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Peter Benner

Publications and source records attributed to Peter Benner.

At least 19 recordsLinked to original sources
arXiv2026

Reduced and Mixed-Precision Block Householder QR Factorization

Recent advances in hardware architecture have highlighted the importance of reduced-precision arithmetic for improving performance and reducing energy consumption in large-scale scientific applications. However, uniformly low-precision methods can suffer from numerical instability, especially in fundamental linear algebra routines. Mixed-precision strategies therefore play an important role in balancing efficiency, stability, and accuracy. In this paper, we focus on the block Householder QR decomposition using the WY representation and study its behavior in reduced- and mixed-precision settings. Starting from a uniform half-precision variant, we show that performing norm computations with fp32 accumulation is necessary to avoid overflow and underflow on large matrices. Based on this observation, we introduce mpWYQR, a mixed-precision algorithm that increases the precision of only selected computations and avoids most of the precision conversions required by the existing mixed-precision blocked QR algorithm (mpBQR3) introduced in [L. M. Yang, A. Fox, and G. Sanders, SIAM J. Sci. Comp., 43.3 (2021), pp. A1723--A1753]. Our analysis shows that mpWYQR accuracy is governed by low-precision unit roundoff, and that its error bound exceeds that of mpBQR3 by only a constant factor, with identical heuristic bounds. In experiments across a range of matrix sizes and condition numbers, we show that mpWYQR retains mpBQR3 accuracy while running up to $18\times$ faster and using less memory. We also discuss the performance cost of computing inner products in higher precision, and the library support needed for an efficient GPU implementation.

math.NA↗
arXiv2026

Reduced-Order Inference with Structure-Preserving Parametrization for Bending and Rotating Systems

Mechanical systems are often characterized only by their response to certain loads known from experiments or simulations. The obtained data can be used for various purposes: system analysis, design of mathematical models, or construction of reduced-order models for further simulations under different loading conditions. The use of data for reduced-order modeling is an important developing research direction, especially when the high-dimensional system operators are unknown and their low-dimensional approximation is required for accurate but fast simulations. Our goal is to obtain the low-dimensional surrogate model from the available input signal and deformation trajectory data, capturing the correct system behavior for the basic deformation cases, namely bending and rotation, which are present in almost every complex mechanical system. In this work, we propose a methodology to infer the system operators by solving a nonlinear unconstrained optimization problem. The methodology is based on the operator inference approach for second-order systems and includes a parametrization of the unknown operators that preserves their symmetric positive definite or skew-symmetric structure. We demonstrate the performance of the novel approach for three numerical examples that are used to simulate basic bending and rotating.

math.DS↗
arXiv2026

SIGGRAPH 23 Extended article: Improved Projective Dynamics Global Using Snapshots-based Reduced Bases

We propose a snapshots-based method to compute reduction subspaces for physics-based simulations. Our method is applicable to any mesh with some artistic prior knowledge of the solution and only requires a record of existing solutions during, for instance, the range-of-motion test that is required before approving a mesh character for an application. Our subspaces span a wider range of motion, especially large deformations, and rotations by default. Compared to the state-of-the-art, we achieve improved numerical stability, computational efficiency, and more realistic simulations with a smaller sub-space.

math.DS↗
arXiv2026

Robust forms for passive descriptor systems

We show how to compute the passivity radius of a given generalized state-space model of a rational transfer function that represents a passive system, both in the continuous-time and discrete-time cases. We then use this to construct robust descriptor realizations. For the continuous-time case, these realizations have a port-Hamiltonian descriptor form. For the discrete-time case a similar result holds for the so-called normalized realizations. The computation of such robust realizations is linked in both these cases to a particular solution of a linear matrix inequality that characterizes the passivity of the transfer function.

math.OC↗
arXiv2026

On the Compatibility of DEIM Subspaces for Nonlinear Forces Reduction in Projective Dynamics

