Space Discretisation with Sparse Grids
Sam Maurus
Faculty of Informatics, Technical University of Munich
Boltzmannstr. 3, D-85748 Garching, Germany
maurus@in.tum.de
Abstract—Many problems in scientific computing exhibit high
dimensionality. Attempts to solve such problems using conventional
discretisation techniques are typically hampered by the so-called
‘curse of dimensionality’. Sparse grids provide a general and
flexible means for discretising a high-dimensional problem space
that remedies the curse at least to some extent. This paper outlines
the theoretical fundamentals of sparse grids and presents some
approaches in which sparse grids can be constructed and applied
in various applications of scientific computing.
Index Terms—sparse grids, nodal basis, hierarchical basis,
direct sparse grid approach, sparse grid combination technique,
curse of dimensionality
1. INTRODUCTION
In 1961 Richard Bellman coined the term curse of
dimensionality, describing it as ‘a malediction that has
plagued the scientist from the earliest days’ [2]. In the
context of scientific computing, the term refers to the
tractability problems caused by the exponential dependence
between the computational cost of approximating (and
representing) a fixed-accuracy solution and the problem
dimensionality 𝑑 . That is, increasing the number of
dimensions in the problem exponentially increases the cost
of solving the problem to the same level of accuracy.
Depending on the nature of the problem, tractability issues
can typically be seen starting at three, four or five
dimensions [17]. Attempting to apply classical discretisation
techniques to problems with a higher number of dimensions
results in computational requirements that quickly
overwhelm even the most state-of-the-art machines.
Examples of problems with a high dimension count are
found in finance (pricing of options/derivatives) [3], data
mining [4], stochastic differential equations [10] and
quantum mechanics (e.g. the electronic Schrödinger’s
equation for molecules like water or ammonia, which
contains 30 dimensions) [5].
In this paper I provide the reader with detail in a number
of sparse grid fundamentals. I begin in Section 2 by
describing the curse of dimensionality and how it is
inevitably encountered when using naive discretisation
approaches in high-dimensional problem spaces. A simple
example is provided to give a real-world perspective on its
effects.
In Section 3 I illustrate the concepts of nodal and
hierarchical bases. The formulas used to construct both types
of bases in an arbitrary number of dimensions are presented.
I also make a comparison between the two types of bases.
I give a detailed description of two common methods for
direct sparse grid construction in Section 4, showing that the
resulting points from the second method are a subset of the
resulting points from the first method, which are in turn a
subset of the classical regular grid points. The methods are
shown to differ by the norms considered in a strategy used to
optimise a cost/benefit relationship. A comparison between
the construction techniques is made taking into account
accuracy and cost.
In Section 5 I present the sparse grid combination
technique, showing that it offers an alternative method for
arriving at a sparse grid solution through the superposition of
multiple smaller (and hence more tractable) full-grid
solutions. In Section 6 I discuss the practical advantages of
this technique in comparison to the normal ‘direct’ sparse
grid approach.
Finally, I present a short discussion on the applications of
sparse grids in state-of-the-art financial and insurance
problems in Section 7. Section 8 gives concluding remarks.
2. THE CURSE OF DIMENSIONALITY
To provide a sense of purpose and scope for the
remainder of the paper, I will first provide a motivating
example of how traditional methods suitable for solving lowdimensional problems become intractable for approaching
high-dimensional problems.
Problems from classical physics typically consider only a
maximum of three spatial dimensions and possibly one time
dimension. The stationary heat equation is one such example.
To solve this equation, one might discretise a threedimensional unit cube with a classical Cartesian grid
employing 𝑝 = 100 points in each dimension. The resulting
number of grid points would be 𝑃 = 𝑝𝑑 = 1003 = 106 .
Assuming ideal sequential execution and a single floating
point operation per grid point, the time required to solve this
problem would be of the order of nanoseconds on the modest
63-TFLOPS HLRB2 supercomputer in Munich.
