arXiv subjects

Andreas Frommer

Publications and source records attributed to Andreas Frommer.

At least 19 recordsLinked to original sources

arXiv2026

Variance reduction with probing and Multilevel Monte Carlo in Lattice QCD

Trace estimation is central in many lattice QCD computations, but the accuracy of the standard, stochastic Hutchinson method improves only with the square root of the sample size, making precise results expensive. We investigate two complementary variance reduction strategies. First, multigrid multilevel Monte Carlo uses a multigrid hierarchy to construct an unbiased multilevel estimator via recursive coarse grid corrections available from the multigrid hierarchy of the solver. Second, stochastic probing uses distance-$d$ graph colorings; we propose a torus based coloring that requires substantially fewer colors than hierarchical probing at the same distance. We test these approaches on two representative problems: the connected pseudoscalar correlator and disconnected fermion loops. For the connected pseudoscalar two-point function, the multilevel decomposition yields a variance reduction of up to $\mathcal{O}(10^5)$ at large time separations and translates into a clear cost reduction at fixed accuracy, thus confirming earlier results of arXiv:2412.06347. For the disconnected loops, in contrast, the multilevel decomposition provides only moderate gains, whereas probing combined with dilution delivers a substantial cost reduction that improves as the number of probing vectors is increased. Overall, the results highlight a pronounced complementarity: deflation schemes are most effective for observables dominated by long distance propagation, while probing is most effective for localized quantities.

hep-lat↗

arXiv2026

Exploiting Task-Based Parallelism for the Red-Black Gauss-Seidel Method on 2D Grids

Gauss-Seidel is a well-established iterative method for the solution of linear systems, and multicoloring has been widely used to increase parallelism in iterative solution techniques. Implementing multi-color Gauss-Seidel with conventional divide-and-conquer parallelization strategies, however, may be inefficient due to global synchronization requirements and load imbalances. Task-based programming models can mitigate these issues by enabling fine-grained parallelism, removing global barriers and allowing updates of different colors to partially overlap in time. In this work, we implement the red-black Gauss-Seidel method using two task-based programming models and compare them with a classical divide-and-conquer parallel implementation to evaluate the impact of fine-grained parallelism on execution efficiency. The red-black scheme serves as a representative example, as task-based approaches naturally extend to more general multi-color schemes arising from unstructured grids and wider stencils. Using the solve of the 2D Poisson equation as benchmark, our results show that task-based implementations can achieve performance comparable to conventional divide-and-conquer parallelization while providing greater resilience to hardware-level asynchronicity.

cs.DC↗

arXiv2026

Probing and graph coloring techniques for trace estimation in Lattice QCD

The computation of $\mathrm{Tr}[D^{-1}]$, where $D$ is the Wilson-Dirac matrix of Lattice QCD, is a fundamental and computationally demanding task with applications to disconnected hadronic correlation functions. Since $D^{-1}$ is a dense matrix of prohibitive size, its trace cannot be computed exactly, and one must resort to stochastic estimation via the Hutchinson estimator. The variance of the resulting estimation, however, can be large, as it is dominated by the off-diagonal entries of $D^{-1}$. We review the stochastic probing technique, which reduces the variance by constructing structured sampling vectors from distance-$d$ colorings of the graph associated with $D$, exploiting the exponential off-diagonal decay of $D^{-1}$ to eliminate dominant short-range contributions to the variance. We then present a novel multiplier-based coloring scheme, which achieves valid distance-$d$ colorings at arbitrary distances with significantly fewer colors than the established hierarchical probing construction. We prove that at any intermediate coloring falling between two consecutive hierarchical levels, the multiplier-based estimator achieves strictly lower variance than the partial hierarchical estimator, for large enough $d$. This is confirmed by numerical experiments showing that the multiplier-based variance decreases smoothly and monotonically with the number of colors, avoiding the irregular behavior affecting hierarchical probing at intermediate colorings, and achieving a substantial improvement in relative accuracy.

hep-lat↗

arXiv2026

Performance-Portable Optimization and Analysis of Multiple Right-Hand Sides in a Lattice QCD Solver

