arXiv subjects

Tom-Christian Riemer

Publications and source records attributed to Tom-Christian Riemer.

6 recordsLinked to original sources

arXiv2026

Projection-based low-rank assembly in IgA

Isogeometric Analysis (IgA) uses the same spline functions to represent the computational domain and to approximate the solution. This allows exact geometry descriptions, but the resulting mass and stiffness matrices are expensive to assemble and to store, especially in three dimensions. We present a projection-based low-rank approach for assembling the mass and stiffness tensors of orientation-preserving tensor-product B-spline geometries. For the mass tensor, we exploit the polynomial structure of the determinant of the Jacobian of the geometry map and represent it in reduced spline product spaces by univariate coefficient transfer operators; with exact quadrature and without truncation, the resulting low-rank tensor is an exact reformulation of the standard Galerkin mass tensor. For the stiffness tensor, we split the rational weight function into a polynomial numerator, again represented in reduced spline product spaces, and the reciprocal determinant, which is in general not a spline function and is therefore approximated by an $L^2$-projection onto a tensor-product spline space. Both constructions are carried out entirely in the tensor-train (TT) format, with the projection system solved by the alternating minimal energy (AMEn) method, so that full high-order coefficient tensors are never formed and the multidimensional integrals reduce to univariate integrals and contracted products. The method is implemented in MATLAB using GeoPDEs and the TT-Toolbox. Numerical experiments show that it is competitive with full assembly and with the interpolation-based low-rank method, and that it applies in two situations in which interpolation is problematic: a singular interpolation system and nearly singular geometries. The construction is restricted to orientation-preserving tensor-product B-spline geometries and does not cover NURBS.

math.NA↗

arXiv2026

Adaptive Krylov Methods for Low-Rank Exponential Integrators

Differential equations arise in numerous applications, particularly within scientific and technical contexts. Systems of stiff, time-dependent ordinary differential equations constitute the focus of this work. Exponential integrators are designed to solve such equations by integrating the linear part exactly, while simultaneously approximating the nonlinear part through a linear combination of $φ$-functions. By utilizing an augmented stiffness matrix, state-of-the-art methods like KIOPS and RK2EXPINT solve the linear part and evaluate linear combinations of $φ$-functions for the nonlinear part in a single step, effectively reducing the computational effort to a single matrix exponential evaluation. However, these classical approaches assume that the system is represented by matrices and vectors, potentially not utilizing the underlying high-dimensional structure. Tensors address this limitation and offer significant storage efficiency through well-established decompositions like the Tensor Train (TT) format. This work provides a general framework for solving stiff, time-dependent systems directly within the TT format. Specifically, KIOPS-TT and RK2EXPINT-TT are developed as extensions of the original KIOPS and RK2EXPINT algorithms. This involves reformulating the scheme of explicit exponential Runge-Kutta integrators for tensors and augmenting the stiffness tensor to compute linear combinations of $φ$-functions acting on tensors through a single evaluation of the exponential function using Krylov subspace methods. Furthermore, it is shown that the underlying theory of the matrix methods remains valid, thereby enabling the transfer of key theorems to the tensor case. Numerical experiments confirm significant speed-ups for KIOPS-TT and RK2EXPINT-TT in low-rank scenarios compared to their classical counterparts.

