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Sparse tensor phase space Galerkin approximation for radiative transport
SpringerPlus volume 3, Article number: 230 (2014)
Abstract
We develop, analyze, and test a sparse tensor product phase space Galerkin discretization framework for the stationary monochromatic radiative transfer problem with scattering. The mathematical model describes the transport of radiation on a phase space of the Cartesian product of a typically threedimensional physical domain and twodimensional angular domain. Known solution methods such as the discrete ordinates method and a spherical harmonics method are derived from the presented Galerkin framework. We construct sparse versions of these wellestablished methods from the framework and prove that these sparse tensor discretizations break the “curse of dimensionality”: essentially (up to logarithmic factors in the total number of degrees of freedom) the solution complexity increases only as in a problem posed in the physical domain alone, while asymptotic convergence orders in terms of the discretization parameters remain essentially equal to those of a full tensor phase space Galerkin discretization. Algorithmically we compute the sparse tensor approximations by the combination technique. In numerical experiments on 2+1 and 3+2 dimensional phase spaces we demonstrate that the advantages of sparse tensorization can be leveraged in applications.
Introduction
In this paper, we consider the numerical solution of the radiative transfer problem (RTP). This transport problem is stated on the phase space $\Omega =D\times \mathcal{S}$ as the Cartesian product of a bounded physical domain $D\subset {\mathbb{R}}^{d}$, where d=2,3, and the unit ${d}_{\mathcal{S}}$sphere as the parameter domain $\mathcal{S}$ of dimension ${d}_{\mathcal{S}}=d1=1,2$. The RTP (see e.g. Modest 2003) is then given as the task of finding the unknown radiative intensity$u:\Omega \to \mathbb{R}$, a real function over the phase space satisfying
We refer to Eq. (1a) as the stationary monochromatic radiative transfer equation (RTE), while Eq. (1b) constitutes inflow boundary conditions. A ray of light of direction s is attenuated by absorption and scattering with the medium. In (1a), κ≥0 is the absorption coefficient, σ≥0 the scattering coefficient, and Φ>0 the scattering kernel or scattering phase function. The scattering phase function is normalized to ${\int}_{\mathcal{S}}\Phi (\mathit{s},{\mathit{s}}^{\prime})d\phantom{\rule{0.3em}{0ex}}{\mathit{s}}^{\prime}=1$ for each direction s. Sources inside the domain D are modeled by the blackbody intensity I_{ b }≥0, radiation from sources outside of the domain or from its enclosings is prescribed by the boundary data g≥0. The vector n(x) denotes the outward unit normal vector which is defined in (almost every) point x on the boundary ∂ D of the physical domain.
Due to the high dimension of the phase space, the nonlocality of the scattering operator, and the hyperbolic nature of the PDE, the efficient numerical simulation of radiative transfer is a challenging computational task even today. Still, radiative transfer as such or as a means of energy transfer among others is of interest in many applications, e.g. in the fields of heat transfer (Modest 2003), neutron transport (Hébert 2010), atmospheric sciences (Evans 1998), medical imaging (e.g. Peng et al. 2011), or other areas where transported particles interact with a background medium, but only negligibly with each other.
In this paper, we extend the range of sparse tensor product discretization methods for the RTP investigated before (Grella and Schwab 2011a; Grella and Schwab 2011b; Grella 2013; Widmer et al. 2008) by a new phase space Galerkin framework.
Apart from Monte Carlo methods for raytracing, the most popular deterministic approaches for the radiative transfer problem are the discrete ordinates method and the spherical harmonics approximation. We quote a brief overview of their properties from (Grella 2013).
In the discrete ordinate method (DOM) or S_{ N }approximation, the angular domain is discretized by a number of fixed directions, which are inserted into Eq. (1) so that a system of spatial PDEs results. Without scattering the equations for single directions are independent of each other, with scattering, however, they are coupled through the scattering integral. After the straightforward discretization of the angular domain, the spatial PDEs are typically solved using finite differences, finite elements, or finite volume methods.
The DOM is popular as it is simple to implement, offers straightforward parallelization, and can capture directed radiation relatively well as some of the ordinates can usually be chosen freely.
On the downside, the method can suffer from socalled “ray effects” (Lathrop 1968): due to the point evaluation in the angular domain, the scalar flux or incident radiation from small isotropic sources may appear starlike with rays emanating from the source into the chosen angular directions (Stone 2007, p. 2 and following). These effects occur especially pronounced in settings with low scattering and absorption, i.e. in optically thin media.
An example for truncated series expansion is the method of spherical harmonics or P_{ N }approximation. The solution of Eq. (1) is replaced by a series of spherical harmonics up to some order N with spatially dependent coefficients. Due to orthogonality relations, the scattering part often decouples or couples only few terms depending on the scattering kernel. However, the system of PDEs for the spatial coefficient functions is always coupled by the transport part s·∇_{ x }u.
As low order series expansions in spherical harmonics do not permit a very localized resolution of the angular variable, the method performs best when the solutions are nearly isotropic in angle, which is the case in diffusive, socalled “optically thick” media. Then, rather low order spherical harmonics approximations might suffice for a good approximation. Indeed, the P_{1} method can be formulated as a diffusion equation for the incident radiation (Modest 2003, Sec. 15.4). For smooth solutions, the spherical harmonics method exhibits spectral convergence in angle (Grella and Schwab 2011a).
On the other hand, beamlike solutions require a high spectral order to be resolved appropriately, leading to high computational complexity. In general, higher spectral orders also lead to a sharp increase in computational complexity when boundary conditions are to be satisfied (Modest and Yang 2008).
When combined with a standard finite element or finite volume discretization in the physical domain D, the deterministic, numerical S_{ N } and P_{ N }approximations exhibit the socalled “curse of dimensionality”: the error (typically the L^{2}error of the solution) with respect to the numbers of degrees of freedom (DoF) M_{ D } and ${M}_{\mathcal{S}}$ on the component domains D and $\mathcal{S}$ scales with the dimension d and ${d}_{\mathcal{S}}$ of the application problem as $O\left({M}_{D}^{s/d}+{M}_{\mathcal{S}}^{t/{d}_{\mathcal{S}}}\right)$ with constants s and t.
The first sparse finite element approximation method was proposed in (Zenger 1991) for the solution of Laplace equation in the unit square and cube. In this paper, Zenger developed the (direct) sparse grid approximation method which alleviates this curse of dimensionality: the computational complexity is reduced, up to logarithmic terms, to that of a onedimensional problem.
