Numerics and Fractals††thanks: This research was partially supported under Australian Research Council’s Discovery Projects funding scheme (project number DP130101738) and the Technische Universität München â Institute for Advanced Study, funded by the German Excellence Initiative.
Local iterated function systems are an important generalisation of the standard (global) iterated function systems (IFSs). For a particular class of mappings, their fixed points are the graphs of local fractal functions and these functions themselves are known to be the fixed points of an associated Read-Bajactarević operator. This paper establishes existence and properties of local fractal functions and discusses how they are computed. In particular, it is shown that piecewise polynomials are a special case of local fractal functions. Finally, we develop a method to compute the components of a local IFS from data or (partial differential) equations.
Key words. Iterated function system, local iterated function system, attractor, code space, fractal function, fractal imaging, fractal compression, subdivision schemes
AMS subject classifications. 28A80, 33F05, 41A05, 65D05
Contractive operators on function spaces are important for the development of both the theory and algorithms for the solution of integral and differential equations. They are used in the theory of elliptic partial differential equations, Fredholm integral equations of the second kind, Volterra integral equations, and ordinary differential equations. This is just a small selection of instances were they appear in mathematics. Contractive operators are fundamental for the development of iterative solvers in general and wavelet-based solvers for elliptic problems  in particular.
One class of contractive operators is defined on the graphs of functions using a special kind of iterated function system (IFS). The fixed point of such an IFS is the graph of a fractal function. There is a vast literature on IFSs, see for example the recent review by the first author . Computationally, IFSs are used in Computer Graphics in refinement methods which effectively compute points on curves and surfaces . They are also used to compute function values of piecewise polynomial functions and wavelets. In fact, it can be shown that these applications use a variant of IFSs where the iterated functions are defined locally . These local IFSs and, in particular, their computational application are the topic of the following discussion. In this first manuscript we will mostly consider functions of one real variable in the examples. Functions of multiple variables are planned to be covered in a future paper.
The remaining part of this introduction will provide some further background and motivation for our approach to utilise IFSs or local IFSs in computations. In the second section we introduce and review local IFSs. The third section applies local IFSs to graphs of functions to define local fractal functions. It will be seen that these functions are the fixed points of a Read–Bajactarević (RB) operator. (See also  for the use of such operators in the theory of (global) fractal functions.) Section 4 provides a reformulation of the RB operator in terms of matrices acting on vectors of function values over grids. Several examples of local fractal functions are then displayed. In Section 5 we discuss the important case of polynomials and their RB operators. In a penultimate section we discuss the determination of (approximate) iterated function systems both from data and from functional equations such as partial differential equations. We conclude this discussion with some general remarks and in particular with a connection between fractals and the active research area of tensor approximation.
1.1 Fractals and numerics
One can show that graphs of piecewise polynomial functions can be written as the fixed points of local IFSs. Thus the popular finite element method approximates solutions of PDEs with particular fractal functions. However, numerical methods do not usually use IFSs directly. Exceptions are the subdivision schemes used in computer graphics where (local) IFSs are employed mostly for the representation of smooth curves and surfaces.
We suggest the construction and use of IFSs for the solution of PDEs. This is done by choosing an initial IFS and then changing it iteratively until it approximates a desired function given by either data or functional (e.g. partial differential) equations. We use ideas based on the collage theorem to fit a given function class and refine the domains of the IFS if necessary.
In the following we will discuss the numerical application of local IFSs which is not based on a basis of a linear space but on the IFS itself. We anticipate that this approach has the following advantages over approaches that are based on a linear basis:
The same approach can be used to approximate and solve PDEs on very general grids defined by IFSs including fractal sets.
Visualisation and numerical solutions are computed simultaneously and can be done on the same or on neighboring processors of a multiprocessor system such that communication overhead may be reduced.
Dimensionality is handled much more flexibly in fractals – for example, one may use 1D solvers for higher-dimensional problems.
We can at the same time adapt the basis functions (or frames) as well as solving the problem. Searches over large collections of dictionaries of finite dimensional approximation spaces can be done locally during the solution.
The computational complexity is bounded by the resolution one requires.
Adaptivity is naturally included as in wavelet-based methods and is a consequence of the iteration – one application of the IFS reduces the finest scale.
Convergence of the method can be controlled with few parameters and is driven by the convergence of the IFS.
The theory is based on the theory for fractals and IFSs which is well established. In addition, there has been a lot of work on wavelets and subdivision schemes which provides further firm foundations.
1.2 The Collage Theorem
While it is usually assumed that the iterated function system (IFS) is given, a very important class of methods used in image processing determines the IFS from its fixed point. An important result used here is the Collage Theorem . For the purposes of self-containment, we state this theorem below.
