293 research outputs found
Numerical methods for boundary value problems on random domains
No full textIn this thesis, we consider the numerical solution of
elliptic boundary value problems on random domains.
The underlying domain is modelled
via a random vector field which is given by its mean
and its covariance.
Having these statistics of the random perturbation at
hand, we aim at determining the related statistics of
the random solution.
To that end, we propose the domain mapping method
on the one hand and the perturbation method on the
other hand.
For the domain mapping method, we have to compute the
random vector field's Karhunen-Loève expansion.
For this purpose, we compare cluster methods, namely
the adaptive cross approximation and the fast multipole
method, and the pivoted Cholesky decomposition.
After this, we show regularity results for the random
solution dependent on the decay of the random vector
field's Karhunen-Loève expansion. These results are used
to employ a Quasi-Monte Carlo quadrature for the
approximation of mean and variance.
For the perturbation method, we linearize the random
solution's dependence on the vector field by means of
a shape Taylor expansion. This approach yields a single
partial differential equation for the approximation of
the mean and a tensor product partial differential
equation for the approximation of the covariance. The latter
is solved efficiently with the aid of the sparse tensor
product combination technique
Preconditioning of wavelet BEM by the incomplete Cholesky factorization
No full textThe present paper is dedicated to the preconditioning of boundary element matrices which are given in wavelet coordinates. We investigate the incomplete Cholesky factorization for a pattern which includes also the coefficients of all off-diagonal bands associated with the level-level-interactions. The pattern is chosen in such a way that the incomplete Cholesky factorization is computable in log-linear complexity. Numerical experiments are performed to quantify the effects of the proposed preconditioning. The present paper is dedicated to the preconditioning of boundary element matrices which are given in wavelet coordinates. We investigate the incomplete Cholesky factorization for a pattern which includes also the coefficients of all off-diagonal bands associated with the level-level-interactions. The pattern is chosen in such a way that the incomplete Cholesky factorization is computable in log-linear complexity. Numerical experiments are performed to quantify the effects of the proposed preconditioning
Trial methods for Bernoulli's free boundary problem
No full textFree boundary problems deal with solving partial differential equations in a domain,
a part of whose boundary is unknown – the so-called free boundary. Beside
the standard boundary conditions that are needed in order to solve the partial
differential equation, an additional boundary condition is imposed at the free
boundary. One aims thus to determine both, the free boundary and the solution
of the partial differential equation.
This thesis is dedicated to the solution of the generalized exterior Bernoulli
free boundary problem which is an important model problem for developing algorithms
in a broad band of applications such as optimal design, fluid dynamics,
electromagnentic shaping etc. Due to its various advantages in the analysis and
implementation, the trial method, which is a fixed-point type iteration method,
has been chosen as numerical method.
The iterative scheme starts with an initial guess of the free boundary. Given
one boundary condition at the free boundary, the boundary element method is
applied to compute an approximation of the violated boundary data. The free
boundary is then updated such that the violated boundary condition is satisfied
at the new boundary. Taylor’s expansion of the violated boundary data around
the actual boundary yields the underlying equation, which is formulated as an
optimization problem for the sought update function. When a target tolerance is
achieved the iterative procedure stops and the approximate solution of the free
boundary problem is detected.
How efficient or quick the trial method is converging depends significantly
on the update rule for the free boundary, and thus on the violated boundary
condition. Firstly, the trial method with violated Dirichlet data is examined and
updates based on the first and the second order Taylor expansion are performed.
