arXiv subjects

Matthias Kabel

Publications and source records attributed to Matthias Kabel.

7 recordsLinked to original sources

arXiv2025

Performance Benchmarking of Tensor Trains for accelerated Quantum-Inspired Homogenization on TPU, GPU and CPU architectures

Recent advances in high-resolution CT-imaging technology are creating a new class of ultra-high resolved microstructural datasets that challenge the limits of traditional homogenization approaches. While state-of-the-art FFT-based homogenization techniques remain effective for moderate datasets, their memory footprint and computational cost grow rapidly with increasing resolution, making them progressively inefficient for industrial-scale problems. To address these challenges, the recently developed Superfast-Fourier Transform (SFFT)-based homogenization algorithm leverages the memory-efficient low-rank representations of Tensor Trains (TTs), which reduce the storage and computational requirements of large-scale homogenization problems. Developed for CPU usage, SFFT-based Homogenization efficiently handles high-resolution datasets, assuming the underlying data is well-behaved. In this work, we investigate the performance of fundamental TT operations on modern hardware accelerators using the JAX framework. A benchmarking study across CPUs, GPUs, and TPUs evaluates execution times and computational efficiency, highlighting the strengths and limitations of TT operations on different architectures and motivating future hybrid approaches. Building on these insights, we adapt the SFFT-based homogenization algorithm for accelerator execution, enabling homogenization at high resolutions ranging from 300 million to 70 billion grid points, which are infeasible for the best available GPU-based FFT reference implementation. While the observed scaling behavior is geometry-dependent, the results demonstrate the potential of accelerator-based quantum-inspired homogenization for high-performance multiscale simulations.

cond-mat.mtrl-sci↗

arXiv2024

SFFT-Based Homogenization: Using Tensor Trains to Enhance FFT-Based Homogenization

Homogenization is a fundamental technique for estimating the macroscopic properties of materials with microscale heterogeneity. Among Homogenization methods, the FFT-based Homogenization algorithm has become widely used due to its computational efficiency and ability to handle complex microstructures. Nevertheless, even with GPU acceleration, FFT-based Homogenization for industrial applications remains excessively time-consuming, particularly when generating elastic training data for AI models. This is due to the curse of dimensionality, which arises from the algorithms reliance on the Fast Fourier Transform, creating a fundamental bottleneck. In this paper, we propose a quantum-inspired SFFT-based Homogenization algorithm that leverages the improved time complexity of a Tensor Train variant of the Quantum Fourier Transform. By additionally exploiting structural properties of the underlying microstructure, our method achieves exponential improvements in time complexity and memory efficiency compared to the traditional FFT-based technique - all while remaining executable on classical hardware. We evaluate the performance of our algorithm across increasingly complex microstructures, demonstrating its potential advantages and limitations.

cond-mat.mtrl-sci↗

arXiv2024

Piecewise Polynomial Tensor Network Quantum Feature Encoding

This work introduces a novel method for embedding continuous variables into quantum circuits via piecewise polynomial features, utilizing low-rank tensor networks. Our approach, termed Piecewise Polynomial Tensor Network Quantum Feature Encoding (PPTNQFE), aims to broaden the applicability of quantum algorithms by incorporating spatially localized representations suited for numerical applications like partial differential equations and function regression. We demonstrate the potential of PPTNQFE through efficient point evaluations of solutions of discretized differential equations and in modeling functions with localized features such as jump discontinuities. While promising, challenges such as unexplored noise impact and design of trainable circuits remain. This study opens new avenues for enhancing quantum models with novel feature embeddings and leveraging TN representations for a wider array of function types in quantum machine learning.

quant-ph↗

arXiv2022

A Performance Study of Variational Quantum Algorithms for Solving the Poisson Equation on a Quantum Computer

