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Martin Stoll

Publications and source records attributed to Martin Stoll.

At least 73 records · Page 4Linked to original sources
arXiv2019

Semi-Supervised Classification on Non-Sparse Graphs Using Low-Rank Graph Convolutional Networks

Graph Convolutional Networks (GCNs) have proven to be successful tools for semi-supervised learning on graph-based datasets. For sparse graphs, linear and polynomial filter functions have yielded impressive results. For large non-sparse graphs, however, network training and evaluation becomes prohibitively expensive. By introducing low-rank filters, we gain significant runtime acceleration and simultaneously improved accuracy. We further propose an architecture change mimicking techniques from Model Order Reduction in what we call a reduced-order GCN. Moreover, we present how our method can also be applied to hypergraph datasets and how hypergraph convolution can be implemented efficiently.

cs.LG↗
arXiv2019

Interior Point Methods and Preconditioning for PDE-Constrained Optimization Problems Involving Sparsity Terms

PDE-constrained optimization problems with control or state constraints are challenging from an analytical as well as numerical perspective. The combination of these constraints with a sparsity-promoting $\rm L^1$ term within the objective function requires sophisticated optimization methods. We propose the use of an Interior Point scheme applied to a smoothed reformulation of the discretized problem, and illustrate that such a scheme exhibits robust performance with respect to parameter changes. To increase the potency of this method we introduce fast and efficient preconditioners which enable us to solve problems from a number of PDE applications in low iteration numbers and CPU times, even when the parameters involved are altered dramatically.

math.OC↗
arXiv2018

NFFT meets Krylov methods: Fast matrix-vector products for the graph Laplacian of fully connected networks

The graph Laplacian is a standard tool in data science, machine learning, and image processing. The corresponding matrix inherits the complex structure of the underlying network and is in certain applications densely populated. This makes computations, in particular matrix-vector products, with the graph Laplacian a hard task. A typical application is the computation of a number of its eigenvalues and eigenvectors. Standard methods become infeasible as the number of nodes in the graph is too large. We propose the use of the fast summation based on the nonequispaced fast Fourier transform (NFFT) to perform the dense matrix-vector product with the graph Laplacian fast without ever forming the whole matrix. The enormous flexibility of the NFFT algorithm allows us to embed the accelerated multiplication into Lanczos-based eigenvalues routines or iterative linear system solvers and even consider other than the standard Gaussian kernels. We illustrate the feasibility of our approach on a number of test problems from image segmentation to semi-supervised learning based on graph-based PDEs. In particular, we compare our approach with the Nyström method. Moreover, we present and test an enhanced, hybrid version of the Nyström method, which internally uses the NFFT.

cs.LG↗
arXiv2018

Efficient Numerical Methods for Gas Network Modeling and Simulation

We study the modeling and simulation of gas pipeline networks, with a focus on fast numerical methods for the simulation of transient dynamics. The obtained mathematical model of the underlying network is represented by a nonlinear differential algebraic equation (DAE). With our modeling, we reduce the number of algebraic constraints, which correspond to the $(2,2)$ block in our semi-explicit DAE model, to the order of junction nodes in the network, where a junction node couples at least three pipelines. We can furthermore ensure that the $(1, 1)$ block of all system matrices including the Jacobian is block lower triangular by using a specific ordering of the pipes of the network. We then exploit this structure to propose an efficient preconditioner for the fast simulation of the network. We test our numerical methods on benchmark problems of (well-)known gas networks and the numerical results show the efficiency of our methods.

math.NA↗
arXiv2018

A low-rank tensor method for PDE-constrained optimization with isogeometric analysis

Isogeometric analysis (IGA) has become one of the most popular methods for the discretization of partial differential equations motivated by the use of NURBS for geometric representations in industry and science. A crucial challenge lies in the solution of the discretized equations, which we discuss in this talk with a particular focus on PDE-constrained optimization discretized using IGA. The discretization results in a system of large mass and stiffness matrices, which are typically very costly to assemble. To reduce the computation time and storage requirements, low-rank tensor methods have become a promising tool. We present a framework for the assembly of these matrices in low-rank form as the sum of a small number of Kronecker products. For assembly of the smaller matrices only univariate integration is required. The resulting low rank Kronecker product structure of the mass and stiffness matrices can be used to solve a PDE-constrained optimization problem without assembling the actual system matrices. We present a framework which preserves and exploits the low-rank Kronecker product format for both the matrices and the solution. We use the block AMEn method to efficiently solve the corresponding KKT system of the optimization problem. We show several numerical experiments with 3D geometries to demonstrate that the low-rank assembly and solution drastically reduces the memory demands and computing times, depending on the approximation ranks of the domain.