Now consider the task of approximating the price 𝑢 of a
financial option using a multi-asset Black-Scholes parabolic
partial differential equation (PDE). The number of
dimensions 𝑑 is related to the number of assets in such PDEs.
For example, dimensional counts of 𝑑 ≈ 30 are common in
problems dealing with stock indices [1]. Taking 𝑑 = 10 as
an example and using the same Cartesian grid discretisation
approach results in 𝑃 = 𝑝𝑑 = 10010 = 1020 . A solution to
this equation on the same supercomputer would require a
number of weeks to complete, and further doubling the
number of dimensions to 20 would result in an ideal
computation time of approximately 1018 years. In addition,
the storage requirements quickly become unmanageable.
These effects are the typical of the manifests of the curse of
dimensionality.
The curse can be mitigated by choosing a bounded
number of points in the problem space based on random
sampling, as is done with Monte Carlo methods. The sparse
grid approach provides however a more deterministic and
structured way to tame the curse, and it is this approach that
is the focus of the remainder of this paper. A brief
comparison between the sparse grid and Monte Carlo
approaches is given in Section 7.
3. BASIS FUNCTIONS AND SPACES
In this section I present the concepts of nodal and
hierarchical bases and make a comparison between them.
3.1. Nodal Basis
In one dimension, the nodal basis can be defined on the
unit interval Ω = [0,1] for a dyadic discretisation level1 𝑙 ∈
ℕ0 . The corresponding isotropic grid Ω𝑙 has mesh width
ℎ𝑙 = 2−𝑙 with boundary points 𝑥0 = 0 and 𝑥(2𝑙 ) = 1, where
𝑥𝑗 = 𝑗 ∙ ℎ𝑙 . The nodal basis for level 𝑙 is then defined as in
[25] by the function space 𝑉𝑙 of piecewise one-dimensional
linear functions
𝑉𝑙 ∶= span{𝜙𝑙,𝑗 | 𝑗 = 0, … , 2𝑙 }
(1)
Each of the one-dimensional functions 𝜙𝑙,𝑗 in the space 𝑉𝑙
is defined as
𝑥
1 − | − 𝑗| ,
ℎ
𝜙𝑙,𝑗 (𝑥) = {
𝑙
0,
𝑥 ∈ [𝑥𝑗 − ℎ𝑙 , 𝑥𝑗 + ℎ𝑙 ]
(2)
otherwise.
Fig. 1 shows 𝜙3,5 (𝑥) as an example. This function is the
sixth member of the 𝑉3 function space, having support on the
1 3
5
2 4
8
interval [ , ] and a maximum value satisfying 𝜙3,5 ( ) = 1.
Fig. 1: Nodal basis function 𝝓𝟑,𝟓 (𝒙)
1
In this definition, ℕ0 is the set of natural numbers with 0
included
The nodal basis 𝑉3 is shown fully in Fig. 2.
Fig. 2: Nodal basis space 𝑽𝟑 , consisting of nine onedimensional functions 𝝓𝟑,𝒋 (𝒙)
One can generalise the nodal basis to an arbitrary number
of dimensions 𝑑 supporting anisotropic mesh widths across
dimensions (i.e. the mesh width is constant along any one
dimension, but may not be equal to the mesh width on other
dimensions). We first extend the scalar discretisation level 𝑙
to a multi-index vector 𝑙 = (𝑙1 , … , 𝑙𝑑 ) ∈ ℕ0 𝑑 as defined in
[25]. For example, the vector 𝑙 = (1,4, 2) refers to the
discretised three-dimensional unit cube with mesh widths in
the respective dimensions defined by ℎ𝑙 ∶= (ℎ𝑙1 , … , ℎ𝑙𝑑 )
∶= (2−𝑙1 , … , 2−𝑙𝑑 ) and calculated as ℎ𝑙 = (2−1 , 2−4 , 2−2 ) ∈
ℝ3 for this example.
