arXiv subjects

Monika Eisenmann

Publications and source records attributed to Monika Eisenmann.

14 recordsLinked to original sources

arXiv2026

Error Bound and Stability Analysis for a Randomized Singly Diagonally Implicit Runge-Kutta Method

A randomized Singly Diagonally Implicit Runge-Kutta (SDIRK) method, based on the randomized trapezoidal rule as the underlying quadrature scheme, is proposed. Every realization of the scheme is an algebraically stable SDIRK method of at least second order. The main result is the proof that the randomized scheme converges with order 2.5 in the root mean square sense under low regularity assumptions. Numerical experiments illustrate the robustness of the new scheme when applied to nonsmooth problems.

math.NA↗

arXiv2026

Error bounds for full space-time splitting discretizations of semi-linear SPDEs -- with a focus on dG domain decompositions

We consider a fully discretized numerical scheme for parabolic stochastic partial differential equations with multiplicative noise. Our abstract framework can be applied to formulate a non-iterative domain decomposition approach. Such methods can help to parallelize the code and therefore lead to a more efficient implementation. The domain decomposition is integrated through the Douglas-Rachford splitting scheme, where one split operator acts on one part of the domain. For an efficient space discretization of the underlying equation, we chose the discontinuous Galerkin method as this suits the parallelization strategy well. For this fully discretized scheme, we provide a strong space-time convergence result. We conclude the manuscript with numerical experiments validating our theoretical findings.

math.NA↗

arXiv2026

Convergence analysis of Schwarz-like methods for degenerate elliptic-parabolic equations

Convergence is proven for Schwarz-like methods applied to degenerate elliptic-parabolic equations with a $p$-structure. This family of PDEs, e.g., arises when modelling nonlinear diffusion processes. The Schwarz-like approximation methods are based on decomposing the space-time domain into overlapping subdomains, which enables parallel implementations. The methods are derived by introducing a pseudo-time component and applying time integrators of splitting type, which are time stepped towards infinity. This approach of decomposing the space-time domain is related to Schwarz waveform relaxation methods, but the methods considered here have the advantage that they can be proven to converge when applied to nonlinear parabolic, or even degenerate elliptic-parabolic, PDEs. We prove convergence by deriving a nonlinear framework based on the abstract theory for monotone operators and the existence theory for degenerate elliptic-parabolic equations.

math.NA↗

arXiv2024

Analysis of a Class of Stochastic Component-Wise Soft-Clipping Schemes

Choosing the optimization algorithm that performs best on a given machine learning problem is often delicate, and there is no guarantee that current state-of-the-art algorithms will perform well across all tasks. Consequently, the more reliable methods that one has at hand, the larger the likelihood of a good end result. To this end, we introduce and analyze a large class of stochastic so-called soft-clipping schemes with a broad range of applications. Despite the wide adoption of clipping techniques in practice, soft-clipping methods have not been analyzed to a large extent in the literature. In particular, a rigorous mathematical analysis is lacking in the general, nonlinear case. Our analysis lays a theoretical foundation for a large class of such schemes, and motivates their usage. In particular, under standard assumptions such as Lipschitz continuous gradients of the objective function, we give rigorous proofs of convergence in expectation. These include rates in both the convex and the non-convex case, as well as almost sure convergence to a stationary point in the non-convex case. The computational cost of the analyzed schemes is essentially the same as that of stochastic gradient descent.

math.OC↗

arXiv2024

A domain decomposition method for stochastic evolution equations

In recent years, SPDEs have become a well-studied field in mathematics. With their increase in popularity, it becomes important to efficiently approximate their solutions. Thus, our goal is a contribution towards the development of efficient and practical time-stepping methods for SPDEs. Operator splitting schemes are a powerful tool for deterministic and stochastic differential equations. An example is given by domain decomposition schemes, where we split the domain into sub-domains. Instead of solving one expensive problem on the entire domain, we deal with cheaper problems on the sub-domains. This is particularly useful in modern computer architectures, as the sub-problems may often be solved in parallel. While splitting methods have already been used to study domain decomposition methods for deterministic PDEs, this is a new approach for SPDEs. We provide an abstract convergence analysis of a splitting scheme for stochastic evolution equations and state a domain decomposition scheme as an application of the setting. The theoretical results are verified through numerical experiments.

math.NA↗

arXiv2022

A randomized operator splitting scheme inspired by stochastic optimization methods

