AN EFFICIENT MULTIGRID CALCULATION OF THE FAR FIELD

AN EFFICIENT MULTIGRID CALCULATION OF THE FAR FIELDMAP FOR HELMHOLTZ AND SCHRODINGER EQUATIONS

SIEGFRIED COOLS∗, BRAM REPS∗† , AND WIM VANROOSE∗

Abstract. In this paper we present a new highly efficient calculation method for the far fieldamplitude patterns that arise from scattering problems governed by the d-dimensional Helmholtzequation and, by extension, Schrodinger’s equation. The method is based upon a reformulationof the classical real-valued Green’s function integral expression for the far field amplitude to anintegral over a complex contour. On this complex contour (or manifold, in multiple dimensions)the scattered wave can be calculated very efficiently using an iterative multigrid method as a solverfor the discretized scattered wave system, resulting in a fast and scalable calculation of the far fieldmap. The so-called ‘full complex contour’ approach is successfully validated on model problems intwo and three spatial dimensions, and multigrid convergence results are provided to demonstrate thewavenumber scalability and overall performance of the proposed method.

Key words. Far field map, Helmholtz equation, Schrodinger equation, multigrid, complexcontour, cross sections.

AMS subject classifications.

DOI.

1. Introduction. Scattering problems are of key importance in many areas ofscience and engineering since they carry information about an object of interest overlarge distances, remote from the given target. Consequently, ever since their originalstatement a variety of applications of scattering problems have arisen in many differentscientific subdomains. In chemistry and quantum physics, for example, virtuallyall knowledge about the inner workings of a molecule has been obtained throughscattering experiments [36]. Similarly, in many real-life electromagnetic or acousticscattering problems information about a far away object is obtained through radar orsonar [16], intrinsically requiring the solution of 2D or 3D wave equations.

New state-of-the-art experimental techniques measure the full 4π angular depen-dency of multiple particles escaping from a molecular reaction [35]. Through theseexperiments, the cross section of reactions involving multiple particles can be detectedin coincidence. Many experiments are being planned at facilities such as e.g. theDESY Free-electron laser (FLASH) in Hamburg or the Linac Coherent Light Source(LCLS) in Stanford. The accurate prediction of the corresponding amplitudes start-ing from first principles requires the use of efficient numerical methods to solve thehigh-dimensional Helmholtz or Schrodinger problems, which can scale up to 6D or 9Din this context. Indeed, after discretization one generally obtains a large, sparse andindefinite system of equations in the unknown scattered wave. Direct solution of thissystem is usually prohibited due to the massive size of the problem in higher spatialdimensions.

Preconditioned Krylov subspace methods are currently among the most efficientnumerical algorithms for the solution of general high-dimensional equations that arisefrom discretizations of partial differential equations, as they exploit the sparsity struc-ture of the discretized system of equations and allow for reasonably good scaling withrespect to the number of unknowns. Indeed, preconditioned Krylov subspace methodsare able to solve some symmetric positive definite systems in only O(n) iterations,

∗Dept. Math. & Comp. Sc., University of Antwerp, Middelheimlaan 1, 2020 Antwerp, Belgium†Intel R© ExaScience Lab, Kapeldreef 75, B-3001 Leuven, Belgium

1

2 S. COOLS, B. REPS AND W. VANROOSE

where n is the number of unknowns in the system [49]. However, scattering problemsare often described by Helmholtz equations, which after discretization lead to highlyindefinite linear systems that are notoriously hard to solve using the current gener-ation of iterative methods. Moreover, the highly efficient iterative multigrid method[10, 12, 13, 47, 48] is known to break down when applied to these type of problems[18, 24].

Over the past decade significant research has been performed on the constructionof good preconditioners for Helmholtz problems. Recent work includes the wave-rayapproach [11], the idea of separation of variables [38], algebraic multilevel methods[9], multigrid deflation [44, 45] and a transformation of the Helmholtz equation intoan advection-diffusion-reaction problem [26]. In 2004 the Complex Shifted Laplacian(CSL) was proposed by Erlangga, Vuik and Oosterlee [21, 22, 23] as an effectiveKrylov subspace method preconditioner for Helmholtz problems. The key idea be-hind this preconditioner is to formulate a perturbed Helmholtz problem that includesa complex-valued wave number. Given a sufficiently large complex shift, this im-plies a damping in the problem, thus making the perturbed problem solvable usingmultigrid in contrast to the original Helmholtz problem with real-valued wavenum-bers. By introducing the complex shifted problem as a preconditioner, the resultingKrylov method profits from advantageous spectral properties, leading to a reasonableconvergence rate. The concept of CSL has been further generalized in a variety ofpapers among which [2, 19, 20, 37]. The choice of a sufficiently large complex shiftparameter, commonly denoted by β, is vital to the stability of the multigrid solutionmethod. The general rule of thumb for the choice of the complex shift suggestedin the literature is β = 0.5 [22, 39]. This experimental guideline was recently con-firmed through a rigorous LFA analysis for the constant-k problem in [17], provingthe multigrid correction scheme to be generally stable for shifts β larger than 0.5.

Recently a variation on the Complex Shifted Laplacian scheme by the name ofComplex Stretched Grid (CSG) was proposed in [40, 41], introducing a complex-valued grid distance instead of a complex-valued wavenumber in the preconditioningsystem. It was furthermore shown in [39] that the resulting Krylov subspace methodhas very similar convergence properties. Indeed, the CSG preconditioner can beshown to be generally equivalent to the CSL scheme, and thus can be solved equallyefficiently using multigrid. However, despite its overall qualitative performance, theCSL/CSG preconditioned Krylov subspace solution method suffers from a significantwavenumber dependency of the convergence rate [39]. Additionally, convergence ratesquickly deteriorate in the presence of evanescent waves in the Helmholtz solution.

This paper focuses on calculating the far field map resulting from Helmholtz andSchrodinger type scattering problems [3], which yields a 360 representation of thescattered wave amplitude at large distances from the object of interest. The calcu-lation of the far field map can typically be considered a two-step process. First aHelmholtz problem with absorbing boundary conditions is solved on a finite numer-ical box covering the object of interest. In the second step a volume integral overthe Green’s function and the numerical solution is calculated to obtain the angular-dependent far field amplitude map. This strategy was successfully applied to cal-culate impact ionization in hydrogen [42] and double photo-ionization in molecules[51, 52] described by the Schrodinger equation, which in this case translates into a6-dimensional Helmholtz problem.

In this paper we propose a new method for the calculation of the far field map.The method reformulates the integral over the Green’s function on a complex con-

MULTIGRID CALCULATION OF THE FAR FIELD MAP 3

tour. This modified approach requires the solution of the Helmholtz equation on acomplex contour. It is shown that the latter problem is equivalent to a ComplexShifted Laplacian problem that can be solved very efficiently by a multigrid method.To validate our approach, the method is successfully illustrated on both 2D and 3DHelmholtz and Schrodinger equations for a variety of discretization levels. The ab-sorbing boundary conditions used in this paper are based on the principle of ExteriorComplex Scaling (ECS) that was introduced in the 1970’s [1, 4, 46], and is nowadaysfrequently used in scattering applications. This method is equivalent to a complexstretching implementation of Perfectly Matched Layers (PML) [7, 15].

The outline of the article is the following. In Section 2 we introduce the nota-tion and terminology that will be used throughout the text. Additionally, we brieflyillustrate the classical calculation of the far field map for Helmholtz type scatteringproblems. In the second part of this section we introduce an alternative way of calcu-lating the far field mapping based upon a reformulation of the integral over a complexcontour, for which the corresponding Helmholtz system is very efficiently solved it-eratively. The new technique is validated and convergence results are shown for avariety of Helmholtz type model scattering problems in both 2D and 3D in Section3, where it is found that the method allows for a very fast and scalable far field mapcalculation. In Section 4, the method is validated and extensively tested on severalSchrodinger type model problems in two and three spatial dimensions respectively.Benchmark problems include quantum-mechanical model problem situations in whichsingle, double and/or triple ionization occur. Finally, along with a discussion on thetopic, conclusions are drawn in Section 5.

2. The Helmholtz equation and the far field map. In this section we in-troduce the general notation used throughout the text and we illustrate the classicalderivation of the far field scattered wave solution and calculation of its amplitudefrom a general Helmholtz type scattering problem.