The idea of sparse tensorization of finite element and finite difference methods was generalized by (Bungartz and Griebel 2004; Hegland 2003; Garcke 2007), and others, for the numerical solution of PDEs as well as for other applications where standard numerical methods are obstructed by the curse of dimensionality.
Sparse tensor methods were first applied to radiative transfer by (Widmer et al. 2008). In that paper, the authors formulated a least squares phase space Galerkin sparse tensor approximation with hierarchical finite elements as discretization of the physical domain and wavelets in the angular domain. They proved that sufficient regularity of the solution provided, their method breaks the curse of dimensionality: the problem complexity reduces to loglinear in the number of degrees of freedom, while convergence rates deteriorate only by a logarithmic factor. However, the discretization of the scattering operator had not been addressed in that work.
In earlier work (Grella and Schwab 2011a), we showed that the sparse tensor product method of (Widmer et al. 2008) can also be combined with a spectral discretization involving spherical harmonics, resulting in a sparse P_{ N }method which also treats scattering. Boundary conditions were satisfied in a strong sense by introducing piecewise spectral functions in angle.
Secondly we presented a sparse tensor version of the DOM as a sparse collocation method with a Galerkin ansatz in the physical domain and strong enforcement of the boundary conditions, while not yet accounting for scattering (Grella and Schwab 2011b). This sparse tensor S_{ N }method was realized computationally with the sparse grid combination technique (Griebel et al. 1992) to construct a sparse approximation to the radiative transfer solution.
The sparse DOM was subsequently reformulated as a phase space Galerkin method with quadrature in angle (Grella 2013) in order to treat sparse P_{ N } and sparse S_{ N }method in a more uniform manner. In this reformulation, we employed streamline upwind Petrov Galerkin (SUPG) stabilization and weak satisfacion of boundary conditions. Sparse S_{ N }methods were derived as a direct sparse tensor method and implemented algorithmically via the combination technique.
In the present paper, we derive a sparse P_{ N } and sparse S_{ N }method from the same phase space Galerkin framework with transport stabilization and scattering. Boundary conditions are satisfied in a weak sense. In doing so we close a gap in the list of conceivable combinations of formulations regarding stabilization and type of angular approximation. In contrast to the previous approach (Grella 2013), we stabilize the formulation in a different way and the analytical focus will be on the direct sparse approach. With transport stabilization and direct sparse approach we follow (Widmer et al. 2008) more closely, extending their work by treatment of scattering and weak satisfaction of the boundary conditions.
Similar savings in computational effort are realized with other variational formulations, such as PetrovGalerkin saddle point formulations (see e.g. Dahmen et al. (2012) and the references there).
The outline of this paper is as follows. In Section ‘Phase space Galerkin method’ we formulate the phase space Galerkin framework in operator form and outline how P_{ N } and S_{ N }methods can be derived from it. Then we develop full tensor and sparse tensor discretizations based on the framework and analyze and compare their convergence properties.
Section ‘Numerical experiments’ presents several basic numerical experiments designed with the purpose of validating and illustrating the theoretical convergence results.
Finally we conclude this work in Section ‘Conclusion’ by summarizing and reviewing the results.
Phase space Galerkin method
We begin by introducing the radiative transfer problem in operator form. Using this compact notation we then state the variational formulation of our phase space Galerkin method and proceed to discretizations of the method.
Operator formulation
Problem (1) reads in operator form: Find the intensity $u(\mathit{x},\mathit{s}):D\times \mathcal{S}\to \mathbb{R}$ such that
In this, ∂ Ω_{−} represents the inflow part of the boundary $\mathrm{\partial \Omega}=\mathrm{\partial D}\times \mathcal{S}$ of the computational domain or phase space$\Omega =D\times \mathcal{S}$. The inflow boundary is defined by
with the physical part of the inflow boundary
Correspondingly we define the physical part of the outflow boundary as
The radiative transfer operator A=T+Q consists of the transport operator T,
and the scattering operator Q,
Here, Q_{1}=Id−Σ is the unity scattering operator, and Σ is the scattering integral operator, the integral of Φ and u. The source function f contains the sources of radiation in the domain,
and g is the incoming radiation on the boundary ∂ Ω_{−}, as in Sec. ‘Introduction’.
Properties of the scattering operator
Aside from the positivity and normalization requirements already mentioned in Sec. ‘Introduction’, we assume an isotropic medium, i.e. Φ does not depend on x. As Φ models the type of scattering, this assumption can safely be made for most applications (cf. Modest 2003, p. 268). Variations in the strength of scattering due to e.g. varying spatial density of the medium are modeled by the scattering coefficient σ. As long as the following properties hold for almost every x, the complexity and convergence analysis later on could also be conducted without this assumption.
Furthermore, if spherical scatterers are assumed, the scattering phase function does not vary with the azimuthal angle so that Φ only depends on the inner product of s and s^{′}. From this it follows immediately that Φ(s,s^{′})=Φ(s·s^{′})=Φ(s^{′},s).
From here on, we shall take Φ (cf. Kanschat to be forward dominant2008, Def. 1) if $\Phi (\mathit{s},{\mathit{s}}^{\prime})=\sum _{k=0}^{\infty}{a}_{k}cos(karccos(\mathit{s}\xb7{\mathit{s}}^{\prime}\left)\right)$ with all a_{ k }≥0. Then, one can show that Σ is positive semidefinite (Kanschat 2008, Lemmata 2 and 3), i.e.
Normalization and symmetry of Φ with respect to its arguments leads to normalization of the operator norm $\left\right\Sigma {}_{{L}^{2}\left(\mathcal{S}\right)\to {L}^{2}\left(\mathcal{S}\right)}=1$ (Kanschat 2008, Lemma 5).
From these properties and a HilbertSchmidt theorem for integral operators (e. g. Knapp 2005, Thm. 2.4), one can derive that the spectrum of Q_{1} lies in [ 0,1] with an isolated eigenvalue λ_{0}=0, from which the next largest eigenvalue λ_{1} differs by a positive constant (Ávila et al. 2011, Sec. 2.2).
With the previous considerations, one arrives at the following properties of Q:
Lemma 1.
For any u∈L^{2}(Ω), the scattering operator Q as defined by Eq. (7) satisfies (cf. Ávila et al. 2011, Eq. (11))
in which the projector P^{⊥} maps $u(\mathit{x},\xb7)\in {L}^{2}\left(\mathcal{S}\right)$ to (kerQ)^{⊥}, the space orthogonal to the kernel of Q, and λ_{1}∈(0,1] is the smallest nonzero eigenvalue of Q_{1}.