math.NA↗
arXiv2018

Fast iterative solvers for an optimal transport problem

Optimal transport problems pose many challenges when considering their numerical treatment. We investigate the solution of a PDE-constrained optimisation problem subject to a particular transport equation arising from the modelling of image metamorphosis. We present the nonlinear optimisation problem, and discuss the discretisation and treatment of the nonlinearity via a Gauss--Newton scheme. We then derive preconditioners that can be used to solve the linear systems at the heart of the (Gauss--)Newton method. With the optical flow in mind, we further propose the reduction of dimensionality by choosing a radial basis function discretisation that uses the centres of superpixels as the collocation points. Again, we derive suitable preconditioners that can be used for this formulation.

math.NA↗
arXiv2017

An inexact Newton-Krylov method for stochastic eigenvalue problems

This paper aims at the efficient numerical solution of stochastic eigenvalue problems. Such problems often lead to prohibitively high dimensional systems with tensor product structure when discretized with the stochastic Galerkin method. Here, we exploit this inherent tensor product structure to develop a globalized low-rank inexact Newton method with which we tackle the stochastic eigenproblem. We illustrate the effectiveness of our solver with numerical experiments.

math.NA↗
arXiv2017

Solving optimal control problems governed by random Navier-Stokes equations using low-rank methods

Many problems in computational science and engineering are simultaneously characterized by the following challenging issues: uncertainty, nonlinearity, nonstationarity and high dimensionality. Existing numerical techniques for such models would typically require considerable computational and storage resources. This is the case, for instance, for an optimization problem governed by time-dependent Navier-Stokes equations with uncertain inputs. In particular, the stochastic Galerkin finite element method often leads to a prohibitively high dimensional saddle-point system with tensor product structure. In this paper, we approximate the solution by the low-rank Tensor Train decomposition, and present a numerically efficient algorithm to solve the optimality equations directly in the low-rank representation. We show that the solution of the vorticity minimization problem with a distributed control admits a representation with ranks that depend modestly on model and discretization parameters even for high Reynolds numbers. For lower Reynolds numbers this is also the case for a boundary control. This opens the way for a reduced-order modeling of the stochastic optimal flow control with a moderate cost at all stages.

math.NA↗
arXiv2017

Low-rank computation of posterior covariance matrices in Bayesian inverse problems

We consider the problem of estimating the uncertainty in statistical inverse problems using Bayesian inference. When the probability density of the noise and the prior are Gaussian, the solution of such a statistical inverse problem is also Gaussian. Therefore, the underlying solution is characterized by the mean and covariance matrix of the posterior probability density. However, the covariance matrix of the posterior probability density is full and large. Hence, the computation of such a matrix is impossible for large dimensional parameter spaces. It is shown that for many ill-posed problems, the Hessian matrix of the data misfit part has low numerical rank and it is therefore possible to perform a low-rank approach to approximate the posterior covariance matrix. For such a low-rank approximation, one needs to solve a forward partial differential equation (PDE) and the adjoint PDE in both space and time. This in turn gives $\mathcal{O}(n_x n_t)$ complexity for both, computation and storage, where $n_x$ is the dimension of the spatial domain and $n_t$ is the dimension of the time domain. Such computations and storage demand are infeasible for large problems. To overcome this obstacle, we develop a new approach that utilizes a recently developed low-rank in time algorithm together with the low-rank Hessian method. We reduce both the computational complexity and storage requirement from $\mathcal{O}(n_x n_t)$ to $\mathcal{O}(n_x + n_t)$. We use numerical experiments to illustrate the advantages of our approach.

math.NA↗
arXiv2017

Preconditioning of a coupled Cahn--Hilliard Navier--Stokes system

Recently, Garcke et al.[Garcke, Hinze, Kahle, A stable and linear time discretization for a thermodynamically consistent model for two-phase incompressible flow, Applied Numerical Mathematics 99, pp. 151-171, 2016] developed a consistent discretization scheme for a thermodynamically consistent diffuse interface model for incompressible two-phase flows with different densities. At the heart of this method lies the solution of large and sparse linear systems that arise in a semismooth Newton method. We propose the use of preconditioned Krylov subspace solvers using effective Schur complement approximations. Numerical results illustrate the efficiency of our approach. In particular, our preconditioner is shown to be robust with respect to parameter changes.

math.NA↗
arXiv2016

Preconditioning PDE-constrained optimization with $\rm L^1$-sparsity and control constraints

PDE-constrained optimization aims at finding optimal setups for partial differential equations so that relevant quantities are minimized. Including sparsity promoting terms in the formulation of such problems results in more practically relevant computed controls but adds more challenges to the numerical solution of these problems. The needed $\rm L^1$-terms as well as additional inclusion of box control constraints require the use of semismooth Newton methods. We propose robust preconditioners for different formulations of the Newton's equation. With the inclusion of a line-search strategy and an inexact approach for the solution of the linear systems, the resulting semismooth Newton's method is feasible for practical problems. Our results are underpinned by a theoretical analysis of the preconditioned matrix. Numerical experiments illustrate the robustness of the proposed scheme.

