Efficient numerical approximation of solutions to high-dimensional partial differential equations

Subtitle:with applications in option pricing and scattering problems

A lot of phenomena observed around us can be described in terms of mathematical problems or equations. Although the computational power to numerically solve these problems has extensively increased over the past decades, the mathematical equations to solve have become more and more complicated. The aim of this thesis is to study and develop efficient numerical methods to approximate solutions to high-dimensional partial differential equations (PDEs). The approximation techniques under consideration are based upon two, distinct approaches. The first idea replaces a single high-dimensional partial differential operator by a linear combination of multiple low-dimensional partial differential operators. Inspired by the principal component analysis (PCA), Reisinger and Wittum suggest a transformation of the covariance matrix that appear in the multi-dimensional Black-Scholes equation. In financial applications the eigenvalue corresponding to the first principal component is often dominant and this observation will be exploited in this first approximation approach. It turns out that neglecting all other principal components does not yield a good approximation but adding first-order corrections yields a good PCA-based approximation for the Black-Scholes operator. The main advantage of this PCA-based approximation approach is that an analytical approximation to the solution of the Black-Scholes PDE is obtained in terms of solutions to only one- and two-dimensional PDEs. These PDEs are independent of each other and can therefore be solved in parallel. The second approach restricts the rank of the solution of a differential equation and derives a differential equation for the low-rank components. For example, a numerical representation of the solution of a two-dimensional problem on a certain grid can be represented by a matrix. From that matrix a singular value decomposition can be computed to obtain the singular values with the left- and right singular vectors. It is observed that the solution of certain Helmholtz problems that appear in scattering problems are of low rank. Thus instead of solving a differential equation on a full grid the differential equation is projected on the space spanned by the other factor matrices. This leads to an equation for the remaining low-rank factor of the solution. The equation for this low-rank factor can be related to equations the arise in the coupled channel technique. This idea for two-dimensional problems can be extended to larger dimensional problems where we obtain a low-rank Tucker tensor representation of the solution.

Publication year:2022

Keywords:Doctoral thesis

Accessibility:Open