2.1. Derivation of the far field mapping. The Helmholtz equation is a simplemathematical representation of the physics behind a wave scattering at an objectdefined on a compact support area O located within a domain Ω ⊂ Rd. The equationis given by (

−∆− k2(x))u(x) = f(x), x ∈ Rd, (2.1)

with dimension d ≥ 1, where ∆ is the Laplace operator, f designates the right handside or source term, and k is the (spatially dependent) wavenumber, representing thematerial properties inside the object of interest. Indeed, the wavenumber function kis defined as

k(x) =

k1(x), for x ∈ O,k0, for x ∈ Ω \O, (2.2)

where k0 ∈ R is a scalar constant denoting the wave number outside the object ofinterest. The scattered wave solution is given by the unknown function u. Throughoutthe text we will use the following convenient notation

χ(x) :=k2(x)− k2

0

k20

, (2.3)

such that k2(x) = k20 (1 + χ(x)). Note that the function χ is trivially zero outside

the object of interest O where the space-dependent wavenumber k(x) is reduced to


k0. Defining the incoming wave as uin(x) = eik0η·x, where η is the unit vector thatdefines the direction, the right-hand side is typically given by f(x) = k2

0χ(x)uin(x).Reformulating (2.1), we obtain(

−∆− k2(x))u(x) = k2

0χ(x)uin(x), x ∈ Rd. (2.4)

This equation is typically formulated on the domain Ω with outgoing wave boundaryconditions on ∂Ω, and can in principle be solved on a numerical box (i.e. a discretizedsubset of ΩN ⊂ Ω) covering the support of χ, with absorbing boundary conditionsalong all edges. Let us assume that the numerical solution satisfying (2.4) on this boxhas been calculated and is denoted by uN .

In order to calculate the far field scattered wave pattern the above equation isreorganized as(

−∆− k20

)u(x) = k2

0χ(x) (uin(x) + u(x)) , x ∈ Rd. (2.5)

Note that we can replace the function u(x) in the right hand side of this equation withthe numerical solution uN (x) obtained from equation (2.4). In doing so, the aboveequation becomes an inhomogeneous Helmholtz equation with constant wavenumber(

−∆− k20

)u(x) = g(x), x ∈ Rd, (2.6)

where the short notation g(x) := k20χ(x)(uin(x)+uN (x)) is introduced for readability

and notational convenience. It holds that g(x) = 0 for x ∈ Rd\O. The above equationcan easily be solved analytically using the Helmholtz Green’s function G(x,x′), i.e.

u(x) =

∫RdG(x,x′) g(x′) dx′, x ∈ Rd. (2.7)

Since the function g is only non-zero inside the numerical box that was used to solveequation (2.4), the above integral over Rd can be replaced by a finite integral over Ω

u(x) =

∫Ω

G(x,x′)k20χ(x′)

(uin(x′) + uN (x′)

)dx′, x ∈ Rd. (2.8)

In practice, this expression allows us to calculate the scattered wave solution u in anypoint x ∈ Rd \ ΩN outside the numerical box, using only the information x ∈ ΩN

inside the numerical box.Given the integral expression (2.8), the asymptotic form of the Green’s function

can be used to compute the far field map of the scattered wave u. In the followingthis will be illustrated for a 2D model example where the Green’s function is givenexplicitly by

G(x,x′) =i

4H

(1)0 (k0|x− x′|), x,x′ ∈ Rd. (2.9)

where i represents the imaginary unit and H(1)0 is the 0-th order Hankel function of

the first kind. An analogous derivation can be performed in 3D, where we mentionfor completeness that the Green’s function is given by

G(x,x′) =eik0|x−x

′|

4π|x− x′| , x,x′ ∈ Rd. (2.10)


To calculate the angular dependence of the far field map, the direction of the unitvector α is introduced that is in 2D defined by a single angle α with the positivehorizontal axis, i.e. α = (cosα, sinα)T . Rewriting the spatial coordinates x in polar

coordinates as x = (ρ cosα, ρ sinα)T

the asymptotic form of the Green’s function for|x| 1 (ρ→∞) is given by

i

4H

(1)0 (k0|x− x′|) =

i

4

√2

πe−iπ/4

1√k0ρ

eik0ρe−ik0x′ cosα−ik0y′ sinα

=i

4

√2

πe−iπ/4

1√k0ρ

eik0ρe−ik0x′·α (2.11)

where we have used the fact that the Hankel function H(1)0 is asymptotically given by

H(1)0 (r) =

√2

πrexp

(i(r − π

4)), r ∈ R, r 1. (2.12)

This leads to the following asymptotic form of the 2D scattered wave solution

u(ρ, α) =i

4

√2

πe−iπ/4

eik0ρ√k0ρ

∫Ω

e−ik0x′·αg(x′) dx′, (2.13)

for ρ → ∞. The above expression is called the 2D far field wave pattern of u, withthe integral being denoted as the far field (amplitude) map

F (α) =

∫Ω

e−ik0x′·αg(x′) dx′. (2.14)

The value of the integral depends only on the direction α (or, in 2D, on the angleα) and the wave number k0. Expression (2.13) readily extends to the d-dimensionalcase, where it holds more generally that

limρ→∞

u(ρ,α) = D(ρ)F (α), α ∈ Rd, (2.15)

for a function D(ρ) which is known explicitly and a far field map F (α) given by (2.14).Note that this far field map is in fact a Fourier integral of the function g.

Summarizing, we conclude that the calculation of the far field wave pattern ofthe scattered wave u consists of two main steps. First, one has to solve a Helmholtzequation with a spatially dependent wave number on a numerical box with absorbingboundary conditions as in (2.4). Once the numerical solution is obtained, it is followedby the calculation of a Fourier integral (2.14) over the aforementioned numericaldomain. The main computational bottleneck of this calculation generally lies withinthe first step, since it requires an efficient and computationally inexpensive methodfor the solution of the high dimensional indefinite Helmholtz system with absorbingboundary conditions.

The statement of the far field map presented in this section relies on the factthat the object of interest, represented by the function χ, is compactly supported. Inparticular, this is used when computing the numerical solution uN to equation (2.4)on a bounded numerical box that covers the support of χ. The above reasoning canhowever be readily extended to the more general class of analytical object functions χthat vanish at infinity, i.e. χ ∈ V where V = f : Rd → R analytical | ∀ε > 0, ∃K ⊂


<z

=z

γ

Fig. 2.1. Schematic representation of the complex contour for the far field integral calculationillustrated in 1D. The full line represents the real domain Ω, the dotted and dashed lines representthe subareas Z1 = xeiγ : x ∈ Ω ⊂ R and Z2 = beiθ : b ∈ ∂Ω, θ ∈ [0, γ] of the complex contourrespectively.

Rd compact, ∀x ∈ Rd \ K : |f(x)| < ε. Indeed, due to the existence of smoothbump functions [29, 33], functions with compact support can be shown to be densewithin the space of functions that vanish at infinity. Consequently, every analyticalfunction χ ∈ V can be arbitrarily closely approximated by a series of compactlysupported functions χnn. This in turn implies that the corresponding solutionsuNn n on a limited computational box can be arbitrarily close to the solution of theHelmholtz equation generated with the analytical object of interest χ ∈ V . Intuitively,this means that if χ is analytical but sufficiently small everywhere outside O, thecomputational domain may be retricted to a numerical box covering O as if χ wascompactly supported. Hence, the far field map (2.14) is well-defined for analyticalfunctions χ that vanish at infinity. This observation will prove particularly useful inthe next section.

2.2. Calculation on a complex contour. In this section we will illustrate howthe integral (2.14) can be reformulated on a complex contour and why this is usefulin terms of numerical computation. First, we note that the integral can be split intoa sum of two contributions F (α) = I1 + I2 with

I1 =

∫Ω

e−ik0x·αχ(x)uin(x)dx and I2 =

∫Ω

e−ik0x·αχ(x)uN (x)dx. (2.16)

Calculation of first integral I1 is generally easy, since it only requires the expressionfor the incoming wave, which is known analytically. The second integral howeverrequires the solution of the Helmholtz equation on the numerical box, which is knownto be notoriously hard to obtain using iterative methods. In particular, the highlyefficient multigrid solution method is unable to solve these type of indefinite Helmholtzequations due to instability in both the coarse grid correction and relaxation scheme.This divergence is due to close-to-zero eigenvalues of the discretized operator on someintermediate multigrid levels [18].