The multi-dimensional nodal basis is then defined by the
function space 𝑉𝑙 as the natural extension of (1):
𝑉𝑙 ∶= span {𝜙𝑙,𝑗 | 𝑗𝑡 = 0, … , 2𝑙𝑡 , 𝑡 = 1, … , 𝑑}
(3)
The basis functions for the 𝑑-dimensional nodal basis on
the hypercube [0,1]𝑑 can be defined as in [11] as the tensor
product of the one-dimensional nodal basis functions defined
in (2), that is
𝜙𝑖,𝑗 (𝑥1 , … 𝑥𝑑 ) ∶= 𝜙𝑙1 ,𝑗1 (𝑥1 ) ∙ … ∙ 𝜙𝑙𝑑 ,𝑗𝑑 (𝑥𝑑 )
(4)
Fig. 3 shows 𝜙𝑙,𝑗 for the case where 𝑙 = (2,1) and 𝑗 =
(1,1) . The projections of the one-dimensional basis
functions involved in the tensor product are shown on the
respective axis planes.
increment subspaces of all discretisation levels 𝑘 that are
smaller than or equal to 𝑙.
An example of the one-dimensional hierarchical basis 𝐻𝟑
(consisting of the hierarchical increment spaces 𝑊0 through
𝑊3 ) is given in Fig. 4. Labels for the single nodal basis
functions 𝜙𝑙,𝑗 (𝑥) that comprise each hierarchical increment
space are given in the figure.
For the multi-dimensional case where 𝑙1 =. . . = 𝑙𝑑 = 𝑛 we
can define 𝐻𝑛 = 𝐻(𝑛,…,𝑛) [14] and hence the hierarchical
basis for level 𝑛 as:
𝑛
𝑛
𝑛
𝐻𝑛 ∶= ⊕ ⊕ ⋯ ⊕ 𝑊𝑙 = ⊕ 𝑊𝑙
𝑙1 =0 𝑙2 =0
𝑙𝑑 =0
|𝑙|∞ ≤𝑛
(8)
with the max norm defined as |𝑢|∞ ∶= max{𝑢1 , … , 𝑢𝑑 }.
Fig. 3: Nodal basis function 𝝓𝒍,𝒋 with 𝒍 = (𝟐, 𝟏) and 𝒋 =
(𝟏, 𝟏)
3.2. Hierarchical Basis
As will be seen later, hierarchical bases are an important
building block for sparse grids. To see how a hierarchical
basis is constructed, we first introduce the idea of the
hierarchical increment space 𝑊𝑙 as in [25]
𝑊𝑙 ∶= span {𝜙𝑙,𝑗 | 𝑗 ∈ 𝐵𝑙 }
(5)
with 𝜙𝑙,𝑗 defined as in (4) and the index set 𝐵𝑙 given in
the one-dimensional case by
𝑗 𝑒 𝑗 = 1 … 2𝑙1 − 1, 𝑗 odd
𝐵𝑙1 = {𝑗 ∈ ℕ |
𝑘 𝑗 = 0,1
if 𝑙1 > 0
}
if 𝑙1 = 0
(6)
and analogously for the 𝑑 -dimensional case by the
definition in [25] for example. Note from (6) that the
hierarchical increment spaces for 𝑙1 > 0 consist of only
those basis functions that are centred at odd grid points.
Alternatively, one can see the hierarchical increment space
𝑊𝑙 as consisting of all those functions of 𝑉𝑙 which vanish at
all grid points of coarser grids [14]. This concept is shown in
two dimensions in Fig. 5.
In comparison to the nodal basis, the hierarchical basis for
𝑙 = (3,3) for example is produced in a multi-level fashion
by combining all 16 hierarchical increment subspaces shown
in Fig. 5. The hierarchical basis [15] is defined in general by
𝐻𝑙 ∶= span{𝜙𝑘,𝑖 ∶ 𝑖 ∈ 𝐵𝑘 , 𝑘 ≤ 𝑙}
(7)
where the ‘less-than or equal to’ relationship (≤) is elementwise.