For a proof of (11) we refer to (Grella 2013).
Variational formulation
Our variational formulation will be based on a Galerkin finite element framework over the phase space Ω with stabilization applied to the operator RTP (2).
A generic stabilized phase space variational formulation
To begin with, we define the Hilbert space
with the usual L^{2}(Ω) inner product
and the triple bar norm
For the weak enforcement of boundary conditions, we define the boundary form
in which we have omitted the dependence of the outward unit normal n on the position x. This boundary form was introduced by Manteuffel et al. (2000, Eq. (2.16)). It is welldefined for functions v∈L^{2}(Ω) with finite inflow norm
Combining (14) and (16) yields the new norm
which gives rise to the closed, linear subspace of ,
which, with the inner product related to ∥v∥_{1}, is a Hilbert space. For functions $u,v\in {\mathcal{V}}_{1}$, we define the bilinear form
where R is a stabilization operator on the test function side yet to be specified. Together with the linear form
the bilinear form constitutes the following variational problem: Find $u\in {\mathcal{V}}_{1}$ such that
Different ways of stabilization are conceivable and have been used in the literature, e. g. the least squares approach by (Manteuffel et al. 2000), or SUPG introduced by (Brooks and Hughes 1982). For our purposes here, we will choose the Tstabilized formulation (Grella and Schwab 2011a) to avoid meshdependent quantities and the square of the scattering operator. More precisely, we set R=ε T with a stabilization parameter ε that depends on the absorption coefficient κ.
Properties of the variational formulation
At this point, we introduce the anisotropic or mixed Sobolev spaces${H}^{s,t}\left(\Omega \right)={H}^{s}\left(D\right)\otimes {H}^{t}\left(\mathcal{S}\right)$ as
with the corresponding mixed Sobolev norms${\u2225\xb7\u2225}_{{H}^{s,t}\left(\Omega \right)}$, given by
Here, ${\mathrm{D}}_{\mathit{s}}^{\mathit{\beta}}{\mathrm{D}}_{\mathit{x}}^{\mathit{\alpha}}v$ denotes the weak derivative of $v:D\times \mathcal{S}\to \mathbb{R}$ of order α w. r. t. x∈D and order β w. r. t. $\mathit{s}\in \mathcal{S}$, with the multiindices $\mathit{\alpha}\in {\mathbb{N}}_{0}^{d}$ and $\mathit{\beta}\in {\mathbb{N}}_{0}^{{d}_{\mathcal{S}}+1}$.
The following lemma collects auxiliary results which will become helpful later.
Lemma 2(Auxiliary results).

1.
Let $v\in \mathcal{V}$. Then ${\left(v,\mathit{s}\xb7{\nabla}_{x}v\right)}_{{L}^{2}\left(\Omega \right)}\ge \frac{1}{2}\underset{\mathcal{S}}{\int}\underset{{\Gamma}_{\left(\mathcal{S}\right)}}{\int}{v}^{2}\mathit{s}\xb7\mathit{n}\left(\mathit{x}\right)\mathrm{d}\mathit{x}d\phantom{\rule{0.3em}{0ex}}\mathit{s}$. If furthermore $v\in {\mathcal{V}}_{0}$, then ${\left(v,\mathit{s}\xb7{\nabla}_{x}v\right)}_{{L}^{2}\left(\Omega \right)}\ge 0$.

2.
For v∈H ^{1,0}(Ω), it holds $\u2225\mathit{s}\xb7{\nabla}_{x}v\u2225\le \sqrt{d}\parallel v{\parallel}_{{H}^{1,0}\left(\Omega \right)}$.
Proof
1. A proof is given by (Manteuffel et al. 2000, Thm. 2.1). It uses the divergence theorem and exploits the fact that $v{}_{\partial {\Omega}_{}}=0$ for s·n(x)<0 if $v\in {\mathcal{V}}_{0}$, where n(x) is the outward unit normal on the boundary ∂ D:
As s·n≥0 in the second integral, we obtain the first assertion. If additionally $v\in {\mathcal{V}}_{0}$, then the first integral vanishes, and the second assertion follows.
2. We again quote Manteuffel et al. (2000, Lemma 4.1 (i)):
In order to establish wellposedness of the variational formulation (21), we prove continuity and coercivity of the bilinear form (19) and continuity of the linear form (20) in the following.
Lemma 3
(Continuity of bilinear form). Let σ, κ, ε∈L^{∞}(D) with ${\u2225\sigma \u2225}_{{L}^{\infty}\left(D\right)}=:{\sigma}_{max}$, ${\u2225\kappa \u2225}_{{L}^{\infty}\left(D\right)}=:{\kappa}_{max}$, ${\u2225\epsilon \u2225}_{{L}^{\infty}\left(D\right)}=:{\epsilon}_{max}$, then there is a constant 0<c_{ c }<∞ such that for all $u,v\in {\mathcal{V}}_{1}$
Proof
We proceed analogously to (Manteuffel et al. 2000, Thm. 3.3). To begin with, we estimate for all $u,v\in \mathcal{V}$
Using the CauchySchwarz inequality as well as estimates (24) and (25) it holds
Lemma 4
(Continuity of linear form). Given the assumptions of Lemma 3 on κ, σ, ε, and additionally f∈L^{2}(Ω), $g:\partial {\Omega}_{}\to \mathbb{R}$ with ∥g∥_{−}<∞, there is a constant 0<c_{ l }<∞ such that for $v\in {\mathcal{V}}_{1}$ it holds
Proof
The proof is analogous to that of Lemma 3:
Next, we show coercivity of the bilinear form. For ease of exposition we shall assume ε and κ to be constant on the physical domain. Coercivity can also be obtained for nonconstant coefficients (see Widmer 2009, Thm. 2.2, for an example). Coercivity of the SUPG variational formulation for the RTP has also been proved by (Ávila et al. 2011, Lemma 2), although in a different norm. Previously, we had proved coercivity of the Tstabilized variational formulation without the boundary form b(·,·) (Grella 2013, Lemma 4.1), here we include this boundary form in the formulation, which will motivate the choice of the stabilization parameter ε.
