhμ

∂U

∂μ

)+ 1

sin θ

∂

∂θ

(sin θ

∂U

∂θ

)+ ( 1

sin2 θ+ 1

sinh2 μ

)∂2U

∂ϕ2

)

.

Using the method of separation of variables we obtain that a harmonic function U

can be represented as a linear combination of the countable family

�m� (μ, θ, ϕ) = Qm

� (coshμ)Y�,m(θ, ϕ), � = 0, 1, 2, . . . , m = −�, . . . , �,

where Y�,m(θ, ϕ) =√

(2�+1)(�−m)!4π(�+m)! Pm

� (cos θ)eimϕ are the spherical harmonics and

Pm� and Qm

� are the associated Legendre functions of the first and the second kind,see Ref. [1] (Chapter 8).

Suppose u(θ, ϕ) := U(F (θ, ϕ)) is represented in the series form

u(θ, ϕ) =∞∑

�=0

�∑

m=−�

u�,mY�,m(θ, ϕ), (3.23)

where u�,m are the Fourier coefficients of u given by

u�,m =∫

S

uY�,m dσ =∫ 2π

0

∫ π

0u(θ, ϕ)Y�,m(θ, ϕ) sin θ dθdϕ.

Then the solution of the Laplace equation in the unbounded domain Dc is

U(μ, θ, ϕ) =∞∑

�=0

�∑

m=−�

Qm� (coshμ)

Qm� (coshμ0)

u�,mY�,m(θ, ϕ), μ ≥ μ0. (3.24)

We note here that the normal vector to the spheroid � is

∂x

∂θ× ∂x

∂ϕ= c2

0 sinhμ0 sin θ (coshμ0 sin θ cosϕ, coshμ0 sin θ sin ϕ, sinhμ0 cos θ) ,

(3.25)whose Euclidean norm is

∣∣∣∂x

∂θ× ∂x

∂ϕ

∣∣∣ = c2

0 sinhμ0 sin θ√

cosh2 μ0 − cos2 θ. (3.26)

By elementary calculation, we obtain

∂νU(μ0, θ, ϕ) = − 1

c0

√cosh2 μ0 − cos2 θ

∂U

∂μ(μ0, θ, ϕ),

where ∂νU is the outward normal derivative of U on the spheroid �. Hence, Eq. 3.24implies

∂νU(μ0, θ, ϕ) = − 1

c0


∞∑

�=0

�∑

m=−�

ddμ

Qm� (coshμ0)

Qm� (coshμ0)

u�,mY�,m(θ, ϕ). (3.27)

A. Chernov, D. Pham

The exterior Neumann problem (1.1) is uniquely solvable and its solution admits therepresentation (3.24) once the Fourier coefficients u�,m of the Dirichlet boundary dataare known. By Eq. 3.27, computation of u�,m is equivalent to solution of the equation

Ku = g on �, (3.28)

where K is the Dirichlet-to-Neumann map given by

Kv(θ, ϕ) := − 1

c0


∞∑

�=0

�∑

m=−�

ddμ

Qm� (coshμ0)

Qm� (coshμ0)

v�,mY�,m(θ, ϕ);(3.29)

see e.g. Ref. [22]. In this paper, we are dealing with the Eq. 3.28 in which the right-hand side is random g = g(ω).

In the case of oblate spheroid or a sphere, similar argument can be used to convertthe exterior Neumann problem (1.1) to the pseudodifferential Eq. 3.28. The deter-ministic method described in Section 3.2 for computation of statistical moments ofthe solution can be used in the same manner. In this paper, we consider only the casefor the prolate spheroid.

The following lemma proved in Ref. [13] (Lemma 3.2) shows that K is boundedbijective linear operator from H 1/2(�) to H−1/2(�).

Lemma 3.1 Let � ∈ N0 and |m| ≤ � be integers. For any nonzero μ there holds

(�2 + 1)1/2 sinh2 μ

coshμ< −

ddμ

Qm� (coshμ)

Qm� (coshμ)

sinhμ <√

2 (�2 + 1)1/2 coshμ. (3.30)

By this, we can recast Eq. 3.28 in the variational form: Find u ∈ H 1/2(�)

satisfying

D(u, v) = 〈g, v〉 ∀v ∈ H 1/2(�), (3.31)

with the bilinear form D(·, ·) and the duality pairing 〈·, ·〉

D(v,w) :=∫

�

(Kv)w dσ, 〈g, v〉 :=∫

�

gv dσ ∀v,w ∈ H 1/2(�), g ∈ H−1/2(�).

(3.32)

In view of Eqs. 3.26, 3.29 and 3.32, the bilinear form D(·, ·) has the seriesrepresentation

D(v, w) =∫ 2π

0

∫ π

0(Kv)w

∣∣∣∂x

∂θ× ∂x

∂ϕ

∣∣∣ dθ dϕ = c0

∞∑

�=0

�∑

m=−�

Hm� v�,mw�,m,

where

Hm� = −

ddμ

Qm� (coshμ0)

Qm� (coshμ0)

sinhμ0 � (�2 + 1)1/2. (3.33)

Lemma 3.1 implies well-posedness of problem (3.31).