Equation (7) describes the hierarchical basis as the
combination of basis functions that belong to the hierarchical
Fig. 4: One-dimensional hierarchical basis 𝑯𝟑
hence be less sparse if based on a hierarchical basis than it
would be if based on a nodal basis [18].
4. SPARSE GRIDS
Sparse grids are a general, multi-dimensional
discretisation technique which, at least to some extent, cures
the curse of dimensionality [8][25]. This section presents
two construction techniques for sparse grids and compares
them to the classical Cartesian discretisation technique.
4.1. Principle of Construction
Two sparse grid construction techniques are discussed and
compared here: the 𝐿2 -norm and energy-norm based
methods.
Fig. 5: Decomposition of the two-dimensional space 𝑯𝟑
into its hierarchical increment spaces 𝑾𝒍 (adapted from
[25]). Selection of only the spaces above the dashed
separator results in the two-dimensional 𝑳𝟐 -norm based
sparse grid space 𝑽𝟑 (𝟏) .
3.3. Comparison of Nodal and Hierarchical Basis
The one-dimensional case is considered here for
simplicity, but the comparisons made can be generalised to
higher dimensions in a straight-forward manner.
A comparison between the number of nodes and basis
functions is presented in Table I.
4.1.1. 𝐿2 -norm based sparse grids
The classical method for constructing a sparse grid for a
given level is based on optimising the cost-benefit
relationship using the 𝐿2 -norm and involves cutting a
diagonal in the tableau of hierarchical increment spaces and
taking the spaces that lie above this diagonal (in Fig. 5 these
spaces lie above the dashed separator) [14][16]. More
formally, the 𝐿2 -norm based sparse grid space 𝑉𝑛 (1) for level
𝑛 can be written as in [13]:
𝑉𝑛 (1) ∶=
⊕
|𝑙|1 ≤ 𝑛+𝑑−1
𝑊𝑙
(9)
For example, the sparse grid space 𝑉3 (1) in two dimensions
is built by combining all spaces {𝑊𝑙 ∈ 𝑊(𝛼,𝛽) , 𝛼 + 𝛽 ≤ 4 },
as illustrated in Fig. 5. The points from this sparse grid space
are shown in Fig. 6.
TABLE I
COMPARISON OF NODAL AND HIERARCHICAL BASIS FOR LEVEL 𝑙
Number of
nodes
Basis
functions
Refinement
flexibility
Nodal Basis
(2𝑙 + 1)
Hierarchical Basis
(2𝑙 + 1)
(2𝑙 − 1) interior hat
functions with
support 2ℎ, and two
boundary functions of
support ℎ [8].
Two boundary
functions of support 1,
one global function of
support 1, two
functions of support ½,
four functions of
support ¼ and so on
until the required level
is reached [8].
Incremental refinement
possible (simple
addition of extra
increment spaces).
Refinement requires a
new setup.
For comparison it is also worthy to note that the supports
of hierarchical basis functions of different levels can overlap
(visible in Fig. 4). If used generally to generate a problem
stiffness matrix 𝐴 for a linear system, the matrix 𝐴 will
Fig. 6: The two-dimensional 𝑳𝟐 -norm based sparse grid
space 𝑽𝟑 (𝟏)
4.1.2. Energy-norm based sparse grids
Another method for constructing a sparse grid (as
described in [16]) involves consideration of the energy norm
𝑑
2
1/2
𝜕𝑢(𝑥)
‖𝑢‖𝐸 ∶= (∫ ∑ (
) 𝑑𝑥 )
𝜕𝑥𝑗
Ω
(10)
𝑗=1
A selection of hierarchical increment spaces can be
defined based on this norm which optimise the cost/benefit
relationship [16], resulting in the energy-norm based sparse
grid space
𝑂(2𝑛∙𝑑 ) = 𝑂(ℎ𝑛 −𝑑 )
to
𝑑−1
𝑉𝑛 (𝐸) ∶=
⊕
1
1
𝑙𝑗
𝑛
|𝑙|1 − ∙log2 (∑𝑑
𝑗=1 4 )≤ (𝑛+𝑑−1)−5∙log2 (4 +4𝑑−4)
5
𝑊𝑙
𝑂(2𝑛 ∙ 𝑛𝑑−1 ) = 𝑂 (ℎ𝑛 −1 ∙ log (ℎ𝑛 −1 )
(11)
It can be shown (e.g. in [16]) that 𝑉𝑛 (𝐸) is a subspace of
𝑉𝑛 .
