arXiv subjects

Melven Röhrig-Zöllner

Publications and source records attributed to Melven Röhrig-Zöllner.

9 recordsLinked to original sources

arXiv2026

A Discontinuous Galerkin Method for the Intrinsic Beam Model with Kelvin-Voigt Damping

We present an energy-stable discontinuous Galerkin (DG) spatial discretization of the intrinsic beam equations with Kelvin-Voigt damping. We reformulate the governing equations by splitting the total sectional resultants into elastic and viscous parts, thereby exposing a mixed system without mixed space-time derivatives. For homogeneous cantilever boundary data, the continuous system obeys an energy dissipation identity. With characteristic upwinding and alternating auxiliary traces the DG discretization is energy stable at the semidiscrete level for the stated homogeneous split boundary closure. The method is implemented in the Julia-based simulation framework Trixi.jl. A physically compatible manufactured solution with all state and viscous components active assesses convergence, while a rotating-beam test verifies the analytic constant-spin branch and the actuated energy balance.

math.NA↗

arXiv2026

Algorithmic algorithm development with LLMs: A Case Study on LLM-Usage for Contraction Order Optimization in Tensor Networks

We consider LLM-based algorithm development through a case study on contractionorder optimisation for tensor networks with OpenEvolve. We pay particular attention to the choice of the LLM as well as design choices such as evaluation metric and test instances. Our results highlight both the promise of verifier-guided evolutionary coding agents for algorithm development/improvement and the continuing importance of evaluation, validation, and interpretation -- and corresponding challenges -- by the human scientist.

cs.AI↗

arXiv2026

Implementation of QR factorization of tall and very skinny matrices on current GPUs

We consider the problem of computing a QR (or QZ) decomposition of a real, dense, tall and very skinny matrix. That is, the number of columns is tiny compared to the number of rows, rendering most computations completely or partially memory-bandwidth limited. The paper focuses on recent NVIDIA GPGPUs still supporting 64-bit floating-point arithmetic, but the findings carry over to AMD GPUs as well. We discuss two basic algorithms: Methods based on the normal equations (Gram matrix), in particular Cholesky-QR2 and SVQB, and the "tall-skinny QR" (TSQR), based on Householder transformations in a tree-reduction scheme. We propose two primary optimization techniques: Avoiding the write-back of the Q factor ("Q-less QR"), and exploiting fast local memory (shared memory on GPUs). We compare a straight-forward implementation of Gramian-based methods, and a more sophisticated TSQR implementation, in terms of performance achieved, time-to-solution, and implementation complexity. By performance modelling and numerical experiments with our own code and a vendor-optimized library routine, we demonstrate the crucial need for specialized methods and implementations in this memory-bound to transitional (memory/compute-bound) regime, and that TSQR is competitive in terms of time-to-solution, but at the cost of an investment in low-level code optimization.

cs.MS↗

arXiv2024

Performance of linear solvers in tensor-train format on current multicore architectures

Tensor networks are a class of algorithms aimed at reducing the computational complexity of high-dimensional problems. They are used in an increasing number of applications, from quantum simulations to machine learning. Exploiting data parallelism in these algorithms is key to using modern hardware. However, there are several ways to map required tensor operations onto linear algebra routines ("building blocks"). Optimizing this mapping impacts the numerical behavior, so computational and numerical aspects must be considered hand-in-hand. In this paper we discuss the performance of solvers for low-rank linear systems in the tensor-train format (also known as matrix-product states). We consider three popular algorithms: TT-GMRES, MALS, and AMEn. We illustrate their computational complexity based on the example of discretizing a simple high-dimensional PDE in, e.g., $50^{10}$ grid points. This shows that the projection to smaller sub-problems for MALS and AMEn reduces the number of floating-point operations by orders of magnitude. We suggest optimizations regarding orthogonalization steps, singular value decompositions, and tensor contractions. In addition, we propose a generic preconditioner based on a TT-rank-1 approximation of the linear operator. Overall, we obtain roughly a 5x speedup over the reference algorithm for the fastest method (AMEn) on a current multicore CPU.

math.NA↗

arXiv2024

Quantum Software Ecosystem Design

The rapid advancements in quantum computing necessitate a scientific and rigorous approach to the construction of a corresponding software ecosystem, a topic underexplored and primed for systematic investigation. This chapter takes an important step in this direction: It presents scientific considerations essential for building a quantum software ecosystem that makes quantum computing available for scientific and industrial problem solving. Central to this discourse is the concept of hardware-software co-design, which fosters a bidirectional feedback loop from the application layer at the top of the software stack down to the hardware. This approach begins with compilers and low-level software that are specifically designed to align with the unique specifications and constraints of the quantum processor, proceeds with algorithms developed with a clear understanding of underlying hardware and computational model features, and extends to applications that effectively leverage the capabilities to achieve a quantum advantage. We analyze the ecosystem from two critical perspectives: the conceptual view, focusing on theoretical foundations, and the technical infrastructure, addressing practical implementations around real quantum devices necessary for a functional ecosystem. This approach ensures that the focus is towards promising applications with optimized algorithm-circuit synergy, while ensuring a user-friendly design, an effective data management and an overall orchestration. Our chapter thus offers a guide to the essential concepts and practical strategies necessary for developing a scientifically grounded quantum software ecosystem.

