BZPEERPapers, people, evidence and research paths
Menu

Discover

SearchFind papers, people, topics, institutions and opportunities from one place.GraphSee how papers, authors, topics and evidence connect.

Research tools

HomeContinue from the papers, people and topics that matter now.CollectionsTurn saved papers into reading lists, evidence maps and shareable context.ResearchReconstruct the path that led you from search to paper to question.CollaborationMove from a paper, topic or expert request into a focused thread.PublishCreate full-text research articles with metadata, formulas and versions.

Network

ExpertsFind specialists whose work and reviews explain why they can help.CommunitiesJoin research groups organized around papers, topics and shared evidence.

Opportunities

ListingsBrowse roles, calls and collaborations connected to real research context.

Account

Log inReturn to your saved papers, graph views and active threads.Create accountStart saving papers, building a research trail and joining teams.
Log inCreate account

Source author record

Rolf Krause

Rolf Krause appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

ResearcherUnclaimed source record
math.NANumerical AnalysisDistributed, Parallel, and Cluster ComputingComputational Engineering, Finance, and Sciencemath.OCMachine Learningphysics.geo-phApplicationsComputer Visioneess.IVmath.APmath.PRPerformancephysics.comp-phQuantitative Methods

Catalog footprint

What is connected

25works
15topics
4close collaborators

Actions

Connect this record

Log in to claim
Open graphBrowse works

Research graph

See the researcher in context

Open full explorer

Inspect adjacent papers, topics, institutions and collaborators without losing the researcher page.

Building this map preview

BZPEER is loading the nearby papers, people, topics and institutions for this page.

Published work

25 published item(s)

preprint2026arXiv

A Non-Monotone Preconditioned Trust-Region Method for Neural Network Training

Training deep neural networks at scale can benefit from domain decomposition, where the network is split into subdomains trained in parallel and coupled by a global trust-region mechanism. Building on the Additively Preconditioned Trust-Region Strategy (APTS), we propose a non-monotone variant with a nonlinear additive Schwarz preconditioner that combines parallel subdomain corrections with global coarse-space directions. A windowed acceptance criterion allows controlled objective increases, avoiding needless rejection of effective coarse steps. The resulting non-monotone APTS (NAPTS) preserves accuracy while reducing CPU time by 30\% and cutting rejected steps to one third of those in APTS.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Cell Behavior Video Classification Challenge, a benchmark for computer vision methods in time-lapse microscopy

The classification of microscopy videos capturing complex cellular behaviors is crucial for understanding and quantifying the dynamics of biological processes over time. However, it remains a frontier in computer vision, requiring approaches that effectively model the shape and motion of objects without rigid boundaries, extract hierarchical spatiotemporal features from entire image sequences rather than static frames, and account for multiple objects within the field of view. To this end, we organized the Cell Behavior Video Classification Challenge (CBVCC), benchmarking 35 methods based on three approaches: classification of tracking-derived features, end-to-end deep learning architectures to directly learn spatiotemporal features from the entire video sequence without explicit cell tracking, or ensembling tracking-derived with image-derived features. We discuss the results achieved by the participants and compare the potential and limitations of each approach, serving as a basis to foster the development of computer vision methods for studying cellular dynamics.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Globally Convergent Multilevel Training of Deep Residual Networks

We propose a globally convergent multilevel training method for deep residual networks (ResNets). The devised method can be seen as a novel variant of the recursive multilevel trust-region (RMTR) method, which operates in hybrid (stochastic-deterministic) settings by adaptively adjusting mini-batch sizes during the training. The multilevel hierarchy and the transfer operators are constructed by exploiting a dynamical system's viewpoint, which interprets forward propagation through the ResNet as a forward Euler discretization of an initial value problem. In contrast to traditional training approaches, our novel RMTR method also incorporates curvature information on all levels of the multilevel hierarchy by means of the limited-memory SR1 method. The overall performance and the convergence properties of our multilevel training method are numerically investigated using examples from the field of classification and regression.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

A Generalized Multigrid Method for Solving Contact Problems in Lagrange Multiplier based Unfitted Finite Element Method

