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Mario Ohlberger

Publications and source records attributed to Mario Ohlberger.

At least 19 recordsLinked to original sources
arXiv2026

MaRDI Open Interfaces for Interoperable Nonlinear Optimization

MaRDI Open Interfaces is a software package that aims to improve interoperability in scientific computing, particularly, for nonlinear optimization. To this end, this package holds two main characteristics. First, it provides unified interfaces for typical numerical problems to help switching between solvers for the same problem type. Second, it automates data marshalling between programming languages. Hence, computational scientists can conduct experiments faster by using the package, with fewer code-modification and testing efforts. In this work we describe the general structure of the software package and show examples with the interface for nonlinear optimization.

cs.MS↗
arXiv2026

Software package MaRDI Open Interfaces for improved interoperability in numerical optimization

To address the challenges of interoperability in computational science, we present the latest updates to the software package MaRDI Open Interfaces. This software package aims to decrease the time and coding/testing efforts spent by computational scientists on tasks such as writing bindings to numerical solvers and adapting experiment codes to the varying interfaces of solvers for the same problem type (e.g., for benchmarking, which solver is better). By streamlining these tasks, this software package helps researchers focus on the actual essence of their computational projects. Here, we demonstrate a recently developed interface for nonlinear optimization and illustrate how it can be applied for computational experiments with optimization problems. As an example of such problem, we consider training of physics-informed neural networks to predict the solutions of viscous Burgers' equation.

cs.MS↗
arXiv2026

Adaptive Reduced-Basis Trust-Region Methods for Defect Identification in Elastic Materials

Monitoring the integrity of elastic structures using ultrasonic waves requires the efficient identification of material parameters from measured surface displacements. The displacement field is governed by Cauchy's equation of motion, i.e., an elastic wave equation. Consequently, defect localization leads to a high-dimensional spatial parameter identification problem for a hyperbolic system with given initial and boundary conditions. Stable parameter reconstructions typically rely on regularization techniques such as the iteratively regularized Gauss-Newton method (IRGNM). However, its practical application is computationally demanding due to the high-dimensional nature of the problem. To address this bottleneck, we propose a reduced-order modeling approach that simultaneously reduces the state and parameter spaces using adaptively constructed reduced-basis spaces. This yields online-efficient surrogate models for both the forward and adjoint evaluations required in derivative-based optimization. To ensure reliability, the IRGNM iteration is embedded into an adaptive, trust-region framework that provides accuracy of the reduced-order approximations. The approach extends our recent contributions, which focus on elliptic and parabolic problems, to the hyperbolic setting. We demonstrate the reliability and effectiveness of the method for defect detection through numerical experiments.

math.NA↗
arXiv2026

A New Adaptive Deep Learning based Reduced Order Model for Hybrid-Type Parabolic PDEs: Rigorous Error Analysis and Applications

This contribution proposes novel data-driven surrogate modeling approaches for parameterized parabolic PDEs, where the parameter dependence can be split into two parts with different decay behavior of the Kolmogorov $N$-width. Such problems naturally arise in many industrial flow processes with dominant advection or traveling fronts in the solution trajectories. To tackle this challenge, we extend the Deep Orthogonal Decomposition (DOD) method, recently introduced for related stationary problems, to the time-dependent setting. We introduce and rigorously analyze two DOD based approaches: Our approach is based on two novel adaptive deep learning-based surrogate models: The DOD-DL-ROM method which is a Reduced Order Model (ROM) that leverages the adaptive nature of DOD, and the DOD+DFNN method, which combines DOD with a generic Deep Feed-Forward Neural Network (DFNN). On the theory side, we generalize data-driven POD-based ROM arguments to the DOD setting, establishing a quantitative link between online performance and the regularity of an associated optimal map. Furthermore, we identify specific problem size and error tolerance requirements for DOD-based ROMs to outperform POD-based ROMs in hybrid-type problem classes, which is crucial for efficient computation. The significance of this work lies in its potential to accelerate the solution of complex PDEs, enabling faster design and optimization of industrial processes. The proposed approaches are demonstrated on a catalyst filter benchmark problem, showcasing their effectiveness and comparing favorably to traditional POD-based methods.

math.NA↗
arXiv2025

A Parareal Algorithm with Low-Rank Coarse Solvers

We consider a new class of Parareal algorithms, which use ideas from localized reduced basis methods to construct the coarse solver from truncated SVD approximations of the transfer operators mapping initial values for a given time interval to the solution at the end of the interval. By leveraging randomized singular value decompositions, these low-rank approximations are obtained embarrassingly parallel by computing local fine solutions for random initial values. We show a priori and a posteriori error bounds in terms of the computed singular values of the transfer operators. Our numerical experiments demonstrate that our approach can significantly outperform Parareal with single-step coarse solvers. At the same time, it permits to further increase parallelism in Parareal by trading global iterations for a larger number of independent local solves.