Lemma 3.2 The bilinear form D(·, ·) : H 1/2(�) ×H 1/2(�) → R is bounded

|D(v, w)| ≤ C1 ‖v‖H 1/2(�) ‖w‖H 1/2(�) ∀v, w ∈ H 1/2(�), (3.34)

and H 1/2(�)-elliptic

D(v, v) ≥ C2 ‖v‖2H 1/2(�)

∀v ∈ H 1/2(�). (3.35)

The constants C1 and C2 are explicit: C1 = √2c0 coshμ0 and C2 = sinh2 μ0

coshμ0.

3.2 Problems with random data

In what follows we consider the Eq. 3.28 for random loading g, which leads torandom solution u. Let (�,�,P) be a probability space consisting of the space ofelementary events �, σ -algebra of its subsets � and the probability measure P on�. Suppose g is a random field, i.e. a measurable mapping satisfying (1.4). Sincethe solution operator K−1 : H−1/2(�) → H 1/2(�) is continuous, the solutionu := K−1g is measurable and satisfies (1.5).

In order to introduce the notion of a k-th statistical moment of a random field wedefine, following Ref. [26, 31], for an positive integer k and a separable Hilbert spaceX a Bochner space

Lk(�,X) := {v : � → X :∫

�

‖v(ω)‖kX dP(ω) < +∞}equipped with the norm

‖v‖Lk(�,X) :=(∫

�

‖v(ω)‖kX dP(ω)

)1/k

.

Definition 3.3 Let v ∈ Lk(�,X). The k-th order moment Mkv ∈ X(k) of v is givenby

Mkv :=∫

�

(v(ω)⊗ · · · ⊗ v(ω)︸︷︷︸

k times

)dP(ω).

By Eq. 2.20, the k-th order moment Mkv is well defined for any v ∈ Lk(�,X).Let u(ω) be a random solution of Eq. 3.28 with a random right hand side g(ω). Weconsider the tensor product operator K(k) := K⊗ · · · ⊗K (k times) which is a linearmapping

K(k) : H 1/2mix(�

k) → H−1/2mix (�k), (3.36)

see [31, Proposition 2.4] for more details. The k-th moment equation can be nowderived by the general procedure described in [31, Section 2]. Tensorization ofKu(ω) = g(ω) yields for every fixed elementary event ω ∈ �

K(k)(⊗k

i=1 u(ω)) = ⊗k

i=1g(ω) in H−1/2mix (�k). (3.37)

Taking the mean of Eq. 3.37 and observing that E[K(k)

( ⊗ki=1 u(ω)

)] = K(k)Mku

since E and K(k) commute we obtain the deterministic k-th moment problem:

A. Chernov, D. Pham

Given Mkg ∈ H−1/2mix (�k), find Mku ∈ H

1/2mix(�

k) satisfying

K(k)Mku = Mkg. (3.38)

Recalling the mapping properties (3.36) of the product operator Kk , the variationalcounterpart of Eq. 3.38 can be derived. For this, we denote by 〈〈·, ·〉〉 the H−1/2

mix (�k)−H

1/2mix(�

k) duality pairing. Then the variational form of Eq. 3.38 reads:

Given Mkg ∈ H−1/2mix (�k), find Mku ∈ H

1/2mix(�

k) satisfying

D(Mku, v) =⟨⟨Mkg, v

⟩⟩∀v ∈ H

1/2mix(�

k), (3.39)

with the bilinear form D(·, ·) = ⟨⟨K(k)·, ·⟩⟩. Proposition 2.4 from Ref. [31] implies

Lemma 3.4 The bilinear form D(·, ·) : H 1/2mix(�

k)×H1/2mix(�

k) → R is bounded and

H1/2mix(�

k)-elliptic, i.e.,

D(v, w) ≤ Ck1 ‖v‖H 1/2

mix (�k)‖w‖

H1/2mix (�

k), (3.40)

and

Ck2 ‖v‖2

H1/2mix(�

k)≤ D(v, v) (3.41)

for all v, w ∈ H1/2mix(�

k).

By Lemma 3.4 there exists a unique solution of Eq. 3.39.

4 Sparse tensor product spectral Galerkin method

The choice of suitable finite dimensional approximation subspaces to solve theproblem (3.38) numerically is one of the main ingredients deciding the efficiencyof computation. Typically, large subspaces provide a better approximation. How-ever, taking unnecessarily many degrees of freedom requires a larger computationaleffort without significant improvement of the approximation quality. Therefore, theapproximation subspaces must be adapted to the solution of the problem [23].