However, if both u and χ are analytical functions the integral can be calculatedover a complex contour rather than the real axis as follows. Let us define a complexcontour along the rotated real domain Z1 = z ∈ C | z = xeiγ : x ∈ Ω, where γ isa fixed rotation angle, followed by the curved segment Z2 = z ∈ C | z = beiθ : b ∈∂Ω, 0 ≤ θ ≤ γ, as presented schematically on Figure 2.1. The integral I2 can thenbe written as

I2 =

∫Z1

e−ik0z·αχ(z)uN (z)dz +

∫Z2

e−ik0z·αχ(z)uN (z)dz. (2.17)

The second term in the above expression however vanishes, as the function χ is per


<z

=z

γ θ

θ

Fig. 2.2. Schematic representation of a real grid with ECS boundaries vs. a full complex grid,illustrated in 1D.

definition zero everywhere outside the object of interest O, thus notably in all pointsz ∈ Z2. Hence one ultimately obtains

I2 =

∫Z1

e−ik0z·αχ(z)uN (z)dz =

∫Ω

e−ik0eiγx·αχ(xeiγ)uN (xeiγ)eiγ dx. (2.18)

Note that for 0 < γ < π/2 the exponential of xeiγ is increasing in all directions. Atthe same time, the scattered wave solution uN , which consists of outgoing waves onthe complex domain Z1, is decaying in all directions. Additionally, the function χ ispresumed to have a bounded support (or vanish at infinity, see Section 2.1) makingthe above integral calculable on a limited numerical domain.

Expression (2.18) for the integral I2 indicates that the far field map can (atleast partially) be computed over the full complex contour Z1, i.e. a rotation of theoriginal real domain Ω over an angle γ in all spatial dimensions. The advantage of thisapproach is that we only need the value of uN evaluated along this complex contour;thus we now have to solve the Helmholtz equation (2.4) on a complex contour. Onthis contour it is a damped equation which is much easier to solve than the Helmholtzequation along the real axis. Indeed, given a sufficiently large value of γ, it has beenshown in the literature [23, 40] that the multigrid scheme is a very effective solutionmethod for the Helmholtz equation on a complex domain.

2.3. Solving the Helmholtz equation on a complex contour. We nowshow that the Helmholtz problem on the complex domain Z1 is very similar to acomplex shifted Laplacian system [21], and can as such be solved efficiently using amultigrid solver. Consider the Helmholtz problem with a complex shifted wavenumber(

−∆− (1 + iβ)k2(x))u(x) = f(x) (2.19)

with Dirichlet boundary conditions u(x|∂Ω) = 0 and a complex shift parameter β ∈ R.After finite difference discretization on a d-dimensional Cartesian grid with fixed griddistance h in every spatial dimension, one typically obtains a linear system

−(

1

h2L+ (1 + iβ)k2

)uh = bh (2.20)

where L is the matrix operator expressing the stencil structure of the Laplacian. In2D, for example L = kron(I,diag(−1, 2,−1)+kron(diag(−1, 2,−1), I), where the sizeof L intrinsically depends on h. After dividing both sides in linear system (2.20) by(1 + iβ), we immediately obtain the equivalent system

−(

1

(1 + iβ)h2L+ k2

)uh =

bh1 + iβ

, (2.21)


θ (rad.) π/8 π/7 π/6 π/5 π/4 π/3γ (deg.) 7.5 8.5 9.9 11.8 14.6 19.1

Table 2.1ECS angle θ and corresponding rotation angle γ for the full complex grid. Values based on (2.24).

which is identical to the discretization of the original Laplacian with grid distanceh =√

1 + iβ h. This scheme is known as Complex Stretched Grid, and it was shownin [40] to yield exactly the same convergence as the Complex Shifted Laplacian whenboth are used as a preconditioner for a general Krylov method.

It is known that problem (2.20), or equivalently (2.21), can be solved efficientlywith multigrid for values of the complex shift β > 0.5, see [17, 21]. Note that thisrequirement is based on a multigrid cycle with standard weighted Jacobi or Gauss-Seidel smoothing. This rule of thumb can easily be translated into an angle γ for thecomplex scaling. Writing (1 + iβ) = ρ exp(iϕ) with ρ =

√1 + β2 and ϕ = arctanβ,

one readily obtains

h =√

1 + iβ h =√ρ exp(iϕ/2)h (2.22)

Consequently, as the shift β is required to be larger than 0.5, the grid rotation angleγ = ϕ/2 must satisfy

γ >arctan(0.5)

2= 0.2318 ≈ 13.28 (2.23)

Note that when substituting the standard multigrid relaxation schemes like ω-Jacobior Gauss-Seidel by a more robust iterative scheme like e.g. GMRES(m), the rotationangle γ may be chosen even smaller, up to a minimum of approximately 9.5 (see[41]).

In this paper we have chosen to link the grid rotation angle γ to the standardECS absorbing layer angle θ, see Figure 2.2. This is in no way imperious for thefunctionality of the method, but it appears quite naturally from the fact that bothangles perturb (part of) the grid into the complex plane. Suppose the ECS boundarylayer measures one quarter of the length of the entire real domain in every spatialdimension, which is a common choice, we readily derive that the relation between therotation angle γ and the ECS angle θ is given by

γ = arctan

(sin θ

2 + cos θ

). (2.24)

Table 2.1 shows some standard values of the ECS angle θ and corresponding γ valuesaccording to (2.24). Note that for a multigrid scheme with ω-Jacobi or Gauss-Seidelsmoothing to be stable, θ should be chosen no smaller than π/4, according to (2.23).Using the more efficient GMRES(3) method as a smoother replacement, the ECS anglecan be chosen somewhat smaller, i.e. an angle around θ = π/6 suffices to guarantee astable multigrid solution.

3. Numerical results for 2D and 3D Helmholtz problems. In this sec-tion, we validate the theoretical result presented above by a number of numericalexperiments in both two and three spatial dimensions. It will be shown that the pro-posed method results in a very fast and wavenumber independent solution method


Fig. 3.1. Top: 2D object of interest |χ(x)| given by (3.1). Mid: solution to the Helmholtzproblem (2.4) (in modulus) on a nx × ny = 256 × 256 grid solved using LU factorization (left) ona double ECS contour with θ = π/4, and using a series of multigrid V-cycles (right) with ω-Jacobismoother on the corresponding full complex contour up to a residual reduction tolerance of 1e-6.Bottom: resulting 2D Far field maps F (α) calculated following (2.14). Normalized errors with respectto a nx ×ny = 1024× 1024 grid benchmark Far field map solution Fex(α): ‖Fre −Fex‖2/‖Fex‖2 =9.37e-5 (left), ‖Fco − Fex‖2/‖Fex‖2 = 1.39e-4 (right), ‖Fre − Fco‖2/‖Fex‖2 = 1.77e-4.

for the scattered wave system, hence yielding a remarkably efficient method for thecalculation of the far field map.

The model problem used throughout this section is a Helmholtz equation of theform (2.4) with k2(x) = k2

0(1 + χ(x)). The equation is discretized on a nd-pointuniform mesh covering a square numerical domain Ω = [−20, 20]d using second orderfinite differences. In the 2D case the space-dependent wavenumber is defined as

k20 χ(x, y) = −1/5

(e−(x2+(y−4)2) + e−(x2+(y+4)2)

), (x, y) ∈ [−20, 20]2, (3.1)

i.e. the object of interest takes the form of two circular point-like objects with mass


nx × ny × nz 163 323 643 1283 2563

k0

1/410 (10) 9 (59) 9 (560) 9 (4456) 9 (35165)

0.24 0.20 0.21 0.20 0.20

1/212 (12) 10 (63) 10 (611) 10 (4937) 9 (35405)

0.31 0.24 0.22 0.23 0.21

17 (8) 13 (83) 11 (691) 10 (4899) 10 (38975)0.13 0.32 0.27 0.24 0.24

22 (4) 8 (54) 13 (809) 11 (5418) 10 (38051)0.01 0.14 0.33 0.27 0.24

41 (3) 2 (17) 7 (457) 13 (6337) 11 (41848)0.01 0.01 0.12 0.33 0.26

Table 3.13D Helmholtz problem (2.4) solved on a full complex grid with θ = π/6 using a series of

multigrid V(1,1)-cycles with GMRES(3) smoother up to residual reduction tolerance 1e-6. Displayedare the number of V-cycle iterations, number of work units and average convergence factor forvarious wavenumbers k0 and different discretizations. 1 WU is calibrated as the cost of 1 V(1,1)-cycle on the 163-points grid k0 = 1/4 problem. Discretizations respecting the k0h < 0.625 criterionfor a minimum of 10 grid points per wavelength are indicated by a bold typesetting.