Internal interfaces in a domain could exist as a material defect or they can appear due to propagations of cracks. Discretization of such geometries and solution of the contact problem on the internal interfaces can be computationally challenging. We employ an unfitted Finite Element (FE) framework for the discretization of the domains and develop a tailored, globally convergent, and efficient multigrid method for solving contact problems on the internal interfaces. In the unfitted FE methods, structured background meshes are used and only the underlying finite element space has to be modified to incorporate the discontinuities. The non-penetration conditions on the embedded interfaces of the domains are discretized using the method of Lagrange multipliers. We reformulate the arising variational inequality problem as a quadratic minimization problem with linear inequality constraints. Our multigrid method can solve such problems by employing a tailored multilevel hierarchy of the FE spaces and a novel approach for tackling the discretized non-penetration conditions. We employ pseudo-$L^2$ projection-based transfer operators to construct a hierarchy of nested FE spaces from the hierarchy of non-nested meshes. The essential component of our multigrid method is a technique that decouples the linear constraints using an orthogonal transformation of the basis. The decoupled constraints are handled by a modified variant of the projected Gauss-Seidel method, which we employ as a smoother in the multigrid method. These components of the multigrid method allow us to enforce linear constraints locally and ensure the global convergence of our method. We will demonstrate the robustness, efficiency, and level independent convergence property of the proposed method for Signorini's problem and two-body contact problems.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

An experimental comparison of a space-time multigrid method with PFASST for a reaction-diffusion problem

We consider two parallel-in-time approaches applied to a (reaction) diffusion problem, possibly non-linear. In particular, we consider PFASST (Parallel Full Approximation Scheme in Space and Time) and space-time multilevel strategies. For both approaches, we start from an integral formulation of the continuous time-dependent problem. Then, a collocation form for PFASST and a discontinuous Galerkin discretization in time for the space-time multigrid are employed, resulting in the same discrete solution at the time nodes. Strong and weak scaling of both multilevel strategies is compared for varying order of the temporal discretization. Moreover, we investigate the respective convergence behavior for non-linear problems and highlight quantitative differences.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Space-time multilevel Monte Carlo methods and their application to cardiac electrophysiology

We present a novel approach aimed at high-performance uncertainty quantification for time-dependent problems governed by partial differential equations. In particular, we consider input uncertainties described by a Karhunen-Loeeve expansion and compute statistics of high-dimensional quantities-of-interest, such as the cardiac activation potential. Our methodology relies on a close integration of multilevel Monte Carlo methods, parallel iterative solvers, and a space-time discretization. This combination allows for space-time adaptivity, time-changing domains, and to take advantage of past samples to initialize the space-time solution. The resulting sequence of problems is distributed using a multilevel parallelization strategy, allocating batches of samples having different sizes to a different number of processors. We assess the performance of the proposed framework by showing in detail its application to the solution of nonlinear equations arising from cardiac electrophysiology. Specifically, we study the effect of spatially-correlated perturbations of the heart fibers conductivities on the mean and variance of the resulting activation map. As shown by the experiments, the theoretical rates of convergence of multilevel Monte Carlo are achieved. Moreover, the total computational work for a prescribed accuracy is reduced by an order of magnitude with respect to standard Monte Carlo methods.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

A Multilevel Approach to Training

We propose a novel training method based on nonlinear multilevel minimization techniques, commonly used for solving discretized large scale partial differential equations. Our multilevel training method constructs a multilevel hierarchy by reducing the number of samples. The training of the original model is then enhanced by internally training surrogate models constructed with fewer samples. We construct the surrogate models using first-order consistency approach. This gives rise to surrogate models, whose gradients are stochastic estimators of the full gradient, but with reduced variance compared to standard stochastic gradient estimators. We illustrate the convergence behavior of the proposed multilevel method to machine learning applications based on logistic regression. A comparison with subsampled Newton's and variance reduction methods demonstrate the efficiency of our multilevel method.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Modelling of hydro-mechanical processes in heterogeneous fracture intersections using a fictitious domain method with variational transfer operators

Fluid flow in rough fractures and the coupling with the mechanical behaviour of the fractures pose great difficulties for numerical modeling approaches, due to complex fracture surface topographies, the non-linearity of hydromechanical processes and their tightly coupled nature. To this end, we have adapted a fictitious domain method to enable the simulation of hydromechanical processes in fracture-intersections. The main characteristic of the method is the immersion of the fracture, modelled as a linear elastic solid, in the surrounding computational fluid domain, modelled with the incompressible Navier Stokes equations. The fluid and the solid problems are coupled with variational transfer operators. Variational transfer operators are also used to solve contact within the fracture using a dual mortar approach and to generate problem specific fluid meshes. With respect to our applications, the key features of the method are the usage of different finite element discretizations for the solid and the fluid problem and the automatically generated representation of the fluid-solid boundary. We demonstrate that the presented methodology resolves small-scale roughness on the fracture surface, while capturing fluid flow field changes during mechanical loading. Starting with 2D/3D benchmark simulations of intersected fractures, we end with an intersected fracture composed of complex fracture surface topographies, which are in contact under increasing loads. The contributions of this article are: (1) the application of the fictitious domain method to study flow in fractures with intersections, (2) a mortar based contact solver for the solid problem, (3) generation of problem specific grids using the geometry information from the variational transfer operators.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Multilevel Minimization for Deep Residual Networks