math.NA↗
arXiv2025

Adaptive Reduced Basis Trust Region Methods for Parabolic Inverse Problems

We consider nonlinear inverse problems arising in the context of parameter identification for parabolic partial differential equations (PDEs). For stable reconstructions, regularization methods such as the iteratively regularized Gauss-Newton method (IRGNM) are commonly used, but their application is computationally demanding due to the high-dimensional nature of PDE discretizations. To address this bottleneck, we propose a reduced-order modeling approach that accelerates both the state and adjoint evaluations required for derivative-based optimization. Our method builds on the recent contribution [Kartmann et al. Adaptive reduced basis trust region methods for parameter identification problems. Comput. Sci. Eng. 1, 3 (2024)] for elliptic forward operators and constructs the reduced forward operator adaptively in an online fashion, combining both parameter and state space reduction. To ensure reliability, we embed the IRGNM iteration within an adaptive, error-aware trust-region framework that certifies the accuracy of the reduced-order approximations. We demonstrate the effectiveness of the proposed approach through numerical results for both time-dependent and time-independent parameter identification problems in dynamic reaction-diffusion systems. The implementation is made available for reproducibility and further use.

math.NA↗
arXiv2025

Sectional Kolmogorov N-widths for parameter-dependent function spaces: A general framework with application to parametrized Friedrichs' systems

We investigate parametrized variational problems where for each parameter the solution may originate from a different parameter-dependent function space. Our main motivation is the theory of Friedrichs' systems, a large abstract class of linear PDE-problems whose solutions are sought in operator- (and thus parameter-)dependent graph spaces. Other applications include function spaces on parametrized domains or discretizations involving data-dependent stabilizers. Concerning the set of all parameter-dependent solutions, we argue that in these cases the interpretation as a "solution manifold" widely adopted in the model order reduction community is no longer applicable. Instead, we propose a novel framework based on the theory of fiber bundles and explain how established concepts such as approximability generalize by introducing a Sectional Kolmogorov N-width. Further, we prove exponential approximation rates of this N-width if a norm equivalence criterion is fulfilled. Applying this result to problems with Friedrichs' structure then gives a sufficient criterion that can be easily verified.

math.NA↗
arXiv2025

Improving Interoperability in Scientific Computing via MaRDI Open Interfaces

MaRDI Open Interfaces is a software project aimed at improving reuse and interoperability in Scientific Computing by alleviating the difficulties of crossing boundaries between different programming languages, in which numerical packages are usually implemented, and of switching between multiple implementations of the same mathematical problem. The software consists of a set of formal interface specifications for common Scientific Computing tasks, as well as a set of loosely coupled libraries that facilitate implementing these interfaces or adapting existing implementations for multiple programming languages and handle data marshalling automatically without sacrificing performance, enabling users to use different implementations without significant code efforts. The software has high reuse potential due to aim to solve general numerical problems.

cs.MS↗
arXiv2025

Multi-fidelity Learning of Reduced Order Models for Parabolic PDE Constrained Optimization

This article builds on the recently proposed RB-ML-ROM approach for parameterized parabolic PDEs and proposes a novel hierarchical Trust Region algorithm for solving parabolic PDE constrained optimization problems. Instead of using a traditional offline/online splitting approach for model order reduction, we adopt an active learning or enrichment strategy to construct a multi-fidelity hierarchy of reduced order models on-the-fly during the outer optimization loop. The multi-fidelity surrogate model consists of a full order model, a reduced order model and a machine learning model. The proposed hierarchical framework adaptively updates its hierarchy when querying parameters, utilizing a rigorous a posteriori error estimator in an error aware trust region framework. Numerical experiments are given to demonstrate the efficiency of the proposed approach.

math.OC↗
arXiv2024

Adaptive Model Hierarchies for Multi-Query Scenarios

In this contribution we present an abstract framework for adaptive model hierarchies together with several instances of hierarchies for specific applications. The hierarchy is particularly useful when integrated within an outer loop, for instance an optimization iteration or a Monte Carlo estimation where for a large set of requests answers fulfilling certain criteria are required.

math.NA↗
arXiv2024

A Parareal algorithm without Coarse Propagator?

The Parareal algorithm was invented in 2001 in order to parallelize the solution of evolution problems in the time direction. It is based on parallel fine time propagators called F and sequential coarse time propagators called G, which alternatingly solve the evolution problem and iteratively converge to the fine solution. The coarse propagator G is a very important component of Parareal, as one sees in the convergence analyses. We present here for the first time a Parareal algorithm without coarse propagator, and explain why this can work very well for parabolic problems. We give a new convergence proof for coarse propagators approximating in space, in contrast to the more classical coarse propagators which are approximations in time, and our proof also applies in the absence of the coarse propagator. We illustrate our theoretical results with numerical experiments, and also explain why this approach can not work for hyperbolic problems.

math.NA↗
arXiv2024

Construction of local reduced spaces for Friedrichs' systems via randomized training