concentrated at the Cartesian coordinates (0,−4) and (0, 4) (see Figure 3.1). For the3D model problem, the following straightforward extension of the object is used

k20 χ(x, y, z) = −1/5

(e−(x2+(y−4)2+z2) + e−(x2+(y+4)2+z2)

), (x, y, z) ∈ [−20, 20]3,

(3.2)representing two spherical point-like objects in 3D space (see Figure 3.2). The incom-ing wave scattering at the given object is defined by

uin(x) = eik0η·x, x ∈ Ω, (3.3)

where η is the unit vector in the x-direction.Figure 3.1 illustrates the theoretical result presented in Section 2. The above 2D

Helmholtz model problem with wavenumber given by (3.1) and k0 = 1 is solved foruN using respectively a standard LU factorization method on the real domain Ω withECS complex boundary layers (θ = π/4) along the domain boundary ∂Ω, and a seriesof multigrid V(1,1)-cycles with ω-Jacobi smoothing on the full complex domain (γ ≈14.6) up to residual tolerance of 1e-6. The standard multigrid intergrid operatorsused in this work are bilinear interpolation and full weighting restriction. The moduliof the wavenumber function χ (top) and the resulting solution uN (mid) are shown onFigure 3.1 for both methods. Note how the solution uN on the full complex contour isindeed heavily damped compared to the solution on the real domain. Consequently,using the numerical solution uN , the 2D far field map integral (2.14) can be calculatedusing any numerical integration scheme over the real or complex domain respectively.The resulting far field map F (α) is shown as a function of the angle α on Figure 3.1(bottom). One observes that the mapping is indeed quasi-identical when calculatedon the real and complex domain, conform with the theoretical results. The normalizeddifference between both far field maps does not exceed 0.177% (in norm), which iseffectively of the same order of magnitude as the normalized error. However, the


nx × ny × nz 163 323 643 1283 2563

k0

1/48 (11) 6 (52) 5 (384) 5 (3190) 5 (25241)1.93e-9 1.77e-9 2.46e-9 2.68e-9 3.00e-9

1/29 (12) 8 (68) 6 (452) 6 (3392) 6 (30215)1.27e-8 1.87e-9 3.37e-9 1.30e-9 1.25e-9

15 (8) 9 (68) 8 (572) 7 (4013) 6 (30747)

1.33e-8 1.51e-8 4.07e-9 1.76e-9 3.66e-9

21 (5) 5 (43) 9 (600) 8 (4456) 7 (36367)

5.99e-13 1.18e-8 1.91e-8 5.28e-9 2.67e-9

41 (4) 1 (18) 5 (357) 9 (5038) 8 (39038)

8.90e-20 2.86e-13 5.19e-9 1.97e-8 4.65e-9

Table 3.23D Helmholtz problem (2.4) solved on a full complex grid with θ = π/6 using an FMG cycle

with GMRES(3) smoother up to residual reduction tolerance of 1e-6. Displayed are the number ofV-cycle iterations on the designated finest grid, number of work units and resulting residual normfor various wavenumbers k0 and different discretizations. 1 WU is calibrated as the cost of 1 V(1,1)-cycle on the 163-points grid k0 = 1/4 problem. Discretizations respecting the k0h < 0.625 criterionfor a minimum of 10 grid points per wavelength indicated by a bold typesetting.

computational cost of the real-domain method for calculation of the far field mapis reduced significantly by the ability to apply multigrid to the equivalent complexscaled problem.

In Table 3.1 convergence results are shown for the solution of the 3D scatteredwave equation (2.4) using a series of multigrid V(1,1)-cycles on various grid sizes. Notethat the multigrid method scales perfectly as a function of the number of grid points,as doubling the number of grid points in every spatial dimension does not increasethe number of V-cycles required to reach a fixed residual tolerance of 1e-6. This is astandard result from multigrid theory. Additionally and more importantly, remarkablek-scalability is measured for the multigrid solution method on the complex contour.Indeed, the multigrid convergence factor (and thus the corresponding work unit loadrequired to solve the problem up to a given tolerance) is almost fully independent ofthe wavenumber k0, as can be observed from the table. From a physical-numericalpoint of view it is only meaningful to consider discretizations satisfying the k0h <0.625 criterion for a minimum of 10 grid points per wavelength, cf. [6], for which thecorresponding values are designated in Table 3.1 by a bold typesetting.

Ultimately, the computed scattered wave solution on the complex domain canagain be used to calculate the far field integral (2.14). The resulting 3D far fieldmapping for the model problem with k0 = 1 is plotted in Figure 3.2. The left handside panel shows an iso-surface visualization of the 3D object of interest χ(x) given by(3.2). On the right panel a spherical projection of the resulting 3D far field mappingis shown. The color hue indicates the value of the far field amplitude in each outgoingdirection.

Note that the calculation of the scattered wave solution can be optimized evenfurther by considering the Full Multigrid (FMG) scheme. This is a nested iterationof standard V-cycles, where on each level a series of V(1,1)-cycles is used to approx-imately solve the error equation and supply a corrected initial guess for a finer levelby interpolating the corresponding coarse grid solution.


Fig. 3.2. Left: 3D object of interest |χ(x)| given by (3.2). Shown are the |χ(x)| = c isosurfacesfor c = 1e-1, 1e-2, 1e-10, 1e-100 and 1e-300. Right: 3D Far field map, resulting from Helmholtzproblem (2.4) with k0 = 1 solved on a nx × ny × nz = 64× 64× 64 full complex grid with θ = π/6(γ ≈ 9.9) using a series of multigrid V-cycles with GMRES(3) smoother up to residual reductiontolerance 1e-6.

nx × ny × nz 163 323 643 1283 2563

CPU time 0.20 s. 0.78 s. 6.24 s. 53.3 s. 462 s.‖r‖2 3.3e-5 7.9e-5 2.7e-5 1.1e-5 4.6e-6

Table 3.33D Helmholtz problem (2.4) with wavenumber k0 = 1 solved on a full complex grid with θ = π/6

using one FMG-cycle with GMRES(3) smoother. Displayed are the CPU time (in s.) and theresulting residual norm for various discretizations. System specifications: Intel R©CoreTM i7-2720QM2.20GHz CPU, 6MB Cache, 8GB RAM.

Table 3.2 shows convergence results for the solution of the 3D scattered waveequation (2.4) using an FMG scheme. The setting is comparable to that of Table3.1, as a residual reduction tolerance of 10−6 is imposed for each wavenumber and atevery level of the FMG cycle, yielding a fine nx×ny×nz = 2563 grid residual of orderof magnitude 10−9. Note that the number of V-cycles performed on each level in theFMG cycle is decaying as a function of the growing grid size due to the increasinglyaccurate initial guess, resulting in a relatively small number of V-cycles (five to nine)to be performed on the finest level. Consequently, the number of work units (and thusthe computational time) required to reach the designated residual reduction toleranceis significantly lower than the work unit load of the pure V-cycle scheme displayed inTable 3.1.

Timing and residual results from a standard FMG sweep performing only oneV(1,1)-cycle on each level on the 3D Helmholtz scattering problem with a moderatewavenumber k0 = 1 are shown in Table 3.3 for different discretizations. Note thattimings were generated using a basic non-parallelized Matlab code, using only a singlethread on a simple midrange personal computer (system specifications: see captionTable 3.3) and taking less than 8 minutes to solve a 3D Helmholtz problem with 256gridpoints in every spatial dimension.