Managing the high computational cost of iterative solvers for sparse linear systems is a known challenge in scientific computing. Moreover, scientific applications often face memory bandwidth constraints, making it critical to optimize data locality and enhance the efficiency of data transport. We extend the lattice QCD solver DD-$\alpha$AMG to incorporate multiple right-hand sides (rhs) for both the Wilson-Dirac operator evaluation and the GMRES solver, with and without odd-even preconditioning. To optimize auto-vectorization, we introduce a flexible interface that supports various data layouts and implement a new data layout for better SIMD utilization. We evaluate our optimizations on both x86 and Arm clusters, demonstrating performance portability with similar speedups. A key contribution of this work is the performance analysis of our optimizations, which reveals the complexity introduced by architectural constraints and compiler behavior. Additionally, we explore different implementations leveraging a new matrix instruction set for Arm called SME and provide an early assessment of its potential benefits.

cs.DC↗

arXiv2025

Using orthogonal projectors in multigrid multilevel Monte Carlo for trace estimation in lattice QCD

We introduce a multigrid multilevel Monte Carlo method for stochastic trace estimation in lattice QCD based on orthogonal projections. This formulation extends the previously proposed oblique decomposition and it is assessed on three representative problems: the connected pseudoscalar correlator, the trace of the full Dirac operator's inverse $\mathrm{tr}(D^{-1})$, and disconnected fermion loops. For the connected correlator, variance reductions grow systematically with the time separation and lead to cost savings of up to a factor of 30 at large separations, outperforming both the plain Hutchinson's estimator and the oblique formulation. For $\mathrm{tr}(D^{-1})$, reductions are more modest but remain systematic, with stronger effects on more ill-conditioned systems. Disconnected loops show no improvement, since their variance is dominated by local same-slice contributions not targeted by the decomposition.

hep-lat↗

arXiv2025

Energy-preserving iteration schemes for Gauss collocation integrators

In this work, we develop energy-preserving iterative schemes for the (non-)linear systems arising in the Gauss integration of Poisson systems with quadratic Hamiltonian. Exploiting the relation between Gauss collocation integrators and diagonal Pad\'e approximations, we establish a Krylov-subspace iteration scheme based on a $Q$-Arnoldi process for linear systems that provides energy conservation not only at convergence --as standard iteration schemes do--, but also at the level of the individual iterates. It is competitive with GMRES in terms of accuracy and cost for a single iteration step and hence offers significant efficiency gains, when it comes to time integration of high-dimensional Poisson systems within given error tolerances. On top of the linear results, we consider non-linear Poisson systems and design non-linear solvers for the implicit midpoint rule (Gauss integrator of second order), using the fact that the associated Pad\'e approximation is a Cayley transformation. For the non-linear systems arising at each time step, we propose fixed-point and Newton-type iteration schemes that inherit the convergence order with comparable cost from their classical versions, but have energy-preserving iterates.

math.NA↗

arXiv2024

Splitting Techniques for DAEs with port-Hamiltonian Applications

In the simulation of differential-algebraic equations (DAEs), it is essential to employ numerical schemes that take into account the inherent structure and maintain explicit or hidden algebraic constraints without altering them. This paper focuses on operator-splitting techniques for coupled systems and aims at preserving the structure in the port-Hamiltonian framework. The study explores two decomposition strategies: one considering the underlying coupled subsystem structure and the other addressing energy-associated properties such as conservation and dissipation. We show that for coupled index-$1$ DAEs with and without private index-2 variables, the splitting schemes on top of a dimension-reducing decomposition achieve the same convergence rate as in the case of ordinary differential equations. Additionally, we discuss an energy-associated decomposition for index-1 pH-DAEs and introduce generalized Cayley transforms to uphold energy conservation. The effectiveness of both strategies is evaluated using port-Hamiltonian benchmark examples from electric circuits.

math.NA↗

arXiv2024

Polynomial Preconditioning for the Action of the Matrix Square Root and Inverse Square Root

While preconditioning is a long-standing concept to accelerate iterative methods for linear systems, generalizations to matrix functions are still in their infancy. We go a further step in this direction, introducing polynomial preconditioning for Krylov subspace methods which approximate the action of the matrix square root and inverse square root on a vector. Preconditioning reduces the subspace size and therefore avoids the storage problem together with -- for non-Hermitian matrices -- the increased computational cost per iteration that arises in the unpreconditioned case. Polynomial preconditioning is an attractive alternative to current restarting or sketching approaches since it is simpler and computationally more efficient. We demonstrate this for several numerical examples.