This contribution extends the localized training approach, traditionally employed for multiscale problems and parameterized partial differential equations (PDEs) featuring locally heterogeneous coefficients, to the class of linear, positive symmetric operators, known as Friedrichs' operators. Considering a local subdomain with corresponding oversampling domain we prove the compactness of the transfer operator which maps boundary data to solutions on the interior domain. While a Caccioppoli-inequality quantifying the energy decay to the interior holds true for all Friedrichs' systems, showing a compactness result for the graph-spaces hosting the solution is additionally necessary. We discuss the mixed formulation of a convection-diffusion-reaction problem where the necessary compactness result is obtained by the Picard-Weck-Weber theorem. Our numerical results, focusing on a scenario involving heterogeneous diffusion fields with multiple high-conductivity channels, demonstrate the effectiveness of the proposed method.

math.NA↗
arXiv2024

Local training and enrichment based on a residual localization strategy

To efficiently tackle parametrized multi and/or large scale problems, we propose an adaptive localized model order reduction framework combining both local offline training and local online enrichment with localized error control. For the latter, we adapt the residual localization strategy introduced in [Buhr, Engwer, Ohlberger, Rave, SIAM J. Sci. Comput., 2017] which allows to derive a localized a posteriori error estimator that can be employed to adaptively enrich the reduced solution space locally where needed. Numerical experiments demonstrate the potential of the proposed approach.

math.NA↗
arXiv2023

Model order reduction of an ultraweak and optimally stable variational formulation for parametrized reactive transport problems

This contribution introduces a model order reduction approach for an advection-reaction problem with a parametrized reaction function. The underlying discretization uses an ultraweak formulation with an $L^2$-like trial space and an 'optimal' test space as introduced by Demkowicz et al. This ensures the stability of the discretization and in addition allows for a symmetric reformulation of the problem in terms of a dual solution which can also be interpreted as the normal equations of an adjoint least-squares problem. Classic model order reduction techniques can then be applied to the space of dual solutions which also immediately gives a reduced primal space. We show that the necessary computations do not require the reconstruction of any primal solutions and can instead be performed entirely on the space of dual solutions. We prove exponential convergence of the Kolmogorov $N$-width and show that a greedy algorithm produces quasi-optimal approximation spaces for both the primal and the dual solution space. Numerical experiments based on the benchmark problem of a catalytic filter confirm the applicability of the proposed method.

math.NA↗
arXiv2023

Adaptive Reduced Basis Trust Region Methods for Parameter Identification Problems

In this contribution, we are concerned with model order reduction in the context of iterative regularization methods for the solution of inverse problems arising from parameter identification in elliptic partial differential equations. Such methods typically require a large number of forward solutions, which makes the use of the reduced basis method attractive to reduce computational complexity. However, the considered inverse problems are typically ill-posed due to their infinite-dimensional parameter space. Moreover, the infinite-dimensional parameter space makes it impossible to build and certify classical reduced-order models efficiently in a so-called "offline phase". We thus propose a new algorithm that adaptively builds a reduced parameter space in the online phase. The enrichment of the reduced parameter space is naturally inherited from the Tikhonov regularization within an iteratively regularized Gau{\ss}-Newton method. Finally, the adaptive parameter space reduction is combined with a certified reduced basis state space reduction within an adaptive error-aware trust region framework. Numerical experiments are presented to show the efficiency of the combined parameter and state space reduction for inverse parameter identification problems with distributed reaction or diffusion coefficients.

math.NA↗
arXiv2023

A Relaxed Localized Trust-Region Reduced Basis Approach for Optimization of Multiscale Problems

In this contribution, we are concerned with parameter optimization problems that are constrained by multiscale PDE state equations. As an efficient numerical solution approach for such problems, we introduce and analyze a new relaxed and localized trust-region reduced basis method. Localization is obtained based on a Petrov-Galerkin localized orthogonal decomposition method and its recently introduced two-scale reduced basis approximation. We derive efficient localizable a posteriori error estimates for the optimality system, as well as for the two-scale reduced objective functional. While the relaxation of the outer trust-region optimization loop still allows for a rigorous convergence result, the resulting method converges much faster due to larger step sizes in the initial phase of the iterative algorithms. The resulting algorithm is parallelized in order to take advantage of the localization. Numerical experiments are given for a multiscale thermal block benchmark problem. The experiments demonstrate the efficiency of the approach, particularly for large scale problems, where methods based on traditional finite element approximation schemes are prohibitive or fail entirely.

math.NA↗
arXiv2023

An optimally stable approximation of reactive transport using discrete test and infinite trial spaces

In this contribution we propose an optimally stable ultraweak Petrov-Galerkin variational formulation and subsequent discretization for stationary reactive transport problems. The discretization is exclusively based on the choice of discrete approximate test spaces, while the trial space is a priori infinite dimensional. The solution in the trial space or even only functional evaluations of the solution are obtained in a post-processing step. We detail the theoretical framework and demonstrate its usage in a numerical experiment that is motivated from modeling of catalytic filters.

math.NA↗