DFG-Schwerpunktprogramm 1324

”Extraktion quantifizierbarer Information aus komplexen Systemen”

Effective discretization of direct reconstructionschemes for photoacoustic imaging in spherical

geometries

F. Filbir, S. Kunis, R. Seyfried

Preprint 150

Edited by

AG Numerik/OptimierungFachbereich 12 - Mathematik und InformatikPhilipps-Universitat MarburgHans-Meerwein-Str.35032 Marburg

DFG-Schwerpunktprogramm 1324

”Extraktion quantifizierbarer Information aus komplexen Systemen”

Effective discretization of direct reconstructionschemes for photoacoustic imaging in spherical

geometries

F. Filbir, S. Kunis, R. Seyfried

Preprint 150

The consecutive numbering of the publications is determined by theirchronological order.

The aim of this preprint series is to make new research rapidly availablefor scientific discussion. Therefore, the responsibility for the contents issolely due to the authors. The publications will be distributed by theauthors.

Effective discretization of directreconstruction schemes for photoacoustic

imaging in spherical geometries

Frank Filbir † Stefan Kunis‡ Ruben Seyfried§

Recently, kernel methods for the recovery of a function from its spherical meansin spherical acquisition geometry have been proposed. We present efficient algo-rithms for these formulas in the two- and three-dimensional case. Our schemeapplies Fourier techniques for certain convolution type integrals and discretizesphysical space on a polar and spherical grid, respectively.

Key words and phrases : Radon transform, spherical means, fast Fourier trans-form.

2010 AMS Mathematics Subject Classification : 44A12, 65R32, 65T50, 92C55.

1 Introduction

Analogously to the inversion of the Radon transform in computerized tomography, recov-ering a function from its mean values over a family of spheres is relevant in photoacoustictomography. The spherical mean operator R : C(Rd) → C(Rd × [0,∞)) is defined by

Rf(ξ, t) =�

Sd−1

f(ξ + tu) dσ(u),

where Sd−1 =�u ∈ Rd : |u| = 1

�denotes the unit sphere and σ its surface measure with

σ(S1) = 2π and σ(S2) = 4π. The variable t ≥ 0 is called measurement time and thevariable ξ ∈ Rd detector position or center point. In all practical applications these cen-ter points are located on a curve or surface and we consider the classical case ξ ∈ Sd−1

here. Moreover, we restrict ourselves to functions f : Rd → R with support suppf ⊂ B,where B =

�x ∈ Rd : |x| < 1

�denotes the open unit ball, such that the spherical mean values

Rf(ξ, t) vanish for t ≥ 2.As outlined in [3, 1], the function f can be approximated from the spherical mean values

Rf(ξ, t), ξ ∈ Sd−1, t ∈ [0, 2], using specific summability kernels. These kernels are families of

§Technische Universitat Munchen and Helmholtz Zentrum Munchen, frank.filbir@helmholtz-muenchen.de‡University Osnabruck and Helmholtz Zentrum Munchen, stefan.kunis@math.uos.de§Helmholtz Zentrum Munchen, ruben.seyfried@helmholtz-muenchen.de

1

integrable functions Kε : B× B → R, where ε ∈ (0, 1) is a regularization parameter, with theproperties

Kε(x, y) =�

Sd−1

kε(x, ξ, |y − ξ|) dσ(ξ),

f(x) = limε→0

�

Bf(y)Kε(x, y) dy,

and kε : B × Sd−1 × [0, 2] → R denotes some auxiliary function. Then a simple calculationshows, see [1, eq. (3)],

f(x) = limε→0

� 2

0

�

Sd−1

kε(x, ξ, t)Rf(ξ, t) dσ(ξ) td−1 dt.

Our main objective now is an effective discretization of this reconstruction formula for spacedimensions d = 2 and d = 3. Towards this goal, we integrate over the sphere first andunderstand this step as a convolution for which Fourier techniques are applicable. This isfollowed by the integration over time and finally, an interpolation step on the reconstructionis performed.

Similar approaches can be found in [7, 6, 10] and references therein. The papers [7, 6] con-sider approximate reconstruction formulas in d = 3 and d = 2, respectively. The developedalgorithm has three major steps: a filtering step that integrates the data Rf(ξ, t) over timet against a kernel that depends only on the radial coordinate of the reconstruction position,a linear interpolation step on the intermediate data, and a subsequent backprojection stepintegrating over the sphere. Besides technicalities, this algorithm is efficient since the inter-polation step basically decouples radial and angular coordinates. In contrast, our algorithmdiscretizes physical space on a polar or spherical grid, computes the inner integral of thereconstruction formula efficiently, and interpolates only the final result.

The algorithms in [10] implement exact reconstruction formulas by spectral methods whichseparate the radial and angular variables. After careful discretization and truncation ofinvolved series, these schemes achieve optimal arithmetic complexity. The algorithms havefour major steps: Fourier transforms with respect to time and also with respect to the angularcomponent, a multiplication step, inverse Fourier transforms with respect to the angularcomponent, an interpolation step from the polar or spherical grid to a Cartesian grid, anda final Fourier transform on the Cartesian grid. In contrast, we implement an approximatereconstruction formula and discretize on a polar or spherical in the original domain rather thanin the frequency domain. Unfortunately, we do not achieve the optimal orders in arithmeticcomplexity, but as pointed out above our algorithms avoids interpolation of intermediate data.

The paper is organized as follows: We consider the two- and the three-dimensional casein Sections 2 and 3, respectively. After introducing the necessary notation, we first presentthe continuous version of the reconstruction formulas when considered in polar or sphericalcoordinates. Subsequently, we discretize these formulas on a polar or spherical grid, chooseinvolved parameters and analyze the arithmetic complexity of the obtained algorithms. Alltheoretical results are illustrated by a couple of numerical experiments in Section 4 and wefinally conclude our findings in Section 5.

2

2 Circular means

For the two-dimensional case d = 2, we consider detector positions on the unit circle , i.e. ξ ∈S1, surrounding the support of the function f . For each detector position and measurementtime t ∈ [0, 2], the spherical mean is just the integral of f over a circular arc with midpointξ and radius t, see also Figure 2.1(left). An appropriate choice of the function kε now is

h : R → R, h(t) =12π

1− t2

(1 + t2)2,

hε : [−2, 2] → R, hε(t) =1ε2

h(t

ε),

kε : B× S1 × [0, 2] → R, kε(x, ξ, t) =2π(1− |x|2)hε(|x− ξ|2 − t2).

The result that this indeed produces a summability kernel is given in [1, Corollary 2, Section(3.2)]. As a consequence we get an approximation to the function f by

fε(x) =2π(1− |x|2)

� 2

0

�

S1

hε(|x− ξ|2 − t2)Rf(ξ, t) dσ(ξ) t dt. (2.1)

Subsequently, we show that this reconstruction can be understood as a convolution whenwe use polar coordinates for the function f . Discretization leads to a polar grid as depictedin Figure 2.1(middle) and a bilinear interpolation on each polar wedge, cf. Figure 2.1(right)finally gives the reconstruction of f on a Cartesian grid.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0.35 0.4 0.45 0.5 0.55 0.6 0.650.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Figure 2.1: Measurement and reconstruction geometry.

Using the standard parameterization ξ = (cosψ, sinψ)�, ψ ∈ [0, 2π), of the circle S1 andexpressing x ∈ B in polar coordinates x = r(cosϕ, sinϕ)�, r ∈ [0, 1), ϕ ∈ [0, 2π), yields

|x− ξ|2 = 1 + r2 − 2r cos(ψ − ϕ).

For notational convenience, we denote fixed arguments of functions as superscript and skippingthe parameter ε completely, i.e.,

hr,t(ψ) = hε(1 + r2 − 2r cosψ − t2),

3

gt(ψ) = t · Rf (cosψ, sinψ, t) ,

Hence, the approximation (2.1) can be written as a periodic convolution with respect to theangular component

f r,t(ϕ) =�hr,t ∗ gt

�(ϕ) =

� 2π

0hr,t(ϕ− ψ)gt(ψ) dψ,

fε (r cosϕ, r sinϕ) =2π(1− r2)

� 2

0f r,t(ϕ) dt.

Typically, the measurement times t ∈ [0, 2] are equidistant and the detector positions areequiangular ξn = (cosψn, sinψn)� ∈ S1,

tm =2mM

, m = 0, . . . ,M − 1,

ψn =2πnN

, n = 0, . . . , N − 1.

Thus, the integrals in (2.1) are discretized via composite quadrature rules with equidistantnodes. In the angular variable, constant weights give the highest trigonometric degree ofexactness, in the time variable, constant weights yield a midpoint rule. We discretize thespatial variable x ∈ B accordingly on a polar grid x�,j = rj(sinϕ�, cosϕ�)�,

rj =j

J, j = 0, . . . , J − 1,

ϕ� =2π�N

, � = 0, . . . , N − 1,

which leads for fixed ε ∈ (0, 1) to the discrete reconstruction formula

fε(x�,j) ≈ f j� :=

8(1− |x�,j |2)MN

M−1�

m=0

f j,m� (2.2)

f j,m� :=

N−1�

n=0

hε(1 + r2j − t2m − 2rj cosψn−�) tm · Rf(ξn, tm)

For fixed indices j,m, the second sum is a multiplication with a circulant matrix, i.e., f j,m =Hj,mgm, where

f j,m := (f j,m� )�=0,...,N−1 ∈ RN ,

gm := (tm · Rf(ξn, tm))n=0,...,N−1 ∈ RN ,

Hj,m := (hε(1 + r2j − t2m − 2rj cosψn−�))�,n=0,...,N−1 ∈ RN×N .

We diagonalize Hj,m = 1N F ∗ diag h

j,mF by a discrete Fourier transform, where

F := (e−2πikn/N )k,n=0,...,N−1,

hj,m

:= Fhj,m,

hj,m := (hε(1 + r2j − t2m − 2rj cosψn))n=0,...,N−1.