In this section we construct an efficient priori approximation of u ∈ Hsmix(�

k) bypolynomials in the Ht

mix(�k)-norm. The standard full tensor product approximation

of u ∈ Hsmix(�

k) yields the convergence rate (L + 2)−(s−t ) and demands O(L2k

)

unknowns, where L is the degree of polynomials in the approximation space. Extend-ing the approach in Ref. [30] (see also Ref. [3, 4]) we develop a hyperbolic crossapproximation, which provides the same approximation order with O(

L2(logL)k−1)

unknowns.Let L be a positive integer. We introduce the index set

δL :={

� = (�1, . . . , �k) ∈ Nk :

k∏

i=1

(1 + �i) ≤ 1 + L

}

, (4.42)


and the associated finite element space

SδL := span

{Y�,m : � ∈ δL,mi = −�i, . . . , �i

}, (4.43)

where

Y�,m((θ1, ϕ1), . . . , (θk, ϕk)

) := Y�1,m1(θ1, ϕ1) · · ·Y�k,mk (θk, ϕk).

Note here that the set{Y�,m : � ∈ N

k,m = −�, . . . , �}

forms an orthogonal basis ofthe Sobolev space Hs

mix(�k). For any

v ◦ F =∞∑

�=0

�∑

m=−�

v�,mY�,m,

we define by (P δLv) the truncated series

(P δLv) ◦ F =

∑

�∈δL

�∑

m=−�

v�,mY�,m

which corresponds to the orthogonal projection into SδL w.r.t. the inner product

〈·, ·〉H 0mix (�

k). The following theorem allows to quantify the projection error.

Theorem 4.1 Let s and t be real numbers such that t ≤ s. For any v ∈ Hsmix(�

k),there holds∥∥v − P δ

Lv∥∥Htmix (�

k)≤ 1

(L+ 2)s−t

∥∥v − P δ

Lv∥∥Hsmix(�

k)≤ 1

(L+ 2)s−t‖v‖Hs

mix (�k) .

(4.44)

Proof We have

∥∥v − P δ

Lv∥∥2Htmix(�

k)=

∑

�/∈δL

�∑

m=−�

( k∏

i=1

(1 + �i))2t ∣

∣v�,m∣∣2

≤ 1

(L+ 2)2(s−t )

∑

�/∈δL

�∑

m=−�

( k∏

i=1

(1 + �i))2s ∣

∣v�,m∣∣2

= 1

(L+ 2)2(s−t )

∥∥v − P δ

Lv∥∥2Hsmix (�

k)

≤ 1

(L+ 2)2(s−t )‖v‖2

Hsmix (�

k).

The results in the previous section show that the sparse tensor product space SδL

can be used to solve the problem (3.38) approximately. We consider the followingGalerkin formulation: Given Mkg ∈ H

−1/2mix (�k), find μL ∈ Sδ

L satisfying

D(μL, v) =⟨⟨Mkg, v

⟩⟩∀v ∈ Sδ

L. (4.45)

A. Chernov, D. Pham

By Lemma 3.4, Galerkin formulation Eq. 4.45 is well posed. Recalling Eqs. 3.39and 4.45, we have

D(Mku− μL, v) = 0 ∀v ∈ SδL. (4.46)

We have the following a priori convergence estimate.

Theorem 4.2 Let s and t be real numbers satisfying t ≤ 1/2 ≤ s. Assume that Mku

is the solution of Eq. 3.39 and μL ∈ SδL is the sparse tensor Galerkin solution of

Eq. 4.45 for the data Mkg ∈ Hs−1mix (�

k). Then∥∥∥Mku− μL

∥∥∥Htmix (�

k)≤ C

(L+ 2)s−t

∥∥∥Mkg

∥∥∥Hs−1mix (�

k),

where the constant C is independent of L.

Proof Continuity and H1/2mix(�

k)-ellipticity of D (Lemma 3.4) imply

∥∥∥Mku− μL

∥∥∥H

1/2mix (�

k)≤

(C1

C2

)k

infv∈SδL

∥∥∥Mku− v

∥∥∥H

1/2mix (�

k).

This together with Theorem 4.1 yields∥∥∥Mku− μL

∥∥∥H

1/2mix (�

k)� 1

(L+ 2)s− 12

∥∥∥Mku

∥∥∥Hsmix (�

k). (4.47)

Assume that t < 1/2. Duality Eq. 2.22, Galerkin orthogonality (4.46) and Lemma3.4 imply

∥∥∥Mku− μL

∥∥∥Htmix (�

k)= sup

v∈H1−tmix

(�k)

v �=0

⟨⟨Mku− μL, v⟩⟩

H1/2mix (�

k)

‖v‖H 1−tmix (�

k)

� supv∈H1−t

mix(�k)

v �=0

D(Mku− μL, v)


k)

= supv∈H1−t

mix(�k)

v �=0

D(Mku− μL, v − P δLv)


k)

�∥∥∥Mku− μL

∥∥∥H

1/2mix(�

k)sup

v∈H1−tmix

(�k)

v �=0

∥∥v − P δ

Lv∥∥H

1/2mix(�

k)


k)

.