Recent advances in quantum computing and their increased availability has led to a growing interest in possible applications. Among those is the solution of partial differential equations (PDEs) for, e.g., material or flow simulation. Currently, the most promising route to useful deployment of quantum processors in the short to near term are so-called hybrid variational quantum algorithms (VQAs). Thus, variational methods for PDEs have been proposed as a candidate for quantum advantage in the noisy intermediate scale quantum (NISQ) era. In this work, we conduct an extensive study of utilizing VQAs on real quantum devices to solve the simplest prototype of a PDE -- the Poisson equation. Although results on noiseless simulators for small problem sizes may seem deceivingly promising, the performance on quantum computers is very poor. We argue that direct resolution of PDEs via an amplitude encoding of the solution is not a good use case within reach of today's quantum devices -- especially when considering large system sizes and more complicated non-linear PDEs that are required in order to be competitive with classical high-end solvers.

quant-ph↗

arXiv2022

QFT-based Homogenization

Efficient numerical characterization is a key problem in composite material analysis. To follow accuracy improvement in image tomography, memory efficient methods of numerical characterization have been developed. Among them, an FFT based solver has been proposed by Moulinec and Suquet (1994,1998) bringing down numerical characterization complexity to the FFT complexity. Nevertheless, recent development of tomography sensors made memory requirement and calculation time reached another level. To avoid this bottleneck, the new leaps in the field of Quantum Computing have been used. This paper will present the application of the Quantum Fourier Transform (QFT) to replace the Fast Fourier Transform (FFT) in Moulinec and Suquet algorithm. It will mainly focused on how to read out Fourier coefficients stored in a quantum state. First, a reworked Hadamard test algorithm applied with Most likelihood amplitude estimation (MLQAE) is used to determine the quantum coefficients. Second, an improvement avoiding Hadamard test is presented in case of Material characterization on mirrored domain. Finally, this last algorithm is applied to Material homogenization to determine effective stiffness of basic geometries.

math.NA↗

arXiv2022

FFT-based Homogenization at Finite Strains using Composite Boxels (ComBo)

Computational homogenization is the gold standard for concurrent multi-scale simulations (e.g., FE2) in scale-bridging applications. Experimental and synthetic material microstructures are often represented by 3D image data. The computational complexity of simulations operating on such three-dimensional high-resolution voxel data comprising billions of unknowns induces the need for algorithmically and numerically efficient solvers. The inability of voxelized 3D geometries to capture smooth material interfaces accurately, along with the necessity for complexity reduction, motivates a special local coarse-graining technique called composite voxels [Kabel,M. et al. (2015)]. Composite voxels condense multiple fine-scale voxels into a single voxel obeying a theory-inspired constitutive model by employing laminate theory. Composite voxels enhance local field quality at a modest computational cost. Our contribution comprises the generalization towards composite boxels (ComBo) that are nonequiaxed, a feature that can pay off for materials with a preferred direction. A novel image-based normal detection algorithm is devised which improves the accuracy by around 30\% against the orientation cf. [Kabel,M. et al. (2015) ]. Further, the use of ComBo for finite strain simulations is studied in detail. An efficient implementation is proposed, and an essential back-projection algorithm preventing physically inadmissible states is developed, which improves robustness. Various examples show the efficiency of ComBo and the proposed algorithmic enhancements for nonlinear mechanical problems. The general usability is emphasized by examining and comparing the performance of myriad Fast Fourier Transform (FFT) based solvers including a detailed description of the new Doubly-Fine Material Grid (DFMG). All of the employed schemes benefit from the ComBo discretization.

math.NA↗

arXiv2020

AutoMat -- Automatic Differentiation for Generalized Standard Materials on GPUs

We propose a universal method for the evaluation of generalized standard materials that greatly simplifies the material law implementation process. By means of automatic differentiation and a numerical integration scheme, AutoMat reduces the implementation effort to two potential functions. By moving AutoMat to the GPU, we close the performance gap to conventional evaluation routines and demonstrate in detail that the expression level reverse mode of automatic differentiation as well as its extension to second order derivatives can be applied inside CUDA kernels. We underline the effectiveness and the applicability of AutoMat by integrating it into the FFT-based homogenization scheme of Moulinec and Suquet and discuss the benefits of using AutoMat with respect to runtime and solution accuracy for an elasto-viscoplastic example.

cs.CE↗