We bring the inverse Fourier transform in front of the outer summation in (2.2) and have thefollowing Algorithm 1.

4

Algorithm 1Input : discretization parameter N,M, J ∈ N,

measurement times tm = 2mM , m = 0, . . . ,M − 1,

detector positions ξn = (cosψn, sinψn), ψn = 2πnN , n = 0, . . . , N − 1,

data Rf(ξn, tm), n = 0, . . . , N − 1, m = 0, . . . ,M − 1,radii rj = j

J , j = 0, . . . , J − 1,angles ϕ� = 2π�

N , � = 0, . . . , N − 1.Output : function values f jl ≈ fε(rj cosϕ�, rj sinϕ�), � = 0, . . . , N − 1, j = 0, . . . , J

for j = 0, . . . , J − 1 dofor m = 0, . . . ,M − 1 do

set gm = (tm · Rf(ξn, tm))n=0,...,N−1

compute fj,m

= diag(hj,m

)Fgm

end for

compute fj=

8(1−r2j )

MN2

�M−1m=0 f

j,m

compute f j = F ∗fj

end for

Remark 2.1. Finally, the function f needs to be evaluated on a Cartesian grid and since thereconstruction yields function values on the polar grid, we employ the following interpolationscheme, see also Figure 2.1(middle). Let the discretization parameter L ∈ N and nodeszs,t = (s/L, t/L), s, t = −L, . . . , L, be given. For ease of notation, consider some fixed nodezs,t in the positive quadrant with �zs,t�2 < 1, define indices and weights

j =

�J√s2 + t2

L

�, ws,t,j = J

�√s2 + t2

L− rj

�,

� =�N

2πarctan

t

s

�, vs,t,� =

N

2π

�arctan

t

s− ϕ�

�,

and interpolate bilinearly in the polar grid by

fε(zs,t) = (1− ws,t,j) (1− vs,t,�) fjl + ws,t,j (1− vs,t,�) f

j+1l (2.3)

+ (1− ws,t,j) vs,t,�fjl+1 + ws,t,jvs,t,�f

j+1l+1 .

Figure 2.1(right) illustrates part of the polar grid and one evaluation node zs,t in its polarwedge. Alternatively, one might use a nearest neighbor interpolation in the polar grid or aconstant or linear interpolation over some triangulation of the polar grid.

2.1 Parameter choice and computational complexity

An important question concerns the choice of the parameter ε, see also [6, Section 5.1]. Whilethe approximation (2.1) becomes better for smaller ε, the discretization of the outer integralby a composite midpoint rule produces reasonable results only if the integrand is smooth withrespect to the mesh size M−1 in (2.2). Since the function hε has its main lobe in the interval[−ε, ε] and a constant number C ≥ 1 of samples should lie inside this interval, we set

ε =C

M,

5

which can further decreased by an artificial increase of the resolution of the measurements,e.g., by interpolation.

The inner sum in (2.2) is a discrete and cyclic convolution and can be realized by meansof fast Fourier transforms in O(N logN) floating point operations. Taking into account theouter summation over time in (2.2) for all radii, this leads to O(JMN logN) floating pointoperations. Interpolation of the result in (2.3) is a local operation and takes only O(JN+L2)floating point operations. Assuming finally O(J) = O(L) = O(M) = O(N) and consideringthe total problem size n = N2, our algorithm has complexityO(n1.5 logn). We note in passing,that the polar grid discretization might be coarsened near the origin saving a fraction of thetotal computations and that a generalization to nonequally spaced detectors on the unit circleis straightforward using the nonequispaced FFT [9].

3 Spherical means

For the three-dimensional case d = 3, we consider detector positions on the unit sphere, i.e.ξ ∈ S2, surrounding the support of the function f , see Figure 3.1(left). For each detectorposition and measurement time t ∈ [0, 2], the spherical mean is just the integral of f over aspherical cap with midpoint ξ and radius t, see also Figure 3.1(right).

Figure 3.1: Measurement geometry.

Now the results in [1, Theorem 4, Section (4.2)] give rise to a whole family of reconstruction

formulas. Let q ∈ N, q ≥ 2, be given and define cq =4Γ(q+ 5

2)√πΓ(q+1)

. Let (t)+ = max {t, 0},

hq : R → R, hq(t) = cq

��1− t2

�q+− 2qt2

�1− t2

�q−1

+

�,

hε,q : [−2, 2] → R, hε,q(t) =1ε3

hq(t

ε), (3.1)

kε,q : B× S2 × [0, 2] → R, kε,q(x, ξ, t) =(1− |x|2)

2π2hε,q(|x− ξ|2 − t2).

The conditions of [1, Theorem 4] are satisfied since the function hq is even, locally integrable,

6

hq(t) = cqddt t(1− t2)q+, and the function Hq : R3 → R,

Hq(x) :=14π

�

S2

hq(�x, ξ�) dσ(ξ) =1|x|

� |x|

−|x|hq(t) dt =

cq4π

(1− |x|2)q+,

is a radial,�

R3 Hq(x) dx = 1, and |Hq(x)| ≤ 16(1 + |x|)−4. Hence, the function f can beapproximately reconstructed by

fε(x) =(1− |x|2)

2π2

� 2

0

�

S2

hε,q(|x− ξ|2 − t2)Rf(ξ, t) dσ(ξ) t2 dt. (3.2)

Subsequently, we propose a reconstruction scheme generalizing the convolution type ideas.We express the spatial variable in spherical coordinates, cf. Figure 3.2(left), generalize theconvolution on the circle S1 to the sphere S2, and diagonalize the convolutions by meansof appropriate fast spherical Fourier transform. Two other approaches, one based on a dis-cretization in cylinder coordinates, cf. Figure 3.2(right), and another one using the compactsupport of the function hε,q, are discussed in Section 3.2.

−1−0.5

00.5

1

−1

−0.5

0

0.5

1−1

−0.5

0

0.5

1

Figure 3.2: Reconstruction geometries, spherical (left) and cylindrical coordinates (right).

Expressing x ∈ B in spherical coordinates x = rη, r ∈ [0, 1), η ∈ S2, cf. Figure 3.2(left),yields

|x− ξ|2 = 1 + r2 − 2rη · ξ.We write fixed arguments of functions as superscript and suppress the parameters ε and qcompletely, i.e.,

hr,t : [−1, 1] → R, hr,t(y) = hε,q(1 + r2 − t2 − 2ry), (3.3)

gt : S2 → R, gt(ξ) = t2Rf(ξ, t),

the approximation (3.2) can be written as a convolution of a function on the sphere with azonal kernel and a subsequent integration

f r,t(η) =�hr,t ∗ gt

�(η) =

�

S2

hr,t(η · ξ)gt(ξ) dσ(ξ), (3.4)

7

fε(x) = fε(rη) =(1− r2)

2π2

� 2

0f r,t(η) dt.

In order to get a fast algorithm for the convolution (3.4), we follow [8] and expand thefunction hr,t in a Legendre series. The Legendre-Polynomials Pk : [−1, 1] → R are defined as

Pk(x) =1

2kk!ddx

�x2 − 1

�k,

by Y nk : S2 → C we denote the spherical harmonic of degree k ∈ N0 and order n = −k, . . . , k.

We use the addition theorem for spherical harmonics, which separates the dependencies ofthe variables η and ξ in

hr,t(η · ξ) =∞�

k=0

hr,tk Pk(η · ξ) =∞�

k=0

4πhr,tk2k + 1

k�

n=−k

Y nk (η)Y n

k (ξ), (3.5)

where the Fourier Legendre coefficients are given by

hr,tk =2k + 1

2

� 1

−1hr,t(x)Pk(x) dx. (3.6)

Since the sum in (3.5) converge absolutely, which follows from Theorem 3.4, we can inter-change the order of summation and integration, this yields

fε(rη) =2π(1− r2)

∞�

k=0

k�

n=−k

Y nk (η)

� 2

0

hr,tk2k + 1

�

S2

gt(ξ)Y nk (ξ) dσ(ξ) dt. (3.7)

Both integrals are discretized at the locations where the data Rf(ξ, t) is given, i.e., the innerintegral at the detector positions and the outer integral at the measurement times

ξi ∈ S2, i = 0, . . . , I2 − 1,

tm =2mM

, m = 0, . . . ,M − 1.

We evaluate the function fε at points xj,l = rjηl, where

ηl ∈ S2, l = 0, . . . , L2 − 1,

rj =j

J, j = 0, . . . , J − 1.

Moreover, we approximate the series expansion of hrj ,tm as given in (3.5) at a fixed cut-offdegree N ∈ N and compute the Fourier coefficients in this series by a quadrature,

hrj ,tmk ≈ hj,mk :=

N−1�

ν=0

µνPk(λν)hrj ,tm(λν).

Provided N > 2q and the Gauß Legendre nodes and weights (λν , µν) are chosen for thesupport of hr,t, this quadrature is exact since hr,t is a polynomial of degree 2q within itssupport. The resulting truncation error is discussed in Section 3.1.

8

Our algorithm now works as follows. In a first step and for each measurement time tmindividually, we compute discrete spherical Fourier coefficients

gmn,k :=I2−1�

i=0

ωiY nk (ξi)gtm(ξi), k = 0, . . . , N − 1, n = −k, . . . , k,

approximating the inner integral in (3.7) by numerical quadrature on the nodes ξi ∈ S2 andwith some weights ωi > 0, i = 0, . . . , I2 − 1. These computations are realized via an adjointnonequispaced fast spherical Fourier transform (adjoint NFSFT), see e.g. [9], the accuracy ofthis approximation and a precomputation of the weights is discussed in [5].