In physics-based simulations, the evaluation of nonlinear constraint forces represents a primary source of computational complexity. Within the framework of projective dynamics, the solver alternates between estimating these forces and updating vertex positions for the subsequent frame. This work investigates the potential of snapshot-based subspace methods to accelerate the computation of constraint terms while maintaining the fidelity of nonlinear dynamics. We focus specifically on the Discrete Empirical Interpolation Method (DEIM), examining whether it can identify a reduced set of constrained mesh elements that enable accurate interpolation and approximation of forces across the full system. Our experiments demonstrate that DEIM is neither intuitive nor consistently effective for constraint projection reduction. In particular, its performance degrades when constraints involve multiple vertices. To validate these observations, we present comparisons between full-order and reduced-order simulations on both surface meshes embedded in three dimensions and volumetric meshes. Furthermore, we report reconstruction errors for various classes of constraints, providing quantitative evidence of the method limitations in this context.

math.DS↗
arXiv2026

An Iterative Active Subspace Approach for Model Order Reduction of Parametric Systems with High-Dimensional Parameter Spaces

The increasing complexity in design and manufacturing has driven the need for advanced techniques for fast modeling problems with large-dimensional parameter spaces. Avoiding high-fidelity finite element models while achieving fast and accurate simulations in such contexts is challenging. Parametric model order reduction (pMOR) has drawn significant attention in recent years. Nevertheless, the curse of dimensionality in parameter spaces has severely limited its effectiveness. The active subspace (AS) approach has been successfully applied to pMOR for systems with many parameters. However, the balance between accuracy and compactness of the reduced model remains problematic for such systems with high-dimensional parameter spaces. It often results in models that are either not small enough or not accurate enough. In this paper, we propose an iterative active subspace (IAS) approach for parametric model order reduction, which, to some extent, addresses the trade-off between accuracy and reduced model size and achieves substantial computational gains compared to the original active subspace method.

math.NA↗
arXiv2026

Tensor-based Approximation of Molecular Kinetics: Generator Learning, Reaction Coordinates and Incremental Updating

We present an approach to analyze long-timescale kinetics of molecular dynamics simulations - meta-stable states and transition timescales - by a tensor-based approximation of the infinitesimal generator. We start from the variational approximation of low-lying generator eigenvalues, and discretize the associated eigenvalue problem using a tensor-product basis. We derive analytical expressions for the resulting tensor operators in tensor train (TT) format, and provide efficient algorithms to solve the corresponding linear problems. We also treat the case of a state-dependent diffusion field, which is relevant when working in reaction coordinates. We show how the number of reaction coordinates can be gradually increased without having to solve the variational problem from scratch. Using MD simulations of fast-folding proteins and TICA coordinates as reaction coordinates, we demonstrate the effectiveness of the proposed method at identifying long-timescale transitions and meta-stable states.

physics.chem-ph↗
arXiv2026

Exponential Decay for a Boundary-Controlled Nonlinear Parabolic Reactor Model

We study an axial dispersion tubular reactor model governed by a nonlinear parabolic equation with Robin-type boundary conditions and boundary feedback control. We derive sufficient conditions for the exponential stability of the steady-state solution of the closed-loop system and provide an explicit estimate of the decay rate. In addition, numerical simulations are presented to illustrate the sharpness of the obtained decay rate for different choices of the feedback gain parameter.

math.AP↗
arXiv2026

Spatially-Enhanced Temporal Fusion Transformer: Interpretable Multi-Output Prediction for Parametric Dynamical Systems with Time-Varying Inputs

We explore the promising performance of a transformer model in predicting outputs of parametric dynamical systems with external time-varying input signals. The outputs of such systems vary not only with physical parameters but also with external time-varying input signals. Accurately catching the dynamics of such systems is challenging. We have adapted and extended an existing transformer model, called temporal fusion transformer (TFT), for single-output prediction to a multiple-output transformer, named as Spatially-Enhanced Temporal Fusion Transformer (SE-TFT), which is able to predict multiple output responses of these systems. The SE-TFT generalizes the interpretability of the original TFT model. The generalized interpretable attention weight matrix explores not only the temporal correlations in the sequence, but also the interactions between the multiple outputs, providing explanation for the spatial correlation in the output domain. This proposed SE-TFT accurately predicts the sequence of multiple outputs, regardless of the nonlinearity of the system and the dimensionality of the parameter space.

cs.LG↗
arXiv2026

Boundary control and periodic trajectories of semilinear Euler equations with application to hydrogen transport