4. Application to Schrodinger equations. This section illustrates the ap-plication of the proposed complex contour method to high-dimensional Schrodingerequations that are used to describe quantum mechanical scattering problems. The


d-dimensional time-independent Schrodinger equation for a system with unit mass isgiven by (

−1

2∆ + V (x)− E

)ψ(x) = φ(x), for x ∈ Rd, (4.1)

where ∆ is the d-dimensional Laplacian, V (x) is the potential, ψ is the wave functionand φ is the right-hand side, which is often related to the ground state of the system.Depending on the total energy E, the above system allows for scattering solutions, inwhich case the equation can be reformulated as a Helmholtz equation of the form

(−∆− k2(x))ψ(x) = 2φ(x), for x ∈ Rd, (4.2)

where the spatially dependent wavenumber k(x) is defined by k2(x) = 2(E − V (x)).The experimental observations from this type of quantum mechanical systems aretypically far field maps of the solution [51, 52]. Indeed, in an experimental setup,detectors are typically placed at large distances from the object compared to thesize of the system. These detectors consequently measure the probability of parti-cles escaping from the system in certain directions. In many quantum mechanicalsystems the potential V (x) is an analytical function, which suggests analyticity ofthe wavenumber k(x) in the above Helmholtz equation. Additionally, the potential isoften exponentially decaying in function of the spatial coordinates. Hence, for thesetypes of problems, the wavenumber naturally satisfies all conditions for the use of theproposed complex contour method to efficiently calculate the corresponding far fieldmap.

In the paragraphs below, we first discuss a 2D model problem in which singleand double ionization occur, corresponding to waves describing respectively a singleparticle or two particles escaping from the quantum mechanical system. The firstleads to very localized evanescent waves that propagate along the boundaries of thecomputational domain, while the latter gives rise to waves traversing the full domain.The corresponding 2D Schrodinger problem will be solved on a discretized numericaldomain for a range of energies E below and above the double ionization threshold.For each level of energy, we extract the single and double ionization cross sections,which correspond to probabilities of particles escaping from the system, with the helpof an integral of the Green’s function over the numerical box. The cross sections arecalculated using both a traditional method, where the Helmholtz equation is solved ona standard ECS-bounded grid [31], and the new complex contour method, introducedin Section 2.2.

The main purpose of the calculations in Sections 4.1 to 4.4 is to validate the resultsobtained by the complex contour method when applied to Schrodinger’s equation. InSection 4.5 the multigrid performance for solving the 2D complex-valued scatteredwave system is benchmarked. Note, however, that these 2D problems essentially donot require a multigrid solver, since a direct sparse solver performs well for theserelatively small-size problems.

In Section 4.6, we illustrate the convergence of a multigrid solver on a 3D Schro-dinger equation, with energies that allow for triple ionization as well as double andsingle ionization. The previous attempts to solve these problems with the help of thecomplex shifted Laplacian as a preconditioner to a general Krylov method showed anotable deterioration in the convergence behavior in function of the total energy E[40, 8]. However, it will be shown that the new complex contour method, which allowsmultigrid to be used as a solver, performs well for these problems.


Although the benchmark problems considered in this paper mainly use modelpotentials, we believe that the calculations presented below are an important steptowards the application of the method on realistic quantum mechanical systems.

4.1. Cross sections of the 2D Schrodinger problem. Our primary aim is tovalidate the applicability of the new complex contour method on the 2D Schrodingerequation describing a quantum mechanical scattering problem. This problem orig-inates from the expansion of a 6D scattering problem in spherical harmonics, see[3, 51, 8], in which each particle is expressed in terms of its spherical coordinates.The resulting partial waves fit the two-dimensional Schrodinger equation(

−1

2∆ + V1(x) + V2(y) + V12(x, y)− E

)u(x, y) = φ(x, y), x, y ≥ 0, (4.3)

with boundary conditionsu(x, 0) = 0 for x ≥ 0

u(0, y) = 0 for y ≥ 0

outgoing for x→∞ or y →∞,(4.4)

where ∆ represents the 2D Laplacian, V1(x) and V2(y) are the one-body potentials,V12(x, y) is a two-body potential and E is the total energy of the system. Sincethe arguments x and y are in fact radial coordinates in the partial wave expansion,homogeneous Dirichlet boundary conditions are implied at the x = 0 and y = 0boundaries. The potentials V1, V2 and V12 are generally analytical functions thatdecay (exponentially) as the radial coordinates x and y become large.

Depending on the strength of the one-body potentials V1 and V2, the problemallows for so-called single ionization waves, which are localized evanescent waves thatpropagate along the edges of the domain. We refer the reader to Sections 4.2 and 4.3for a more detailed physical clarification. We expound on the situation with a strongpotential V1 in the x-direction; the case with a strong V2 potential is completelyanalogous. If V1 is strong enough, there exists a one-dimensional eigenstate φn(x)for every negative eigenvalue λn < 0, characterized by a one-dimensional Helmholtzequation (

−1

2

d2

dx2+ V1(x)

)φn(x) = λn φn(x), for x ≥ 0. (4.5)

Note that φn(0) = 0 and φn(x→∞) = 0.The far field maps of this system are then again Green’s integrals over the solu-

tion, see [31]. Indeed, the single ionization amplitude or cross section sn(E), whichrepresents the total number of single ionized particles, is given by

sn(E) =

∫Ω

φkn(x)φn(y) (φ(x, y)− V12(x, y)u(x, y)) dx dy, (4.6)

where kn =√

2(E − λn), φn is a one-body eigenstate which is the solution of equation(4.5) with a corresponding eigenvalue λn, and the function φkn is a regular, normalizedsolution of the homogeneous Helmholtz equation(

−1

2

d2

dx2+ V1(x)− 1

2k2

)φk = 0, (4.7)


where k = kn and φkn is normalized with 1/√kn.

Similarly, the double ionization cross section f(k1, k2), which measures the totalnumber of double ionized particles, is defined by the integral

f(k1, k2) =

∫Ω

φk1(x)φk2(y) (φ(x, y)− V12(x, y)u(x, y)) dx dy, x, y ≥ 0, (4.8)

where both φk1(x) and φk2(y) are solutions to (4.7), with k1 =√

2E sin(α) andk2 =

√2E cos(α) respectively, i.e. such that k2

1 +k22 = 2E. The total double ionization

cross section is defined as the integral

σtot(E) =

∫ E

0

σ(√

2ε,√

2(E − ε)) dε, (4.9)

where

σ(k1, k2) =8π2

k20

1

k1k2|f(k1, k2)|2. (4.10)

The above integral expressions are obtained through a reorganization similar to theone performed in Section 2, see (2.5)–(2.6). For example, to calculate the singleionization cross section, equation (4.3) is reorganized as(−1

2∆ + V1(x)− E

)u(x, y) = φ(x, y)−(V2(y)+V12(x, y))u(x, y), x, y ≥ 0. (4.11)

Since the left hand side is separable, the corresponding Green’s function allows us towrite

u(x, y) =

∫Ω

G(x, y|x′, y′)(φ(x′, y′)− (V2(y′) + V12(x′, y′))uN (x′, y′)

)dx′ dy′.

(4.12)Using the asymptotic form of the Green’s function, the above ultimately results inintegral formulation (4.6). The double ionization integral expression (4.8) can bederived in a similar way.

4.2. Spectral Properties. To obtain more insight in the numerical solution ofthe two-dimensional Schrodinger problem, we briefly discuss the spectral propertiesof the discretized Schrodinger operator. The discretized 2D Hamiltonian H2d corre-sponding to equation (4.3) can be written as a sum of two Kronecker products and atwo-body potential, i.e.

H2d = H1d ⊗ I + I ⊗H1d + V12(x, y), (4.13)

where H1d = −1/2∆ + Vi (i = 1, 2) is the one-dimensional Hamiltonian, discretizedusing finite differences. When the two-body potential V12(x, y) is weak relative to theone-body potentials, the eigenvalues of the 2D Hamiltonian can be approximated by

λ2d ≈ λ1di + λ1d

j , 1 ≤ i, j,≤ n. (4.14)

Hence, to form a better understanding of the spectral properties of H2d, let us firstconsider the eigenvalues of H1d. After discretization using second order finite differ-ences, the one-dimensional Hamiltonian can be written as a tridiagonal matrix, wherethe stencil

1

h2

[−1 2 −1

](4.15)


−2 0 2 4 6 8 10−8

−6

−4

−2

0

real

imag

−2 0 2 4 6 8 10−8

−6

−4

−2

0

real

imag

−2 0 2 4 6 8−8

−6

−4

−2

0

real

imag

−2 0 2 4 6 8−8

−6

−4

−2

0

real

imag

Fig. 4.1. Spectrum (close-up) of the discretized 1D (top) and 2D (bottom) Hamiltonian (4.14),i.e. wih E = 0. Left: standard discretization with real-valued grid distance h. Right: complexcontour discretization with complex-valued grid distance h = e−iγh, with γ = π/6. The spectrum isrotated down into the complex plane over 2γ = π/3.

approximates the second derivatives, and the potential is a diagonal matrix evaluatedin the grid points. The spectrum of H1d closely resembles the spectrum of the Lapla-cian (−1/2)∆, however the presence of the potential modifies the smallest eigenvalues.The resulting spectrum is shown on the top left panel of Figure 4.1, which presents aclose-up of the eigenvalues near the origin. A single negative eigenvalue λ1d

0 = −1.0215can be observed, which is due to the attractive potential. The remaining spectrumconsists of a series of positive eigenvalues which are located along the positive realaxis. The top right panel of Figure 4.1 shows the eigenvalues of H1d, discretized alonga complex-valued contour, i.e. the real grid rotated by eiγ . The grid distance used isnow h = heiγ , which results in the following stencil for the second derivative

e−2iγ 1

h2

[−1 2 −1

]. (4.16)

This implies that the spectrum of the Laplacian is rotated down into the complex planeby an angle 2γ. Figure 4.1 shows that most of the eigenvalues are rotated downwardsover 2γ, with the exception of the bound state eigenvalue λ1d

0 , which remains on thenegative real axis. These spectral properties are well known in the physics literature,see for example [34].