Lemma 5
(Coercivity of bilinear form). Let κ, ε be positive functions which are constant on the physical domain D. Assume min x∈D σ=:σ_{min}>0 and ${\sigma}_{max}={\u2225\sigma \u2225}_{{L}^{\infty}\left(D\right)}$, and additionally that
Then the bilinear form a(·,·) from (19) is coercive on ${\mathcal{V}}_{1}\times {\mathcal{V}}_{1}$: there is a constant c_{ e }>0 such that for all $v\in {\mathcal{V}}_{1}$ it holds
Proof
For an overview of the involved terms we split the bilinear form into separate inner products:
As we assumed ε and κ to be constant, we can factor these coefficients out of the inner products. We begin by analyzing the sum of first and fifth inner product.
Applying statement 1 of Lemma 2 yields
Together with the boundary term, we obtain
The second inner product is bounded from below by
To estimate the third inner product, we use property (11) of the scattering operator:
The fourth inner product in Eq. (27) is
For the sixth inner product we apply CauchySchwarz inequality and Young’s inequality with a parameter θ>0:
Combining all estimates yields the result:
By eliminating θ we obtain the condition ${\sigma}_{max}^{2}<4\kappa {\sigma}_{min}^{2}$. The condition ε<2/κ results from the last of the terms over which the minimum is taken.
The previous condition on the stabilization parameter leads to a choice of ε=1/κ. Wellposedness of the variational formulation now follows directly.
Theorem 6
(Existence and uniqueness of solution to variational formulation). Provided that f∈L^{2}(Ω) and ∥g∥_{−}<∞ there exists a unique solution $u\in {\mathcal{V}}_{1}$ to the variational formulation (21).
Proof
Since $({\mathcal{V}}_{1},{\u2225\xb7\u2225}_{1})$ is a Hilbert space and Lemmata 3–5 guarantee continuity of the augmented SUPG bilinear form and linear form as well as coercivity of the bilinear form, the LaxMilgram theorem (Brenner and Scott 2008, Thm. 2.7.7) ensures existence and uniqueness of the solution to (21).
Discretization
For the discretization of the variational problem (21), we restrict the space ${\mathcal{V}}_{1}$ in the variational formulation (21) to tensor products of hierarchic, finite dimensional approximation spaces over the component domains D and $\mathcal{S}$.
Full tensor discretization
In the standard full tensor approximation, we choose a full tensor product space V^{L,N} to approximate ${\mathcal{V}}_{1}$:
As ${H}^{1,0}\left(\Omega \right)\cong {H}^{1}\left(D\right)\otimes {L}^{2}\left(\mathcal{S}\right)$ is a dense subspace of ${\mathcal{V}}_{1}$, we define the family of physical approximation spaces as
the spaces of continuous, piecewise linear functions on a dyadically refined mesh ${\mathcal{T}}_{D}^{{l}_{D}}$ over D. Here, the parameter l_{ D } stands for the physical resolution. It is related to the mesh width h in ${\mathcal{T}}_{D}^{{l}_{D}}$ by $h=O\left({2}^{{l}_{D}}\right)$. With respect to the resolution l_{ D }=0,…,L, the spaces ${V}_{D}^{{l}_{D}}$ form a nested sequence
Let ${M}_{D}:=dim\underset{D}{\overset{L}{V}}$ denote the number of degrees of freedom for the FE space ${V}_{D}^{L}$ in the physical domain D. Then
with the dimension d of the physical domain. The exact number will depend on the geometry of the domain. For a square or cube D=[0,1]^{d}, respectively, we obtain
In the angular domain, we distinguish between the P_{ N }method and the S_{ N }method.
S_{ N }method
Here, the family of approximation spaces is given by
the spaces of piecewise constant functions on a dyadically refined mesh ${\mathcal{T}}_{\mathcal{S}}^{{l}_{\mathcal{S}}}$. As the physical spaces, these spaces exhibit a nested structure. The angular resolution N and the dimension of ${V}_{\mathcal{S}}^{N}$ are related by
P_{ N }method To define the angular approximation spaces of the P_{ N }method, we first introduce the spaces of spectral functions of the ${d}_{\mathcal{S}}$sphere,
where ${Y}_{n,m}^{\left({d}_{\mathcal{S}}\right)}$ are the spherical harmonics of the ${d}_{\mathcal{S}}$sphere, and ${m}_{n,{d}_{\mathcal{S}}}$ is the largest value of the secondary index m depending on the value of the primary index n and the dimension. These spaces offer an inherent nested structure. To obtain the same relation (33) between resolution level and degrees of freedom as in the S_{ N }method, we connect the resolution level N and $\xd1$ by $\xd1={2}^{N}1$.
Then, the angular approximation spaces are
and relation (33) also holds here. Up to the index relabeling and the additional boundary form, we obtain the spherical harmonics method already analyzed by (Grella and Schwab 2011a).
In both methods, the full tensor approximation space consequently has the dimension
The full tensor approximate solution can be expressed by means of a physical basis ${\left\{{\alpha}_{i}\left(\mathit{x}\right)\right\}}_{i=1}^{{M}_{D}}$ of ${V}_{D}^{L}$ and an angular basis ${\left\{{\beta}_{j}\right\}}_{j=1}^{{M}_{\mathcal{S}}}$ of ${V}_{\mathcal{S}}^{N}$ as
with solution coefficients ${u}_{\mathit{\text{ij}}}\in \mathbb{R}$. The discrete variational formulation finally reads: Find u_{L,N}∈V^{L,N} such that
with the bilinear form a(·,·) from (19) and the linear form l(·) from (20). As V^{L,N} is a subspace of ${\mathcal{V}}_{1}$ wellposedness ensured by Thm. 6 for the continuous problem follows also for this discrete problem.
By choosing a subset of ${H}^{1}\left(D\right)\otimes {L}^{2}\left(\mathcal{S}\right)$ as trial space we effectively assume a slightly higher regularity on the solution than what is guaranteed by the definition (12) of ${\mathcal{V}}_{1}$. For instance, solutions with line discontinuities due to the transport of discontinuous boundary data into the domain are not included in V^{L,N}. However, since V^{L,N} is dense in ${\mathcal{V}}_{1}$, even discontinuous solutions will be approximated with increasing resolution. Furthermore, in order to leverage the advantages of a sparse tensor approximation, a higher regularity of the solution will be required in any case.