(4.48)

Recalling Eqs. 4.44 and 4.47 we obtain∥∥∥Mku− μL

∥∥∥Htmix(�

k)� 1

(L + 2)s−t

∥∥∥Mku

∥∥∥Hsmix(�

k)(4.49)

The assertion follows then by continuity of the operator K−1 : Hs−1mix (�

k) →Hs

mix(�k).


To estimate the dimension SδL adopt the approach in Ref. [9] and obtain

dim(SδL) =

∑

�∈δL

k∏

i=1

(2�i + 1) �∫

x∈�(k,L)

k∏

i=1

(2xi + 1) dx � C(k)L

∫

x∈�(k,L)

dx

(4.50)where

�(k, L) ={

x ≥ 0 :k∏

i=1

max(xi , 1) ≤ L+ 1

}

. (4.51)

By Ref. [9] (Corollary 2), the last integral in Eq. 4.50 behaves as O(L(logL)k−1).The lower bound for dim(Sδ

L) can be proved to the same manner, deducing thatdim(Sδ

L) = O(L2(logL)k−1). This yields

Corollary 4.3 Under assumptions of Theorem 4.2 there holds

∥∥∥Mku− μL

∥∥∥Htmix(�

k)≤ C(k)

((logN)k−1

N

) s−t2 ∥

∥∥Mkg

∥∥∥Hs−1mix (�

k)(4.52)

where N is the dimension of the sparse Galerkin space SδL.

Analogously to Eq. 4.45, we introduce the full tensor product Galerkin schemebased on the finite dimensional spaces

SL := span{Y�,m : 0 ≤ �i ≤ L,mi = −�i, . . . , �i

}(4.53)

as a problem of finding μL ∈ SL satisfying

D(μL, v) =⟨⟨Mkg, v

⟩⟩∀v ∈ SL. (4.54)

The convergence estimate for Eq. 4.54 follows naturally for the smoothness scale [5]

Hsiso(�

k) := (Hs(�)⊗H 1/2(�)⊗ . . .⊗H 1/2(�)

) ∩ . . .∩ (H 1/2(�)⊗ . . .⊗H 1/2(�)⊗Hs(�)

)

and is collected in the following theorem.

Remark 4.4 The mixed regularity of the solution Mku takes after that of the righthand side Mkg due to the strong ellipticity of the operator K; see Eq. 3.30. This sug-gests that the sparse hyperbolic cross tensor Galerkin approximation will be robustwhen we solve strongly elliptic pseudodifferential equations on spheroids or sphereswith mixed smooth right hand sides; see [10, 12, 32] for more on pseudifferentialoperators on spheres.

Theorem 4.5 Suppose s ≥ 1/2. Assume that Mku is the solution of Eq. 3.39 andμL ∈ SL is the full tensor Galerkin solution of Eq. 4.54 for sufficiently smooth thedata Mkg. Assume that Mku ∈ Hs

iso(�k). Then

∥∥∥Mku− μL

∥∥∥H

1/2mix (�

k)≤ C(k)

(1

N

) s−1/22k ∥

∥∥Mku

∥∥∥Hsiso(�

k), (4.55)

A. Chernov, D. Pham

where N is the dimension of the full tensor product Galerkin space SL.

Proof Estimate Eq. 4.55 follows by the product argument.

We observe that the convergence rate in Eq. 4.52 deteriorates quite mildly withgrowing k as k appears only in the logarithmic term. Dependence on k is muchheavier in Eq. 4.55 since k appears in the exponent of the leading order term. Thiseffect is known as the curse of dimensionality. We mention, however, that regularityassumptions in Theorem 4.2 are significantly more restrictive, since they imply

Mku ∈ Hsmix(�

k) � Hsiso(�

k), s ≥ 1

2.

By definition of Hsmix(�

k)

v ∈ Hsmix(�

k) ∼�∑

m=−�

|v�,m|2 �(

k∏

i=1

(1 + �i)

)−2s−1−ε

, ε > 0.

(4.56)In other words, Fourier coefficients of the functions v ∈ Hs

mix(�k) possess an

asymptotic decay pattern, and the definition (4.42) of the hyperbolic cross set δL isasymptotically adjusted to the level sets of the Fourier coefficients. In this case sparsetensor product spectral Galerkin solution μL ∈ Sδ

L behaves asymptotically as the bestN -term approximation. This considerations imply that if the level sets of the Fouriercoefficients do not agree with δL, sparse tensor product spectral Galerkin methodwill not be robust. This important situation will be considered in Section 5 (the caseof analytic regularity). Section 6 contains two related numerical examples.

5 Analytic regularity

In this section, we consider the case when the solution Mku belongs to the classof analytic functions. This class is highly demanded in probability theory, e.g. thewell–known Gaussian distribution belongs to this group. We will prove regularitytheorems showing that analytic data Mkg imply analytic solutions Mku of Eq. 3.38.In particular, we establish asymptotic bounds for the Fourier coefficients in the caseof analytic smoothness, being analogous to Eq. 4.56 for functions of finite mixedregularity.