We present a new multilevel minimization framework for the training of deep residual networks (ResNets), which has the potential to significantly reduce training time and effort. Our framework is based on the dynamical system's viewpoint, which formulates a ResNet as the discretization of an initial value problem. The training process is then formulated as a time-dependent optimal control problem, which we discretize using different time-discretization parameters, eventually generating multilevel-hierarchy of auxiliary networks with different resolutions. The training of the original ResNet is then enhanced by training the auxiliary networks with reduced resolutions. By design, our framework is conveniently independent of the choice of the training strategy chosen on each level of the multilevel hierarchy. By means of numerical examples, we analyze the convergence behavior of the proposed method and demonstrate its robustness. For our examples we employ a multilevel gradient-based methods. Comparisons with standard single level methods show a speedup of more than factor three while achieving the same validation accuracy.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

PIEMAP: Personalized Inverse Eikonal Model from cardiac Electro-Anatomical Maps

Electroanatomical mapping, a keystone diagnostic tool in cardiac electrophysiology studies, can provide high-density maps of the local electric properties of the tissue. It is therefore tempting to use such data to better individualize current patient-specific models of the heart through a data assimilation procedure and to extract potentially insightful information such as conduction properties. Parameter identification for state-of-the-art cardiac models is however a challenging task. In this work, we introduce a novel inverse problem for inferring the anisotropic structure of the conductivity tensor, that is fiber orientation and conduction velocity along and across fibers, of an eikonal model for cardiac activation. The proposed method, named PIEMAP, performed robustly with synthetic data and showed promising results with clinical data. These results suggest that PIEMAP could be a useful supplement in future clinical workflows of personalized therapies.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

3D non-conforming mesh model for flow in fractured porous media using Lagrange multipliers

This work presents a modeling approach for single-phase flow in 3D fractured porous media with non-conforming meshes. To this end, a Lagrange multiplier method is combined with a parallel $L^2$-projection variational transfer approach. This Lagrange multiplier method enables the use of non-conforming meshes and depicts the variable coupling between fracture and matrix domain. The $L^2$-projection variational transfer allows general, accurate, and parallel projection of variables between non-conforming meshes (i.e. between fracture and matrix domain). Comparisons of simulations with 2D benchmarks show good agreement, and the method is further validated on 3D fracture networks by comparing it to results from conforming mesh simulations which were used as a reference. Application to realistic fracture networks with hundreds of fractures is demonstrated. Mesh size and mesh convergence are investigated for benchmark cases and 3D fracture network applications. Results demonstrate that the Lagrange multiplier method, in combination with the $L^2$-projection method, is capable of modeling single-phase flow through realistic 3D fracture networks.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

Recursive multilevel trust region method with application to fully monolithic phase-field models of brittle fracture

The simulation of crack initiation and propagation in an elastic material is difficult, as crack paths with complex topologies have to be resolved. Phase-field approach allows to simulate crack behavior by circumventing the need to explicitly model crack paths. However, the underlying mathematical model gives rise to a non-convex constrained minimization problem. In this work, we propose a recursive multilevel trust region (RMTR) method to efficiently solve such a minimization problem. The RMTR method combines the global convergence property of the trust region method and the optimality of the multilevel method. The solution process is accelerated by employing level dependent objective functions, minimization of which provides correction to the original/fine-level problem. In the context of the phase-field fracture approach, it is challenging to design efficient level dependent objective functions as the underlying mathematical model relies on the mesh dependent parameters. We introduce level dependent objective functions that combine fine level description of the crack path with the coarse level discretization. The overall performance and the convergence properties of the proposed RMTR method are investigated by means of several numerical examples in three dimensions.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Towards a multigrid method for the minimum-cost flow problem