math.NA↗

arXiv2025

A Low-Rank tensor framework for THB-Splines

We introduce a low-rank framework for adaptive isogeometric analysis with truncated hierarchical B-splines (THB-splines) that targets the main bottleneck of local refinement: memory- and time-intensive matrix assembly once the global tensor-product structure is lost. The method interpolates geometry-induced weight and source terms in separable spline spaces and computes their tensor-train (TT) representations via the alternating minimal energy (AMEn) solver, enabling level-wise assembly of system operators using univariate quadrature. To recover separability in the adaptive setting, we reduce the active basis to tensor-product domains and partition active/non-active cells into a small number of Cartesian cuboids, so each contributes a Kronecker factor that is accumulated and rounded in TT. We realize the two-scale relation with truncation in low rank and assemble the global hierarchical operators in a block TT format suitable for iterative solvers. A prototype MATLAB implementation built on the GeoPDEs package and the TT-Toolbox demonstrates that, for model problems with moderately complex refinement regions, the approach reduces memory footprint and assembly time while maintaining accuracy; we also discuss limitations when ranks grow with geometric or refinement complexity. This framework advances scalable adaptive IgA with THB-splines, particularly in three dimensions.

math.NA↗

arXiv2025

Physics-Informed DeepONets for drift-diffusion on metric graphs: simulation and parameter identification

We develop a novel physics informed deep learning approach for solving nonlinear drift-diffusion equations on metric graphs. These models represent an important model class with a large number of applications in areas ranging from transport in biological cells to the motion of human crowds. While traditional numerical schemes require a large amount of tailoring, especially in the case of model design or parameter identification problems, physics informed deep operator networks (DeepONet) have emerged as a versatile tool for the solution of partial differential equations with the particular advantage that they easily incorporate parameter identification questions. We here present an approach where we first learn three DeepONet models for representative inflow, inner and outflow edges, resp., and then subsequently couple these models for the solution of the drift-diffusion metric graph problem by relying on an edge-based domain decomposition approach. We illustrate that our framework is applicable for the accurate evaluation of graph-coupled physics models and is well suited for solving optimization or inverse problems on these coupled networks.

cs.LG↗

arXiv2024

IETI-based Low-Rank method for PDE-constrained optimization

Isogeometric Analysis (IgA) is a versatile method for the discretization of partial differential equations on complex domains, which arise in various applications of science and engineering. Some complex geometries can be better described as a computational domain by a multi-patch approach, where each patch is determined by a tensor product Non-Uniform Rational Basis Splines (NURBS) parameterization. This allows on the one hand to consider the problem of the complex assembly of mass or stiffness matrices (or tensors) over the whole geometry locally on the individual smaller patches, and on the other hand it is possible to perform local mesh refinements independently on each patch, allowing efficient local refinement in regions of high activity where higher accuracy is required, while coarser meshes can be used elsewhere. Furthermore, the information about differing material models or properties that are to apply in a subdomain of the geometry can be included in the patch in which this subdomain is located. For this it must be ensured that the approximate solution is continuous over the entire computational domain and therefore at the interfaces of two (or more) patches. The most promising approach for this problem, which transfers the idea of Finite Element Tearing and Interconnecting (FETI) methods into the isogeometric setup, was the IsogEometric Tearing and Interconnecting (IETI) method, where by introducing a constraints matrix and associated Lagrange multipliers and formulating it into a dual problem, depending only on the Lagrange multipliers, continuity at the interfaces was ensured in solving the resulting system. In this paper we illustrate that low-rank methods based on the tensor-train format can be generalised for a multi-patch IgA setup, which follows the IETI idea.

math.NA↗

arXiv2022

A comparison of PINN approaches for drift-diffusion equations on metric graphs

In this paper we focus on comparing machine learning approaches for quantum graphs, which are metric graphs, i.e., graphs with dedicated edge lengths, and an associated differential operator. In our case the differential equation is a drift-diffusion model. Computational methods for quantum graphs require a careful discretization of the differential operator that also incorporates the node conditions, in our case Kirchhoff-Neumann conditions. Traditional numerical schemes are rather mature but have to be tailored manually when the differential equation becomes the constraint in an optimization problem. Recently, physics informed neural networks (PINNs) have emerged as a versatile tool for the solution of partial differential equations from a range of applications. They offer flexibility to solve parameter identification or optimization problems by only slightly changing the problem formulation used for the forward simulation. We compare several PINN approaches for solving the drift-diffusion on the metric graph.

stat.ML↗