A thorough analysis of the convergence of the trial method in combination with
results from shape sensitivity analysis motivates the development of higher order
convergent versions of the trial method. Finally, the gained experience is
exploited to draw very important conclusions about the trial method with violated
Neumann data, which is until now poorly explored and has never been
numerically implemented
A Note on the Construction of L -Fold Sparse Tensor Product Spaces
No full textIn the present paper, we consider the construction of general sparse tensor product spaces in arbitrary space dimensions when the single subdomains are of different dimensionality and the associated ansatz spaces possess different approximation properties. Our theory extends the results from Griebel and Harbrecht (Math. Comput., 2013) for the construction of two-fold sparse tensor product space to arbitrary L-fold sparse tensor product space
A note on the construction of L-fold sparse tensor product spaces
No full textIn the present paper, we consider the construction of general sparse tensor product spaces in arbitrary space dimensions when the single subdomains are of different dimensionality and theassociated ansatz spaces possess different approximation properties. Our theory extends the results from Griebel and Harbrecht [Math. Comput., 82(282):975-994, 2013] for the construction of two-fold sparse tensor product space to arbitrary L-fold sparse tensor product spaces
A second order convergent trial method for a free boundary problem in three dimensions
No full textThe present article is concerned with the solution of a generalized Bernoulli free boundary problem in three spatial dimensions. We parametrize the free boundary under consideration over the sphere and apply a trial method which updates the free boundary into the normal direction. At the free boundary, we prescribe the Neumann boundary condition and update the free boundary with the help of the remaining Dirichlet boundary condition. An inexact Newton update is employed which leads to a novel second order convergent trial method. Numerical examples show the feasibility of the present approach. In particular, a parametric representation is utilized which imposes no restriction to the free boundary under consideration except for its genus
Biorthogonal wavelet bases for the boundary element method
No full textAs shown by Dahmen, Harbrecht and Schneider, the fully discrete wavelet Galerkin scheme for boundary integral equations scales linearly with the number of unknowns without compromising the accuracy of the underlying Galerkin scheme. The supposition is a wavelet basis with a sufficiently large number of vanishing moments. In this paper we present several constructions of appropriate wavelet bases on manifolds based on the biorthogonal spline wavelets of A. Cohen, I. Daubechies and J.-C. Feauveau. By numerical experiments we demonstrate that it is worthwhile to spent effort on their construction to increase the performance of the wavelet Galerkin scheme considerably
Biorthogonal Wavelet Bases for the Boundary Element Method
No full textAs shown by Dahmen, Harbrecht and Schneider [7, 23, 32], the fully discrete wavelet Galerkin scheme for boundary integral equations scales linearly with the number of unknowns without compromising the accuracy of the underlying Galerkin scheme. The supposition is a wavelet basis with a sufficiently large number of vanishing moments. In this paper we present several constructions of appropriate wavelet bases on manifolds based on the biorthogonal spline wavelets of A. Cohen, I. Daubechies and J.-C. Feauveau [4]. By numerical experiments we demonstrate that it is worthwhile to spent effort on their construction to increase the performance of the wavelet Galerkin scheme considerably
On the convergence of the combination technique
No full textSparse tensor product spaces provide an efficient tool todiscretize higher dimensional operator equations. The direct Galerkin method in such ansatz spaces may employ hierarchical bases, interpolets, wavelets or multilevel frames. Besides, an alternative approach is provided by the so-called combination technique. It properly combines the Galerkin solutions of the underlying problem on certain full (but small) tensor product spaces. So far, however, the combination technique has been analyzed only for special model problems. In the present paper, we provide now the analysis of the combination technique for quite general operator equations in sparse tensor product spaces. We prove that the combination technique produces the same order of convergence as the Galerkin approximation with respect to the sparse tensor product space. Furthermore, the order of the cost complexity is the same as for the Galerkin approach in the sparse tensor product space. Our theoretical findings are validated by numerical experiments
LOW-RANK APPROXIMATION OF CONTINUOUS FUNCTIONS IN SOBOLEV SPACES WITH DOMINATING MIXED SMOOTHNESS
No full text17291746Let Ωi ⊂ Rni, i = 1,…,m, be given domains. In this article, we study the low-rank approximation with respect to L2(Ω1 × ·· · × Ωm) of functions from Sobolev spaces with dominating mixed smoothness. To this end, we first estimate the rank of a bivariate approximation, i.e., the rank of the continuous singular value decomposition. In comparison to the case of functions from Sobolev spaces with isotropic smoothness, compare Griebel and Harbrecht [IMA J. Numer. Anal. 34 (2014), pp. 28–54] and Griebel and Harbrecht [IMA J. Numer. Anal. 39 (2019), pp. 1652–1671], we obtain improved results due to the additional mixed smoothness. This convergence result is then used to study the tensor train decomposition as a method to construct multivariate low-rank approximations of functions from Sobolev spaces with dominating mixed smoothness. We show that this approach is able to beat the curse of dimension9234
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