(1)
4.2. Comparison of construction approaches
The hierarchical increment spaces that form part of the
𝐿2 -norm and energy-norm based sparse grids are compared
for a two-dimensional case in Fig. 7 (reproduced from [16]).
The aforementioned subset relationship between the two sets
is noteworthy.
with ℎ𝑛 = 2 .
Although a dependence on 𝑑 is still present in the
asymptotic complexity for sparse grids, it is much weaker.
This hence represents a considerable reduction in the number
of grid points and thus improves the tractability of solving
high-dimensional problems in the sense of the involved
computational work and storage. The use of 𝐿2 -norm based
sparse grids hence alleviate to some extent the curse of
dimensionality.
Turning our attention now to energy-norm based sparse
grids, we recall from Fig. 7 that 𝑉𝑛 (𝐸) is a subspace of 𝑉𝑛 (1) ,
and thus |𝑉𝑛 (𝐸) | < |𝑉𝑛 (1) |. In fact, it can be shown (e.g. in
[16]) that the number of points in an energy-norm based
sparse grid satisfies
|𝑉𝑛 𝐸 | ≤ 2𝑛 ∙
Fig. 7: Comparison of the hierarchical increment spaces
used in the 𝑳𝟐 -norm based (left) and energy-norm based
(right) sparse grid construction approaches for 𝒏 = 𝟑𝟎
and 𝒅 = 𝟐.
Comparing the number of grid points (including the
boundary points) between a regular grid and an 𝐿2 -norm
based sparse grid, we first note that the number of points in
an 𝐿2 -norm based sparse grid is given by [16]
|𝑉𝑛 (1) | = 2𝑛 (
𝑛𝑑−1
+ 𝑂(𝑛𝑑−2 ))
(𝑑 − 1)!
+ 2𝑑−1 (2𝑛 − 1)𝑑 + 2𝑑
)
−𝑛
𝑑 𝑑
∙ e = 𝑂(ℎ𝑛 −1 )
2
(14)
Equation (14) represents an important result: There is no
asymptotic dependence of |𝑉𝑛 (𝐸) | on 𝑑 (although dependence
does exist in hidden constants [15]). The curse of
dimensionality is hence (at least in an asymptotic sense)
overcome [16].
A natural question that one might pose at this stage is:
What is the deterioration in accuracy that results from using
sparse grids based on such norms? Assuming that the
solution is sufficiently smooth, it can be shown (e.g. in [16])
that the interpolation error (measured by the energy norm)
for both approaches is in 𝑂(ℎ𝑛 ) . There is hence no
deterioration in complexity and accuracy for higherdimensional problems when using energy-norm based sparse
grids instead of 𝐿2 -norm based sparse grids (ignoring the
constant factors that are hidden in the Landau relations).
(12)
TABLE II
COMPARISON OF THE NUMBER OF GRID POINTS BETWEEN CARTESIAN AND
SPARSE GRIDS IN TWO AND THREE DIMENSIONS
Table II shows the number of grid points for two and
three dimensions (including the boundary points) for a 𝐿2 norm based sparse grid compared to a standard Cartesian
grid. Already at three dimensions we can see a significant
reduction in the number of points required to represent a
given discretisation level. The reduction becomes more
pronounced with increasing 𝑑.
From (12) we can express the general asymptotic
complexity of the grid-point count for 𝐿2 -norm based sparse
grids.