Equivalence of collocation DOM and phase space Galerkin DOM with quadrature
Ordinarily the discrete ordinates method is presented as a collocation method in angle: Fixed directions ${\mathit{s}}_{j}\in \mathcal{S}$, $j=1,\dots ,{M}_{\mathcal{S}}$, are inserted into the RTE (1a), and for each direction, the intensity ${u}_{j}\left(\mathit{x}\right):=u(\mathit{x},{\mathit{s}}_{j})\in {V}_{D}^{L}$ is sought as the solution to a purely spatial PDE. In these PDEs, the scattering integral is replaced by a quadrature rule
with weights w_{ m }>0. By applying a Galerkin ansatz with stabilization in the physical domain to the PDEs, a system of coupled variational formulations
results with directional stabilization and transport operators
In the phase space Galerkin approach, variational formulation (38) is discretized further by substituting the angular quadrature rule (39) for all angular integrals so that the bilinear form (19) is approximated by
Let the linear functional l(·) from (20) be approximated by a functional $\stackrel{~}{l}(\xb7)$ with angular quadrature correspondingly, then the directional solutions u_{ j } are determined from the variational formulation with angular quadrature
Since this formulation has to hold for all v∈V^{L,N}, it follows that for test functions which vanish at every angular quadrature node s_{ i }, $i=1,\dots ,{M}_{\mathcal{S}}$ except one s_{ j }, formulation (42) can be reduced to the variational formulation (40) from the collocation discretization. This condition on the test functions is satisfied e. g. for a basis of the test space of characteristic functions on the angular mesh if each mesh cell contains exactly one angular quadrature node. With such a onepoint quadrature rule and characteristic basis functions of ${V}_{\mathcal{S}}^{N}$, the phase space Galerkin DOM is therefore equivalent to the collocation DOM after discretization.
Sparse tensor discretization
The full tensor approach presented before shows the typical complexity for full tensor approximations: The number of degrees of freedoms increases exponentially with the dimension and the resolution levels in a dyadically refined scheme.
A way to counter this exponential increase is found in sparse tensorization. Using the same approximation spaces on the component domains ${V}_{D}^{{l}_{D}}$ and ${V}_{\mathcal{S}}^{{l}_{\mathcal{S}}}$ as for the full tensor approximation we define a sparse tensor approximation space ${\widehat{V}}^{L,N}$ by
where the sparsity profile$f:\{0,\dots ,L\}\times \{0,\dots ,N\}\to \mathbb{R}$ determines which tensor product subspaces ${V}_{D}^{{l}_{D}}\otimes {V}_{\mathcal{S}}^{{l}_{\mathcal{S}}}$ are to be included in the approximation. The sparsity profile usually depends on N as well. Here, we employ a linear profile
which is normally chosen if the component complexities M_{ D } and ${M}_{\mathcal{S}}$ depend on the resolution parameters L and N in the same way and identical order of approximation is sought over both component domains (cf. Zenger 1991; Bungartz and Griebel 2004; Griebel and Harbrecht 2013a).
If direct sum decompositions of the component approximation spaces ${V}_{D}^{{l}_{D}}$ and ${V}_{\mathcal{S}}^{{l}_{\mathcal{S}}}$ into detail spaces ${W}_{D}^{{l}_{D}}$ and ${W}_{\mathcal{S}}^{{l}_{\mathcal{S}}}$, i. e.
are available (correspondingly in the angular domain), then the sparse tensor approximation space ${\widehat{V}}^{L,N}$ can also be written as
By choosing hierarchical bases for ${V}_{D}^{{l}_{D}}$ and ${V}_{\mathcal{S}}^{{l}_{\mathcal{S}}}$, each degree of freedom u_{ i j } can directly be associated with a tensor product detail space ${W}_{D}^{{l}_{D}}\otimes {W}_{\mathcal{S}}^{{l}_{\mathcal{S}}}$. The sparse solution is then given by
Thus, the sparse discrete variational problem reads: Find ${\xfb}_{L,N}\in {\widehat{V}}^{L,N}$ such that
The dimension of the sparse tensor product space ${\widehat{V}}^{L,N}$ depends on the sparsity profile $f({l}_{D},{l}_{\mathcal{S}})$. For a linear sparsity profile as in (44), the following complexity estimate is known (e. g. Bungartz and Griebel 2004, Lemma 3.6), or Griebel and Harbrecht (2013a, Thm. 4.1)).
Lemma 7.
Assuming the dimensions of the detail spaces ${W}_{D}^{{l}_{D}}$ and ${W}_{\mathcal{S}}^{{l}_{\mathcal{S}}}$ scale as $dim\left(\underset{i}{\overset{{l}_{i}}{W}}\right)\le {c}_{i}{2}^{{d}_{i}{l}_{i}}$ with constants c_{ i }>0 and dimensions d_{ i }, $i=D,\mathcal{S}$, with d_{ D }=d, and given a linear sparsity profile $f({l}_{D},{l}_{\mathcal{S}})$ as in (44), the dimension of the sparse tensor product approximation space ${\widehat{V}}^{L,N}$ as defined by (45) is
where θ=1 if $\mathit{\text{dL}}={d}_{\mathcal{S}}N$ and θ=0 otherwise. Relation “ ≲” defines an order up to constants with respect to the relevant scaling parameters L, N: a≲b iff a≤C b with constant C independent of L and N.
Error analysis
In this section, we shall show that the convergence rates of the full tensor and sparse tensor Galerkin methods differ only by a logarithmic factor in the degrees of freedom, provided that somewhat stronger regularity requirements are met for the exact solution.
The analysis will proceed along the usual fashion, cp. (Bungartz and Griebel 2004). We define the Galerkin projector${\mathrm{P}}^{L,N}:{\mathcal{V}}_{1}\to {V}^{L,N}$ into the full tensor product approximation space
Letting L→∞ (N→∞) the fact that the subspaces are closed and dense implies that in the respective limits we obtain semidiscrete Galerkin projectors${\mathrm{P}}_{\mathcal{S}}^{N}:={lim}_{L\to \infty}{\mathrm{P}}^{L,N}$$\left({\mathrm{P}}_{D}^{L}:={lim}_{N\to \infty}{\mathrm{P}}^{L,N}\right)$ on the physical (angular) domain, as the Galerkin projector is stable in the ∥·∥_{1}norm:
Lemma 8
(Stability of the Galerkin projector). Let $v\in {\mathcal{V}}_{1}$. Then there is a constant c_{ P }>0 independent of L and N so that
Proof
With continuity (Lemma 3) of the bilinear form we obtain
Since this holds for all v_{L,N}∈V^{L,N}, we can set v_{L,N}=P^{L,N}v and exploit coercivity of the bilinear form (Lemma 5):
If P^{L,N}v≠0 we obtain the result with c_{ P }=c_{ c }/c_{ e }.
