arXiv subjects

Roland Maier

Publications and source records attributed to Roland Maier.

At least 19 recordsLinked to original sources

arXiv2026

Enriched higher-order multiscale approaches with applications to wave propagation

We consider the numerical solution of partial differential equations with coefficients that are strongly heterogeneous in space. We provide an overview of higher-order localized orthogonal decomposition (LOD) methods for the elliptic setting, including recent advancements, and then present a generalization of the strategy to linear hyperbolic multiscale problems. We address the limitations of earlier constructions for the wave equation, which only achieve second-order convergence in space, independent of the chosen polynomial degree. Building on the methodology of enriched corrections recently developed for parabolic multiscale problems, we motivate and propose an enriched higher-order LOD method for the wave equation. The enriched corrections exhibit exponential decay and can be computed on patches. Under minimal assumptions on the coefficient and standard well-preparedness conditions on the data, we derive a priori error estimates that achieve optimal high-order convergence rates, thereby overcoming the previously observed saturation of the convergence rate. With the fifth-order Rosenbrock-Wanner (ROW) time integrator, we conduct a series of numerical examples to verify our theoretical results. We provide examples showing the optimal spatial convergence of the method including the localization errors for different polynomial orders. We also present examples showing the optimal convergence rates of the time discretization.

math.NA↗

arXiv2026

A post-processed higher-order multiscale method for nondivergence-form elliptic equations

We study the finite element approximation of linear second-order elliptic partial differential equations in nondivergence form with highly heterogeneous diffusion and drift coefficients. A generalized Cordes condition is imposed to guarantee that a suitably renormalized version of the nondivergence-form differential operator is near the Laplacian. Based on a stabilized symmetric formulation for the gradient that enables the use of $H^1$-conforming approximation spaces, we construct a multiscale method following the methodology of the localized orthogonal decomposition with coarse basis functions tailored to the heterogeneous coefficients. We employ a novel post-processing strategy to obtain higher-order convergence rates, overcoming previous limitations imposed by the low regularity of the load functional. Numerical experiments demonstrate the performance of the method.

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arXiv2026

Complexity bounds on neural networks for the solution of structured linear systems of equations

We derive upper bounds on the complexity of ReLU neural networks approximating the solution of a linear system given the matrix and the right-hand side. We focus on matrices which are symmetric positive definite and sparse, as they appear in the context of finite difference and finite element methods. For such matrices, we extend available results for the matrix inversion to the task of solving a linear system, where we leverage favorable properties of classical methods such as the modified Richardson and the conjugate gradient method. Our bounds on the number of layers and neurons are not only explicit with respect to the size of the matrices, but also with respect to their condition numbers.

math.NA↗

arXiv2026

GLENN: Neural network-enhanced computation of Ginzburg-Landau energy minimizers

In this work, we propose a neural network-enhanced finite element strategy to compute the minimizer of the Ginzburg-Landau energy based on an unsupervised deep Ritz-type strategy. We treat the parameter $\kappa$ as a variable input parameter to obtain possible minimizers for a large range of $\kappa$-values. This allows for two possible strategies: 1) The neural network may be extensively trained to work as a stand-alone solver. 2) Neural network results are used as starting values for a subsequent classical iterative minimization procedure. The latter strategy particularly circumvents the missing reliability of the neural network-based approach. Numerical examples are presented that show the potential of the proposed strategy.

math.NA↗

arXiv2025

PUFM++: Point Cloud Upsampling via Enhanced Flow Matching

Recent advances in generative modeling have demonstrated strong promise for high-quality point cloud upsampling. In this work, we present PUFM++, an enhanced flow-matching framework for reconstructing dense and accurate point clouds from sparse, noisy, and partial observations. PUFM++ improves flow matching along three key axes: (i) geometric fidelity, (ii) robustness to imperfect input, and (iii) consistency with downstream surface-based tasks. We introduce a two-stage flow-matching strategy that first learns a direct, straight-path flow from sparse inputs to dense targets, and then refines it using noise-perturbed samples to approximate the terminal marginal distribution better. To accelerate and stabilize inference, we propose a data-driven adaptive time scheduler that improves sampling efficiency based on interpolation behavior. We further impose on-manifold constraints during sampling to ensure that generated points remain aligned with the underlying surface. Finally, we incorporate a recurrent interface network~(RIN) to strengthen hierarchical feature interactions and boost reconstruction quality. Extensive experiments on synthetic benchmarks and real-world scans show that PUFM++ sets a new state of the art in point cloud upsampling, delivering superior visual fidelity and quantitative accuracy across a wide range of tasks. Code and pretrained models are publicly available at https://github.com/Holmes-Alan/Enhanced_PUFM.

