Tutorial page
To get a good first impression about the possibilities of the WML (Wavelet and Multiscale Library), we suggest to download the GUI (Graphical User Interface) for the MSL (Marburg Software Library). Instructions are coming soon! This page is designed for the student and the scholar. Here, you get first instructions on the mathematical background of ansatz systems with multiscale structure, related topics, and applications. 'Further reading' provides you with a list of research papers that are well suited to start your studies. You may navigate this page by using the directory on the left.
Approximation categories
In many applications, the infinite data of a continuous function has to be approximated by a finite subset of the data or by 'simpler functions'. For instance, a computer can only store a finite amount of data; a radio transmission can only transmit a finite number of frequencies; and many operator equations cannot be solved explicitely, which means they can only be solved approximately with the help of numerical methods that only work on 'simpler functions'.
The choices of permissible 'simpler functions' and the applicable numerical methods determine the so called approximation spaces, i.e., the set of 'simpler functions' and its inherent structure. The approximation method is called linear if the approximation space is a linear space; and the method is called non-linear if the appxomimation space is a non-linear space.
An integral part of approximation theory is to determine the efficiency and compare different approximation categories, which can be done by comparing the approximation rates of the numerical methods, i.e., the ratio between accuracy and complexity of the approximant. Therefore, one investigates the properties of the continuous function or operator equation, e.g., regularity properties in various scales. Especially, when solving operator equations, where the exact solution is only implicitely known, finding estimates of its regularity in specific scales is crucial in order to determine if a specific numerical method even converges to the solution.
Approximation and regularity
To solve operator equations numerically, it is well known that the performance of linear approximation methods depends strongly on the Sobolev regularity of the exact solution and the shape of the domain. If the Sobolev regularity of the given load function of the equation is high, in smooth domains, the Sobolev regularity of the solution will also be high in many cases.
In domains with singularties on the boundary (for example polygonal domains, polyhedral domains, or general Lipschitz domains) this conclusion does not hold anymore: Despite smooth load functions, the Sobolev regularity of the solution might not be very high and therefore the approximation rate of linear schemes will be low. In contrast, the convergence rate of adaptive wavelet schemes depends on the Besov regularity (in specific scales) of the exact solution and, most importantly, this rate is not affected for a broad class of non-smooth domains.
To be precise, the Besov regularity in the scale 1/p = s/d + 1/q (where p is a measure parameter of the Besov space, s is the Besov regularity parameter, and d denotes the dimension of the domain) determines the Lq-convergence rate of adaptive wavelet schemes; which may very well achieve the rate of best n-term approximation, i.e., n^{s/d}. Here, n is the number of degrees of freedom needed to achieve the desired accuracy of the approximant.
Introduction to Wavelets
Consider the space L2 – the space of all square integrable functions over the whole euclidean space.
With the help of a multiresulotion analysis one can construct a basis for this space with a multiscale structure.
First, a finite number (depending on the domain dimension) of functions, called mother wavelets, need to be constructed.
Then, the basis elements – the wavelets – are usually generated by dyadic scalings and integer shifts of the mother wavelets.
For numerical algorithms we apply wavelets basis that are Riesz bases. We require compact support and sufficient differentiability of the wavelets. Such wavelets may also form a basis for other function spaces and can characterize Sobolev or Besov spaces.
Introduction to Shearlets
"Finding optimal representations of signals in higher dimensions is currently the subject of intensive research. An important motivation is to obtain directional representations which capture directional features like orientations of curves in images while providing sparse decompositions. Since wavelets [...] do not provide any directional information, several new representation systems were proposed in the past [...]. The shearlets are an affine system with a single generating mother shearlet function parameterized by a scaling, shear, and translation parameter -- the shear parameter capturing the direction of singularities."
Solving differential equations
Consider an operator mapping from a certain function space into its dual space, that is modelling some real-life dynamics. In many cases, one can only measure the image of a function under the action of the operator, and is then seeking to find suitable preimages. This situation is self-evidently called 'to solve an operator equation'.
As it stands, many operator equations cannot be solved explicitely and therefore one has to apply numerical methods to, at least, compute an approximate solution with a desired accuracy. The first question that arises is: In which function space should one search for suitable preimages, i.e., solutions to the operator equation?
For linear differential operators a common choice are Sobolev spaces. One reason is, the Sobolev smoothness of the (unknown) exact solution determines directly the linear approximation rate of a linear numerical method. Furthermore, the approximation rate of adaptive wavelet schemes can be determined by a specific Besov smoothness of the exact solution. That means, whenever one can deduce information about the structure of the exact solution, e.g., whenever one finds estimates to its regularity in certain scales, one can deduce the approximation rates of related numerical methods.