Let {(Ui, φi)}Mi=1 be an atlas for the unit sphere S, where Ui are open subsets ofS, covering S, and where φi : Ui → B are analytic real mappings, B is the openunit ball in R

2. A function v defined on the unit sphere S is said to be analytic on S

if the compositions v ◦ φ−1i : B → R are analytic for all i; see Ref. [16], page 159.

Recalling that F : S → � is the transform mapping the unit sphere S to the spheroid� given by Eq. 2.8, a function v defined on the spheroid � is said to be analytic if thecomposition v ◦ F is analytic on the unit sphere.


Let {αi}Mi=1 be a partition of unity subordinated to the atlas {Ui, φi}Mi=1, i.e., a setof infinitely differentiable functions αi on S vanishing outside the sets Ui , such that∑M

i=1 αi = 1 on S. The Sobolev space Hs(S) can be defined as follows [19]:

Hs(S) := {v : (αiv) ◦ φ−1i ∈ Hs(B), i = 1, . . . ,M}. (5.57)

A norm on Hs(S) is given by

‖v‖′Hs(S) :=(

M∑

i=1

∥∥∥(αiv) ◦ φ−1

i

∥∥∥

2

Hs(Bi)

)1/2

. (5.58)

We note that the Sobolev space (5.57) with the corresponding norm (5.58) isequivalent to the Sobolev defined by

Hs(S) :={

v ∈ D′(S) :∞∑

�=0

�∑

m=−�

(1 + �)2s∣∣v�,m

∣∣2 < +∞

}

, (5.59)

with the norm

‖v‖Hs(S) :=( ∞∑

�=0

�∑

m=−�

(1 + �)2s∣∣v�,m

∣∣2

)1/2

; (5.60)

see Ref. [19], page 37. The following lemma presents an necessary and sufficientcondition for analyticity of a function on the unit sphere.

Lemma 5.1 A function v is analytic on S if and only if v ∈ Hn(S) and there existconstants C1 and C2 such that

‖v‖Hn(S) ≤ C1Cn2 n! ∀n ≥ 0. (5.61)

Proof We note that a function v is analytic on S if and only if v ∈ Hn(S) and thereexist constants C1 and C2 such that

‖v‖′Hn(S) ≤ C1Cn2 n!, (5.62)

for all nonnegative integers n, see Ref. [8], page 26. The inequality (5.61) is thenderived from the equivalence between the two norms Eqs. 5.58 and 5.60.

We proceed with the definition of analytic functions on a k-dimensional sphereS. Let {(Ui, φi)}Mi=1 be an atlas for the unit sphere S as before. A function v

defined on Sk = S × · · · × S is said to be analytic on S

k if the compositionsv(φ−1

i1(·), . . . , φ−1

ik(·)) : Bk → R are analytic for all i1, . . . , ik ∈ {1, . . . ,M}. A

function v defined on the k-dimensional spheroid �k is said to be analytic if the com-position v ◦ F is analytic on S

k . For any n ≥ 0, the Sobolev space Hn(Sk) can bedefined analogously to Eq. 5.58. A function v is then analytic on the k-tensor productSk of the sphere S if and only if the following inequality holds [8]:

‖v‖Hn(Sk) ≤ C1Cn2 n! ∀n ≥ 0. (5.63)

A. Chernov, D. Pham

The following theorem gives an equivalence relation between the analyticity ofa function defined on S

k and the exponential decay condition on the Fouriercoefficients of the function.

Theorem 5.1 A function v is analytic on the k-tensor product Sk of the sphere S ifand only if there exists a real number ρ > 1 such that

∣∣v�,m

∣∣ ≤ Cρ−|�| ∀� ∈ N

k and m = −�, . . . , �. (5.64)

Proof Suppose that v is a function defined on Sk such that inequality (5.64) holds

and let us prove that Eq. 5.63 is satisfied. Note that

Hn(Sk) =⋂

n1+...+nk=n

Hn1(S)⊗ . . .⊗Hnk(S),

in which

‖v‖Hn(Sk) �∑

n1+...+nk=n

‖v‖Hn1 (S)⊗...⊗Hnk (S) ∀v ∈ Hn(Sk). (5.65)

We have

‖v‖2Hn1 (S)⊗...⊗Hnk (S) =

∞∑

�=0

�∑

m=−�

(k∏

i=1

(1 + �i)2ni

)∣∣v�,m

∣∣2 (5.66)

�∞∑

�=0

�∑

m=−�

(k∏

i=1

(1 + �i)2ni

)

ρ−2|�|

=k∏

i=1

⎛

⎝∞∑

�i=0

�i∑

mi=−�i

(1 + �i)2ni ρ−2�i

⎞

⎠

�k∏

i=1

⎛

⎝∞∑

�i=0

(1 + �i)2ni+1ρ−2�i

⎞

⎠ . (5.67)

Omitting the subscript i, i.e., n and � stand for ni and �i , respectively, the sum∑∞�=0(1 + �)2n+1ρ−2� is proportionally bounded by the improper integral

∫∞0 (1 +

x)2n+1(ρ2)−x dx. Using integration by parts 2n+ 1 times, we obtain

∫ ∞

0(1 + x)2n+1(ρ2)−x dx = (2n+ 1)!