cs.CV↗

arXiv2025

Optimal higher-order convergence rates for parabolic multiscale problems

In this paper, we introduce a higher-order multiscale method for time-dependent problems with highly oscillatory coefficients. Building on the localized orthogonal decomposition (LOD) framework, we construct enriched correction operators to enrich the multiscale spaces, ensuring higher-order convergence without requiring assumptions on the coefficient beyond boundedness. This approach addresses the challenge of a reduction of convergence rates when applying higher-order LOD methods to time-dependent problems. Addressing a parabolic equation as a model problem, we prove the exponential decay of these enriched corrections and establish rigorous a priori error estimates. Numerical experiments confirm our theoretical results.

math.NA↗

arXiv2025

A Generalized Framework for Higher-Order Localized Orthogonal Decomposition Methods

We introduce a generalized framework for studying higher-order versions of the multiscale method known as Localized Orthogonal Decomposition. Through a suitable reformulation, we are able to accommodate both conforming and nonconforming constraints in the construction process. In particular, we offer a new perspective on localization strategies. We fully analyze the strategy for linear elliptic problems and discuss extensions to the Helmholtz equation and the Gross--Pitaevskii eigenvalue problem. Numerical examples are presented that particularly provide valuable comparisons between conforming and nonconforming constraints.

math.NA↗

arXiv2024

Neural numerical homogenization based on Deep Ritz corrections

Numerical homogenization methods aim at providing appropriate coarse-scale approximations of solutions to (elliptic) partial differential equations that involve highly oscillatory coefficients. The localized orthogonal decomposition (LOD) method is an effective way of dealing with such coefficients, especially if they are non-periodic and non-smooth. It modifies classical finite element basis functions by suitable fine-scale corrections. In this paper, we make use of the structure of the LOD method, but we propose to calculate the corrections based on a Deep Ritz approach involving a parametrization of the coefficients to tackle temporal variations or uncertainties. Numerical examples for a parabolic model problem are presented to assess the performance of the approach.

math.NA↗

arXiv2024

Multiscale Corrections by Continuous Super-Resolution

Finite element methods typically require a high resolution to satisfactorily approximate micro and even macro patterns of an underlying physical model. This issue can be circumvented by appropriate multiscale strategies that are able to obtain reasonable approximations on under-resolved scales. In this paper, we study the implicit neural representation and propose a continuous super-resolution network as a correction strategy for multiscale effects. It can take coarse finite element data to learn both in-distribution and out-of-distribution high-resolution finite element predictions. Our highlight is the design of a local implicit transformer, which is able to learn multiscale features. We also propose Gabor wavelet-based coordinate encodings, which can overcome the bias of neural networks learning low-frequency features. Finally, perception is often preferred over distortion, so scientists can recognize the visual pattern for further investigation. However, implicit neural representation is known for its lack of local pattern supervision. We propose to use stochastic cosine similarities to compare the local feature differences between prediction and ground truth. It shows better performance on structural alignments. Our experiments show that our proposed strategy achieves superior performance as an in-distribution and out-of-distribution super-resolution strategy.

math.NA↗

arXiv2024

A Higher-Order Multiscale Method for the Wave Equation

In this paper we propose a multiscale method for the acoustic wave equation in highly oscillatory media. We use a higher-order extension of the localized orthogonal decomposition method combined with a higher-order time stepping scheme and present rigorous a-priori error estimates in the energy-induced norm. We find that in the very general setting without additional assumptions on the coefficient beyond boundedness, arbitrary orders of convergence cannot be expected but that increasing the polynomial degree may still considerably reduce the size of the error. Under additional regularity assumptions, higher orders can be obtained as well. Numerical examples are presented that confirm the theoretical results.

math.NA↗

arXiv2024

Localized implicit time stepping for the wave equation

This work proposes a discretization of the acoustic wave equation with possibly oscillatory coefficients based on a superposition of discrete solutions to spatially localized subproblems computed with an implicit time discretization. Based on exponentially decaying entries of the global system matrices and an appropriate partition of unity, it is proved that the superposition of localized solutions is appropriately close to the solution of the (global) implicit scheme. It is thereby justified that the localized (and especially parallel) computation on multiple overlapping subdomains is reasonable. Moreover, a re-start is introduced after a certain amount of time steps to maintain a moderate overlap of the subdomains. Overall, the approach may be understood as a domain decomposition strategy in space on successive short time intervals that completely avoids inner iterations. Numerical examples are presented.

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arXiv2023

An algebraic multiscale method for spatial network models

In this work, we present a multiscale approach for the reliable coarse-scale approximation of spatial network models represented by a linear system of equations with respect to the nodes of a graph. The method is based on the ideas of the Localized Orthogonal Decomposition (LOD) strategy and is constructed in a fully algebraic way. This allows to apply the method to geometrically challenging objects such as corrugated cardboard. In particular, the method can also be applied to finite difference or finite element discretizations of elliptic partial differential equations, yielding an approximation with similar properties as the LOD in the continuous setting. We present a rigorous error analysis of the proposed method under suitable assumptions on the network. Moreover, numerical examples are presented that underline our theoretical results.

math.NA↗

arXiv2023

A localized orthogonal decomposition strategy for hybrid discontinuous Galerkin methods