We consequently turn to the two-dimensional problem setting, where the eigenval-ues of the Hamiltonian H2d are the sums of the one-dimensional operator eigenvaluesH1d ⊗ I + I ⊗ H1d. The resulting eigenvalues are shown on the bottom two panels


of Figure 4.1. Again, the eigenvalues of the 2D Hamiltonian are rotated down inthe complex plane when the system is discretized along a complex-valued contour.In 2D, an isolated eigenvalue appears around λ1d

0 + λ1d0 = −2.043, and two series of

eigenvalues emerge from the real axis: a first branch of eigenvalues starting at −1.012,which originates from the sum of the negative eigenvalue λ1d

0 of the first 1D Hamil-tonian combined with all the positive eigenvalues of the second 1D Hamiltonian; anda second series of eigenvalues starting at the origin, originating from the sums of thepositive eigenvalues of both one-dimensional Hamiltonians.

4.3. Solution types: single and double ionization. The Schrodinger equa-tion (4.3) can be written shortly as (H−E)u = φ, where H = (−1/2)∆+V1 +V2 +V12

and E is the total energy of the system. Depending on this energy E, the Schrodingersystem has different types of solutions. In this section we briefly expound on thephysical interpretation of these solution types, using a model problem example. Thissection may prove less interesting to readers who are primarily interested in the com-putational aspects of the solution, and as such, can be skipped at will.

When the energy is larger that the smallest (negative) eigenvalue of the Hamilto-nian, i.e. λ0 < E, one-body eigenstate solutions to (4.5) arise. These eigenstates canbe combined into separable waves of the form

un(x, y) = φn(x) exp(ikny), (4.17)

with kn =√

2(E − λn). This expression effectively is a solution to the Schrodingerproblem (4.3) in the region where x is small and y → ∞. Indeed, when y is large,the potentials V2 and V12 are negligibly small and the resulting Schrodinger equationbecomes separable in variables, where φn(x) and exp(ikny) are the solutions of theseparated operators respectively. An analogous argument holds for the case when V2

is large, in which case there exist evanescent waves of the form

un(x, y) = φn(y) exp(iknx). (4.18)

These separable waves are solutions of the Schrodinger system for x→∞ and y small,and can be derived similarly to (4.17). When both V1 and V2 are large, (4.17) and(4.18) exist simultaneously. Note that we can associate such a separable wave witheach eigenstate φn of equation (4.5) that corresponds to a negative eigenvalue of theHamiltonian. These localized waves are called single ionization waves in the physicsliterature, since they correspond to a quantum mechanical system in which a singleparticle is ionized. Single ionization waves are present in the solution as soon as theenergy E is above the λ0 threshold. When there is a second eigenstate with negativeenergy, say φ1 with λ1, an additional single ionization wave appears in the problemas soon as E > λ1. We refer to the specialized literature for a detailed discussion ofthe ionization process, see [36, 25].

The left panel of Figure 4.2 shows the solution to (4.3) for a total energy E = −0.5.The model problem under consideration fits equation (4.3), with a right-hand sidegiven by φ(x, y) = exp(−3(x+ y)2). The one-body potentials are defined by V1(x) =−4.5 exp(−x2) and V2(y) = −4.5 exp(−y2), yielding an eigenstate of equation (4.5)with energy λ0 = −1.0215. The two-body potential equals V12(x, y) = 2 exp(−(x +y)2). The equation is discretized on a [0, 20]2 domain using 500 grid points in everyspatial dimension. An ECS absorbing layer consisting of an additional 250 grid pointsdamps the outgoing waves along the right and top edges of the domain. Note fromFigure 4.2 how the single ionization eigenstate solutions given by (4.17)-(4.18) appearalong the edges of the domain.


Single ionization Double ionization

Fig. 4.2. Scattered wave solutions u(x, y) to model problem (4.3). Model specifications: seeaccompanying text. Left: solution for energy E = −0.5 where only single ionization occurs. Right:solution for energy E = 1.5 where both double and single ionization occur. Single ionization wavesare localized solely along the edges of the domain (left), while double ionization waves appear bothalong the edges and in the middle of the domain (right).

When the total energy E exceeds zero, i.e. E > 0, additional scattering solutionsto the Schrodinger equation appear when both x → ∞ and y → ∞. If all potentialsare asymptotically zero, equation (4.3) boils down to a Helmholtz equation with wavenumber k =

√2E for x → ∞ and y → ∞. The resulting waves are known as

double ionization waves and physically correspond to the simultaneous ejection oftwo particles from the quantum mechanical system. In the far field, i.e. for x → ∞and y → ∞, double ionization waves behave as ei

√2E√x2+y2 . At the same time, it

is still possible to have single ionization, since we have E > λ0. The right panel ofFigure 4.2 shows the solution to the above model problem with a total energy E = 1.5,clearly displaying the double ionization waves. Note how single and double ionizationwaves coexist in the solution. Additionally, one observes that double ionization wavesoscillate faster in the x- or y-direction than a free wave with wave number k =

√2E,

since kn ≥ k.

4.4. Validation of the complex contour method on the 2D Schrodingerproblem. In physical experiments, the total number of single ionized or double ion-ized particles is typically observed for a range of energy levels E using advanced detec-tors. These observations are made far away from the object and effectively measurethe far field amplitudes of the solutions. The outcome of this type of experiments canbe predicted by calculating the single (4.6) and double ionization (4.8) cross sections,using the numerical solution of equation (4.3), see [31]. Indeed, in order to calculatethese cross sections, the numerical solution uN (x, y) to (4.3) is required, which isgenerally hard to obtain, especially in higher spatial dimensions. However, since thepotentials V1, V2 and V12 are analytical functions, the integrals for single and doubleionization can be calculated along a complex contour rather than the classical realdomain, in analogy to the discussion in Section 2.1. Consequently, one requires thescattering solution of equation (4.3) on a complex contour, which is generally mucheasier to compute numerically.

In the following, we calculate the single and double ionization cross sections for anumber of energies E between −1 and 3 using both the classical real-valued discretiza-tion and the complex contour approach. The corresponding 2D scattering problems


Fig. 4.3. Scattered wave solutions u(x, y) to model problem (4.3) for E = 1. Left: solutionon a real grid with ECS absorbing boundary layer (θ = π/7 ≈ 25). Right: solution on a straightcomplex scaled contour (γ ≈ 8.5), resulting in a damped scattered wave solution.

(4.3) are solved on a numerical domain Ω = [0, 15]2 covered by a finite difference gridconsisting of 300 grid points in every spatial dimension. Additionally, an ECS absorb-ing boundary layer starting at x = 15 and y = 15 respectively is used to implementthe outgoing boundary conditions, adding an additional 150 grid points in every spa-tial dimension. The ECS angle is θ = π/7 ≈ 25.7. Alternatively, a complex scaledgrid with an overall complex rotation angle γ ≈ 8.5 is used. Solutions uN (x, y) to(4.3) for a total energy E = 1 on both the classical real-valued grid and the complexcontour are presented on Figure 4.3. Note how the solution is damped when evaluatedalong the complex contour.