Let be a complete metric space. Denote by the associated complete metric space based on the hyperspace of nonempty compact subsets of endowed with the Hausdorff metric . Let and be given. Suppose that is a contractive IFS such that
where is the attractor of the IFS and and .
It has been demonstrated that approaches that are based on the Collage Theorem lead to very efficient image compression methods. The interested reader is referred to [6, 13] for methodologies and to  for a summery of fractal-type approaches in an analytical setting. Note that the application of an IFS starts with points on a large scale and then moves to finer scales. This is very similar to some multigrid methods and wavelet methods.
2 Local Iterated Function Systems
In the following, denotes a complete metric space with metric and the set of positive integers.
Let and let . Suppose is a family of nonempty subsets of . Further assume that for each there exists a continuous mapping , . Then is called a local iterated function system (local IFS).
Note that if each , then Definition LABEL:localIFS coincides with the usual definition of a standard (global) IFS on a complete metric space. However, the possibility of choosing the domain for each continuous mapping different from the entire space adds additional flexibility as will be recognized in the sequel.
A mapping is called contractive on or a contraction on if there exists a constant so that
A local IFS is called contractive if there exists a metric equivalent to with respect to which all functions are contractive (on their respective domains).
Let be the power set of . With a local IFS we associate a set-valued operator by setting
Here . By a slight abuse of notation, we use the same symbol for a local IFS and its associated operator.
A subset is called a local attractor for the local IFS if
In (LABEL:attr) we allow for to be the empty set. Thus, every local IFS has at least one local attractor, namely . However, it may also have many distinct ones. In the latter case, if and are distinct local attractors, then is also a local attractor. Hence, there exists a largest local attractor for , namely the union of all distinct local attractors. We refer to this largest local attractor as the local attractor of a local IFS .
We remark that there exists an alternative definition for (LABEL:hutchop). For given functions which are only defined on one could introduce set functions (which will also be called ) which are defined on by
On the left-hand side is the set of values of the original as in the previous definition. This extension of a given function to sets which include elements which are not in the domain of basically just ignores those elements. In the following we will assume this definition of the set function to be used.
In the case where is compact and the , closed, i.e., compact in , and where the local IFS is contractive, the local attractor may be computed as follows. Let and set
Then is a decreasing nested sequence of compact sets. If each is nonempty, then by the Cantor Intersection Theorem,
Using [19, Proposition 3 (vii)], we see that
where the limit is taken with respect to the Hausdorff metric on . This implies that
Thus, . A (mild) condition guaranteeing that each is nonempty is that , . (See also .)
In the above setting where the have been extended to , one can derive a relation between the local attractor of a contractive local IFS and the (global) attractor of the associated (global) IFS where the extensions of to all sets are defined as above. To this end, let the sequence be defined as above. The unique attractor of the IFS is obtained as the fixed point of the set-valued map ,
where . If the IFS is contractive, then the set-valued mapping (LABEL:setvalued) is contractive on (with respect to the Hausdorff metric) and its fixed point can be obtained as the limit of the sequence of sets with and
Note that and, assuming that , , it follows by induction that
Hence, upon taking the limit with respect to the Hausdorff metric as , we obtain . This proves the next result.
Let be a compact metric space and let , , be closed, i.e., compact in . Suppose that the local IFS and the IFS are both contractive. Then the local attractor of is a subset of the attractor of .
Contractive local IFSs are point-fibered if is compact and the , , are closed. To show this, define the code space of a local IFS by and endow it with the product topology . It is known that is metrizable and that is induced by the Fréchet metric ,
where and . (As a reference, see for instance , Theorem 4.2.2.) The elements of are called codes.
Define a set-valued mapping , where denotes the hyperspace of all compact subsets of , by
where . Then is point-fibred, i.e., a singleton. Moreover, in this case, the local attractor equals . (For details regarding point-fibred IFSs, we refer the interested reader to , Chapters 3–5.)
Let and suppose that and . Define
Furthermore, let , , be given by
respectively, where .
The (global) IFS has as its unique attractor the line segment . The local attractor of the local IFS is the union of the fixed point of and the fixed point of .
3 Local Fractal Functions
In this section, we exhibit a class of special attractors of local IFSs, namely local attractors that are the graphs of bounded functions. These functions will be called local fractal functions. We prove that the set of discontinuities of these bounded functions is countably infinite and we derive conditions under which local fractal functions are elements of the Lebesgue spaces .
To this end, we assume that and set . Let be a nonempty connected set and a family of nonempty connected subsets of . Suppose is a family of bijective mappings with the property that
forms a (set-theoretic) partition , i.e., and , for all .
Now suppose that is a complete metric space with metric . A mapping is called bounded (with respect to the metric ) if there exists an so that for all , .