A mathematical model of gas flow in a pipeline controlled by the inlet pressure and the outlet mass flow flux is considered in the form of the isothermal Euler equations with an appropriate equation of state. Within the framework of boundary control systems, this model is transformed into a nonlinear abstract differential equation using an appropriate lifting operator. For this abstract equation, a representation in terms of Fourier coefficients is derived analytically. Conditions for the existence of periodic solutions to a broad class of nonlinear control systems with continuously differentiable controls are established in abstract spaces. This framework is applied to a realistic hydrogen transport model to evaluate periodic operating regimes under periodic fluctuations in supply and demand.

math.OC↗
arXiv2026

Computation of structured stability radii for Dissipative-Hamiltonian systems

We study linear time-invariant Dissipative Hamiltonian (DH) systems arising in energy-based modeling of dynamical systems. An advantage of DH systems is that they are always stable due to the structure of their coefficient matrices, and, under further weak conditions, even asymptotically stable. In this paper, we discuss the computation of the stability radii for a given asymptotically stable DH system; i.e., the smallest structured perturbation that puts a DH system on the boundary of the region of asymptotic stability, so that it has purely imaginary eigenvalues. We obtain explicit computable formulas for various structured stability radii. For this, the problem of computing stability radii is reformulated in terms of minimizing the Rayleigh quotient of a Hermitian matrix or the sum of two generalized Rayleigh quotients of Hermitian semidefinite matrices. This reformulation results in the problem of minimizing the largest eigenvalue of an eigenvector-dependent Hermitian matrix or minimizing the smallest eigenvalue of a Hermitian matrix which depends on the eigenvector. It is also demonstrated (via numerical experiments) that, under structure-preserving perturbations, the asymptotic stability of a DH system is much more robust than under general perturbations, since the distance to instability is typically much larger when structure-preserving perturbations are considered. Finally, similar results are obtained for optimally robust representations of stable systems.

math.OC↗
arXiv2026

A Local Discontinuous Galerkin Method for Dirichlet Boundary Control Problems

In this paper, we consider control constrained $L^2-$Dirichlet boundary control of a convection-diffusion equation on a two dimensional convex polygonal domain. We discretize the control problem based on the local discontinuous Galerkin method with piecewise linear ansatz functions for the flux and potential. We derive a priori error estimates for the full as well as for the variational discrete control approximation. We present a selection of numerical results to demonstrate the performance of our approach and to underpin the theoretical findings.

math.OC↗
arXiv2026

Partial Observation of Linear Systems with the Mori-Zwanzig Formalism

The Mori-Zwanzig formalism provides a systematic framework for deriving reduced-order model of dynamical systems when only part of the state is observed, but its practical use is often limited by the complexity of the resulting computations. This paper develops an explicit formulation of the Mori-Zwanzig equation for linear time-invariant systems under partially observed observables. By expressing the dynamics in terms of observables, the Koopman generator, and projections onto resolved and unresolved components, we derive closed-form representations of the Markovian, noise, and memory contributions that arise in the Mori-Zwanzig identity. For the linear setting, the resulting formulas recover the reduced dynamics obtained from the variation-of-constants formula while retaining the operator-based structure of the Mori-Zwanzig approach. This makes the derivation a transparent reference case for reduced-order modelling with memory and clarifies how unresolved variables influence the observed dynamics through history-dependent terms. The analysis also identifies the ingredients needed for extensions to nonlinear systems and more general projections, including spectral filtering and data-driven approximations of memory effects. Analytical and numerical examples involving the harmonic oscillator and wave equations illustrate the construction and demonstrate how the formalism can be used to obtain interpretable reduced-order models for partially observed systems.

math.DS↗
arXiv2026

Low Precision Fortran -- Enabling Low Precision Floating Point Arithmetic in Modern Fortran