Figure 4.4 shows the rate of single and double ionization as a function of the totalenergy E. The dashed and dotted lines represent the single and double ionizationamplitudes calculated using the traditional real-valued method with ECS absorbingboundary conditions [31]. The solid line is the total cross section and is calculatedusing the optical theorem, see [36]. One observes that single ionization occurs startingfrom E > −1.0215. Double ionization only occurs when E > 0, and comprises only afraction of the single ionization cross section (cf. Figure 4.2). Note how the energy ofthe single ionized bound states rises as the total energy grows, and remains presenteven when E > 0. Results obtained using the complex contour approach are indicatedby the N and • symbols on Figure 4.4. In this case, the Schrodinger equation (4.3)is first solved on a complex contour, yielding a damped solution as shown by Figure4.3 (right panel), followed by the calculation of the integrals (4.6) and (4.8) along thiscomplex contour. Identical results are obtained by both calculation methods, thusvalidating the applicability of the complex contour approach on Schrodinger-typeproblems.

4.5. Multigrid performance on the 2D Schrodinger problem. In this sec-tion, we benchmark the performance of multigrid as a solver for the 2D Schrodingerscattering problem on a complex-valued grid. It appears that the multigrid conver-gence rate critically depends on the value of the total energy E. Indeed, Figure 4.5shows the convergence rate of a standard multigrid V(1,1)-cycle for the 2D modelproblem described in the sections above. We observe that the multigrid scheme failsto converge for total energies E between −1 and 0. Note that this corresponds pre-cisely to the energy range where only single ionization occurs. However, for energylevels E > 0, where both single and double ionization occur, multigrid succesfully


−1 0 1 2 3

0

1

2

3

4

·10−3

E

Cro

ssse

ctio

n

Single ionization (real)

Single ionization (complex)

Double ionization (real)

Double ionization (complex)Total cross section

Fig. 4.4. Comparison of the single and double ionization total cross sections calculated usingthe scattered wave solution uN of (4.3) calculated on (a) a traditional real-valued ECS grid withθ = π/7 and (b) a full complex contour with γ = 8.5. The energy range starts at the singleionization threshold E = −1, corresponding to a strictly positive cross section. Double ionizationoccurs for energy levels E > 0.

converges.

The observed convergence behaviour can be explained using the spectral proper-ties of the Hamiltonian operator which were presented in Section 4.2. The bottomright panel of Figure 4.1 shows the eigenvalues of H1d⊗I+I⊗H1d, discretized alonga complex contour, which is an approximate representative of the spectrum of H2d.Changing the total energy E will shift the spectrum of the Hamiltonian to the left orright. For energy levels −1 < E < 0 the spectrum partially shifts to the right, result-ing in a spectrum with real-valued eigenvalues both to the left and right of the origin.This indeed implies difficulties for both the smoother and the coarse grid correctionscheme to work efficiently. However, when 0 < E, the spectrum shifts to the left,moving all eigenvalues away from zero. This results in a spectrum that is distinctlyseparated from the origin, corresponding to a problem which is much more amenableto iterative solution.

Although it might appear to the reader that multigrid is not generally efficientas a Schrodinger solver due to the poor convergence in the −1 < E < 0 region, itis important to note that multigrid is performant in the region of physical interest.Indeed, the double ionization problem requires a full two-dimensional description asstated above, requiring multigrid to converge for energy levels E > 0. In contrast,the purely single ionized problem (with V12 = 0) can be solved in a one-dimensionalHelmholtz setting, see (4.5), where multigrid can indeed be shown to perform well forenergy levels −1 < E < 0, cf. Section 3.

Although single ionization waves are present in the solution for energy levelsE > 0, they do not undermine the multigrid convergence in this regime. This isremarkable, because single ionization waves are very localized evanescent waves alongthe edges of the domain, which generally cannot be represented efficiently on coarsergrids, since there might not be enough grid points covering these regions. However,


−2 −1 0 1 2 3

0

0.2

0.4

0.6

0.8

1

E

Mu

ltig

rid

Con

ver

gen

cera

te

Fig. 4.5. 2D Schrodinger problem (4.3) for a total energy range E ∈ [−2, 3] solved on a fullcomplex grid with γ ≈ 8.5. Displayed is the average multigrid convergence rate of a V(1,1)-cyclewith GMRES(3) smoother as a function of the energy E. Average (‖rk‖/‖r0‖)1/k calculated fromexperimental results based upon k = 4 consecutive V-cycles.

despite the fact that the coarsening strategy of the multigrid method used in thiswork is not adapted to evanescent waves, the natural damping implied by the complexcontour evaluation ensures good multigrid performance.

4.6. Solution of a 3D Schrodinger equation. As demonstrated on a 2Dmodel problem in the previous sections, the far field map (cross section) of a gen-eral Schrodinger problem can be accurately calculated using the complex contourapproach. In this section, we focus on the numerical solution of the 3D Schrodingerequation on the complex contour using multigrid. In the three-dimensional case, theuse of a direct solver is prohibited due to the size of the problem.

We consider the 3D Schrodinger equation, modelling a realistic scattering problemthat includes single, double and triple ionization. As discussed above, this problemfeatures very localized waves, which require sufficiently high-resolution solution meth-ods. The model problem for the 3D partial wave expansion is(− 1

2∆ + V1(x) + V2(y) + V3(z)

+ V12(x, y) + V23(y, z) + V31(z, x)− E)u(x, y, z) = φ(x, y, z), x, y, z ≥ 0,

(4.19)

with boundary conditionsu(x, y, 0) = 0 for x, y ≥ 0

u(0, y, z) = 0 for y, z ≥ 0

u(x, 0, z) = 0 for x, z ≥ 0

outgoing for x→∞ or y →∞ or z →∞,

(4.20)

Let us discuss a system in which V1, V2 and V3 are identical one-body potentials andV12, V23 and V31 are, similarly, identical two-body potentials. Let the strength of


the one-body potential be such that there is a single negative eigenvalue for the 1Dsubsystem (

−1

2

d2

dx2+ V (x)

)φ0(x) = λ0 φ0(x), x ≥ 0, (4.21)

with λ0 < 0, where we have dropped the subscript on the 1D potential V . If thetwo-body potential V12(x, y) is negligibly small, then there automatically exists abound state of the 2D subsystem. Indeed, the state φ0(x)φ0(y) is an eigenstate of theseparable Hamiltonian (−1/2)∆ + V (x) + V (y) with eigenvalue 2λ0. In the presenceof a small but non-negligible two-body potential, this state will be slightly perturbed,resulting in an eigenstate φ0(x, y) that fits the 2D subsystem(

−1

2∆ + V (x) + V (y) + V12(x, y)

)φ0(x, y) = µ0φ0(x, y), x, y ≥ 0. (4.22)

The corresponding eigenvalue is µ0 ≈ 2λ0 < λ0 < 0. This ordering is typical forrealistic atomic and molecular systems [4]. Similarly, the 3D system will have aneigenstate that looks approximately like φ0(x, y)φ0(z), or any of its coordinate per-mutations. This 3D eigenstate φ0(x, y, z) fits the equation(− 1

2∆ + V1(x) + V2(y) + V3(z)

+ V12(x, y) + V23(y, z) + V31(z, x))φ0(x, y, z) = ν0 φ0(x, y, z), x, y, z ≥ 0,

(4.23)

where ν0 ≈ µ0 + λ0 ≈ 3λ0.Assuming that the potentials are such that ν0 < µ0 < λ0 < 0, there are now

four possible regimes of interest in equation (4.19), depending on the total energyE. First, for E < ν0, the problem is positive definite, and hence easy to solvenumerically. However, in this regime no interesting physical reactions occur. Similarly,for ν0 < E < µ0, there are no scattering states in the solution. For energy levelsµ0 < E < λ0, single ionization scattering occurs. Consequently, in this regime, thereexist scattering solutions that are localized along one of the three axes in the 3Ddomain. These solutions take the form v(z)φ0(x, y) as z → ∞, where φ0(x, y) is theeigenstate of (4.22) and v(z) is a scattering solution satisfying outgoing wave boundaryconditions. Similar solutions are found for the respective coordinate permutations.For energy levels λ0 < E < 0, both single and double ionization occurs. For theseenergy levels, the solution contains — besides single ionization waves — also doubleionization waves of the form w(y, z)φ0(x), where φ0(x) is an eigenstate of (4.21) andw(y, z) is a 2D scattering state satisfying the outgoing wave boundary conditions.Together with coordinate permutations, these waves are localized along the faces ofthe 3D domain, where one of the three coordinates, x, y or z, is small. Finally, forE > 0, the solution additionally contains triple ionization waves. These are waves thatdescribe a quantum mechanical system that is fully broken up into its sub-particles.In this case, all three relative coordinates x, y and z can become large, resulting in awave which extends to the entire domain.