(2n+1∑

i=0

1

i! (2 lnρ)2n+2−i

)

. (5.68)

Denote ai := 1/(i! (2 lnρ)2n+2−i

)for i = 0, . . . , 2n + 1 and i0 := �2 lnρ�. Then

for all i ≥ i0, we have

ai

ai0= 2 lnρ

i0 + 1

2 lnρ

i0 + 2. . .

2 ln ρ

i≤ 1,


thus ai ≤ ai0 for all i ≥ i0. For any i < i0, we have

ai = 1

i! (2 lnρ)2n+2−i= (2 ln ρ)i

i! a0 ≤ (2 ln ρ)i0 a0.

Hence, we have ai � a0 for all i = 0, . . . , 2n+1. This together with Eq. 5.68 implies

∫ ∞

0(1 + x)2n+1(ρ2)−x dx � (2n+ 2)!

(2 lnρ)2n+2 . (5.69)

We note that for any nonnegative integersm and n, there holds (m+n)! < 2m+n m! n!.Applying this inequality twice we deduce

(2n+ 2)! ≤ 22n+2 n! (n + 2)! ≤ 22n+2 n! 2n+2 n! 2! = 23n+5 (n!)2.

This together with Eq. 5.69 yields

∫ ∞

0(1 + x)2n+1(ρ2)−x dx � 2n+3 (n!)2

(ln ρ)2n+2� 2n (n!)2

(ln ρ)2n. (5.70)

Applying Eq. 5.70 into Eq. 5.67 with n = ni , we obtain

‖v‖2Hn1 (S)⊗...⊗Hnk (S) �

2n1+...+nk (n1! . . . nk !)2

(ln ρ)2(n1+...+nk)�

2n1+...+nk((n1 + . . .+ nk)!

)2

(ln ρ)2(n1+...+nk).

This and Eq. 5.65 yield

‖v‖Hn(Sk) �(√

2

lnρ

)n

n!.

By Eq. 5.63, the function v is analytic on Sk .

Conversely, suppose v is an analytic function defined on Sk . Then,

by Eqs. 5.63, 5.65 and 5.66, we deduce

∞∑

�=0

�∑

m=−�

(k∏

i=1

(1 + �i)2ni

)∣∣v�,m

∣∣2 � C2n (n!)2 ∀n1 + . . .+ nk = n.

This implies that for all � ∈ Nk and m = −�, . . . , �, it holds

(k∏

i=1

(1 + �i)ni

)∣∣v�,m

∣∣ � Cn n! ∀n1 + . . .+ nk = n. (5.71)

A. Chernov, D. Pham

Note that

|�|n <

⎛

⎝k∑

i+1

(1 + �i)

⎞

⎠

n

=∑

n1+...+nk=n

(n

n1, . . . , nk

) k∏

i=1

(1 + �i)ni

≤∑

n1+...+nk=n

knk∏

i=1

(1 + �i)ni

≤ #

{

(n1, . . . , nk) :k∑

i=1

ni = n

}

kn maxn1+...+nk=n

{k∏

i=1

(1 + �i)ni

}

≤(n+ k − 1

n

)

kn maxn1+...+nk=n

{k∏

i=1

(1 + �i)ni

}

� nk−1kn maxn1+...+nk=n

{k∏

i=1

(1 + �i)ni

}

.

This together with Eq. 5.71 yields

|�|n ∣∣v�,m

∣∣ � nk−1 (Ck)nn!.

Since limn→∞ (nk−1)/(Cn) = 0 for any C > 1 and any fixed moment order k, thereexists a constant C′ independent of n such that nk−1(Ck)n ≤ (C′)n for all n ≥ 0.Then we have

|�|n ∣∣v�,m

∣∣ � (C′)nn! ∀� ∈ N

k, m = −�, . . . , �.

The Stirling estimate gives n! ≤ e (n/e)n√

2πn for all n ≥ 1. Thus,

∣∣v�,m

∣∣ � (e

√2π)

(C′ne |�|

)n √n. (5.72)

Bound Eq. 5.72 holds for all n ≥ 1. Next, we choose n = n(�) so that (C′n)/(e |�|)< 1. Later on this will imply exponential convergence of the coefficients.

Let �0 be an integer such that e�0/C′ > 1. We choose a real number δ ∈ (e−1, 1)

satisfying (δe�0)/C′ > 1. Then we have

(δe |�|)/C′ > 1 ∀ |�| ≥ �0.