math.NA↗

arXiv2023

Krylov subspace restarting for matrix Laplace transforms

A common way to approximate $F(A)b$ -- the action of a matrix function on a vector -- is to use the Arnoldi approximation. Since a new vector needs to be generated and stored in every iteration, one is often forced to rely on restart algorithms which are either not efficient, not stable or only applicable to restricted classes of functions. We present a new representation of the error of the Arnoldi iterates if the function $F$ is given as a Laplace transform. Based on this representation we build an efficient and stable restart algorithm. In doing so we extend earlier work for the class of Stieltjes functions which are special Laplace transforms. We report several numerical experiments including comparisons with the restart method for Stieltjes functions.

math.NA↗

arXiv2023

Operator splitting for semi-explicit differential-algebraic equations and port-Hamiltonian DAEs

Operator splitting methods allow to split the operator describing a complex dynamical system into a sequence of simpler subsystems and treat each part independently. In the modeling of dynamical problems, systems of (possibly coupled) differential-algebraic equations (DAEs) arise. This motivates the application of operator splittings which are aware of the various structural forms of DAEs. Here, we present an approach for the splitting of coupled index-1 DAE as well as for the splitting of port-Hamiltonian DAEs, taking advantage of the energy-conservative and energy-dissipative parts. We provide numerical examples illustrating our second-order convergence results.

math.DS↗

arXiv2023

Analysis of stochastic probing methods for estimating the trace of functions of sparse symmetric matrices

We consider the problem of estimating the trace of a matrix function $f(A)$. In certain situations, in particular if $f(A)$ cannot be well approximated by a low-rank matrix, combining probing methods based on graph colorings with stochastic trace estimation techniques can yield accurate approximations at moderate cost. So far, such methods have not been thoroughly analyzed, though, but were rather used as efficient heuristics by practitioners. In this manuscript, we perform a detailed analysis of stochastic probing methods and, in particular, expose conditions under which the expected approximation error in the stochastic probing method scales more favorably with the dimension of the matrix than the error in non-stochastic probing. Extending results from [E. Aune, D. P. Simpson, J. Eidsvik, Parameter estimation in high dimensional Gaussian distributions, Stat. Comput., 24, pp. 247--263, 2014], we also characterize situations in which using just one stochastic vector is always -- not only in expectation -- better than the deterministic probing method. Several numerical experiments illustrate our theory and compare with existing methods.

math.NA↗

arXiv2023

Operator splitting for port-Hamiltonian systems

The port-Hamiltonian approach presents an energy-based modeling of dynamical systems with energy-conservative and energy-dissipative parts as well as an interconnection over the so-called ports. In this paper, we apply an operator splitting that treats the energy-conservative and energy-dissipative parts separately. This paves the way for linear equation solvers to exploit the respective special structures of the iteration matrices as well as the multirate potential in the different right-hand sides. We illustrate the approach using test examples from coupled multibody system dynamics.

math.NA↗

arXiv2023

Coarsest-level improvements in multigrid for lattice QCD on large-scale computers

Numerical simulations of quantum chromodynamics (QCD) on a lattice require the frequent solution of linear systems of equations with large, sparse and typically ill-conditioned matrices. Algebraic multigrid methods are meanwhile the standard for these difficult solves. Although the linear systems at the coarsest level of the multigrid hierarchy are much smaller than the ones at the finest level, they can be severely ill-conditioned, thus affecting the scalability of the whole solver. In this paper, we investigate different novel ways to enhance the coarsest-level solver and demonstrate their potential using DD-$α$AMG, one of the publicly available algebraic multigrid solvers for lattice QCD. We do this for two lattice discretizations, namely clover-improved Wilson and twisted mass. For both the combination of two of the investigated enhancements, deflation and polynomial preconditioning, yield significant improvements in the regime of small mass parameters. In the clover-improved Wilson case we observe a significantly improved insensitivity of the solver to conditioning, and for twisted mass we are able to get rid of a somewhat artificial increase of the twisted mass parameter on the coarsest level used so far to make the coarsest level solves converge more rapidly.

math.NA↗

arXiv2023

MG-MLMC++ as a Variance Reduction Method for Estimating the Trace of a Matrix Inverse