Note that in fact only the latter problem, when E > 0 and triple ionization ispresent, requires a full 3D description. For the regimes in which only single ionizationoccurs, a 1D description should be sufficient, due to the separated character of thesolution. Similarly, for problems with both single and double ionization, but no triple


E < ν0 ν0 < E < µ0 µ0 < E < λ0 λ0 < E < 0 0 < E

Indefinite No Yes Yes Yes YesSingle ion. No No Yes Yes YesDouble ion. No No No Yes YesTriple ion. No No No No Yes

Table 4.1Schematic overview of the different scattering regimes in the 3D model problem described in

Section 4.6. Depending on the value of the total energy E with respect to the eigenvalues ν0, µ0 andλ0 of the 3D, 2D and 1D (sub-)system respectively, different types of scattering occur. The systemis indefinite as soon as ν0 < E. Single ionization waves emerge as soon as µ0 < E and doubleionization occurs for λ0 < E. Triple ionization waves are present only when 0 < E.

ionization, a simpler 2D description such as the one given by (4.3) can be used to fullydescribe the physics behind the problem. Hence, in view of the efficient solution ofthe 3D Schrodinger problem (4.23), our main interest goes out to the E > 0 regime.

4.7. Multigrid performance on the 3D Schrodinger problem. We nowstudy the convergence of a multigrid solver for the 3D Schrodinger equation (4.19)for energies E that cover all possible scattering regimes. The model problem underconsideration is a straightforward generalization of the 2D model presented in Section4.3, featuring one-body potentials V1(x) = −4.5 exp(−x2), V2(y) = −4.5 exp(−y2)and V3(z) = −4.5 exp(−z2), and two-body potentials V12(x, y) = 2 exp(−(x + y)2),V23(y, z) = 2 exp(−(y+z)2) and V31(x, z) = 2 exp(−(x+z)2). These potentials implythe existence of a 1D eigenstate in (4.21) with corresponding energy λ0 = −1.0215,a 2D eigenstate solution of (4.22) with energy µ0 = −1.841, and an additional 3Deigenstate in (4.23), which has energy ν0 = −2.751. The problem is solved for arange of different total energies E, using an identical right-hand side φ(x, y, z) =exp(−3(x+y+z)2) for all energies. The discretization comprises 2553 points, coveringthe complex-valued cube domain [0, 15eiπ/12]3.

The 3D model problem described above is now solved using a full multigrid F(5)-cycle [48]. This implies that the problem is first discretized on a 73-point grid, whereit is solved exactly. The solution obtained on this level is consecutively interpolatedand used as an initial guess to the same problem discretized using 153 grid points,after which 5 V(1,1)-cycles are applied. This process is repeated recursively until wearrive at the finest level in the multigrid hierarchy, which consists of 2553 grid points.On this level the V-cycle convergence rate is measured by averaging the residualreduction rate over three consecutive V-cycles. Figure 4.6 shows the convergence rateas a function of the total energy E. We see acceptable convergence behavior for energylevels E < −3.0, where the problem is positive definite. However, for energy levelsbetween −3.0 < E < 0, where single and double ionization occur, unacceptably slowconvergence is measured. For energy levels E > 0, where single, double and tripleionization waves coexist, multigrid convergence is again optimal.

In analogy to the 2D problem, the observed lack of convergence for a limited rangeof energies can be understood by analyzing the spectral properties of the discrete3D Schrodinger operator. The discrete operator comprises a Laplacian operator,discretized along a complex contour, which results in multiple series of eigenvaluesstretching deep into the bottom half of the complex plane. A more detailed discussionon the eigenvalues of the discrete Helmholtz operator can be found in [40]. Forthe total Schrodinger operator, the eigenvalues differ slightly. For each subsystem


−4 −2 0 2 4 6 80

0.2

0.4

0.6

0.8

1

E

Mu

ltig

rid

Conver

gen

cera

te

Fig. 4.6. 3D Schrodinger problem (4.19) for a total energy range E ∈ [−4, 8] solved on a fullcomplex grid with γ = π/12 = 15. Displayed is the average multigrid convergence rate of a V(1,1)-cycle with GMRES(3) smoother as a function of the energy E. Average (‖rk‖/‖r0‖)1/k calculatedfrom experimental results based upon k = 3 consecutive V-cycles.

eigenvalue, either 2D or 1D, a series of eigenvalues arises from just below the real axisinto the negative half of the complex plane, cf. Figure 4.1. Changing the energy E willshift the distribution of these eigenvalues in the direction of the real axis. For energylevels ν0 < E < 0, there are series of eigenvalues both in the third and the fourthquadrant in the complex plane. Both series start close to the real axis, resulting in anindefinite problem with eigenvalues closely near the origin, causing poor convergenceof the multigrid method. Contrarily, in the 0 < E regime, all eigenvalues are boundedaway from the origin. Indeed, in this case all eigenvalue series start from eigenvaluesalong the negative real axis. The largest real-valued eigenvalue lies at a distance |E|to the left of origin, implying the entire spectrum can be distinctly separated fromthe origin by a virtual straight line, which results in good multigrid convergence.

Note, however, that from a physical point of view, the lack of convergence forenergy levels E < 0 is not a concern, since the solutions in this energy regime canbe described either by a 1D or 2D equation (see higher), and a full 3D description isgenerally not required.

5. Conclusions and discussion. In this paper we have developed a novelhighly efficient method for the calculation of the far field map resulting from d-dimensional Helmholtz and Schrodinger type scattering problems where the wavenum-ber is an analytical function. Our approach is based on the reformulation of theclassically real-valued Green’s function volume integral for the far field map to anequivalent volume integral over a complex-valued domain.

The advantage of the proposed reformulation lies in the scattered wave solutionof the Helmholtz problem on a complex domain, which can be calculated efficientlyusing a multigrid method, in particular for high-dimensional problems. Indeed, thereformulation of the Helmholtz forward problem on the full complex contour is shownto be equivalent to a Complex Shifted Laplacian problem, for which multigrid hasbeen proven in the literature to be a fast and scalable solver. However, whereas the


Complex Shifted Laplacian was previously only used as a preconditioner for highlyindefinite Helmholtz problems, the complex-valued far field map calculation proposedin this paper effectively allows for multigrid to be used as a solver on the perturbedproblem.

The functionality of the method is primarily validated on 2D and 3D Helmholtztype model problems. It is confirmed that the values of the far field map calculated onthe full complex grid exactly matches the values of the classical real-valued integral.Furthermore, the number of multigrid iterations is shown to be largely wavenumberindependent, yielding a fast overall far field map calculation.

One area of scientific computing where the proposed technique might be particu-larly valuable is in the numerical solution of quantum mechanical scattering problems.These are generally high-dimensional scattering problems where the wavenumber isindeed an analytical function, and where 6D or 9D problems are common. We havevalidated that for a 2D Schrodinger type model problem the proposed method canaccurately calculate the cross sections that are measured in physical experiments. Inaddition, we have studied the convergence rate for a 3D Schrodinger equation. Resultsshow reasonable multigrid convergence rates for the energy range of interest.

Despite their use as benchmark problems, the model problems described in thispaper form an important testing framework for more realistic applications. However,further analysis of the convergence rates are necessary for realistic Coulomb potentialsto make the method more robust, and eventually usable by computational physicistsand chemists.

Finally, we note that a number of modifications can be made to improve theefficiency of the method even further, like e.g. choosing the shape of the complexcontour for the integral based on a steepest descent scheme, as proposed in [28].

6. Acknowledgments. This research was partly funded by the Fonds voorWetenschappelijk Onderzoek (FWO) project G.0.120.08 and Krediet aan navorserproject number 1.5.145.10. Additionally, this work was partly funded by Intel R© andby the Institute for the Promotion of Innovation through Science and Technology inFlanders (IWT). The authors would like to thank Hisham bin Zubair for sharing amultigrid implementation and D. Huybrechs, C.W. McCurdy and D.J. Haxton forfruitful discussions on the subject.

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AN EFFICIENT MULTIGRID CALCULATION OF THE FAR FIELD