Although Fortran is almost 70 years old, the language continues to evolve in order to keep pace with developments in computer science. In particular, a flexible type system was introduced that allows developers to specify the sizes of floating-point numbers and integers. In the latest revisions of the Fortran standard, portable type variants for IEEE 754 binary64 (double precision, real64) and binary32 (single precision, real32) were added. However, the rapid development of AI toolkits and accelerator hardware has created a strong focus on floating-point types of lower precision and lower memory usage than binary32. While the IEEE 754-2019 standard defines the binary16 type for representing half-precision numbers, the Fortran standard does not provide the real16 variant in the type system. In contrast, most C compilers support such a data type. In numerical linear algebra, there is strong interest in exploiting the high performance of accelerator devices for core algorithms like matrix decompositions or iterative solvers. Especially when the performance ratio between double, single, and half precision is on the order of 1:2:20, as on current NVidia H100 accelerators, it becomes highly beneficial to use lower-precision types. Yet, before performance can be targeted, correctness and accuracy must be verified when operating below single precision. In this article, we present our Low Precision Fortran (LPF) library that enables the use of low-precision types -- binary16, bfloat16, fp8_e4m3, and fp8_e5m2 -- just like any other floating-point type in Fortran. Furthermore, we introduce extensions that support BLAS operations in low precision and show how easily existing routines can be rewritten to use these data types.

cs.MS↗
arXiv2026

Periodic solutions of nonlinear control systems with switching: a Lie-algebraic and contraction approach

This paper is devoted to the analysis of periodic solutions of nonlinear control-affine systems with bang-bang controls. Such problems naturally arise in periodic optimal control with constrained inputs, which have, in particular, important applications in the performance optimization of chemical reactions. We reduce the problem of constructing a periodic solution to that of finding a fixed point of a composition of exponential maps. The latter problem is then addressed using the Baker-Campbell-Hausdorff-Dynkin (BCHD) formula. We establish the equivalence between periodic solutions of the original control system and those of an associated autonomous system involving iterated Lie brackets. Applying incremental stability arguments allows us to further simplify the problem to finding the equilibria of this autonomous system. The developed theory is then applied to nonlinear chemical reaction models with constrained controls.

math.OC↗
arXiv2026

A hybrid Chebyshev-Tucker tensor format for approximation of multivariate functions

We introduce and analyze a mesh-free two-level hybrid Chebyshev-Tucker tensor representation for approximating multivariate functions, which combines tensor-product Chebyshev interpolation with the low-rank Tucker decomposition of the tensor of Chebyshev coefficients. This construction allows to avoid the expensive rank-structured grid-based approximation of function-related tensors on large spatial grids, while benefiting from the Tucker decomposition of the moderate-sized core tensor of Chebyshev coefficients. Thus, we can compute the nearly optimal Tucker decomposition of the 3D function with controllable accuracy $\varepsilon >0$ without discretizing the function on a full fine grid in the domain, but only using its values at a small set of Chebyshev nodes computed either from the explicit analytic expression of the target function or from its data-sparse representation in a rank-structured tensor format with moderate rank parameter. Finally, we can represent the function in the algebraic Tucker format with optimal $\varepsilon$-rank on an arbitrarily large 3D tensor grid in the computational domain by discretizing the Chebyshev polynomials on that grid. The rank parameters of the nonlinear Tucker-ALS decomposition of the coefficient tensor can be much smaller than the polynomial degrees of the initial Chebyshev linear interpolation in the function independent polynomial basis set. It is shown that our techniques can be gainfully applied to the long-range part of the singular electrostatic potential of multi-particle systems represented on a fine grid in the range-separated (RS) tensor format. We provide error and complexity estimates and demonstrate the computational efficiency of the proposed techniques on challenging examples, including the collective electrostatic potential for large bio-molecular systems and lattice-type compounds.

math.NA↗
arXiv2026

Mixed-precision iterative refinement for low-rank Lyapunov equations

We develop a mixed-precision iterative refinement framework for solving low-rank Lyapunov matrix equations $AX + XA^T + W =0$, where $W=LL^T$ or $W=LSL^T$. Via rounding error analysis of the algorithms we derive sufficient conditions for the attainable normwise residuals in different precision settings and show how the algorithmic parameters should be chosen. These conditions are independent of the choice of inner solver, provided that the prescribed residual accuracy is attained in the inner solves. Using the sign-function Newton iteration as the solver, we demonstrate that reduced precisions, such as half precision with unit roundoff $u_s$, can be used efficiently for Lyapunov equations with condition numbers of order $1/u_s$ without compromising the attainable solution quality. This provides an algorithmic framework towards exploiting native low-precision hardware to accelerate Lyapunov solvers without sacrificing accuracy.

math.NA↗