Hutchinson's method estimates the trace of a matrix function $f(D)$ stochastically using samples $τ^Hf(D)τ$, where the components of the random vectors $τ$ obey an isotropic probability distribution. Estimating the trace of the inverse of a discretized Dirac operator or variants thereof have become a major challenge in lattice QCD simulations, as they represent the disconnected contribution to certain observables. The Hutchinson Monte Carlo sampling, however, suffers from the fact that its accuracy depends quadratically on the sample size, making higher precision estimation very expensive. Meyer, Musco, Musco and Woodruff recently proposed an enhancement of Hutchinson's method, termed \texttt{Hutch++}, in which the sample space is enriched by several vectors of the form $f(D)ζ$, $ζ$ a random vector as in Hutchinson's method. Theoretical analyses show that under certain circumstances the number of these added sample vectors can be chosen in a way to reduce the dependence of the variance of the resulting estimator from the number $N$ of samples from $\mathcal{O}(1/N)$ to $\mathcal{O}(1/N^2)$. In this study we combine \texttt{Hutch++} with our recently suggested multigrid multilevel Monte Carlo approach. We present results for the Schwinger discretization of the $2$-di\-men\-si\-onal Dirac operator, revealing that the two approaches contribute additively to variance reduction.

hep-lat↗

arXiv2022

A flexible short recurrence Krylov subspace method for matrices arising in the time integration of port Hamiltonian systems and ODEs/DAEs with a dissipative Hamiltonian

For several classes of mathematical models that yield linear systems, the splitting of the matrix into its Hermitian and skew Hermitian parts is naturally related to properties of the underlying model. This is particularly so for discretizations of dissipative Hamiltonian ODEs, DAEs and port Hamiltonian systems where, in addition, the Hermitian part is positive definite or semi-definite. It is then possible to develop short recurrence optimal Krylov subspace methods in which the Hermitian part is used as a preconditioner. In this paper we develop new, right preconditioned variants of this approach which as their crucial new feature allow the systems with the Hermitian part to be solved only approximately in each iteration while keeping the short recurrences. This new class of methods is particularly efficient as it allows, for example, to use few steps of a multigrid solver or a (preconditioned) CG method for the Hermitian part in each iteration. We illustrate this with several numerical experiments for large scale systems.

math.NA↗

arXiv2022

Deflated Multigrid Multilevel Monte Carlo

In lattice QCD, the trace of the inverse of the discretized Dirac operator appears in the disconnected fermion loop contribution to an observable. As simulation methods get more and more precise, these contributions become increasingly important. Hence, we consider here the problem of computing the trace $\mathrm{tr}(D^{-1})$, with $D$ the Dirac operator. The Hutchinson method, which is very frequently used to stochastically estimate the trace of a function of a matrix, approximates the trace as the average over estimates of the form $x^{H} D^{-1} x$, with the entries of the vector $x$ following a certain probability distribution. For $N$ samples, the accuracy is $\mathcal{O}(1/\sqrt{N})$. In recent work, we have introduced multigrid multilevel Monte Carlo: having a multigrid hierarchy with operators $D_{\ell}$, $P_{\ell}$ and $R_{\ell}$, for level $\ell$, we can rewrite the trace $\mathrm{tr}(D^{-1})$ via a telescopic sum with difference-levels, written in terms of the aforementioned operators and with a reduced variance. We have seen significant reductions in the variance and the total work with respect to exactly deflated Hutchinson. In this work, we explore the use of exact deflation in combination with the multigrid multilevel Monte Carlo method, and demonstrate how this leads to both algorithmic and computational gains.

hep-lat↗

arXiv2022

Krylov Subspace Recycling For Matrix Functions

We derive an augmented Krylov subspace method with subspace recycling for computing a sequence of matrix function applications on a set of vectors. The matrix is either fixed or changes as the sequence progresses. We assume consecutive matrices are closely related, but make no assumptions on the relationship between the vectors. We present three versions of the method with different practical implementations. We demonstrate the effectiveness of the method using a range of numerical experiments with a selection of functions and matrices. We primarily focus our attention on the sign function arising in the overlap formalism of lattice QCD.

math.NA↗

arXiv2022

On the Convergence of Randomized and Greedy Relaxation Schemes for Solving Nonsingular Linear Systems of Equations

We extend results known for the randomized Gauss-Seidel and the Gauss-Southwell methods for the case of a Hermitian and positive definite matrix to certain classes of non-Hermitian matrices. We obtain convergence results for a whole range of parameters describing the probabilities in the randomized method or the greedy choice strategy in the Gauss-Southwell-type methods. We identify those choices which make our convergence bounds best possible. Our main tool is to use weighted l1-norms to measure the residuals. A major result is that the best convergence bounds that we obtain for the expected values in the randomized algorithm are as good as the best for the deterministic, but more costly algorithms of Gauss-Southwell type. Numerical experiments illustrate the convergence of the method and the bounds obtained. Comparisons with the randomized Kaczmarz method are also presented.

math.NA↗