Note that the inequality (5.72) is true for all n ≥ 1. We choose n = �δ e|�|C ′ �. Noting

that the function f (x) = (ax)x is increasing if 1/e < ax < 1, we obtain

∣∣v�,m

∣∣ � δ

δe|�|C′

√

δe |�|C′ �

(δδ eC′

)|�| √|�| ∀ |�| ≥ �0, m = −�, . . . , �.

Recalling that δδe

C′ < 1, we can choose a ρ > 1 such that(δδ eC′

)|�| √|�| � ρ−|�| ∀� ∈ Nk.

This implies∣∣v�,m

∣∣ � ρ−|�| ∀ |�| ≥ �0, m = −�, . . . , �,


which completes the proof of the theorem.

The analyticity of a function v defined on the spheroid �k is defined via the analyt-icity of the composition v◦F on the sphere Sk . Noting that the Fourier coefficients ofa function v defined on �k are given by Eq. 2.17, Theorem 5.2 implies the followingcorollary.

Corollary 5.3 A function v defined on the k-dimensional spheroid �k is analytic ifand only if

∣∣v�,m

∣∣ ≤ Cρ−|�| ∀� ∈ N

k and m = −�, . . . , �,

where the constant C is independent of � and m.

The following theorem shows that the analyticity of the right hand side of thestatistical Dirichlet to Neumann moment Eq. 3.38 implies the analyticity of thecorresponding solution and vice versa.

Theorem 5.4 The right hand side Mkg of Eq. 3.38 is analytic if and only if thesolution Mku is analytic.

Proof If the right hand side Mkg is analytic, Corollary 5.3 gives∣∣∣Mkg�,m

∣∣∣ ≤ Cρ−|�| ∀� ∈ N

k, m = −�, . . . , �

for some ρ > 1. Recalling Eq. 3.33 this bound is equivalent to the estimate∣∣∣Mku�,m

∣∣∣ ≤ C′ρ−|�| ∀� ∈ N

k, m = −�, . . . , �.

The analyticity of Mku follows by Corollary 5.3. The reverse statement followsanalogously.

Remark 5.5

(i) Corollary 5.3 indicates that the index set based in the total degree selection

δTDL := {� ∈ N

k : |�| ≤ L}

(5.73)

will be asymptotically optimal in the case of analytic regularity.(ii) The analyticity of the solution Mku takes after that of the right hand side

Mkg thanks to the strong ellipticity of the operator K, which suggests a similarimplication for other strongly elliptic pseudodifferential equations.

6 Numerical experiments

The representation (3.29) of the action of the Dirichlet-to-Neumann mapK by an infi-nite series of spherical harmonics suggests (finite subsets of) the space of sphericalharmonics as the basis and test functions for numerical implementation. Notice thatcoefficients of the expansion (3.29) are θ -dependent, namely they all are scaled with

A. Chernov, D. Pham

(cosh2 μ0 − cos2 θ)− 12 . The reciprocal of this factor is contained in the expression of

the surface element (3.26) of �. This implies the orthogonality relation

⟨KY�,m, Y�′,m′⟩ = −c0

ddμ

Qm� (coshμ0)

Qm� (coshμ0)

sinhμ0δ�,�′δm,m′ . (6.74)

Therefore, the stiffness matrix of the Dirichlet-to-Neumann operator K on � in thebasis of spherical harmonics is diagonal and, by Lemma 3.1, positive definite. Thisproperty extends to the stiffness matrix of the k-th moment equation discretized byproducts of spherical harmonics. Diagonality of the stiffness matrix makes the solu-tion of linear system trivial and therefore is of great advantage in practice. Noticethat this favorable diagonality property of the stiffness matrix vanishes if some other,e.g. locally supported basis is used. Solution of the linear system will be thereforesignificantly more time consuming.

In this section, we present results of two numerical experiments illustrating con-vergence of sparse approximation for the second moment problem on the prolatespheroid with parameters c0 = 4 and μ0 = 1. For the sake of simplicity, we restrictto the second moment problem, i.e. Eq. 3.39 with k = 2, see Section 3.2 for defi-nition of the statistical moments and the related probability spaces. We assume thatsecond moments of the data M2g are known from elsewhere and report on conver-gence behavior of the spectral approximation to second moment of the solution to theexact function M2u. In particular, in two numerical experiments below we choosetwo different functions M2g with different smoothness properties (of mixed andanalytic smoothness) and compare performance of the sparse and full tensor productapproximations analyzed in Sections 4 and 5. We emphasize that the data M2g inboth numerical tests are artificial. Particularly, they do not come from a mathemati-cal model and are chosen to illustrate convergence of the numerical schemes for two

1 2 3 4 6 9 13 161910

−4

10−3

10−2

10−1

100

polynomial degree

sparse tensor productfull tensor productexpected order (−2.5)

Fig. 3 Relative error in energy norm for the right-hand side M2g from Eq. 6.75 vs. the maximalpolynomial degree


Table 1 Numbers of unknownsfor full and sparse tensorproduct approximation

Degree full sparse

1 16 7

4 625 58

8 6561 267

12 28561 633

16 83521 1180

20 194481 2066

classes of smoothness analyzed in Sections 4 and 5. An application of the suggestedmethod to real models and data, e.g. in Earth observation, is outside the scope of thispaper.

