arXiv subjects

Harri Hakula

Publications and source records attributed to Harri Hakula.

At least 19 recordsLinked to original sources

arXiv2026

Copula Transformations for Data-Consistent Inversion

Data-consistent inversion (DCI) constructs probability measures whose push-forward distributions agree with observed data, while iterative data-consistent inversion (iDCI) extends this framework to generalized stochastic inverse problems by enforcing multiple push-forward constraints sequentially. Although iDCI avoids the direct approximation of high-dimensional joint densities, its relationship to the original joint DCI solution has remained unclear. In this work, we establish this relationship through copula theory. Using Sklar's theorem, we derive a factorization of the DCI update into separate marginal and dependence transformations and show that the discrepancy remaining after convergence of the iDCI algorithm is entirely characterized by the copulas associated with the observed and predicted joint distributions. This characterization motivates a copula-transformed iDCI solution, and we prove that an exact copula transformation recovers the original DCI solution. We further establish convergence results for approximate copula transformations under converging sequences of reference measures and progressively enriched feasible sets. Numerical examples demonstrate how the geometry induced by the quantity-of-interest map governs the importance of the copula transformation, illustrate an adaptive reference-measure refinement strategy for improving computational accuracy under a fixed sampling budget, and demonstrate the progressive refinement of generalized stochastic inverse problems through heterogeneous, asynchronously acquired experiments.

stat.ML↗

arXiv2026

A boundary integral equation method for Steklov eigenvalue problems for smooth planar domains

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.

math.NA↗

arXiv2025

Fast capacity computation for maze-like configurations

We study the conformal capacity ${\rm cap}(\Omega,K)$ where $\Omega$ is a bounded domain of $\mathbb{R}^2$ and $K$ is a compact connected set in $\Omega$. Because the exact numerical value of the capacity is known only in a handful of special cases, it is important to find estimates for the capacity in terms of domain functionals, simpler than the capacity itself. Here, we study condensers of maze-like structure and compute their capacity by means of a high-order $hp$- finite element method. We compare these numerical results to the estimates given by the quasihyperbolic length and perimeter of the compact set. In particular, we consider the behaviour of these value pairs, numerical results and estimates, when the structure parameters vary and the walls of the maze approach the compact set. Over the configurations covered in the numerical experiments, the quasihyperbolic estimates are shown to have the desired asymptotic properties and superior computational efficiencies once the case-specific analysis is completed.

math.NA↗

arXiv2024

Constrained maximization of conformal capacity

We consider constellations of disks which are unions of disjoint hyperbolic disks in the unit disk with fixed radii and unfixed centers. We study the problem of maximizing the conformal capacity of a constellation with a fixed number of disks under constraints on the centers in two cases. In the first case the constraint is that the centers are at most at distance $R \in(0,1)$ from the origin and in the second case it is required that the centers are on the subsegment $[-R,R]$ of a diameter of the unit disk. We study also similar types of constellations with hyperbolic segments instead of the hyperbolic disks. Our computational experiments suggest that a dispersion phenomenon occurs: the disks/segments go as close to the unit circle as possible under these constraints and stay as far as possible from each other. The computation of capacity reduces to the Dirichlet problem for the Laplace equation which we solve using two methods: a fast boundary integral equation method and a high-order finite element method.

math.CV↗

arXiv2024

Laplace--Beltrami Equations and Numerical Conformal Mappings on Surfaces

The conjugate function method is an algorithm for numerical computation of conformal mappings for simply and multiply connected domains. In this paper, the conjugate function method is extended to cover conformal mappings between Riemannian surfaces. The main challenge addressed here is the connection between Laplace--Beltrami equations on surfaces and the computation of the conformal modulus of a quadrilateral. We consider mappings of simply, doubly, and multiply connected domains. The numerical computation is based on an $hp$-adaptive finite element method. The key advantage of our approach is that it allows highly accurate computations of mappings on surfaces, including domains of complex boundary geometry involving strong singularities and cusps. The efficacy of the proposed method is illustrated via an extensive set of numerical experiments including error estimates.

math.NA↗

arXiv2024

From Displacements to Distributions: A Machine-Learning Enabled Framework for Quantifying Uncertainties in Parameters of Computational Models

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.

stat.ML↗

arXiv2023

Mobile disks in hyperbolic space and minimization of conformal capacity

Our focus is to study constellations of disjoint disks in the hyperbolic space, the unit disk equipped with the hyperbolic metric. Each constellation corresponds to a set $E$ which is the union of $m>2$ disks with hyperbolic radii $r_j>0, j=1,...,m$. The centers of the disks are not fixed and hence individual disks of the constellation are allowed to move under the constraints that they do not overlap and their hyperbolic radii remain invariant. Our main objective is to find computational lower bounds for the conformal capacity of a given constellation. The capacity depends on the centers and radii in a very complicated way even in the simplest cases when $m=3$ or $m=4$. In the absence of analytic methods our work is based on numerical simulations using two different numerical methods, the boundary integral equation method and the $hp$-FEM method, resp. Our simulations combine capacity computation with minimization methods and produce extremal cases where the disks of the constellation are grouped next to each other. This resembles the behavior of animal colonies minimizing heat flow in arctic areas.

