Andreas Frommer
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View article: Using orthogonal projectors in multigrid multilevel Monte Carlo for trace estimation in lattice QCD
Using orthogonal projectors in multigrid multilevel Monte Carlo for trace estimation in lattice QCD Open
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 repre…
View article: Energy-preserving iteration schemes for Gauss collocation integrators
Energy-preserving iteration schemes for Gauss collocation integrators Open
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 dia…
View article: Splitting Techniques for DAEs with port-Hamiltonian Applications
Splitting Techniques for DAEs with port-Hamiltonian Applications Open
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…
View article: Polynomial Preconditioning for the Action of the Matrix Square Root and Inverse Square Root
Polynomial Preconditioning for the Action of the Matrix Square Root and Inverse Square Root Open
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 preconditio…
View article: Polynomial preconditioning for the action of the matrix square root and inverse square root
Polynomial preconditioning for the action of the matrix square root and inverse square root Open
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 preconditio…
View article: Operator splitting for semi-explicit differential-algebraic equations and port-Hamiltonian DAEs
Operator splitting for semi-explicit differential-algebraic equations and port-Hamiltonian DAEs Open
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) diff…
View article: Analysis of stochastic probing methods for estimating the trace of functions of sparse symmetric matrices
Analysis of stochastic probing methods for estimating the trace of functions of sparse symmetric matrices Open
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 stochasti…
View article: Operator splitting for port-Hamiltonian systems
Operator splitting for port-Hamiltonian systems Open
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 splittin…
View article: MG-MLMC++ as a Variance Reduction Method for Estimating the Trace of a Matrix Inverse
MG-MLMC++ as a Variance Reduction Method for Estimating the Trace of a Matrix Inverse Open
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 …
View article: 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
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 Open
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 discretiz…
View article: Deflated Multigrid Multilevel Monte Carlo
Deflated Multigrid Multilevel Monte Carlo Open
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 imp…
View article: Deflated Multigrid Multilevel Monte Carlo
Deflated Multigrid Multilevel Monte Carlo Open
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 imp…
View article: MGMLMC++ as a Variance Reduction Method for Estimating the Trace of a Matrix Inverse
MGMLMC++ as a Variance Reduction Method for Estimating the Trace of a Matrix Inverse Open
Hutchinson's method estimates the trace of a matrix function $f(D)$ stochastically using samples $\\tau^Hf(D)\\tau$, where the components of the random vectors $\\tau$ obey an isotropic probability distribution. Estimating the trace of the…
View article: On the convergence of randomized and greedy relaxation schemes for solving nonsingular linear systems of equations
On the convergence of randomized and greedy relaxation schemes for solving nonsingular linear systems of equations Open
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 …
View article: Krylov Subspace Recycling For Matrix Functions
Krylov Subspace Recycling For Matrix Functions Open
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 mat…
View article: On the Convergence of Randomized and Greedy Relaxation Schemes for Solving Nonsingular Linear Systems of Equations
On the Convergence of Randomized and Greedy Relaxation Schemes for Solving Nonsingular Linear Systems of Equations Open
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 …
View article: Krylov subspace restarting for matrix Laplace transforms
Krylov subspace restarting for matrix Laplace transforms Open
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 algor…
View article: Coarsest-level improvements in multigrid for lattice QCD on large-scale computers
Coarsest-level improvements in multigrid for lattice QCD on large-scale computers Open
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 standa…
View article: Matrix functions via linear systems built from continued fractions
Matrix functions via linear systems built from continued fractions Open
A widely used approach to compute the action $f(A)v$ of a matrix function $f(A)$ on a vector $v$ is to use a rational approximation $r$ for $f$ and compute $r(A)v$ instead. If $r$ is not computed adaptively as in rational Krylov methods, t…
View article: A Multilevel Approach to Variance Reduction in the Stochastic Estimation of the Trace of a Matrix
A Multilevel Approach to Variance Reduction in the Stochastic Estimation of the Trace of a Matrix Open
The trace of a matrix function f(A), most notably of the matrix inverse, can be estimated stochastically using samples< x,f(A)x> if the components of the random vectors x obey an appropriate probability distribution. However such a Monte-C…
View article: Analysis of Probing Techniques for Sparse Approximation and Trace Estimation of Decaying Matrix Functions
Analysis of Probing Techniques for Sparse Approximation and Trace Estimation of Decaying Matrix Functions Open
The computation of matrix functions $f(A)$, or related quantities like their trace, is an important but challenging task, in particular for large and sparse matrices $A$. In recent years, probing methods have become an often considered too…
View article: Asynchronous Richardson iterations
Asynchronous Richardson iterations Open
We consider asynchronous versions of the first and second order Richardson methods for solving linear systems of equations. These methods depend on parameters whose values are chosen a priori. We explore the parameter values that can be pr…
View article: Block Krylov Subspace Methods for Functions of Matrices II: Modified Block FOM
Block Krylov Subspace Methods for Functions of Matrices II: Modified Block FOM Open
We analyze an expansion of the generalized block Krylov subspace framework of [Electron. Trans. Numer. Anal., 47 (2017), pp. 100--126]. This expansion allows the use of low-rank modifications of the matrix projected onto the block Krylov s…