quant-ph↗

arXiv2022

Performance of the low-rank tensor-train SVD (TT-SVD) for large dense tensors on modern multi-core CPUs

There are several factorizations of multi-dimensional tensors into lower-dimensional components, known as `tensor networks'. We consider the popular `tensor-train' (TT) format and ask: How efficiently can we compute a low-rank approximation from a full tensor on current multi-core CPUs? Compared to sparse and dense linear algebra, kernel libraries for multi-linear algebra are rare and typically not as well optimized. Linear algebra libraries like BLAS and LAPACK may provide the required operations in principle, but often at the cost of additional data movements for rearranging memory layouts. Furthermore, these libraries are typically optimized for the compute-bound case (e.g.\ square matrix operations) whereas low-rank tensor decompositions lead to memory bandwidth limited operations. We propose a `tensor-train singular value decomposition' (TT-SVD) algorithm based on two building blocks: a `Q-less tall-skinny QR' factorization, and a fused tall-skinny matrix-matrix multiplication and reshape operation. We analyze the performance of the resulting TT-SVD algorithm using the Roofline performance model. In addition, we present performance results for different algorithmic variants for shared-memory as well as distributed-memory architectures. Our experiments show that commonly used TT-SVD implementations suffer severe performance penalties. We conclude that a dedicated library for tensor factorization kernels would benefit the community: Computing a low-rank approximation can be as cheap as reading the data twice from main memory. As a consequence, an implementation that achieves realistic performance will move the limit at which one has to resort to randomized methods that only process part of the data.

math.NA↗

arXiv2021

(R)SE challenges in HPC

We discuss some specific software engineering challenges in the field of high-performance computing, and argue that the slow adoption of SE tools and techniques is at least in part caused by the fact that these do not address the HPC challenges `out-of-the-box'. By giving some examples of solutions for designing, testing and benchmarking HPC software, we intend to bring software engineering and HPC closer together.

cs.SE↗

arXiv2018

Benefits from using mixed precision computations in the ELPA-AEO and ESSEX-II eigensolver projects

We first briefly report on the status and recent achievements of the ELPA-AEO (Eigenvalue Solvers for Petaflop Applications - Algorithmic Extensions and Optimizations) and ESSEX II (Equipping Sparse Solvers for Exascale) projects. In both collaboratory efforts, scientists from the application areas, mathematicians, and computer scientists work together to develop and make available efficient highly parallel methods for the solution of eigenvalue problems. Then we focus on a topic addressed in both projects, the use of mixed precision computations to enhance efficiency. We give a more detailed description of our approaches for benefiting from either lower or higher precision in three selected contexts and of the results thus obtained.

physics.comp-ph↗

arXiv2016

GHOST: Building blocks for high performance sparse linear algebra on heterogeneous systems

While many of the architectural details of future exascale-class high performance computer systems are still a matter of intense research, there appears to be a general consensus that they will be strongly heterogeneous, featuring "standard" as well as "accelerated" resources. Today, such resources are available as multicore processors, graphics processing units (GPUs), and other accelerators such as the Intel Xeon Phi. Any software infrastructure that claims usefulness for such environments must be able to meet their inherent challenges: massive multi-level parallelism, topology, asynchronicity, and abstraction. The "General, Hybrid, and Optimized Sparse Toolkit" (GHOST) is a collection of building blocks that targets algorithms dealing with sparse matrix representations on current and future large-scale systems. It implements the "MPI+X" paradigm, has a pure C interface, and provides hybrid-parallel numerical kernels, intelligent resource management, and truly heterogeneous parallelism for multicore CPUs, Nvidia GPUs, and the Intel Xeon Phi. We describe the details of its design with respect to the challenges posed by modern heterogeneous supercomputers and recent algorithmic developments. Implementation details which are indispensable for achieving high efficiency are pointed out and their necessity is justified by performance measurements or predictions based on performance models. The library code and several applications are available as open source. We also provide instructions on how to make use of GHOST in existing software packages, together with a case study which demonstrates the applicability and performance of GHOST as a component within a larger software stack.

cs.DC↗