We formulate and analyze a multiscale method for an elliptic problem with an oscillatory coefficient based on a skeletal (hybrid) formulation. More precisely, we employ hybrid discontinuous Galerkin approaches and combine them with the localized orthogonal decomposition methodology to obtain a coarse-scale skeletal method that effectively includes fine-scale information. This work is the first step in reliably merging hybrid skeletal formulations and localized orthogonal decomposition to unite the advantages of both strategies. Numerical experiments are presented to illustrate the theoretical findings.

math.NA↗

arXiv2023

An improved high-order method for elliptic multiscale problems

In this work, we propose a high-order multiscale method for an elliptic model problem with rough and possibly highly oscillatory coefficients. Convergence rates of higher order are obtained using the regularity of the right-hand side only. Hence, no restrictive assumptions on the coefficient, the domain, or the exact solution are required. In the spirit of the Localized Orthogonal Decomposition, the method constructs coarse problem-adapted ansatz spaces by solving auxiliary problems on local subdomains. More precisely, our approach is based on the strategy presented by Maier [SIAM J. Numer. Anal. 59(2), 2021]. The unique selling point of the proposed method is an improved localization strategy curing the effect of deteriorating errors with respect to the mesh size when the local subdomains are not large enough. We present a rigorous a priori error analysis and demonstrate the performance of the method in a series of numerical experiments.

math.NA↗

arXiv2022

Neural network approximation of coarse-scale surrogates in numerical homogenization

Coarse-scale surrogate models in the context of numerical homogenization of linear elliptic problems with arbitrary rough diffusion coefficients rely on the efficient solution of fine-scale sub-problems on local subdomains whose solutions are then employed to deduce appropriate coarse contributions to the surrogate model. However, in the absence of periodicity and scale separation, the reliability of such models requires the local subdomains to cover the whole domain which may result in high offline costs, in particular for parameter-dependent and stochastic problems. This paper justifies the use of neural networks for the approximation of coarse-scale surrogate models by analyzing their approximation properties. For a prototypical and representative numerical homogenization technique, the Localized Orthogonal Decomposition method, we show that one single neural network is sufficient to approximate the coarse contributions of all occurring coefficient-dependent local sub-problems for a non-trivial class of diffusion coefficients up to arbitrary accuracy. We present rigorous upper bounds on the depth and number of non-zero parameters for such a network to achieve a given accuracy. Further, we analyze the overall error of the resulting neural network enhanced numerical homogenization surrogate model.

math.NA↗

arXiv2022

Adaptive Refinement for Unstructured T-Splines with Linear Complexity

We present an adaptive refinement algorithm for T-splines on unstructured 2D meshes. While for structured 2D meshes, one can refine elements alternatingly in horizontal and vertical direction, such an approach cannot be generalized directly to unstructured meshes, where no two unique global mesh directions can be assigned. To resolve this issue, we introduce the concept of direction indices, i.e., integers associated to each edge, which are inspired by theory on higher-dimensional structured T-splines. Together with refinement levels of edges, these indices essentially drive the refinement scheme. We combine these ideas with an edge subdivision routine that allows for I-nodes, yielding a very flexible refinement scheme that nicely distributes the T-nodes, preserving global linear independence, analysis-suitability (local linear independence) except in the vicinity of extraordinary nodes, sparsity of the system matrix, and shape regularity of the mesh elements. Further, we show that the refinement procedure has linear complexity in the sense of guaranteed upper bounds on a) the distance between marked and additionally refined elements, and on b) the ratio of the numbers of generated and marked mesh elements.

math.NA↗

arXiv2022

Operator Compression with Deep Neural Networks

This paper studies the compression of partial differential operators using neural networks. We consider a family of operators, parameterized by a potentially high-dimensional space of coefficients that may vary on a large range of scales. Based on existing methods that compress such a multiscale operator to a finite-dimensional sparse surrogate model on a given target scale, we propose to directly approximate the coefficient-to-surrogate map with a neural network. We emulate local assembly structures of the surrogates and thus only require a moderately sized network that can be trained efficiently in an offline phase. This enables large compression ratios and the online computation of a surrogate based on simple forward passes through the network is substantially accelerated compared to classical numerical upscaling approaches. We apply the abstract framework to a family of prototypical second-order elliptic heterogeneous diffusion operators as a demonstrating example.

math.NA↗

arXiv2021

Multiscale Scattering in Nonlinear Kerr-Type Media

We propose a multiscale approach for a nonlinear Helmholtz problem with possible oscillations in the Kerr coefficient, the refractive index, and the diffusion coefficient. The method does not rely on structural assumptions on the coefficients and combines the multiscale technique known as Localized Orthogonal Decomposition with an adaptive iterative approximation of the nonlinearity. We rigorously analyze the method in terms of well-posedness and convergence properties based on suitable assumptions on the initial data and the discretization parameters. Numerical examples illustrate the theoretical error estimates and underline the practicability of the approach.

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