We present a first step towards a multigrid method for solving the min-cost flow problem. Specifically, we present a strategy that takes advantage of existing black-box fast iterative linear solvers, i.e. algebraic multigrid methods. We show with standard benchmarks that, while less competitive than combinatorial techniques on small problems that run on a single core, our approach scales well with problem size, complexity, and number of processors, allowing for tackling large-scale problems on modern parallel architectures. Our approach is based on combining interior-point with multigrid methods for solving the nonlinear KKT equations via Newton's method. However, the Jacobian matrix arising in the Newton iteration is indefinite and its condition number cannot be expected to be bounded. In fact, the eigenvalues of the Jacobian can both vanish and blow up near the solution, leading to a significant slow-down of the convergence speed of iterative solvers - or to the loss of convergence at all. In order to allow for the application of multigrid methods, which have been originally designed for elliptic problems, we furthermore show that the occurring Jacobian can be interpreted as the stiffness matrix of a mixed formulation of the weighted graph Laplacian of the network, whose metric depends on the slack variables and the multipliers of the inequality constraints. Together with our regularization, this allows for the application of a black-box algebraic multigrid method on the Schur-complement of the system.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Explicit Parallel-in-time Integration of a Linear Acoustic-Advection System

The applicability of the Parareal parallel-in-time integration scheme for the solution of a linear, two-dimensional hyperbolic acoustic-advection system, which is often used as a test case for integration schemes for numerical weather prediction (NWP), is addressed. Parallel-in-time schemes are a possible way to increase, on the algorithmic level, the amount of parallelism, a requirement arising from the rapidly growing number of CPUs in high performance computer systems. A recently introduced modification of the "parallel implicit time-integration algorithm" could successfully solve hyperbolic problems arising in structural dynamics. It has later been cast into the framework of Parareal. The present paper adapts this modified Parareal and employs it for the solution of a hyperbolic flow problem, where the initial value problem solved in parallel arises from the spatial discretization of a partial differential equation by a finite difference method. It is demonstrated that the modified Parareal is stable and can produce reasonably accurate solutions while allowing for a noticeable reduction of the time-to-solution. The implementation relies on integration schemes already widely used in NWP (RK-3, partially split forward Euler, forward-backward). It is demonstrated that using an explicit partially split scheme for the coarse integrator allows to avoid the use of an implicit scheme while still achieving speedup.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Numerical simulation of skin transport using Parareal

In-silico investigation of skin permeation is an important but also computationally demanding problem. To resolve all scales involved in full detail will not only require exascale computing capacities but also suitable parallel algorithms. This article investigates the applicability of the time-parallel Parareal algorithm to a brick and mortar setup, a precursory problem to skin permeation. The C++ library Lib4PrM implementing Parareal is combined with the UG4 simulation framework, which provides the spatial discretization and parallelization. The combination's performance is studied with respect to convergence and speedup. It is confirmed that anisotropies in the domain and jumps in diffusion coefficients only have a minor impact on Parareal's convergence. The influence of load imbalances in time due to differences in number of iterations required by the spatial solver as well as spatio-temporal weak scaling is discussed.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

A multi-level spectral deferred correction method

The spectral deferred correction (SDC) method is an iterative scheme for computing a higher-order collocation solution to an ODE by performing a series of correction sweeps using a low-order timestepping method. This paper examines a variation of SDC for the temporal integration of PDEs called multi-level spectral deferred corrections (MLSDC), where sweeps are performed on a hierarchy of levels and an FAS correction term, as in nonlinear multigrid methods, couples solutions on different levels. Three different strategies to reduce the computational cost of correction sweeps on the coarser levels are examined: reducing the degrees of freedom, reducing the order of the spatial discretization, and reducing the accuracy when solving linear systems arising in implicit temporal integration. Several numerical examples demonstrate the effect of multi-level coarsening on the convergence and cost of SDC integration. In particular, MLSDC can provide significant savings in compute time compared to SDC for a three-dimensional problem.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

A space-time parallel solver for the three-dimensional heat equation

The paper presents a combination of the time-parallel "parallel full approximation scheme in space and time" (PFASST) with a parallel multigrid method (PMG) in space, resulting in a mesh-based solver for the three-dimensional heat equation with a uniquely high degree of efficient concurrency. Parallel scaling tests are reported on the Cray XE6 machine "Monte Rosa" on up to 16,384 cores and on the IBM Blue Gene/Q system "JUQUEEN" on up to 65,536 cores. The efficacy of the combined spatial- and temporal parallelization is shown by demonstrating that using PFASST in addition to PMG significantly extends the strong-scaling limit. Implications of using spatial coarsening strategies in PFASST's multi-level hierarchy in large-scale parallel simulations are discussed.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

A stencil-based implementation of Parareal in the C++ domain specific embedded language STELLA