In a second step and for each radius rj individually, we compute

f jn,k :=

2M

M−1�

m=0

hj,mk2k + 1

gmn,k, k = 0, . . . , N − 1, n = −k, . . . , k, (3.8)

approximating the outer integral in (3.7) by numerical quadrature at the nodes tm and withconstant weights 2

M . Finally, we evaluate the truncated outer sum in (3.7) at the target nodesηl, l = 0, . . . , L2 − 1, i.e.,

fε(rjηl) ≈ f jl :=

2(1− r2j )

π

N�

k=0

k�

n=−k

f jn,kY

nk (ηl), l = 0, . . . , L2 − 1, (3.9)

by a nonequispaced fast spherical Fourier transform (NFSFT). For notational convenience,we define

gm :=�ωig

tm(ξi)�i=0,...,I2−1

=�ωit

2mRf(ξi, tm)

�i=0,...,I2−1

∈ RI2

Y ξ := (Y nk (ξi))i=0,...,I2−1; k=0,...,N−1, n=−k,...,k ∈ CI2×N2

,

Y η := (Y nk (ηl))l=0,...,L2−1; k=0,...,N−1, n=−k,...,k ∈ CL2×N2

.

and formulate Algorithm 2.

Remark 3.1. The reconstruction yields function values on the spherical grid, see also Figure3.2(left). Analogously to Remark 2.1, we interpolate trilinear in the spherical grid. We assumeη ∈ S2 is given in the form

η�,n = (sin(ψ�) cos(ϕn), sin(ψ�) sin(ϕn), cos(ψ�))� , ϕ� =

2π�L

, ψn =πn

L− 1,

� = 0, . . . , L − 1, n = 0, . . . , L− 1, and Algorithm 2 outputs values f j�,n ≈ fε(rjη�,n). Let thediscretization parameter K ∈ N and Cartesian nodes zs,t,w =

�sK , t

K , pK

�, s, t, p = −K, . . . ,K,

be given. For ease of notation, consider some fixed node zs,t,p with �zs,t,p�2 < 1 and definethe corresponding indices and weights by

j =

�J�s2 + t2 + p2

K

�, wj = J

��s2 + t2 + p2

K− rj

�,

� =�L

2π

�π + sgn(t) arccos

s√s2 + t2

��, v� =

L

2π

�π + sgn(t) arccos

s√s2 + t2

− ϕ�

�,

9

n =

�L− 1π

�arccos

p�s2 + t2 + p2

��, un =

L− 1π

�arccos

p�s2 + t2 + p2

− ψn

�.

Now, interpolate along ϕ, ψ, and r respectively, by

c00 = (1− v�)fj�,n + v�f

j�+1,n, c01 = (1− v�)f

j�,n+1 + v�f

j�+1,n+1,

c10 = (1− v�)fj+1�,n + v�f

j+1�+1,n, c11 = (1− v�)f

j+1�,n+1 + v�f

j+1�+1,n+1,

c0 = (1− un)c00 + unc01, c1 = (1− un)c10 + unc11,

fε(zs,t,p) = (1− wj)c0 + wjc1.

Algorithm 2Input : discretization parameter I,M,L, J,N ∈ N,

measurement times tm = 2mM , m = 0, . . . ,M − 1,

detector positions ξi ∈ S2, i = 0, . . . , I2 − 1,data Rf(ξi, tm), i = 0, . . . , I2 − 1, m = 0, . . . ,M − 1radii rj = j

J , j = 0, . . . , J − 1,angles ηl ∈ S2, l = 0, . . . , L2 − 1.

Output : function values f jl ≈ fε(rjηl), l = 0, . . . , L2 − 1, j = 0, . . . , J − 1.

for m = 0, . . . ,M − 1 doset gm =

�ωit

2mRf(ξi, tm)

�i=0,...,I2−1

compute gm = Y ∗ξg

m

end forfor j = 0, . . . , J − 1 do

for k = 0, . . . , N − 1 dofor m = 0, . . . ,M − 1 do

compute hj,mk =�N−1

ν=0 µνPk(λν)hrj ,tm(λν)end forfor n = −k, . . . , k do

compute f jn,k =

4(1−r2j )

(2k+1)Mπ

�M−1m=0 hj,mk gmn,k

end forend for

compute f j = Y ηfj

end for

3.1 Parameter choice and computational complexity

It remains to choose the parameters q,N ∈ N, ε > 0, and analyze the final computationalcomplexity of Algorithm 2. Similar to Section 2.1, we subsequently argue that the regulariza-tion parameter ε is bounded with respect to the discretization of the measurement time, i.e.,ε ≥ CM−1. Moreover, the parameter q ∈ N determines the smoothness of the function hr,t

and thus the asymptotic decay of its Fourier Legendre coefficients (3.6). Theorem 3.4 makesthis observation precise and allows for an error estimate in Corollary 3.5 which also showsthat the asymptotic decay sets in for k ≥ Cε−1. Hence, the choice N = CM of the cut-offdegree allows for a guaranteed accuracy and a complexity estimate of Algorithm 2.

10

For notational convenience, we drop all parameters from the considered function h : R → R,

h(y) := hr,t(y) = hε,q(1 + r2 − t2 − 2ry).

Since

supp h = [v − u, v + u] , v :=1 + r2 − t2

2r, u :=

ε

2r, (3.10)

we obtain that supp h ∩ [−1, 1] �= ∅ if and only if

t ∈��

((1− r)2 − ε)+,�

(1 + r)2 + ε�.

Consequently, we have hk �= 0 and thus nonzero terms in the sum (3.8) only if m =M1, . . . ,M2, where

M1 =�M

2

�((1− rj)2 − ε)+

�, M2 = min

��M

2

�(1 + rj)2 + ε

�,M − 1

�

and this speeds up the total computations. On the other hand, the sum (3.8) represents theintegration over t and is a reasonable discretization only if at least a constant number ofsamples are taken into account. Considering the critical case r = 0, dropping the rounding,and using

M2 −M1 ≥M

2�√

1 + ε−√1− ε

�≥ Mε

2,

this is the case for ε = C/M .We proceed by estimating the Fourier Legendre coefficients (3.6). Trivially, all coefficients

fulfill hk = 0, k ∈ N0, if supp h∩[−1, 1] = ∅, and we have hk = 0, k ≥ 2q+1, if [−1, 1] ⊂ supp h.Subsequently, we discuss the remaining case and set

[a, b] := supp h ∩ [−1, 1]. (3.11)

We start by bounding the coefficients hk independently of k in Lemma 3.2 and compute thevalues of the function h and its derivatives in the endpoints of its support in Lemma 3.3. Thisallows for the estimate on the decay of the Fourier Legendre coefficients hk in Theorem 3.4.

Lemma 3.2. The Fourier-Legendre coefficients (3.6) fulfill

���hk��� ≤ 4q(q + 1)cq

ε3,

where cq is given in (3.1).

Proof. First note, that h�(a) = h�(b) = 0 if −1 < a < b < 1 and that Pk(1) = 1 andPk(−1) = (−1)k, k ∈ N0. In combination with

(2k + 1)Pk = P �k+1 − P �

k−1, k ∈ N, (3.12)

integration by parts leads to

���hk��� = 2k + 1

2

����� b

ah(s)Pk(s) ds

���� ≤12

� b

a

��h�(s)�� |Pk+1(s)− Pk−1(s)| ds.

11

Using |Pk(s)| ≤ 1 and

|h�(s)| = 2qcq|(v − s)|ε3u2q

(u2− (v− s)2)q−2+

��3u2 − (2q + 1)(v − s)2�� ≤ 4q(q + 1)cq

ε3u≤ 8q(q + 1)cq

ε3(b− a)

which follows from maxs∈[a,b] |v − s| ≤ maxs∈[v−u,v+u] |v − s| = u, (u2 − (v − s)2)+ ≤ u2, and2u ≥ b− a, the assertion follows.

Lemma 3.3. The function h and its derivatives of order p = 0, . . . , 2q fulfill

h(p)(v + u) =

�0 if p = 0, . . . , q − 2,(−1)q22q−1−p(p+2)!cq

ε3up(q+1)

�q+1

p−q+1

�if p = q − 1, . . . , 2q,

and h(p)(v − u) = (−1)ph(p)(v + u), see (3.1) for the definition of cq.

Proof. We consider the auxiliary function gq(s) = (u2 − (v − s)2)q = (u− v + s)q(u+ v − s)q

which fulfills

g(p)q (s) =

p�

i=0

�p

i

�dp−i

dsp−i(u− v + s)q

di

dsi(u+ v − s)q

=

��pi=0

�qi

��q

p−i

�(−1)ip!(u− v + s)q−p+i(u+ v − s)q−i if p < q,

�qi=(p−q)

�qi

��q

p−i

�(−1)ip!(u− v + s)q−p+i(u+ v − s)q−i if p ≥ q,

and as a consequence

g(p)q (v + u) =

�0 if p < q,

(−1)q�

qp−q

�p!(2u)2q−p if p ≥ q.

The assertion follows from g(p)q (v − u) = (−1)pg(p)

q (v + u) and

h(s) =(2q + 1)cq

ε3u−2qgq(s)−

2qcqε3

u−2(q−1)gq−1(s).

Theorem 3.4. The Fourier Legendre coefficients hk of the function h obey the inequality

���hk��� ≤

Cq√u

�(1− b2)

14 + (1− a2)

14

�

ε3σq− 12

�2 +

1σ

�q+1

, k > 2q + 1,

where a < b are given by (3.11), σ = u(k − 2q), Cq = (2q + 1)!cq, and cq is given in (3.1).