𝑑−1
|𝑉𝑛 (1) | = 𝑂(2𝑛 ∙ 𝑛𝑑−1 ) = 𝑂 (ℎ𝑛 −1 ∙ log (ℎ𝑛 −1 )
)
(13)
The use of such a sparse grid in place of a Cartesian grid
𝑉𝑛 hence reduces the number of grid points from
∞
Level
𝑛=1
𝑛=2
𝑛=3
𝑛=4
𝑛=5
𝑛=6
𝑛=7
𝑛=8
𝑛=9
𝑛 = 10
…
𝑛 = 30
𝒅=𝟐
Cartesian
Sparse Grid
Grid
9
9
25
21
81
49
289
113
257
1.1 × 103
577
4.2 × 103
16.6 × 103
1.3 × 103
66.0 × 103
2.8 × 103
3
263 × 10
6.1 × 103
6
1.1 × 10
13.3 × 103
1.2 × 1018
35 × 109
𝒅=𝟑
Cartesian
Sparse Grid
Grid
27
21
125
51
729
123
299
4.9 × 103
731
36.0 × 103
275 × 103
1.8 × 103
2.1 × 106
4.3 × 103
6
17.0 × 10
10.5 × 103
6
135 × 10
25.1 × 103
9
1.1 × 10
59.4 × 103
1.2 × 1027
481 × 109
4.3. Further sparse grid techniques
Although not addressed in this paper, further techniques
exist for the direct construction of sparse grids. The curious
reader is referred to the techniques of delayed construction
[20], generalised construction [21] and dimension-adaptive
construction [22]. An overview of these techniques can be
found in [6].
5. THE COMBINATION TECHNIQUE
The ‘direct’ construction techniques described in Section
4 can be used to discretise a problem space and form a
sparse grid.
Sparse grid spaces can however also be achieved using
non-direct methods. One such popular method is the socalled combination technique. The idea is that a sparse grid
space can be generated through the superposition of various
(coarser) regular-grid spaces having anisotropic mesh widths
[9]. The technique thus uses the combination of certain 𝑉𝑙
instead of 𝑊𝑙 spaces.
The combination technique (as first published in [19])
recognises that the sparse grid space 𝑉𝑛 (1) can be viewed as a
combination of nodal-basis spaces. To build up a sparse grid
space 𝑉𝑛 (1) for dimension 𝑑 the technique considers all of
the regular grids Ω𝑙 with
|𝑙|1 = 𝑙1 + 𝑙2 + ⋯ + 𝑙𝑑 = 𝑛 + 𝑞,
𝑞 = 0, … , 𝑑 − 1
Fig. 8: Construction of the two-dimensional sparse grid
space 𝑽𝟑 (𝟏) using the combination technique
Fig. 9 further visualises the idea formulated in (16): the
combination technique uses full-grid spaces on two
diagonals in order to generate a two-dimensional sparse grid
space. In general, the combination technique uses full-grid
spaces on 𝑑 hyper-planes to generate a 𝑑-dimensional sparse
grid space.
(15)
With these grids we take the corresponding nodal-basis
functions 𝑢𝑙 and combine them to form a sparse grid
function 𝑢𝑛 (1) using the following formula (adapted from
[13]):
𝑢𝑛 (1) =
⊕
𝑛≤|𝑙|1 ≤𝑛+𝑑−1
(−1)𝑛+𝑑−|𝑙|1−1 (
𝑑−1
)𝑢
|𝑙|1 − 𝑛 𝑙
(16)
To illustrate this idea further, Fig. 8 shows how a simple
two-dimensional sparse grid space can be constructed
through the combination of certain regular grid spaces.
Fig. 9: Full-grid (nodal-basis) spaces used in the
construction of the two-dimensional space 𝑽𝟒 (𝟏) using the
combination technique. The green spaces (|𝒍|𝟏 = 𝒏 + 𝟏)
are added; the red spaces (|𝒍|𝟏 = 𝒏) are subtracted.