In view of the rapid rise of the number of cores in modern supercomputers, time-parallel methods that introduce concurrency along the temporal axis are becoming increasingly popular. For the solution of time-dependent partial differential equations, these methods can add another direction for concurrency on top of spatial parallelization. The paper presents an implementation of the time-parallel Parareal method in a C++ domain specific language for stencil computations (STELLA). STELLA provides both an OpenMP and a CUDA backend for a shared memory parallelization, using the CPU or GPU inside a node for the spatial stencils. Here, we intertwine this node-wise spatial parallelism with the time-parallel Parareal. This is done by adding an MPI-based implementation of Parareal, which allows us to parallelize in time across nodes. The performance of Parareal with both backends is analyzed in terms of speedup, parallel efficiency and energy-to-solution for an advection-diffusion problem with a time-dependent diffusion coefficient.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Convergence of Parareal for the Navier-Stokes equations depending on the Reynolds number

The paper presents first a linear stability analysis for the time-parallel Parareal method, using an IMEX Euler as coarse and a Runge-Kutta-3 method as fine propagator, confirming that dominant imaginary eigenvalues negatively affect Parareal's convergence. This suggests that when Parareal is applied to the nonlinear Navier-Stokes equations, problems for small viscosities could arise. Numerical results for a driven cavity benchmark are presented, confirming that Parareal's convergence can indeed deteriorate as viscosity decreases and the flow becomes increasingly dominated by convection. The effect is found to strongly depend on the spatial resolution.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Inexact spectral deferred corrections

Spectral deferred correction (SDC) methods are an attractive approach to iteratively computing collocation solutions to an ODE by performing so-called sweeps with a low-order time stepping method. SDC allows to easily construct high order split methods where e.g. stiff terms of the ODE are treated implicitly. This requires the solution to full accuracy of multiple linear systems of equations during each sweep, e.g. with a multigrid method. In this paper, we present an inexact variant of SDC, where each solve of a linear system is replaced by a single multigrid V-cycle and thus significantly reduces the cost for each sweep. For the investigated examples, this strategy results only in a small increase of the number of required sweeps and we demonstrate that "inexact spectral deferred corrections" can provide a dramatic reduction of the overall number of multigrid V-cycles required to complete an SDC time step.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Parareal for diffusion problems with space- and time-dependent coefficients

For the time-parallel Parareal method, there exists both numerical and analytical proof that it converges very well for diffusive problems like the heat equation. Many applications, however, do not lead to simple homogeneous diffusive scenarios but feature strongly inhomogeneous and possibly anisotropic coefficients. The paper presents results from a numerical study of how space- and time-dependent coefficients in a diffusion setup affect Parareal's convergence behaviour. It is shown that, for the presented examples, non-constant diffusion coefficients have only marginal influence on how fast Parareal converges. Furthermore, an example is shown that illustrates how for linear problems the maximum singular value of the Parareal iteration matrix can be used to estimate convergence rates.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

A Probabilistic Model for LCF

Fatigue life of components or test specimens often exhibit a significant scatter. Furthermore, size effects have a non-negligible influence on fatigue life of parts with different geometries. We present a new probabilistic model for low-cycle fatigue (LCF) in the context of polycrystalline metal. The model takes size effects and inhomogeneous strain fields into account by means of the Poisson point process (PPP). This approach is based on the assumption of independently occurring LCF cracks and the Coffin-Manson-Basquin (CMB) equation. Within the probabilistic model, we give a new and more physical interpretation of the CMB parameters which are in the original approach no material parameters in a strict sense, as they depend on the specimen geometry. Calibration and validation of the proposed model is performed using results of strain controlled LCF tests of specimens with different surface areas. The test specimens are made of the nickel base superalloy RENE 80.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Probabilistic Analysis of LCF Crack Initiation Life of a Turbine Blade under Thermomechanical Loading

An accurate assessment for fatigue damage as a function of activation and deactivation cycles is vital for the design of many engineering parts. In this paper we extend the probabilistic and local approach to this problem proposed in [1,2] and [3] to the case of non-constant temperature fields and thermomechanical loading. The method has been implemented as a finite element postprocessor and applied to an example case of a gas-turbine blade which is made of a conventionally cast nickel base superalloy.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Risk estimation for LCF crack initiation

An accurate risk assessment for fatigue damage is of vital importance for the design and service of today's turbomachinery components. We present an approach for quantifying the probability of crack initiation due to surface driven low-cycle fatigue (LCF). This approach is based on the theory of failure-time processes and takes inhomogeneous stress fields and size effects into account. The method has been implemented as a finite-element postprocessor which uses quadrature formulae of higher order. Results of applying this new approach to an example case of a gas-turbine compressor disk are discussed.

0 citations0 reviewsNo code linkedOpen access