Proof. For notational convenience let [f ]ba := f(b)− f(a). Induction over p = 1, . . . , 2q, usingintegration by parts together with equation (3.12), yields

|hk| ≤12

p−1�

i=0

12i(k − (p− 1))i

i�

l=0

�i

l

� �����h(i)(Pk−i+2l+1 − Pk−i+2l−1)

�ba

����

+12

12p−1(k − (p− 1))p−1

p�

l=0

�p

l

� ����� b

ah(p)(s)Pk−p+2l(s) ds

����

12

≤ 12

2q�

i=0

1(2(k − 2q))i

i�

l=0

�i

l

� �����h(i)(Pk−i+2l+1 − Pk−i+2l−1)

�ba

���� , (3.13)

where the last step for p = 2q is due to h(2q)q being constant. We use the inequality

|Pk+1(x)− Pk−1(x)| ≤2(1− x2)

14√

k, k ≥ 2, x ∈ [−1, 1],

see [2] and [11, p. 172, eq. (7.33.10)] for a related asymptotic statement. Since [a, b] � [−1, 1],we assume that b < 1.Together with the symmetry of the derivatives, this provides

�����h(i)(Pk−i+2l+1 − Pk−i+2l−1)

�ba

���� ≤���h(i)(b)

��� 2(1− b2)14√

k − i+ 2l+

���h(i)(a)��� 2(1− a2)

14√

k − i+ 2l

≤ 2��h(i)(b)

��√k − i

�(1− b2)

14 + (1− a2)

14

�,

where the corresponding term on the right hand side vanishes if a = −1. We proceed in (3.13)by

���hk��� ≤

2q�

i=0

(1− b2)14 + (1− a2)

14

(2(k − 2q))i

��h(i)(b)��

√k − i

i�

l=0

�i

l

�

=�(1− b2)

14 + (1− a2)

14

� 2q�

i=0

1(k − 2q)i

��h(i)(b)��

√k − i

.

Applying Lemma 3.3, we get

2q�

i=0

1(k − 2q)i

��h(i)(b)��

√k − i

≤ cqε3

2q�

i=q−1

22q−1−i

(k − 2q)i+12ui

�q + 1

i− q + 1

�(i+ 2)!q + 1

≤ cq(2q + 1)!ε3

2q�

i=q−1

22q−i

(k − 2q)i+12ui

�q + 1

i− q + 1

�

=cq(2q + 1)!

√u

ε3σq− 12

q+1�

i=0

2q+1−i

σi

�q + 1i

�

and the assertion follows by the binomial theorem.

Corollary 3.5. Let ε ∈ (0, 1), N ∈ N, N ≥ 2/ε, and the approximation fε in (3.2) betruncated by fN

ε : B → R,

fNε (rη) =

(1− r2)2π2

� 2

0

�

S2

hN (ηξ)gt(ξ) dσ(ξ)t2 dt, hN =N+2q−1�

k=0

hkPk,

where r ∈ (0, 1) and η ∈ S2, then

�fε − fNε �∞ ≤ Cq(1− r2)rq−1

ε72

(εN)32−q�f�∞,

where Cq = 32 · 6q+1 · Cq.

13

Proof. Using maxx∈[−1,1] |Pk(x)| = 1, we get

|fε(rη)− fNε (rη)| ≤ 1− r2

2π2

� 2

0

�

S2

��h(ηξ)− hN (ηξ)�� ��gt(ξ)

�� |t|2 dσ(ξ) dt

≤ 1− r2

2π2

∞�

k=N+2q

|hk|� 2

0

�

S2

�

S2

|f(ξ + tu)| |t|2 dσ(u) dσ(ξ) dt

≤ 26(1− r2)3

�f�∞∞�

k=N+2q

|hk|.

Applying Theorem 3.4 together with 2 + 1σ ≤ 3 and u = ε

2r , the sum of the Fourier Legendrecoefficients is bounded by

∞�

k=N+2q

|hk| ≤2 3q+1Cq

ε3uq−1

∞�

k=N

k12−q ≤ 2 3q+1Cq

ε3uq−1

�N

12−q +

� ∞

Nx

12−q dx

�≤ 2 3q+2Cq(2r)q−1

ε72 (εN)q−

32

.

The constants in the above estimate are not optimized in any way and numerical experi-ments suggest that the actual error includes an additional factor N− 1

2 . In total, the truncationerror decays algebraically with a rate depending on the smoothness parameter q and this be-havior sets in for N ≥ ε−1.

Remark 3.6. Regarding the total accuracy of Algorithm 2, we note the following.

i) Approximating the function f by (3.2) induces the approximation error

�f − fε�∞,

whose discussion, in particular its so called degree of approximation with respect to εand q, is beyond the scope of this paper.

ii) Corollary 3.5 discusses the truncation error

�fε − fNε �∞

and suggests a choice of the truncation parameter N with respect to ε.

iii) Finally, the discrete spherical Fourier coefficients gmk,n of the given data are computed bya quadrature rule. Together with the discretization of the integral over the measurementtime, this introduces a discretization error

maxj,l

|fNε (rjηl)− f lj |.

Provided the detector positions are somewhat uniformly distributed on the sphere, weexpect a degree of exactness N ≈ I and a rate of convergence N−q� where q� is related tothe smoothness of the given data, see also [5]. The integral over the measurement timeis computed by a simple trapezoidal rule whose accuracy is discussed e.g. in [12].

The above discussion supports the parameter choice ε = O(1/M), N = O(M), and more-over, we assume O(I) = O(J) = O(N). Each spherical Fourier transforms in Algorithm 2is computed in O(N2 log2 N) floating point operations and thus, the most time consumingparts rely on the two innermost loops. Hence, we have a total complexity of O(n

43 ) floating

point operations with respect to the total problem size n = N3.

14

3.2 Compact support and cylindrical coordinates

Subsequently, we shortly discuss two other approaches to discretize the reconstruction for-mula (3.2). Using the compact support of the function hε,q and after changing the order ofintegration, we obtain

fε(x) =(1− |x|2)

2π2

�

S2

� min{2,√|x−ξ|2+ε}

√(|x−ξ|2−ε)+

hε,q(|x− ξ|2 − t2)Rf(ξ, t) t2 dt dσ(ξ).

For simplicity, we consider the case of a single discretization parameter N ∈ N and reconstructf on the Cartesian grid x�,p,j = 1

N3 (�, p, j)� ∈ [−1, 1]3, �, p, j = −N, . . . , N . Discretizing theouter integral by N2 nodes on S2 and the inner integral over the original interval [0, 2] by Nnodes leads for fixed indices l, p, j to

fε(xl,p,j) ≈2cq(1− |xl,p,j |2)

N3

N�

i=1

N�

n=1

N2�

m=N1

hε,q(|xl,p,j − ξi,n|2 − t2m)Rf(ξi,n, tm) t2m sinψ1,i.

In case |xl,p,j − ξi,n|2 ≥ ε, we have

(N2 −N1)2 ≤ N2

��|xl,p,j − ξi,n|2 + ε−

�|xl,p,j − ξi,n|2 − ε

�2

= 2N2

�|xl,p,j − ξi,n|2 −

�|xl,p,j − ξi,n|4 − ε2

�

≤ 2N2

�|xl,p,j − ξi,n|2 −

�(|xl,p,j − ξi,n|2 − ε)2

�= 2εN2,

in case |xl,p,j−ξi,n|2 < ε even simpler N2−N1 = N�|xl,p,j − ξi,n|2 + ε ≤

√2εN . Assuming as

above ε = CN−1, this yields N2 −N1 = O(√N) and thus, with respect to the total problem

size n = N3, a total complexity of O(n116 ) floating point operations.

The second approach is a direct generalization of the two-dimensional case, where weexpress the spatial variable in cylinder coordinates and thus reconstructs f for each fixedthird Cartesian coordinate separately, cf. Figure 3.2(right). We use the parameterizationξ = (sinψ1 cosψ2, sinψ1 sinψ2, cosψ1)�, ψ1 ∈ [0, π], ψ2 ∈ [0, 2π), of the sphere S2 and expressx ∈ B in cylindrical coordinates x = (r cosϕ, r sinϕ, z)�, r ∈ [0, 1), z ∈ (−1, 1), ϕ ∈ [0, 2π),which yields

|x− ξ|2 = 1 + r2 + z2 − 2r sinψ1 cos(ψ2 − ϕ)− 2z cosψ1.

Denoting fixed arguments of functions as superscript and skipping the parameters ε and qcompletely, i.e.,

hr,t,z,ψ1(ψ2) = hε,q(1 + r2 + z2 − 2r sinψ1 cos(ψ2)− 2z cosψ1 − t2),

gt,ψ1(ψ2) = t2 · Rf(sinψ1 cosψ2, sinψ1 sinψ2, cosψ1, t),

the approximation (3.2) can be written as a periodic convolution with respect to the angularcomponent

f r,t,ψ1,z(ϕ) =�hr,t,ψ1,z ∗ gψ1,t

�(ϕ) =

� 2π

0hr,t,ψ1,z(ϕ− ψ2)gt,ψ1(ψ2) dψ2,

15

fε(r cosϕ, r sinϕ, z) =1

2π2(1− r2 − z2)

� 2

0

� π

0f r,t,ψ1,z(ϕ) sinψ1 dψ1 dt.

We assume, similar to the two-dimensional case, equidistant measurement times t ∈ [0, 2] andequiangular detector positions ξi,n = (sinψ1,i cosψ2,n, sinψ1,i sinψ2,n, cosψ1,i)� ∈ S2,

tm =2mM

, m = 0, . . . ,M − 1,

ψ1,i =πi

I − 1, i = 0, . . . , I − 1,

ψ2,n =2πnN

, n = 0, . . . , N − 1.

Furthermore, we discretize the spatial variable x ∈ B in cylindrical coordinates x�,p,j =(rj sinϕ�, rj cosϕ�, zp)�,

zp =2p+ 1− P

P, p = 0, . . . , P − 1,

rj =j

J, j = 0, . . . , Jp − 1, Jp =

��1− z2

p · J�,

ϕ� =2π�N

, � = 0, . . . , N − 1,

which leads to the discrete reconstruction formula

fε(x�,p,j) ≈ f j,pl :=

2(1− r2j − z2

p)(I − 1)MN

M−1�

m=0

I−1�

i=0

sinψ1,ifj,m,p,i� (3.14)

f j,m,p,i� :=

N−1�

n=0

hε,q(1 + r2j + z2

p − 2zp cosψ1,i − t2m − 2rj sinψ1,i cosψ2,n−l)t2mRf(ξi,n, tm).