Error estimates on the physical domain
To begin with, we require some approximation results in the H^{1}(D)norm on the physical domain. With a Clémenttype quasiinterpolation operator ${\mathrm{P}}_{\mathrm{I}}^{L}$ (Scott and Zhang, 1990, Thm. 4.1 and Cor. 4.1) we obtain
Lemma 9
(Approximation of quasiinterpolation). For polyhedral $D\subset {\mathbb{R}}^{d}$ and a shaperegular triangulation ${\mathcal{T}}_{D}^{L}$ on D with mesh width h=2^{−L}, the quasiinterpolation ${\mathrm{P}}_{\mathrm{I}}^{L}v$ of a function v∈H^{s+1}(D), s∈ [ 0,1], to the space ${V}_{D}^{L}={S}^{0,1}\left(D,{\mathcal{T}}_{D}^{L}\right)$ of piecewise affine functions on ${\mathcal{T}}_{D}^{L}$ satisfies the error estimate
where c_{ H }>0 is a constant independent of L.
Lemma 10
(Stability of quasiinterpolation). Under the assumptions of Lemma 9, quasiinterpolation is H^{1}stable, i.e. there exists a constant c_{ B }>0 independent of L such that for all v∈H^{1}(D) it holds
Next we derive an error estimate for the Galerkin approximation on the physical domain. At this point, the approximation is semidiscrete.
Lemma 11
(Error estimate for Galerkin projection on physical domain). Let u∈H^{s+1,0}(Ω), s∈{0,1}, be the exact solution to problem (21) and ${u}_{L}:={\mathrm{P}}_{D}^{L}u\in {V}_{D}^{L}\otimes {L}^{2}\left(\mathcal{S}\right)$ the Galerkin projected solution to
with a(·,·) from (19) and l(·) from (20). Then, there is a constant c_{ p }>0 independent of L such that
Proof
The proof is standard, and is based on coercivity and Galerkin orthogonality. We proceed analogous to (Ávila et al. 2011, Lemma 3 and Theorem 1). After inserting the quasiinterpolated solution ${\xfb}_{L}:=({\mathrm{P}}_{\mathrm{I}}^{L}\otimes {\text{Id}}_{\mathcal{S}})u$ with ${\mathrm{P}}_{\mathrm{I}}^{L}$ from Lemma 9 the triangle inequality permits us to write
For the first part, we use the fact that there is a constant c_{ n }>0 for all $v\in {H}^{1}\left(D\right)\otimes {L}^{2}\left(\mathcal{S}\right)$ such that
Thus, we can apply Lemma 9:
For the second part in (50), we use coercivity of the bilinear form, then in a second step Galerkin orthogonality, and finally continuity of the bilinear form to write
and therefore with Lemma 9
By inserting into (50) we arrive at the result
Error estimates on the angular domain
On the angular domain, the considerations in the following require an approximation result for L^{2}projections.
Lemma 12
For functions $v\in {H}^{t}\left(\mathcal{S}\right)$, t∈{0,1}, the L^{2}projection to the space ${V}_{\mathcal{S}}^{N}$ satisfies the error estimate
where the constant c_{ l }>0 is independent of N.
This result can be obtained for approximation by spectral functions as in the spherical harmonics method (in which case t≥0 is arbitrary), for instance, as well as for approximation by piecewise constants as in the discrete ordinates method (in which case 0≤t≤1). It allows the derivation of the same approximation rate for the semidiscrete Galerkin projection on the angular domain.
Lemma 13
(Error estimate for angular Galerkin projection). Let u∈H^{1,t}(Ω), t∈{0,1}, be the exact solution to problem (21) and ${u}_{N}:={\mathrm{P}}_{\mathcal{S}}^{N}u\in {H}^{1}\left(D\right)\otimes {V}_{\mathcal{S}}^{N}$ the Galerkin projected solution with angular part from the subspace ${V}_{\mathcal{S}}^{N}$ of ${L}^{2}\left(\mathcal{S}\right)$. Then there is a constant c_{ a }>0 independent of N such that
Proof
The proof proceeds analogously to the one of Lemma 11 while substituting the L^{2}projected solution with Lemma 12 for the quasiinterpolated solution, the details are therefore omitted here.
Error estimate for the full tensor phase space Galerkin method
The following theorem gives an error estimate for the full tensor approximation.
Theorem 14
(Error estimate full tensor Galerkin method). The full tensor Galerkin approximation u_{L,N}=P^{L,N}u of a solution u∈H^{s+1,0}(Ω)∩H^{1,t}(Ω), s∈{0,1}, t∈{0,1}, to the variational problem (21) satisfies the asymptotic error estimate
with relation “ ≲” as in Lemma 7.
Proof
By Céa’s Lemma (Brenner and Scott 2008, Thm. 2.8.1) the Galerkin approximation is quasioptimal in V^{L,N}, its error can therefore be bounded (up to constants) by the error of any other approximation to u in V^{L,N}, for example the quasiinterpolated and L^{2}projected approximation ${\mathrm{P}}_{\mathrm{I}}^{L}\otimes {\mathrm{P}}_{{L}^{2}}^{N}u$:
Here, we used the approximation properties of the quasiinterpolant from Lemma 9 and of the angular L^{2} projection from Lemma 12. The last step is a consequence of the H^{1}stability asserted in Lemma 10 of the quasiinterpolation.
Error estimate for the sparse tensor phase space Galerkin method
After the full tensor approximation properties, we consider the convergence properties of a direct sparse tensor approximation on the sparse tensor product space ${\widehat{V}}^{L,N}$ as defined in (45).
In analogy to the full tensor Galerkin projector P^{L,N}, we can define a sparse tensor Galerkin projector${\widehat{\mathrm{P}}}^{L,N}$ by the orthogonality relation
The error of the sparse tensor solution ${\xfb}_{L,N}={\widehat{\mathrm{P}}}^{L,N}u$ is estimated in the following theorem (see also Widmer 2009, Thm. 2.6) and (Griebel and Harbrecht 2013a, Thms. 4.3 and 7.1)).
Theorem 15
(Error estimate of direct sparse tensor solution). Let the linear sparsity profile as in (44) be given. Assume further that L and N vary such that −s L+t N=ζ=const, then the direct sparse tensor approximation ${\xfb}_{L,N}$ of a function u∈H^{s+1,t}(Ω), s,t∈{0,1}, satisfies the error estimate
where relation “ ≲” is defined as in Lemma 7.