math.CV↗

arXiv2023

Uncertainty quantification for random domains using periodic random variables

We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently introduced by Kaarnioja, Kuo, and Sloan (SIAM J. Numer. Anal., 2020) in which a countably infinite number of independent random variables enter the random field as periodic functions. We develop lattice quasi-Monte Carlo (QMC) cubature rules for computing the expected value of the solution to the Poisson problem subject to domain uncertainty. These QMC rules can be shown to exhibit higher order cubature convergence rates permitted by the periodic setting independently of the stochastic dimension of the problem. In addition, we present a complete error analysis for the problem by taking into account the approximation errors incurred by truncating the input random field to a finite number of terms and discretizing the spatial domain using finite elements. The paper concludes with numerical experiments demonstrating the theoretical error estimates.

math.NA↗

arXiv2022

Conformal capacity and polycircular domains

We study numerical conformal mapping of multiply connected planar domains with boundaries consisting of unions of finitely many circular arcs, so called polycircular domains. We compute the conformal capacities of condensers defined by polycircular domains. Experimental error estimates are provided for the computed capacity and, when possible, the rate of convergence under refinement of discretisation is analysed. The main ingredients of the computation are two computational methods, on one hand the boundary integral equation method combined with the fast multipole method and on the other hand the $hp$-FEM method. The results obtained with these two methods agree with high accuracy.

math.NA↗

arXiv2021

Conformal Moduli of Symmetric Circular Quadrilaterals With Cusps

We investigate moduli of planar circular quadrilaterals symmetric with respect to both the coordinate axes. First we develop an analytic approach which reduces this problem to ODEs and devise a numeric method to find out the accessory parameters. This method uses the Schwarz equation to determine conformal mapping of the unit disk onto a given circular quadrilateral. We also give an example of a circular quadrilateral for which the value of the conformal modulus can be found in the analytic form; this example is used to validate the numeric calculations. We also use another method, so called hpFEM, for the numeric calculation of the moduli. These two different approaches provide results agreeing with high accuracy.

math.NA↗

arXiv2021

Stochastic collocation method for computing eigenspaces of parameter-dependent operators

We consider computing eigenspaces of an elliptic self-adjoint operator depending on a countable number of parameters in an affine fashion. The eigenspaces of interest are assumed to be isolated in the sense that the corresponding eigenvalues are separated from the rest of the spectrum for all values of the parameters. We show that such eigenspaces can in fact be extended to complex-analytic functions of the parameters and quantify this analytic dependence in way that leads to convergence of sparse polynomial approximations. A stochastic collocation method on an anisoptropic sparse grid in the parameter domain is proposed for computing a basis for the eigenspace of interest. The convergence of this method is verified in a series of numerical examples based on the eigenvalue problem of a stochastic diffusion operator.

math.NA↗

arXiv2020

A Posteriori Error Estimates for Elliptic Eigenvalue Problems Using Auxiliary Subspace Techniques

We propose an a posteriori error estimator for high-order $p$- or $hp$-finite element discretizations of selfadjoint linear elliptic eigenvalue problems that is appropriate for estimating the error in the approximation of an eigenvalue cluster and the corresponding invariant subspace. The estimator is based on the computation of approximate error functions in a space that complements the one in which the approximate eigenvectors were computed. These error functions are used to construct estimates of collective measures of error, such as the Hausdorff distance between the true and approximate clusters of eigenvalues, and the subspace gap between the corresponding true and approximate invariant subspaces. Numerical experiments demonstrate the practical effectivity of the approach.

math.NA↗

arXiv2018

On capacity computation for symmetric polygonal condensers

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.

math.NA↗

arXiv2018

Multiparametric shell eigenvalue problems

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.

math.NA↗

arXiv2017

Asymptotic convergence of spectral inverse iterations for stochastic eigenvalue problems

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.

math.NA↗

arXiv2017

Computation of extremal eigenvalues of high-dimensional lattice-theoretic tensors via tensor-train decompositions

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.

math.NA↗

arXiv2016

Spatial Mappings for Planning and Optimization of Cellular Networks

In cellular networks, users are grouped into different cells and served by different access points (base stations) that provide wireless access to services and applications. In general, the service demand is very heterogeneous, non-uniformly distributed, and dynamic. Consequently, radio access networks create very irregular topologies with more access points where service demand is concentrated. While this dynamism requires networks with the ability to adapt to time-varying conditions, the non-uniformity of the service demand makes the planning, analysis, and optimization difficult. In order to help with these tasks, a framework based on canonical domains and spatial mappings (e.g., conformal mapping) have recently been proposed. The idea is to carry out part of the planning in a canonical (perfectly symmetric) domain that is connected to the physical one (real-scenario) by means of a spatial transformation designed to map the access points consistently with the service demand. This paper continues the research in that direction by introducing additional tools and possibilities to that framework, namely the use of centroidal Voronoi algorithms and non-conformal composite mappings. Moreover, power optimization is also introduced to the framework. The results show the usability and effectiveness of the proposed method and its promising research perspectives.

cs.NI↗

arXiv2015

On efficient construction of stochastic moment matrices

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.

math.NA↗