Using the idea of Algorithm 1 for each third spatial coordinate zp individually, this approachis of particular interest if one needs to reconstruct f on a few horizontal planes only. Forfixed j,m, p, i, the inner sum again is a discrete and cyclic convolution and realized by meansof fast Fourier transforms in O(N logN) floating point operations. Taking into account theouter summations over time and angle in (3.14) for all radii and angles, assuming that alldiscretization parameters are of order O(N), and considering the total problem size n = N3

this leads to O(n5/3 logn) floating point operations.

4 Numerical results

The implementation of Algorithm 1 and Algorithm 2 is realized in MATLAB and we use aLenovo Thinkpad T60, 4GByte, Intel(R) Core(TM)2 Duo CPU P8700 2.53GHz for all nu-merical experiments. Besides introductory examples, our interest is the computation time forincreasing discretization parameters and the accuracy with respect to the involved parameters.

4.1 Circular means

We start by some introductory example using the well known Shepp Logan phantom, see Fig-ure 4.1(left). As for the ordinary Radon transform, its circular mean values can be computed

16

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

0 π/2 π 3/2π 2π

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

Figure 4.1: Shepp Logan phantom (left) and circular mean values (middle) together with aprofile for ξ = (0, 1)� (right).

analytically [4], Figure 4.1(middle & right) show the entire data and a profile for ξ = (0, 1)�,respectively.

The input of Algorithm 1 are these spherical means Rf(ξn, tm), n = 0, . . . , N − 1, m =0, . . . ,M − 1, for discretization parameters N = 360 and M = 2000. We reconstruct thephantom f on a polar grid with J = 600 radii and set the regularization parameter toε = 5 · 10−3. Figure 4.2(left & middle) shows the reconstruction on a Cartesian grid and aprofile for x(2) = 0, clearly visible is a smoothing effect on the jump singularities leading alsoto a damping of small details.

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

−1 −0.5 0 0.5 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

4 16 64 256 1024 4096

10−6

10−4

10−2

100

102

Figure 4.2: Reconstruction of the Shepp Logan phantom, on a Cartesian grid (left) and aprofile for x(2) = 0 (middle). Computation times in seconds with respect to thecommon discretization parameter (right).

For the discussion of accuracy and computation time, we consider the function f : R2 → R,

f(x) =�1− |x− a|2

0.62

�3

+

, a =15(1, 1)� ,

We fix the regularization parameter ε = 10−2, choose discretization parameters N = M =J = 2l, l = 1, . . . , 12, and interpolate to a Cartesian grid with L = 2l−1 grid points in eachcoordinate, see Remark 2.1. Figure 4.2(right) shows the computation time for the reconstruc-tion with interpolation together with the estimated order O

�N3

�, where we neglected the

logarithmic term. The total accuracy of Algorithm 1 is measured by

E∞ = max�,j

|f(x�,j)− f jl |, (4.1)

17

and we consider this quantity for fixed parameters N = J = 500 and M = 8000 and adecreasing regularization parameter ε = 2−l, l = 1, . . . , 10. Table 4.1 shows an error behaviorE∞ ≈ 2.8ε until the discretization becomes too coarse at ε ≈ 8

M resulting in an increasingerror.

ε 2−1 2−2 2−3 2−4 2−5

E∞ 7.1 · 10−1 4.9 · 10−1 3.0 · 10−1 1.6 · 10−1 8.6 · 10−2

ε 2−6 2−7 2−8 2−9 2−10

E∞ 4.4 · 10−2 2.2 · 10−2 1.1 · 10−2 5.7 · 10−3 4.9 · 10−2

Table 4.1: Error of the reconstruction with respect to the regularization parameter.

4.2 Spherical means

We start again by some simple test-function as depicted in Figure 4.3(left). The sphericalmeans of this superposition of characteristic functions on balls are computed analytically[4]. Figure 4.3(middle & right) show a equatorial cross section for ξ(3) = 0 and a profile forξ = (1, 0, 0)� of these mean values, respectively.

0 π/2 π 3/2π 2π

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 20

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Figure 4.3: Test-function (left) and spherical mean values values, the equatorial cross sectionfor ξ(3) = 0 (middle) and a profile for ξ = (1, 0, 0)� (right).

The spherical means Rf(ξ, t) are the input of Algorithm 2, they are given on a standardspherical grid ξi1,i2 = (sinψi1 cosϕi2 , sinψi1 cosϕi2 , cosψi1)

�, where

ψi1 =πi1I1

i1 = 0, . . . , I1 − 1,

ϕi2 =2πi2I2

i2 = 0, . . . , I2 − 1,

I = I1I2, and the discretization parameters are I1 = 100, I2 = 200, M = 1500. The remaininginput parameters of Algorithm 2 are set as follows. We choose regularization parameters q = 4and ε = 4 · 10−2, a cut-off degree N = 100, and reconstruct the test-function on a standardspherical grid xj,i1,i2 = rjξi1,i2 with J = 200 radii. The result after interpolating to a Cartesiangrid and a profile for x(2) = x(3) = 0 are shown in Figure 4.4(left & middle).

As for the two-dimensional case, we consider the function f : R3 → R,

f(x) =�1− |x− a|2

0.62

�3

+

, a =15(1, 1, 1)� ,

18

−1 −0.5 0 0.5 1

0

1

2

3

4

25 50 100 200 300

100

101

102

103

Figure 4.4: Reconstruction on a Cartesian grid (left) and a profile (dashed together with theoriginal function) for x(2) = x(3) = 0 (middle). Computation times in seconds (◦)and estimated order O

�N4

�with respect to the common discretization parameter

N (right).

for the discussion of the accuracy and computation time. Figure 4.4(right) shows the es-timated arithmetic complexity O

�N4

�and the actual time usage of Algorithm 2 for fixed

regularization parameters q = 4, ε = 10−2, and increasing discretization parameters N =M = J = I1 = I2 = 25, 50, 75, . . . , 300.

2 4 6 8 10

10−2

10−1

100

10−2

10−1

100

0.7510

0.758

0.756

0.754

0.752

0.750

2 8 32 128 512 2048

10−2

10−1

100

Figure 4.5: Reconstruction error E∞ with respect to the cut-off degree N (left) and the reg-ularization parameters ε (middle) and q (right). In the left diagram, we fixedq = 32 and ε = 1, 0.5, 0.756, 0.1 (◦, ×, +, �). The second diagram shows E∞ forfixed N = 50 and q = 2, 4, 32 (◦, +, ×) together with the rate ε3/2 (solid line).In the right diagram, we consider the error with respect to q and set N = 50 andε = 1, 0.75, 0.756 (◦, +, ×), in addition, the rate q−3/4 is shown (solid line).

Moreover, we consider the total accuracy (4.1) of Algorithm 2 with respect to the cut-off degree N and the regularization parameters ε, q. We fix the discretization parametersI1 = 100, I2 = 200, J = 100, and M = 2000. Figure 4.5(left) shows the reconstruction errorfor fixed regularization parameters q = 32, ε = 1, 0.5, 0.756, 0.1, and increasing cut-off degreeN = 2, . . . , 10. Surprisingly, already a small cut-off degree N = 10 achieves an accuracysmaller than 10−2 for ε = 0.756, 0.1 and we thus fix the cut-off degree N = 50 subsequently.Figure 4.5(middle) shows the reconstruction error for decreasing regularization parameterε = 0.75l, l = 0, . . . , 10 and fixed q = 2, 4, 32. Up to the finally achieved accuracy, dependingmainly on the time discretization M , the error decays at a rate ε3/2. Finally, we consider E∞with respect to the regularization parameter q = 2l, l = 1, . . . , 11, and for fixed N = 50 and

19

ε = 1, 0.75, 0.756 in Figure 4.5(right). Here, the numerical error decays at a rate q−3/4.

5 Summary

We suggested effective discretizations for the recovery of a function from its spherical meansin spherical acquisition geometry. For a total problem size n, the resulting algorithms havecomplexity O(n

32 logn) and O(n

43 ) for the two- and three-dimensional case, respectively.

While this is still slower than the efficient schemes [10] for exact reconstruction formulas,it improves over the best known results [7] for approximate inversion.

Acknowledgment. The authors acknowledge support by the DFG within the project KU2557/1-1 and by the Helmholtz Association within the young investigator group VH-NG-526.

References

[1] M. Ansorg, F. Filbir, W. R. Madych, and R. Seyfried. Summability kernels for circularand spherical mean data. Inverse Problems, 29(1):015002, 2013.

[2] V. A. Antonov, K. V. Kholshevnikov, and V. S. Shaidulin. Estimating the derivative ofthe Legendre polynomial. Vestnik St. Petersburg Univ. Math., 43(4):191–197, 2010.

[3] F. Filbir, R. Hielscher, and W. R. Madych. Reconstruction from circular and sphericalmean data. Appl. Comput. Harmon. Anal., 29(1):111–120, 2010.

[4] T. Gorner and S. Kunis. SMV, Matlab toolbox for computing spherical mean values.http://www.analysis.uos.de/software, 2012.

[5] M. Graf, S. Kunis, and D. Potts. On the computation of nonnegative quadrature weightson the sphere. Appl. Comput. Harmon. Anal., 27:124 – 132, 2009.

[6] M. Haltmeier. A mollification approach for inverting the spherical mean Radon trans-form. SIAM J. Appl. Math., 71(5):1637–1652, 2011.

[7] M. Haltmeier, T. Schuster, and O. Scherzer. Filtered backprojection for thermoacousticcomputed tomography in spherical geometry. Math. Methods Appl. Sci., 28(16):1919–1937, 2005.

[8] J. Keiner, S. Kunis, and D. Potts. Fast summation of radial functions on the sphere.Computing, 78(1):1 – 15, 2006.