6. SPARSE GRID CONSTRUCTION COMPARISON: DIRECT
APPROACH VERSUS COMBINATION APPROACH
The term ‘direct sparse grid approach’ is an umbrella term
used to describe the various methods that arrive at a sparse
grid space and solution through the consideration of
hierarchical increment function spaces. In comparison, the
combination technique discussed in Section 5 considers
Cartesian grids that stem from the nodal basis.
A major benefit of the combination technique is that its
algorithmic implementation is naturally parallel [12]. That is,
each of the Cartesian grid problems that are involved in the
combination process can be processed and solved
independently. Synchronisation between the processes
solving the separate problems need only occur at the stage
where the partial solutions are combined [19], and efficient
distributed storage strategies are possible [26]. For nonstationary problems (e.g. turbulence simulation), the option
exists to isolate each of the partial-solutions for multiple
time steps before combining them. These concepts are
shown in Fig. 10, where a non-stationary problem is solved
using the combination technique on the sparse grid space
𝑉3 (1) . It is also worthy to note here that the power of existing
‘out of the box’ regular-grid tools (e.g. multigrid methods)
can be leveraged to efficiently arrive at each of the partial
solutions [17].
Fig. 10: Parallel implementation concept for the sparse
grid combination technique.
On the other hand, direct approaches in the hierarchical
function space involve algorithms that operate on one sparse
grid. Unlike the combination technique, the process of
computing a solution on this grid is not naturally parallel,
and techniques like multigrid cannot be applied in a straightforward manner. These inherent drawbacks limit the
practical success of direct sparse grid approaches.
The practical advantages of the combination technique
over direct sparse grid approaches have been verified on
different parallel architectures and have seen it successfully
applied to several types of elliptic PDEs including problems
from computational fluid dynamics (see the references in
[27]).
7. APPLICATIONS IN NUMERICAL FINANCE AND
INSURANCE
There exist many problems in practical finance and
insurance that have a dimensionality of greater than three
and are thus (due to the curse of dimensionality) not
treatable with conventional discretisation techniques. Such
problems have traditionally been solved by Monte-Carlo
methods or related approaches [23].
Monte-Carlo methods do however have a number of
practical drawbacks. For example, convergence is typically
considered slow and error bounds can only be specified in a
probabilistic sense [24]. Sparse grids are practically very
attractive in comparison because error bounds can be
determined and the convergence order is higher. Sparse grids
hence have applications in many facets of finance and
insurance [3]. Some concrete examples include pricing of
options and equity baskets [24] in the field of finance and
asset-liability management (ALM) models [6] in the
insurance industry.
8. SUMMARY
Scientific computing is being increasingly applied to
domains in which problems involve many variables and
parameters. Efficient discretisation techniques are mandatory
in order to be able solve such problems using real-world
computational resources.
Built from hierarchical basis spaces, sparse grids provide
a generic and economical means for discretising a problem
space. The corresponding discretization results in a much
weaker dependence between the number of nodes and the
problem dimensionality. Sparse grids are thus practically
able to break the curse of dimensionality and help to pave
the road for solving higher dimensional problems.
The sparse grid combination approach is a commonlyused technique that addresses some of the practical
shortcomings of the so-called ‘direct’ sparse grid solution
methods. Rather than explicitly building a full sparse grid
and attempting to obtain a solution directly on it, the
combination technique recognises and exploits the
possibility to obtain the same solution through the
superposition of multiple (and parallelisable) solutions on
coarser, regular grids.
Now a mature tool for numerical discretisation, sparse
grids offer a useful addition to the scientific computing
toolbox. It is likely that the theoretical advantages and
practical implementation offerings of sparse grids will see
them continue to be used for challenging problems in the
future.
ACKNOWLEDGEMENT
I am grateful to Stefanie Schraufstetter of the Technical
University of Munich (Institut für Informatik) for the helpful
review comments and administrative assistance she provided
during the production of this paper.
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