Harri Hakula

Harri Hakula, Antti Rasila, Matti Vuorinen

We study the problem of computing the exterior modulus of a bounded quadrilateral. We reduce this problem to the numerical solution of the Dirichlet-Neumann problem for the Laplace equation. Several experimental results, with error estimates, are reported. Our main method makes use of an $hp$-FEM algorithm, which enables computations in the case of complicated geometry. For simple geometries, good agreement with computational results based on the SC Toolbox, is observed. We also use the reciprocal error estimation method introduced in our earlier paper to validate our numerical results. In particular, exponential convergence, in accordance with the theory of Babu\vska and Guo, is demonstrated.

Sergei Bezrodnykh, Andrei Bogatyrev, Sergei Goreinov et al.

Making use of two different analytical-numerical methods for capacity computation, we obtain matching to a very high precision numerical values for capacities of a wide family of planar condensers. These two methods are based respectively on the use of the Lauricella function and Riemann theta functions. We apply these results to benchmark the performance of numerical algorithms, which are based on adaptive $hp$--finite element method and boundary integral method.

Harri Hakula, Antti Rasila, Matti Vuorinen

We study the problem of computing the conformal modulus of rings and quadrilaterals with strong singularities and cusps on their boundary. We reduce this problem to the numerical solution of the associated Dirichlet and Dirichlet-Neumann type boundary values problems for the Laplace equation. Several experimental results, with error estimates, are reported. In particular, we consider domains with dendrite like boundaries, in such cases where an analytic formula for the conformal modulus can be derived. Our numerical method makes use of an $hp$-FEM algorithm, written for this very complicated geometry with strong singularities.

Harri Hakula, Tri Quach, Antti Rasila

We present a method for numerical computation of conformal mappings from simply or doubly connected domains onto so-called canonical domains, which in our case are rectangles or annuli. The method is based on conjugate harmonic functions and properties of quadrilaterals. Several numerical examples are given.

Harri Hakula, Mikael Laaksonen

The eigenproblem for thin shells of revolution under uncertainty in material parameters is discussed. Here the focus is on the smallest eigenpairs. Shells of revolution have natural eigenclusters due to symmetries, moreover, the eigenpairs depend on a deterministic parameter, the dimensionless thickness. The stochastic subspace iteration algorithms presented here are capable of resolving the smallest eigenclusters. In the case of random material parameters, it is possible that the eigenmodes cross in the stochastic parameter space. This interesting phenomenon is demonstrated via numerical experiments. Finally, the effect of the chosen material model on the asymptotics in relation to the deterministic parameter is shown to be negligible.

Harri Hakula, Pauliina Ilmonen, Vesa Kaarnioja

This paper lies in the intersection of several fields: number theory, lattice theory, multilinear algebra, and scientific computing. We adapt existing solution algorithms for tensor eigenvalue problems to the tensor-train framework. As an application, we consider eigenvalue problems associated with a class of lattice-theoretic meet and join tensors, which may be regarded as multidimensional extensions of the classically studied meet and join matrices such as GCD and LCM matrices, respectively. In order to effectively apply the solution algorithms, we show that meet tensors have an explicit low-rank tensor-train decomposition with sparse tensor-train cores with respect to the dimension. Moreover, this representation is independent of tensor order, which eliminates the so-called curse of dimensionality from the numerical analysis of these objects and makes the solution of tensor eigenvalue problems tractable with increasing dimensionality and order. For LCM tensors it is shown that a tensor-train decomposition with an a priori known TT rank exists under certain assumptions. We present a series of easily reproducible numerical examples covering tensor eigenvalue and generalized eigenvalue problems that serve as future benchmarks. The numerical results are used to assess the sharpness of existing theoretical estimates.

Jamie Swan, Mohamed M. S. Nasser, Harri Hakula et al.

In this paper, we study the computational question of whether the Steklov spectrum of smooth simply connected planar domains can be approximated accurately by a boundary-only formulation based on harmonic conjugation. For the unit disk, the Dirichlet-to-Neumann operator can be written explicitly in terms of the classical conjugation operator. We show how this viewpoint extends to general bounded and unbounded simply connected domains through the generalized conjugation operator defined through the boundary integral equation with the generalized Neumann kernel. Combined with Fourier differentiation on an equidistant boundary grid, this leads to a dense algebraic eigenvalue problem for the boundary traces of Steklov eigenfunctions. The resulting method uses only boundary data, treats interior and exterior problems in a unified way, and reconstructs eigenfunctions in the domain by harmonic extension. Numerical experiments on benchmark domains and on parameter-dependent smooth families, including ellipses and star-like curves, show high accuracy for smooth boundaries and illustrate how the Steklov spectrum changes with geometry.