[9] J. Keiner, S. Kunis, and D. Potts. Using NFFT 3—a software library for various noneq-uispaced fast Fourier transforms. ACM Trans. Math. Software, 36(4):Art. 19, 30, 2009.

[10] L. Kunyansky. Fast reconstruction algorithms for the thermoacoustic tomography in cer-tain domains with cylindrical or spherical symmetries. Inverse Probl. Imaging, 6(1):111–131, 2012.

[11] G. Szego. Orthogonal polynomials. American Mathematical Society, Providence, R.I.,fourth edition, 1975.

[12] L. N. Trefethen and J. A. C. Weideman. The exponentially convergent trapezoidal rule.SIAM Rev., to appear.

20

Preprint Series DFG-SPP 1324

http://www.dfg-spp1324.de

Reports

[1] R. Ramlau, G. Teschke, and M. Zhariy. A Compressive Landweber Iteration forSolving Ill-Posed Inverse Problems. Preprint 1, DFG-SPP 1324, September 2008.

[2] G. Plonka. The Easy Path Wavelet Transform: A New Adaptive Wavelet Trans-form for Sparse Representation of Two-dimensional Data. Preprint 2, DFG-SPP1324, September 2008.

[3] E. Novak and H. Wozniakowski. Optimal Order of Convergence and (In-)Tractability of Multivariate Approximation of Smooth Functions. Preprint 3,DFG-SPP 1324, October 2008.

[4] M. Espig, L. Grasedyck, and W. Hackbusch. Black Box Low Tensor Rank Ap-proximation Using Fibre-Crosses. Preprint 4, DFG-SPP 1324, October 2008.

[5] T. Bonesky, S. Dahlke, P. Maass, and T. Raasch. Adaptive Wavelet Methodsand Sparsity Reconstruction for Inverse Heat Conduction Problems. Preprint 5,DFG-SPP 1324, January 2009.

[6] E. Novak and H. Wozniakowski. Approximation of Infinitely Differentiable Mul-tivariate Functions Is Intractable. Preprint 6, DFG-SPP 1324, January 2009.

[7] J. Ma and G. Plonka. A Review of Curvelets and Recent Applications. Preprint 7,DFG-SPP 1324, February 2009.

[8] L. Denis, D. A. Lorenz, and D. Trede. Greedy Solution of Ill-Posed Problems:Error Bounds and Exact Inversion. Preprint 8, DFG-SPP 1324, April 2009.

[9] U. Friedrich. A Two Parameter Generalization of Lions’ Nonoverlapping DomainDecomposition Method for Linear Elliptic PDEs. Preprint 9, DFG-SPP 1324,April 2009.

[10] K. Bredies and D. A. Lorenz. Minimization of Non-smooth, Non-convex Func-tionals by Iterative Thresholding. Preprint 10, DFG-SPP 1324, April 2009.

[11] K. Bredies and D. A. Lorenz. Regularization with Non-convex Separable Con-straints. Preprint 11, DFG-SPP 1324, April 2009.

[12] M. Dohler, S. Kunis, and D. Potts. Nonequispaced Hyperbolic Cross Fast FourierTransform. Preprint 12, DFG-SPP 1324, April 2009.

[13] C. Bender. Dual Pricing of Multi-Exercise Options under Volume Constraints.Preprint 13, DFG-SPP 1324, April 2009.

[14] T. Muller-Gronbach and K. Ritter. Variable Subspace Sampling and Multi-levelAlgorithms. Preprint 14, DFG-SPP 1324, May 2009.

[15] G. Plonka, S. Tenorth, and A. Iske. Optimally Sparse Image Representation bythe Easy Path Wavelet Transform. Preprint 15, DFG-SPP 1324, May 2009.

[16] S. Dahlke, E. Novak, and W. Sickel. Optimal Approximation of Elliptic Problemsby Linear and Nonlinear Mappings IV: Errors in L2 and Other Norms. Preprint 16,DFG-SPP 1324, June 2009.

[17] B. Jin, T. Khan, P. Maass, and M. Pidcock. Function Spaces and Optimal Currentsin Impedance Tomography. Preprint 17, DFG-SPP 1324, June 2009.

[18] G. Plonka and J. Ma. Curvelet-Wavelet Regularized Split Bregman Iteration forCompressed Sensing. Preprint 18, DFG-SPP 1324, June 2009.

[19] G. Teschke and C. Borries. Accelerated Projected Steepest Descent Method forNonlinear Inverse Problems with Sparsity Constraints. Preprint 19, DFG-SPP1324, July 2009.

[20] L. Grasedyck. Hierarchical Singular Value Decomposition of Tensors. Preprint 20,DFG-SPP 1324, July 2009.

[21] D. Rudolf. Error Bounds for Computing the Expectation by Markov Chain MonteCarlo. Preprint 21, DFG-SPP 1324, July 2009.

[22] M. Hansen and W. Sickel. Best m-term Approximation and Lizorkin-TriebelSpaces. Preprint 22, DFG-SPP 1324, August 2009.

[23] F.J. Hickernell, T. Muller-Gronbach, B. Niu, and K. Ritter. Multi-level MonteCarlo Algorithms for Infinite-dimensional Integration on RN. Preprint 23, DFG-SPP 1324, August 2009.

[24] S. Dereich and F. Heidenreich. A Multilevel Monte Carlo Algorithm for LevyDriven Stochastic Differential Equations. Preprint 24, DFG-SPP 1324, August2009.

[25] S. Dahlke, M. Fornasier, and T. Raasch. Multilevel Preconditioning for AdaptiveSparse Optimization. Preprint 25, DFG-SPP 1324, August 2009.

[26] S. Dereich. Multilevel Monte Carlo Algorithms for Levy-driven SDEs with Gaus-sian Correction. Preprint 26, DFG-SPP 1324, August 2009.

[27] G. Plonka, S. Tenorth, and D. Rosca. A New Hybrid Method for Image Approx-imation using the Easy Path Wavelet Transform. Preprint 27, DFG-SPP 1324,October 2009.

[28] O. Koch and C. Lubich. Dynamical Low-rank Approximation of Tensors.Preprint 28, DFG-SPP 1324, November 2009.

[29] E. Faou, V. Gradinaru, and C. Lubich. Computing Semi-classical Quantum Dy-namics with Hagedorn Wavepackets. Preprint 29, DFG-SPP 1324, November2009.

[30] D. Conte and C. Lubich. An Error Analysis of the Multi-configuration Time-dependent Hartree Method of Quantum Dynamics. Preprint 30, DFG-SPP 1324,November 2009.

[31] C. E. Powell and E. Ullmann. Preconditioning Stochastic Galerkin Saddle PointProblems. Preprint 31, DFG-SPP 1324, November 2009.

[32] O. G. Ernst and E. Ullmann. Stochastic Galerkin Matrices. Preprint 32, DFG-SPP1324, November 2009.

[33] F. Lindner and R. L. Schilling. Weak Order for the Discretization of the StochasticHeat Equation Driven by Impulsive Noise. Preprint 33, DFG-SPP 1324, November2009.

[34] L. Kammerer and S. Kunis. On the Stability of the Hyperbolic Cross DiscreteFourier Transform. Preprint 34, DFG-SPP 1324, December 2009.

[35] P. Cerejeiras, M. Ferreira, U. Kahler, and G. Teschke. Inversion of the noisy Radontransform on SO(3) by Gabor frames and sparse recovery principles. Preprint 35,DFG-SPP 1324, January 2010.

[36] T. Jahnke and T. Udrescu. Solving Chemical Master Equations by AdaptiveWavelet Compression. Preprint 36, DFG-SPP 1324, January 2010.

[37] P. Kittipoom, G. Kutyniok, and W.-Q Lim. Irregular Shearlet Frames: Geometryand Approximation Properties. Preprint 37, DFG-SPP 1324, February 2010.

[38] G. Kutyniok and W.-Q Lim. Compactly Supported Shearlets are OptimallySparse. Preprint 38, DFG-SPP 1324, February 2010.

[39] M. Hansen and W. Sickel. Best m-Term Approximation and Tensor Products ofSobolev and Besov Spaces – the Case of Non-compact Embeddings. Preprint 39,DFG-SPP 1324, March 2010.

[40] B. Niu, F.J. Hickernell, T. Muller-Gronbach, and K. Ritter. Deterministic Multi-level Algorithms for Infinite-dimensional Integration on RN. Preprint 40, DFG-SPP 1324, March 2010.

[41] P. Kittipoom, G. Kutyniok, and W.-Q Lim. Construction of Compactly SupportedShearlet Frames. Preprint 41, DFG-SPP 1324, March 2010.

[42] C. Bender and J. Steiner. Error Criteria for Numerical Solutions ofBackward SDEs. Preprint 42, DFG-SPP 1324, April 2010.

[43] L. Grasedyck. Polynomial Approximation in Hierarchical Tucker Format byVector-Tensorization. Preprint 43, DFG-SPP 1324, April 2010.

[44] M. Hansen und W. Sickel. Best m-Term Approximation and Sobolev-Besov Spacesof Dominating Mixed Smoothness - the Case of Compact Embeddings. Preprint 44,DFG-SPP 1324, April 2010.

[45] P. Binev, W. Dahmen, and P. Lamby. Fast High-Dimensional Approximation withSparse Occupancy Trees. Preprint 45, DFG-SPP 1324, May 2010.

[46] J. Ballani and L. Grasedyck. A Projection Method to Solve Linear Systems inTensor Format. Preprint 46, DFG-SPP 1324, May 2010.

[47] P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova, and P. Wojtaszczyk.Convergence Rates for Greedy Algorithms in Reduced Basis Methods. Preprint 47,DFG-SPP 1324, May 2010.

[48] S. Kestler and K. Urban. Adaptive Wavelet Methods on Unbounded Domains.Preprint 48, DFG-SPP 1324, June 2010.