Proof
We follow the proof of Thm. 2.6 by (Widmer 2009). First we introduce socalled difference projectors ${\Delta}_{\mathrm{I}}^{{l}_{D}}:={\mathrm{P}}_{\mathrm{I}}^{{l}_{D}}{\mathrm{P}}_{\mathrm{I}}^{{l}_{D}1}$ and ${\Delta}_{{L}^{2}}^{{l}_{\mathcal{S}}}:={\mathrm{P}}_{{L}^{2}}^{{l}_{\mathcal{S}}}{\mathrm{P}}_{{L}^{2}}^{{l}_{\mathcal{S}}1}$ as the difference between projections to two consecutive resolution levels with the convention ${\mathrm{P}}_{\mathrm{I}}^{1}=0={\mathrm{P}}_{{L}^{2}}^{1}$. They project onto the detail spaces ${W}_{D}^{{l}_{D}}$ and ${W}_{\mathcal{S}}^{{l}_{\mathcal{S}}}$, respectively.
With these difference projectors, a sparse quasiinterpolated and L^{2}projected approximation ${\u016b}_{L,N}\in {\widehat{V}}^{L,N}$ to u can be expressed as
where ${l}_{\mathcal{S}}^{max}\left({l}_{D}\right)$ is the largest feasible angular resolution index which results from solving $f({l}_{D},{l}_{\mathcal{S}})\le L$ with respect to ${l}_{\mathcal{S}}$.
Now we exploit quasioptimality of the Galerkin approximation on the sparse tensor product space to replace the Galerkin approximation error by the error of the quasiinterpolated and L^{2}projected approximation. Additionally applying the norm estimate $\parallel v{\parallel}_{1}\lesssim \parallel v{\parallel}_{{H}^{1,0}\left(\Omega \right)}$ yields
The error is split into two terms:
The second term on the right hand side can be estimated by Lemma 9:
This term will not contribute to the asymptotic terms.
The first term on the right hand side of (54) is split up further:
Both norms on the right hand side of (56) can be estimated by Lemma 9 and Lemma 12:
Inserting back into (56) yields
The task is now to estimate the series. Using the assumption ζ=−s+t N/L:
We estimate the sum on the right hand side of (58) by its largest summand. Two cases can be distinguished here:

1.
If ζ≤0, the largest summand occurs for l _{ D }=0:
$$\begin{array}{l}{2}^{\mathit{\text{tN}}}\sum _{{l}_{D}=0}^{L}{2}^{\zeta {l}_{D}}\le L{2}^{\mathit{\text{tN}}}.\end{array}$$ 
2.
If ζ>0, the largest summand occurs for l _{ D }=L:
$$\begin{array}{ll}{2}^{\mathit{\text{tN}}}\sum _{{l}_{D}=0}^{L}{2}^{\zeta {l}_{D}}& \le {2}^{\mathit{\text{tN}}}L{2}^{\mathit{\text{sL}}+\mathit{\text{tN}}}=L{2}^{\mathit{\text{sL}}}.\phantom{\rule{2em}{0ex}}\end{array}$$
In summary, we may write
By combining this estimate with relations (53) to (57), we finally arrive at
In conclusion, we find that the convergence rate of O(2^{−sL−tN}) of the full tensor approximation is maintained up to an additional factor L, which by M_{ D }=O(2^{dL}) is logarithmic in the number of degrees of freedom. This result in conjunction with the greatly reduced complexity of the sparse tensor method (Lemma 7) shows its superior efficiency provided that the function u to be approximated is at least in H^{s+1,t}(Ω), with s,t∈{0,1}.
Numerical experiments
Algorithms
For the numerical experiments we compute a sparse tensor solution with the help of the combination technique. The sparse solution is constructed according to the formula
from a number of solutions ${u}_{{\ell}_{D},{\ell}_{\mathcal{S}}}\in {V}^{{\ell}_{D},{\ell}_{\mathcal{S}}}$ to the full tensor discrete variational formulation (38) of reduced physical resolution ℓ_{ D } and angular resolution ${\ell}_{\mathcal{S}}$.
Clearly ${\stackrel{\u030c}{u}}_{L,N}$ is in the space ${\widehat{V}}^{L,N}=\sum _{{l}_{D}=0}^{L}{V}^{{\ell}_{D},{\ell}_{\mathcal{S}}^{max}\left({\ell}_{D}\right)}$, which is identical to the sparse tensor approximation space from (43). However, in general the combination approximation differs from a direct sparse approximation ${\xfb}_{L,N}$ (see also Grella 2013, Sec. 2.3.1). Due to the quasioptimality of the direct sparse solution as an approximation in ${\widehat{V}}^{L,N}$, the error of the combination approximation can serve as an upper bound (up to factors) for the error ${\u2225u{\xfb}_{L,N}\u2225}_{1}$ of the direct sparse approximation.
Note that the convergence of the combination technique for the radiative transfer problem has not been shown formally yet. A recent proof for elliptic operators by (Griebel and Harbrecht 2013b) would be applicable under certain stability assumptions on the semidiscrete Galerkin projectors (for details we refer to (Grella 2013, Sec. 5.3.7)). However, the use of the combination technique approximation has practical advantages over the direct sparse approximation. First, to construct the subproblem solutions of lower resolution, an existing full tensor solver with standard nonhierarchical FEM bases can be reused, no direct sparse solver needs to be implemented. Second, the splitting into subproblems entails a natural level for parallelism in the algorithm, which can still be combined with parallel solution procedures at the level of each subproblem (an implementation is described in (Grella 2013, Chap. 7)).
Each of the full tensor subproblems is solved by a phase space Galerkin finite element method with nonhierarchical affine hat functions as physical basis and piecewise constants as angular basis. In the experiment of Sec. ‘Experiment 2’, the midpoint rule is used for angular quadrature which corresponds to the S_{ N }method. However, in situations where ray effects (Lathrop 1968) pollute the results, adaptive quadrature may help (Stone 2007). As a simple adaptive rule we link the number of quadrature points n_{ q } per dimension and per mesh element to the resolution levels l_{ D }, ${l}_{\mathcal{S}}$ of the subproblem by ${n}_{q}=max\{{l}_{D}/{l}_{\mathcal{S}},1\}$ in the experiment of Sec. ‘Experiment 1’. Even though the overall computational effort is then not bounded by Lemma 7, the total number of degrees of freedom still is. As the iterative, approximate solution of the linear system constitutes the most time consuming part, the sparse tensor method is, in practice, more efficient than the full tensor method.