Denote by the set
Endowed with the metric
becomes a complete metric space. Similarly, we define , .
Note that under the usual addition and scalar multiplication of functions, the spaces and become metric linear spaces. A metric linear space is a vector space endowed with a metric under which the operations of vector addition and scalar multiplication are continuous.
For , let be a mapping that is uniformly contractive in the second variable, i.e., there exists an so that for all
Define a Read-Bajactarević (RB) operator by
Note that is well-defined and since is bounded and each contractive in the second variable, .
Moreover, by (LABEL:scon), we obtain for all the following inequality:
To simplify notation, we set in the above equation. In other words, is a contraction on the complete metric space and, by the Banach Fixed Point Theorem, has therefore a unique fixed point in . This unique fixed point will be called a local fractal function (generated by ).
Next, we would like to consider a special choice for mappings . To this end, we require the concept of an -space. We recall that a metric is called complete if every Cauchy sequence in converges with respect to to a point of , and translation-invariant if , for all .
A topological vector space is called an -space if its topology is induced by a complete translation-invariant metric .
Now suppose that is an -space. Denote its metric by . We define mappings by
where and is a function.
If in addition we require that the metric is homogeneous, that is,
then given by (LABEL:specialv) satisfies condition (LABEL:scon) provided that the functions are bounded on with bounds in for then
Here, denotes the supremum norm with respect to and .
Thus, for a fixed set of functions and , the associated RB operator (LABEL:RB) has now the form
Let be an -space with homogeneous metric . Let be a nonempty connected set and a family of nonempty connected subsets of . Suppose is a family of bijective mappings satisfying property .
Let , and . Define a mapping by
If then the operator is contractive on the complete metric space and its unique fixed point satisfies the self-referential equation
This fixed point is called a local fractal function.
Proof. The statements follow directly from the considerations preceding the theorem.
Note that the local fractal function generated by the operator defined by (LABEL:eq3.4) does not only depend on the family of subsets but also on the two -tuples of bounded functions , and . The fixed point should therefore be written more precisely as . However, for the sake of notational simplicity, we usually suppress this dependence for both and .
The mapping defines a linear isomorphism from to .
Proof. Let and let . Injectivity follows immediately from the fixed point equation (LABEL:3.4) and the uniqueness of the fixed point: , .
Linearity follows from (LABEL:3.4), the uniqueness of the fixed point and injectivity:
For surjectivity, we define , . Since , we have . Thus, .
We may construct local fractal functions on spaces other than . To this end, we assume again that the functions are given by (LABEL:specialv) and that and . We consider the metric on and as being induced by the -norm. Note that endowed with this norm becomes a Banach space.
We have the following result for RB-operators defined on the Lebesgue spaces , .
Let and suppose that is a family of half-open intervals of . Further suppose that is a partition of and that is a family of affine mappings from onto , , and from onto , where maps onto .
The operator defined by
where , and , , maps into itself. Moreover, if
where denotes the Lipschitz constant of , then is contractive on and its unique fixed point is an element of .
Proof. Note that under the hypotheses on the functions and as well as the mappings , is well-defined and an element of . It remains to be shown that under condition (LABEL:condition), is contractive on .
To this end, let and let . Then
Now let . Then
These calculations prove the claims.
The proof of the theorem shows that the conclusions also hold under the assumption that the family of mappings is generated by the following functions.
Each is a bounded diffeomorphism of class , , from to (obvious modification for ). In this case, the ’s are given by , .
Each is a bounded invertible function in , the class of real-analytic functions from to and its inverse is also in . (Obvious modification for .) The ’s are given as above in item .
Next we investigate the set of discontinuities of the fixed point of the RB-operator (LABEL:Phi).
Let be given as in (LABEL:Phi). Assume that for all the are contractive and the are continuous on . Further assume that condition (LABEL:condition) is satisfied for and that the fixed point is bounded everywhere. Then the set of discontinuities of is at most countably infinite.
Proof. Let be a real-valued function and a nonempty open interval contained in its domain. The oscillation of on is defined as
and the oscillation of a function at a point inside an open interval contained in its domain is defined by
The Banach Fixed Point Theorem implies that we may start with any bounded function, say , to construct a sequence of iterates , , which under the given hypotheses, converge in the –norm to the fixed point .
Each iterate may have finite jump discontinuities at the interior knots of the partition and also at the images of the interior knots. The number of possible discontinuities at level is bounded above by since the sets may only contain a subset of the interior knots. Denote by the finite set of all finite jump discontinuities at level and let . Note that is at most countably infinite.
Let and let . The fixed point equation for ,
implies that for all intervals ,
where and . Hence, for any finite code of length , we have that