[49] H. Yserentant. The Mixed Regularity of Electronic Wave Functions Multiplied byExplicit Correlation Factors. Preprint 49, DFG-SPP 1324, June 2010.

[50] H. Yserentant. On the Complexity of the Electronic Schrodinger Equation.Preprint 50, DFG-SPP 1324, June 2010.

[51] M. Guillemard and A. Iske. Curvature Analysis of Frequency Modulated Manifoldsin Dimensionality Reduction. Preprint 51, DFG-SPP 1324, June 2010.

[52] E. Herrholz and G. Teschke. Compressive Sensing Principles and Iterative SparseRecovery for Inverse and Ill-Posed Problems. Preprint 52, DFG-SPP 1324, July2010.

[53] L. Kammerer, S. Kunis, and D. Potts. Interpolation Lattices for Hyperbolic CrossTrigonometric Polynomials. Preprint 53, DFG-SPP 1324, July 2010.

[54] G. Kutyniok and W.-Q Lim. Shearlets on Bounded Domains. Preprint 54, DFG-SPP 1324, July 2010.

[55] A. Zeiser. Wavelet Approximation in Weighted Sobolev Spaces of Mixed Orderwith Applications to the Electronic Schrodinger Equation. Preprint 55, DFG-SPP1324, July 2010.

[56] G. Kutyniok, J. Lemvig, and W.-Q Lim. Compactly Supported Shearlets.Preprint 56, DFG-SPP 1324, July 2010.

[57] A. Zeiser. On the Optimality of the Inexact Inverse Iteration Coupled with Adap-tive Finite Element Methods. Preprint 57, DFG-SPP 1324, July 2010.

[58] S. Jokar. Sparse Recovery and Kronecker Products. Preprint 58, DFG-SPP 1324,August 2010.

[59] T. Aboiyar, E. H. Georgoulis, and A. Iske. Adaptive ADER Methods Using Kernel-Based Polyharmonic Spline WENO Reconstruction. Preprint 59, DFG-SPP 1324,August 2010.

[60] O. G. Ernst, A. Mugler, H.-J. Starkloff, and E. Ullmann. On the Convergence ofGeneralized Polynomial Chaos Expansions. Preprint 60, DFG-SPP 1324, August2010.

[61] S. Holtz, T. Rohwedder, and R. Schneider. On Manifolds of Tensors of FixedTT-Rank. Preprint 61, DFG-SPP 1324, September 2010.

[62] J. Ballani, L. Grasedyck, and M. Kluge. Black Box Approximation of Tensors inHierarchical Tucker Format. Preprint 62, DFG-SPP 1324, October 2010.

[63] M. Hansen. On Tensor Products of Quasi-Banach Spaces. Preprint 63, DFG-SPP1324, October 2010.

[64] S. Dahlke, G. Steidl, and G. Teschke. Shearlet Coorbit Spaces: Compactly Sup-ported Analyzing Shearlets, Traces and Embeddings. Preprint 64, DFG-SPP 1324,October 2010.

[65] W. Hackbusch. Tensorisation of Vectors and their Efficient Convolution.Preprint 65, DFG-SPP 1324, November 2010.

[66] P. A. Cioica, S. Dahlke, S. Kinzel, F. Lindner, T. Raasch, K. Ritter, and R. L.Schilling. Spatial Besov Regularity for Stochastic Partial Differential Equationson Lipschitz Domains. Preprint 66, DFG-SPP 1324, November 2010.

[67] E. Novak and H. Wozniakowski. On the Power of Function Values for the Ap-proximation Problem in Various Settings. Preprint 67, DFG-SPP 1324, November2010.

[68] A. Hinrichs, E. Novak, and H. Wozniakowski. The Curse of Dimensionality forMonotone and Convex Functions of Many Variables. Preprint 68, DFG-SPP 1324,November 2010.

[69] G. Kutyniok and W.-Q Lim. Image Separation Using Shearlets. Preprint 69,DFG-SPP 1324, November 2010.

[70] B. Jin and P. Maass. An Analysis of Electrical Impedance Tomography withApplications to Tikhonov Regularization. Preprint 70, DFG-SPP 1324, December2010.

[71] S. Holtz, T. Rohwedder, and R. Schneider. The Alternating Linear Scheme forTensor Optimisation in the TT Format. Preprint 71, DFG-SPP 1324, December2010.

[72] T. Muller-Gronbach and K. Ritter. A Local Refinement Strategy for ConstructiveQuantization of Scalar SDEs. Preprint 72, DFG-SPP 1324, December 2010.

[73] T. Rohwedder and R. Schneider. An Analysis for the DIIS Acceleration Methodused in Quantum Chemistry Calculations. Preprint 73, DFG-SPP 1324, December2010.

[74] C. Bender and J. Steiner. Least-Squares Monte Carlo for Backward SDEs.Preprint 74, DFG-SPP 1324, December 2010.

[75] C. Bender. Primal and Dual Pricing of Multiple Exercise Options in ContinuousTime. Preprint 75, DFG-SPP 1324, December 2010.

[76] H. Harbrecht, M. Peters, and R. Schneider. On the Low-rank Approximation bythe Pivoted Cholesky Decomposition. Preprint 76, DFG-SPP 1324, December2010.

[77] P. A. Cioica, S. Dahlke, N. Dohring, S. Kinzel, F. Lindner, T. Raasch, K. Ritter,and R. L. Schilling. Adaptive Wavelet Methods for Elliptic Stochastic PartialDifferential Equations. Preprint 77, DFG-SPP 1324, January 2011.

[78] G. Plonka, S. Tenorth, and A. Iske. Optimal Representation of Piecewise HolderSmooth Bivariate Functions by the Easy Path Wavelet Transform. Preprint 78,DFG-SPP 1324, January 2011.

[79] A. Mugler and H.-J. Starkloff. On Elliptic Partial Differential Equations withRandom Coefficients. Preprint 79, DFG-SPP 1324, January 2011.

[80] T. Muller-Gronbach, K. Ritter, and L. Yaroslavtseva. A Derandomization of theEuler Scheme for Scalar Stochastic Differential Equations. Preprint 80, DFG-SPP1324, January 2011.

[81] W. Dahmen, C. Huang, C. Schwab, and G. Welper. Adaptive Petrov-Galerkinmethods for first order transport equations. Preprint 81, DFG-SPP 1324, January2011.

[82] K. Grella and C. Schwab. Sparse Tensor Spherical Harmonics Approximation inRadiative Transfer. Preprint 82, DFG-SPP 1324, January 2011.

[83] D.A. Lorenz, S. Schiffler, and D. Trede. Beyond Convergence Rates: Exact In-version With Tikhonov Regularization With Sparsity Constraints. Preprint 83,DFG-SPP 1324, January 2011.

[84] S. Dereich, M. Scheutzow, and R. Schottstedt. Constructive quantization: Ap-proximation by empirical measures. Preprint 84, DFG-SPP 1324, January 2011.

[85] S. Dahlke and W. Sickel. On Besov Regularity of Solutions to Nonlinear EllipticPartial Differential Equations. Preprint 85, DFG-SPP 1324, January 2011.

[86] S. Dahlke, U. Friedrich, P. Maass, T. Raasch, and R.A. Ressel. An adaptivewavelet method for parameter identification problems in parabolic partial differ-ential equations. Preprint 86, DFG-SPP 1324, January 2011.

[87] A. Cohen, W. Dahmen, and G. Welper. Adaptivity and Variational Stabilizationfor Convection-Diffusion Equations. Preprint 87, DFG-SPP 1324, January 2011.

[88] T. Jahnke. On Reduced Models for the Chemical Master Equation. Preprint 88,DFG-SPP 1324, January 2011.

[89] P. Binev, W. Dahmen, R. DeVore, P. Lamby, D. Savu, and R. Sharpley. Com-pressed Sensing and Electron Microscopy. Preprint 89, DFG-SPP 1324, March2011.

[90] P. Binev, F. Blanco-Silva, D. Blom, W. Dahmen, P. Lamby, R. Sharpley, andT. Vogt. High Quality Image Formation by Nonlocal Means Applied to High-Angle Annular Dark Field Scanning Transmission Electron Microscopy (HAADF-STEM). Preprint 90, DFG-SPP 1324, March 2011.

[91] R. A. Ressel. A Parameter Identification Problem for a Nonlinear Parabolic Dif-ferential Equation. Preprint 91, DFG-SPP 1324, May 2011.

[92] G. Kutyniok. Data Separation by Sparse Representations. Preprint 92, DFG-SPP1324, May 2011.

[93] M. A. Davenport, M. F. Duarte, Y. C. Eldar, and G. Kutyniok. Introduction toCompressed Sensing. Preprint 93, DFG-SPP 1324, May 2011.

[94] H.-C. Kreusler and H. Yserentant. The Mixed Regularity of Electronic WaveFunctions in Fractional Order and Weighted Sobolev Spaces. Preprint 94, DFG-SPP 1324, June 2011.

[95] E. Ullmann, H. C. Elman, and O. G. Ernst. Efficient Iterative Solvers forStochastic Galerkin Discretizations of Log-Transformed Random Diffusion Prob-lems. Preprint 95, DFG-SPP 1324, June 2011.

[96] S. Kunis and I. Melzer. On the Butterfly Sparse Fourier Transform. Preprint 96,DFG-SPP 1324, June 2011.

[97] T. Rohwedder. The Continuous Coupled Cluster Formulation for the ElectronicSchrodinger Equation. Preprint 97, DFG-SPP 1324, June 2011.

[98] T. Rohwedder and R. Schneider. Error Estimates for the Coupled Cluster Method.Preprint 98, DFG-SPP 1324, June 2011.

[99] P. A. Cioica and S. Dahlke. Spatial Besov Regularity for Semilinear StochasticPartial Differential Equations on Bounded Lipschitz Domains. Preprint 99, DFG-SPP 1324, July 2011.