The proposed method has been implemented in MATLAB 2013 and run on a com-puter equipped with dual i7 Core 2.0 GHz CPU and 4 GB RAM. The associatedLegendre functions of the first and the second kind Pm

� and Qm� were evaluated with

built-in functions in MATLAB.

6.1 Mixed regularity

We first consider the case for the right-hand side

M2g = sin(ρ(x,n)ρ(y,n)

)

c40 sinh2 μ0

√

(cosh2 μ0 − cos2 θx)(cosh2 μ0 − cos2 θy)

, n = (0, 0, c0 coshμ0),

(6.75)

100

101

102

103

104

105

10−4

10−3

10−2

10−1

100

number of unknowns

sparse tensor productfull tensor product

Fig. 4 Relative error in energy norm for the right-hand side M2g defined by Eq. 6.75 vs. the number ofunknowns

A. Chernov, D. Pham

100

101

102

103

104

105

10−4

10−3

10−2

10−1

100

computational time (seconds)


Fig. 5 Relative error in energy norm for the right-hand side M2g defined by Eq. 6.75 vs. the CPU time

for x = (c0 sinhμ0 sin θ cosϕ, c0 sinhμ0 sin θ sin ϕ, c0 coshμ0 cos θ) ∈ � and

ρ(x,n) := √2 − 2 cos θ.

The function M2g belongs to H 2−εmix (�

2) for any ε > 0. Thus, the solution M2u ofthe Eq. 3.38 belongs to H 3−ε

mix (�2).

We compute approximate solutions both using full and sparse tensor productsof polynomials for different discretization levels L. In Fig. 3 we plot the H

1/2mix-

norm of the Galerkin error and observe that the convergence rate with respect to the

0 5 10 150

5

10

15

Coefficient distribution

−15

−10

−5

0

Fig. 6 Coefficient distribution of M2u when M2g is defined by Eq. 6.75


1 2 3 4 6 9 13 1619

10−6

10−4

10−2

100

polynomial degree


Fig. 7 Relative error in energy norm for the smooth right-hand side M2g defined by Eq. 6.76 vs.polynomial degree

polynomial degree is almost the same for both methods: the slope of the conver-gence curve in the semilogarithmic plot is around 2.5. This agrees with the statementof Theorem 4.2. On the other hand, the sparse tensor discretization requires onlyN = O(

L2 logL)

unknowns, which is significantly smaller than N = O(L4

)

unknowns when using the full tensor discretization, see Table 1 and Fig. 4.

100

101

102

103

104

105

10−6

10−4

10−2

100

number of unknowns


Fig. 8 Relative error in energy norm for the smooth right-hand side M2g defined by Eq. 6.76 vs. thenumber of unknowns

A. Chernov, D. Pham

100

101

102

103

104

105

10−6

10−4

10−2

100

computational time (seconds)


Fig. 9 Relative error in energy norm for the smooth right-hand side M2g defined by Eq. 6.76 vs. theCPU time

In terms of computational time, the sparse tensor discretization takes only 8.85minutes which is much smaller than 18.26 hours required by the full tensor dis-cretization when polynomial degree is 20. The plot of the error decay with respectto

0 5 10 150

5

10

15

Coefficient distribution

−10

−5

0

5

Fig. 10 Coefficient distribution of M2u when M2g is defined by Eq. 6.76


the CPU time is given in Fig. 5. The coefficient distribution of the solution is alsoobserved in Fig. 6, in which we present

c�1,�2 =

√√√√√

�1∑

m1=−�1

�2∑

m2=−�2

(M2u)2

�1,�2,m1,m2, �1, �2 = 0, . . . , 15.

In Fig. 6 we also sketch two hyperbolic curves (�1 + 1)(�2 + 1) = L+ 1 for L = 17and 21 to illustrate the hyperbolic decay pattern of the coefficients c�1,�2 .

6.2 Analytic regularity

We consider an analytic right hand side

M2g = exp

(

− |x − y|22(|x − n|2 + |y − n|2 + 1))

)

, x, y ∈ �, (6.76)

where n = (0, 0, 1). In this case, the solution M2u is analytic function by The-orem 5.4. Numerical results do not show advantage of L-sparse discretization (seeFigs. 7, 8) as predicted in Section 5. The Fourier coefficients of M2u satisfy∣∣∣M2u�,m

∣∣∣ � ρ−|�| by Theorem 5.4. In Fig. 10 we show the distribution of the

coefficients c�1,�2 of M2u. This explains why the L-sparse discretization is notadvantageous in the analytic case.

Acknowledgments The authors acknowledge support by the Hausdorff Center for Mathematics.

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Sparse tensor product spectral Galerkin BEM for elliptic problems with random input data on a spheroid