Quantities of interest
In applications, the radiative intensity is often coupled to other modes of energy transport via the net emission (e. g. Larsen et al. 2002, Eq. (1.1a)). The net emission can be computed in turn from the incident radiation
For this reason, we choose the incident radiation as a lowerdimensional variable to visualize results and to analyze errors. The relative L^{2} or H^{1}error of the incident radiation is given by
Numerical experiments
All experiments are set on the domains D= [ 0,1]^{d}, $\mathcal{S}={\mathcal{S}}^{{d}_{\mathcal{S}}}$, with $d={d}_{\mathcal{S}}+1$. We solve the RTP with isotropic scattering $\Phi (\mathit{s},{\mathit{s}}^{\prime})=1/\left\mathcal{S}\right$ and zero inflow boundary conditions g=0.
Experiment 1
We search the solution to the Gaussian blackbody radiation
with absorption and scattering coefficient κ=σ=1.
The H^{1}error of the incident radiation indeed converges faster in the sparse approximation than the full approximation (Figure 1). Note that the L^{2}error of the sparse approximation can be larger than the error of the full approximation because the sparsity profile $f({l}_{D},{l}_{\mathcal{S}})$ has been optimized for essentially undeteriorated convergence in the ∥·∥_{1}norm of the error in the radiative intensity, which is more closely represented by the H^{1}error than the L^{2}error of the incident radiation.
Experiment 2
A blackbody radiation I_{ b }(x,s) corresponding to the exact solution
with fixed ${\mathit{s}}^{\prime}={(1/\sqrt{3},1/\sqrt{3},1/\sqrt{3})}^{\top}$ is inserted into the right hand side functional in (38) (Grella 2013, Sec. 8.2, Exp. 1). The absorption is set to κ=1, the scattering coefficient to σ=0.5.
For this experiment we employed a discrete ordinates solver in which the angular resolution N^{′} is related to the angular degrees of freedom by ${M}_{\mathcal{S}}={({N}^{\prime}+1)}^{2}$ so that N≈⌊log2(N^{′}+1)⌋, where N is the angular resolution used otherwise in this paper.
Figure 2 shows the superior efficiency of the sparse approach with respect to number of degrees of freedom vs. achieved error. The convergence rates indicate that the curse of dimensionality is mitigated by the sparse discrete ordinates method.
For a comparison to other sparse tensor approaches we refer to the numerical experiment section of (Grella 2013), which features a sparse tensor spherical harmonics approximation and a sparse collocation discrete ordinates method realized via the combination technique. We observed that the approach presented here performs similarly to the sparse collocation DOM combination technique as the methods are similar from the point of view of implementation, even though their theoretical derivation is different. The presented approach is somewhat less susceptible to ray effects at the expense of slightly longer computational times as the angular quadrature is adapted to the resolution of the angular mesh. The spherical harmonics method is most effective for solutions with highly regular angular part because of its regularity requirements for spectral convergence. In general, at the same resolution levels L and N, the combination technique approach realizes approximately the same error as the direct sparse approach, while the number of degrees of freedom in the combination technique is larger than in the direct sparse approach because the approximation spaces of different subproblems in the combination technique overlap in the degrees of freedom. It is therefore slightly less efficient than the direct sparse approach, but considerably more efficient than the full tensor approach and advantageous in practice due to faster and simpler implementation and parallelization.
Conclusion
We have shown a direct sparse tensor phase space Galerkin approximation of the radiative intensity in the stationary monochromatic radiative transfer problem can be computed with only $O(log{M}_{D}({M}_{D}+{M}_{\mathcal{S}}\left)\right)$ degrees of freedom as opposed to $O\left({M}_{D}{M}_{\mathcal{S}}\right)$ degrees of freedom for a standard full tensor approximation. Here, M_{ D } is the number of physical degrees of freedom and ${M}_{\mathcal{S}}$ the number of angular degrees of freedom. At the same time, the error of the sparse approximation in the ∥·∥_{1}norm still decreases essentially as the error of the full approximation, namely with the order $O\left(log{M}_{D}\left({M}_{D}^{s/d}+{M}_{\mathcal{S}}^{t/{d}_{\mathcal{S}}}\right)\right)$ as compared to $O\left({M}_{D}^{s/d}+{M}_{\mathcal{S}}^{t/{d}_{\mathcal{S}}}\right)$ in the full tensor approximation. The parameters s,t∈{0,1} indicate the regularity of the exact solution which is required to be in the space of mixed smoothness ${H}^{s+1,t}(D\times \mathcal{S})$ to achieve the sparse convergence rate, whereas ${H}^{s+1,0}(D\times \mathcal{S})\cap {H}^{1,t}(D\times \mathcal{S})$ is sufficient in the full tensor approximation.
To simplify implementation, we realized the sparse tensor approximation algorithmically via the combination technique. Together with suitable quadrature rules, we demonstrated in numerical experiments that this sparse tensor combination approximation retains the analyzed theoretical advantages of the direct sparse tensor method while allowing for straightforward parallelization also at the level of subproblems.
The proposed specialization of the phase space Galerkin framework investigated here has the advantage that both discrete ordinates and spherical harmonics method can be derived from it so that the sparse tensorization benefits hold for the sparse variants of both methods alike.
Therefore, for problems whose solutions exhibit socalled mixed regularity, the sparse tensor product phase space Galerkin approximations realize a significant increase in efficiency, i. e. achievable error per number of degrees of freedom. Even in applications where high numerical accuracy is the main objective, a sparse tensor product approximation might be of value as an initial value for an iterative solver or in a problemadapted preconditilpalitoccheme.
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Acknowledgments
The author wishes to thank Prof. Dr. Ch. Schwab for helpful discussions and valuable suggestions to improvements of this article. Financial support for this work by Schweizerischer Nationalfonds (SNF) under project no. 121892, by Deutsche Forschungsgemeinschaft (DFG) within SPP1324, and by the European Research Council (ERC) under ERC Advanced Grant 247277 is gratefully acknowledged.
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DOI
 35Q79
 65N12
 65N30
 65N35
 Radiative transfer
 Sparse grids
 Discrete ordinates method
 Spherical harmonics method
 Combination technique