[100] L. Grasedyck and W. Hackbusch. An Introduction to Hierarchical (H-) Rank andTT-Rank of Tensors with Examples. Preprint 100, DFG-SPP 1324, August 2011.

[101] N. Chegini, S. Dahlke, U. Friedrich, and R. Stevenson. Piecewise Tensor ProductWavelet Bases by Extensions and Approximation Rates. Preprint 101, DFG-SPP1324, September 2011.

[102] S. Dahlke, P. Oswald, and T. Raasch. A Note on Quarkonial Systems and Multi-level Partition of Unity Methods. Preprint 102, DFG-SPP 1324, September 2011.

[103] A. Uschmajew. Local Convergence of the Alternating Least Squares AlgorithmFor Canonical Tensor Approximation. Preprint 103, DFG-SPP 1324, September2011.

[104] S. Kvaal. Multiconfigurational time-dependent Hartree method for describing par-ticle loss due to absorbing boundary conditions. Preprint 104, DFG-SPP 1324,September 2011.

[105] M. Guillemard and A. Iske. On Groupoid C*-Algebras, Persistent Homology andTime-Frequency Analysis. Preprint 105, DFG-SPP 1324, September 2011.

[106] A. Hinrichs, E. Novak, and H. Wozniakowski. Discontinuous information in theworst case and randomized settings. Preprint 106, DFG-SPP 1324, September2011.

[107] M. Espig, W. Hackbusch, A. Litvinenko, H. Matthies, and E. Zander. EfficientAnalysis of High Dimensional Data in Tensor Formats. Preprint 107, DFG-SPP1324, September 2011.

[108] M. Espig, W. Hackbusch, S. Handschuh, and R. Schneider. Optimization Problemsin Contracted Tensor Networks. Preprint 108, DFG-SPP 1324, October 2011.

[109] S. Dereich, T. Muller-Gronbach, and K. Ritter. On the Complexity of ComputingQuadrature Formulas for SDEs. Preprint 109, DFG-SPP 1324, October 2011.

[110] D. Belomestny. Solving optimal stopping problems by empirical dual optimizationand penalization. Preprint 110, DFG-SPP 1324, November 2011.

[111] D. Belomestny and J. Schoenmakers. Multilevel dual approach for pricing Amer-ican style derivatives. Preprint 111, DFG-SPP 1324, November 2011.

[112] T. Rohwedder and A. Uschmajew. Local convergence of alternating schemes foroptimization of convex problems in the TT format. Preprint 112, DFG-SPP 1324,December 2011.

[113] T. Gorner, R. Hielscher, and S. Kunis. Efficient and accurate computation ofspherical mean values at scattered center points. Preprint 113, DFG-SPP 1324,December 2011.

[114] Y. Dong, T. Gorner, and S. Kunis. An iterative reconstruction scheme for pho-toacoustic imaging. Preprint 114, DFG-SPP 1324, December 2011.

[115] L. Kammerer. Reconstructing hyperbolic cross trigonometric polynomials by sam-pling along generated sets. Preprint 115, DFG-SPP 1324, February 2012.

[116] H. Chen and R. Schneider. Numerical analysis of augmented plane waves methodsfor full-potential electronic structure calculations. Preprint 116, DFG-SPP 1324,February 2012.

[117] J. Ma, G. Plonka, and M.Y. Hussaini. Compressive Video Sampling with Ap-proximate Message Passing Decoding. Preprint 117, DFG-SPP 1324, February2012.

[118] D. Heinen and G. Plonka. Wavelet shrinkage on paths for scattered data denoising.Preprint 118, DFG-SPP 1324, February 2012.

[119] T. Jahnke and M. Kreim. Error bound for piecewise deterministic processes mod-eling stochastic reaction systems. Preprint 119, DFG-SPP 1324, March 2012.

[120] C. Bender and J. Steiner. A-posteriori estimates for backward SDEs. Preprint120, DFG-SPP 1324, April 2012.

[121] M. Espig, W. Hackbusch, A. Litvinenkoy, H.G. Matthiesy, and P. Wahnert. Eff-cient low-rank approximation of the stochastic Galerkin matrix in tensor formats.Preprint 121, DFG-SPP 1324, May 2012.

[122] O. Bokanowski, J. Garcke, M. Griebel, and I. Klompmaker. An adaptive sparsegrid semi-Lagrangian scheme for first order Hamilton-Jacobi Bellman equations.Preprint 122, DFG-SPP 1324, June 2012.

[123] A. Mugler and H.-J. Starkloff. On the convergence of the stochastic Galerkinmethod for random elliptic partial differential equations. Preprint 123, DFG-SPP1324, June 2012.

[124] P.A. Cioica, S. Dahlke, N. Dohring, U. Friedrich, S. Kinzel, F. Lindner, T. Raasch,K. Ritter, and R.L. Schilling. On the convergence analysis of Rothe’s method.Preprint 124, DFG-SPP 1324, July 2012.

[125] P. Binev, A. Cohen, W. Dahmen, and R. DeVore. Classification Algorithms usingAdaptive Partitioning. Preprint 125, DFG-SPP 1324, July 2012.

[126] C. Lubich, T. Rohwedder, R. Schneider, and B. Vandereycken. Dynamical approx-imation of hierarchical Tucker and Tensor-Train tensors. Preprint 126, DFG-SPP1324, July 2012.

[127] M. Kovacs, S. Larsson, and K. Urban. On Wavelet-Galerkin methods for semilinearparabolic equations with additive noise. Preprint 127, DFG-SPP 1324, August2012.

[128] M. Bachmayr, H. Chen, and R. Schneider. Numerical analysis of Gaussian ap-proximations in quantum chemistry. Preprint 128, DFG-SPP 1324, August 2012.

[129] D. Rudolf. Explicit error bounds for Markov chain Monte Carlo. Preprint 129,DFG-SPP 1324, August 2012.

[130] P.A. Cioica, K.-H. Kim, K. Lee, and F. Lindner. On the Lq(Lp)-regularity andBesov smoothness of stochastic parabolic equations on bounded Lipschitz domains.Preprint 130, DFG-SPP 1324, December 2012.

[131] M. Hansen. n−term Approximation Rates and Besov Regularity for Elliptic PDEson Polyhedral Domains. Preprint 131, DFG-SPP 1324, December 2012.

[132] R. E. Bank and H. Yserentant. On the H1 -stability of the L2 -projection ontofinite element spaces. Preprint 132, DFG-SPP 1324, December 2012.

[133] M. Gnewuch, S. Mayer, and K. Ritter. On Weighted Hilbert Spaces and Inte-gration of Functions of Infinitely Many Variables. Preprint 133, DFG-SPP 1324,December 2012.

[134] D. Crisan, J. Diehl, P.K. Friz, and H. Oberhauser. Robust Filtering: CorrelatedNoise and Multidimensional Observation. Preprint 134, DFG-SPP 1324, January2013.

[135] Wolfgang Dahmen, Christian Plesken, and Gerrit Welper. Double Greedy Algo-rithms: Reduced Basis Methods for Transport Dominated Problems. Preprint135, DFG-SPP 1324, February 2013.

[136] Aicke Hinrichs, Erich Novak, Mario Ullrich, and Henryk Wozniakowski. The Curseof Dimensionality for Numerical Integration of Smooth Functions. Preprint 136,DFG-SPP 1324, February 2013.

[137] Markus Bachmayr, Wolfgang Dahmen, Ronald DeVore, and Lars Grasedyck. Ap-proximation of High-Dimensional Rank One Tensors. Preprint 137, DFG-SPP1324, March 2013.

[138] Markus Bachmayr and Wolfgang Dahmen. Adaptive Near-Optimal Rank TensorApproximation for High-Dimensional Operator Equations. Preprint 138, DFG-SPP 1324, April 2013.

[139] Felix Lindner. Singular Behavior of the Solution to the Stochastic Heat Equationon a Polygonal Domain. Preprint 139, DFG-SPP 1324, May 2013.

[140] Stephan Dahlke, Dominik Lellek, Shiu Hong Lui, and Rob Stevenson. AdaptiveWavelet Schwarz Methods for the Navier-Stokes Equation. Preprint 140, DFG-SPP 1324, May 2013.

[141] Jonas Ballani and Lars Grasedyck. Tree Adaptive Approximation in the Hierar-chical Tensor Format. Preprint 141, DFG-SPP 1324, June 2013.

[142] Harry Yserentant. A short theory of the Rayleigh-Ritz method. Preprint 142,DFG-SPP 1324, July 2013.

[143] M. Hefter and K. Ritter. On Embeddings of Weighted Tensor Product HilbertSpaces. Preprint 143, DFG-SPP 1324, August 2013.

[144] M. Altmayer and A. Neuenkirch. Multilevel Monte Carlo Quadrature of Discon-tinuous Payoffs in the Generalized Heston Model using Malliavin Integration byParts. Preprint 144, DFG-SPP 1324, August 2013.

[145] L. Kammerer, D. Potts, and T. Volkmer. Approximation of multivariate functionsby trigonometric polynomials based on rank-1 lattice sampling. Preprint 145,DFG-SPP 1324, September 2013.

[146] C. Bender, N. Schweizer, and J. Zhuo. A primal-dual algorithm for BSDEs.Preprint 146, DFG-SPP 1324, October 2013.

[147] D. Rudolf. Hit-and-run for numerical integration. Preprint 147, DFG-SPP 1324,October 2013.

[148] D. Rudolf and M. Ullrich. Positivity of hit-and-run and related algorithms.Preprint 148, DFG-SPP 1324, October 2013.

[149] L. Grasedyck, M. Kluge, and S. Kramer. Alternating Directions Fitting (ADF) ofHierarchical Low Rank Tensors. Preprint 149, DFG-SPP 1324, October 2013.

[150] F. Filbir, S. Kunis, and R. Seyfried. Effective discretization of direct reconstructionschemes for photoacoustic imaging in spherical geometries. Preprint 150, DFG-SPP 1324, November 2013.