math.NA↗
arXiv2016

Generalizing diffuse interface methods on graphs: non-smooth potentials and hypergraphs

Diffuse interface methods have recently been introduced for the task of semi-supervised learning. The underlying model is well-known in materials science but was extended to graphs using a Ginzburg--Landau functional and the graph Laplacian. We here generalize the previously proposed model by a non-smooth potential function. Additionally, we show that the diffuse interface method can be used for the segmentation of data coming from hypergraphs. For this we show that the graph Laplacian in almost all cases is derived from hypergraph information. Additionally, we show that the formerly introduced hypergraph Laplacian coming from a relaxed optimization problem is well suited to be used within the diffuse interface method. We present computational experiments for graph and hypergraph Laplacians.

stat.ML↗
arXiv2016

A Fast Multipole Method based on Band-limited Approximations for Radial Basis Functions

The meshless/meshfree radial basis function (RBF) method is a powerful technique for interpolating scattered data. But, solving large RBF interpolation problems without fast summation methods is computationally expensive. For RBF interpolation with $N$ points, using a direct method requires $\mathcal{O}(N^2)$ operations. As a fast summation method, the fast multipole method (FMM) has been implemented in speeding up the matrix-vector multiply, which reduces the complexity from $\mathcal{O}(N^2)$ to $\mathcal{O}(N^{1.5})$ and even to $\mathcal{O}(NlogN)$ for the multilevel fast multipole method (MLFMM). In this paper, we present a novel kernel-independent fast multipole method for RBF interpolation, which is used in combination with the evaluation of point-to-point interactions by RBF and the fast matrix-vector multiplication. This approach is based on band-limited approximation and quadrature rules, which extends the range of applicability of FMM.

math.NA↗
arXiv2015

Jumping into buckets, or How to decontaminate overlapping fat jets

At the LHC, tagging boosted heavy particle resonances which decay hadronically, such as top quarks and Higgs bosons, can play an essential role in new physics searches. In events with high multiplicity, however, the standard approach to tag boosted resonances by a large-radius fat jet becomes difficult because the resonances are not well-separated from other hard radiation. In this paper, we propose a different approach to tag and reconstruct boosted resonances by using the recently proposed mass-jump jet algorithm. A key feature of the algorithm is the flexible radius of the jets, which results from a terminating veto that prevents the recombination of two hard prongs if their combined jet mass is substantially larger than the masses of the separate prongs. The idea of collecting jets in "buckets" is also used. As an example, we consider the fully hadronic final state of pair-produced vectorlike top partners at the LHC, $pp\to T\bar{T}\to t\bar{t}HH$, and show that the new approach works better than the corresponding generalized $k_T$ jet clustering algorithm. We also show that tagging and kinematic reconstruction of boosted top quarks and Higgs bosons are possible with good quality even in these very busy final states. The vectorlike top partners are kinematically reconstructed, which allows their direct mass measurement.

hep-ph↗
arXiv2015

Tracking New Physics at the LHC and beyond

Heavy resonances are an integral part of many extensions of the Standard Model. The discovery of such heavy resonances are a primary goal at the LHC and future hadron colliders. When a particle with TeV-scale mass decays into electroweak-scale objects, these objects are highly boosted and their decay products are then strongly collimated, possibly to an extent that they cannot be resolved in the calorimeters of the detectors any more. We develop taggers for electroweak-scale resonances by combining the good energy resolution of the hadronic calorimeter with the superior spatial resolution of the tracking detector. Using track-based techniques we reconstruct heavy $W'$ and $Z'$ bosons and constrain the branching ratio of the rare Higgs boson decay $H \to Z A \to l^+l^-$ jets. The taggers show a good momentum-independent performance up to very large boosts. Using the proposed techniques will allow experiments at the LHC and a future hadron collider to significantly extend its reach in searches for heavy resonances.

hep-ph↗
arXiv2015

Vetoed jet clustering: The mass-jump algorithm

A new class of jet clustering algorithms is introduced. A criterion inspired by successful mass-drop taggers is applied that prevents the recombination of two hard prongs if their combined jet mass is substantially larger than the masses of the separate prongs. This "mass jump" veto effectively results in jets with variable radii in dense environments. Differences to existing methods are investigated. It is shown for boosted top quarks that the new algorithm has beneficial properties which can lead to improved tagging purity.

hep-ph↗
arXiv2014

Reconstruction of Vector-like Top Partner from Fully Hadronic Final States

We investigate the potential to search for the vector-like top partner in fully hadronic final states at the LHC. An algorithm is developed that kinematically reconstructs the vector-like top. We show that for moderate masses and a large branching fraction into the top quark and Higgs boson, the reconstruction works with good quality.

hep-ph↗