Taylor Roper, Harri Hakula, Troy Butler

This work presents novel extensions for combining two frameworks for quantifying both aleatoric (i.e., irreducible) and epistemic (i.e., reducible) sources of uncertainties in the modeling of engineered systems. The data-consistent (DC) framework poses an inverse problem and solution for quantifying aleatoric uncertainties in terms of pullback and push-forward measures for a given Quantity of Interest (QoI) map. Unfortunately, a pre-specified QoI map is not always available a priori to the collection of data associated with system outputs. The data themselves are often polluted with measurement errors (i.e., epistemic uncertainties), which complicates the process of specifying a useful QoI. The Learning Uncertain Quantities (LUQ) framework defines a formal three-step machine-learning enabled process for transforming noisy datasets into samples of a learned QoI map to enable DC-based inversion. We develop a robust filtering step in LUQ that can learn the most useful quantitative information present in spatio-temporal datasets. The learned QoI map transforms simulated and observed datasets into distributions to perform DC-based inversion. We also develop a DC-based inversion scheme that iterates over time as new spatial datasets are obtained and utilizes quantitative diagnostics to identify both the quality and impact of inversion at each iteration. Reproducing Kernel Hilbert Space theory is leveraged to mathematically analyze the learned QoI map and develop a quantitative sufficiency test for evaluating the filtered data. An illustrative example is utilized throughout while the final two examples involve the manufacturing of shells of revolution to demonstrate various aspects of the presented frameworks.

Harri Hakula, Tri Quach, Antti Rasila

The conjugate function method is an algorithm for numerical computation of conformal mappings for simply and doubly connected domains. In this paper the conjugate function method is generalized for multiply connected domains. The key challenge addressed here is the construction of the conjugate domain and the associated conjugate problem. All variants of the method preserve the so-called reciprocal relation of the moduli. An implementation of the algorithm, along with several examples and illustrations are given.

Harri Hakula, Mikael Laaksonen

We consider and analyze applying a spectral inverse iteration algorithm and its subspace iteration variant for computing eigenpairs of an elliptic operator with random coefficients. With these iterative algorithms the solution is sought from a finite dimensional space formed as the tensor product of the approximation space for the underlying stochastic function space, and the approximation space for the underlying spatial function space. Sparse polynomial approximation is employed to obtain the first one, while classical finite elements are employed to obtain the latter. An error analysis is presented for the asymptotic convergence of the spectral inverse iteration to the smallest eigenvalue and the associated eigenvector of the problem. A series of detailed numerical experiments supports the conclusions of this analysis. Numerical experiments are also presented for the spectral subspace iteration, and convergence of the algorithm is observed in an example case, where the eigenvalues cross within the parameter space. The outputs of both algorithms are verified by comparing to solutions obtained by a sparse stochastic collocation method.

Harri Hakula, Matti Leinonen

We consider the construction of the stochastic moment matrices that appear in the typical elliptic diffusion problem considered in the setting of stochastic Galerkin finite element method (sGFEM). Algorithms for the efficient construction of the stochastic moment matrices are presented for certain combinations of affine/non-affine diffusion coefficients and multivariate polynomial spaces. We report the performance of various standard polynomial spaces for three different non-affine diffusion coefficients in a one-dimensional spatial setting and compare observed Legendre coefficient convergence rates to theoretical results.

Harri Hakula, Antti Rasila, Matti Vuorinen

Moduli of rings and quadrilaterals are frequently applied in geometric function theory, see e.g. the Handbook by Kühnau. Yet their exact values are known only in a few special cases. Previously, the class of planar domains with polygonal boundary has been studied by many authors from the point of view of numerical computation. We present here a new $hp$-FEM algorithm for the computation of moduli of rings and quadrilaterals and compare its accuracy and performance with previously known methods such as the Schwarz-Christoffel Toolbox of Driscoll and Trefethen. We also demonstrate that the $hp$-FEM algorithm applies to the case of non-polygonal boundary and report results with concrete error bounds.