In this paper, we combine the operator splitting methodology for abstract evolution equations with that of stochastic methods for large-scale optimization problems. The combination results in a randomized splitting scheme, which in a given time step does not necessarily use all the parts of the split operator. This is in contrast to deterministic splitting schemes which always use every part at least once, and often several times. As a result, the computational cost can be significantly decreased in comparison to such methods. We rigorously define a randomized operator splitting scheme in an abstract setting and provide an error analysis where we prove that the temporal convergence order of the scheme is at least 1/2. We illustrate the theory by numerical experiments on both linear and quasilinear diffusion problems, using a randomized domain decomposition approach. We conclude that choosing the randomization in certain ways may improve the order to 1. This is as accurate as applying e.g. backward (implicit) Euler to the full problem, without splitting.

math.NA↗

arXiv2021

Sub-linear convergence of a stochastic proximal iteration method in Hilbert space

We consider a stochastic version of the proximal point algorithm for optimization problems posed on a Hilbert space. A typical application of this is supervised learning. While the method is not new, it has not been extensively analyzed in this form. Indeed, most related results are confined to the finite-dimensional setting, where error bounds could depend on the dimension of the space. On the other hand, the few existing results in the infinite-dimensional setting only prove very weak types of convergence, owing to weak assumptions on the problem. In particular, there are no results that show convergence with a rate. In this article, we bridge these two worlds by assuming more regularity of the optimization problem, which allows us to prove convergence with an (optimal) sub-linear rate also in an infinite-dimensional setting. In particular, we assume that the objective function is the expected value of a family of convex differentiable functions. While we require that the full objective function is strongly convex, we do not assume that its constituent parts are so. Further, we require that the gradient satisfies a weak local Lipschitz continuity property, where the Lipschitz constant may grow polynomially given certain guarantees on the variance and higher moments near the minimum. We illustrate these results by discretizing a concrete infinite-dimensional classification problem with varying degrees of accuracy.

math.OC↗

arXiv2021

Sub-linear convergence of a tamed stochastic gradient descent method in Hilbert space

In this paper, we introduce the tamed stochastic gradient descent method (TSGD) for optimization problems. Inspired by the tamed Euler scheme, which is a commonly used method within the context of stochastic differential equations, TSGD is an explicit scheme that exhibits stability properties similar to those of implicit schemes. As its computational cost is essentially equivalent to that of the well-known stochastic gradient descent method (SGD), it constitutes a very competitive alternative to such methods. We rigorously prove (optimal) sub-linear convergence of the scheme for strongly convex objective functions on an abstract Hilbert space. The analysis only requires very mild step size restrictions, which illustrates the good stability properties. The analysis is based on a priori estimates more frequently encountered in a time integration context than in optimization, and this alternative approach provides a different perspective also on the convergence of SGD. Finally, we demonstrate the usability of the scheme on a problem arising in a context of supervised learning.

math.OC↗

arXiv2020

A variational approach to the sum splitting scheme

Nonlinear parabolic equations are frequently encountered in applications and efficient approximating techniques for their solution are of great importance. In order to provide an effective scheme for the temporal approximation of such equations, we present a sum splitting scheme that comes with a straight forward parallelization strategy. The convergence analysis is carried out in a variational framework that allows for a general setting and, in particular, nontrivial temporal coefficients. The aim of this work is to illustrate the significant advantages of a variational framework for operator splittings and use this to extend semigroup based theory for this type of scheme.

math.NA↗

arXiv2019

Error estimates of the backward Euler-Maruyama method for multi-valued stochastic differential equations

In this paper, we derive error estimates of the backward Euler-Maruyama method applied to multi-valued stochastic differential equations. An important example of such an equation is a stochastic gradient flow whose associated potential is not continuously differentiable, but assumed to be convex. We show that the backward Euler-Maruyama method is well-defined and convergent of order at least $1/4$ with respect to the root-mean-square norm. Our error analysis relies on techniques for deterministic problems developed in [Nochetto, Savaré, and Verdi, Comm.\ Pure Appl.\ Math., 2000]. We verify that our setting applies to an overdamped Langevin equation with a discontinuous gradient and to a spatially semi-discrete approximation of the stochastic $p$-Laplace equation.

math.NA↗

arXiv2018

Two quadrature rules for stochastic Itô-integrals with fractional Sobolev regularity

In this paper we study the numerical quadrature of a stochastic integral, where the temporal regularity of the integrand is measured in the fractional Sobolev-Slobodeckij norm in $W^{σ,p}(0,T)$, $σ\in (0,2)$, $p \in [2,\infty)$. We introduce two quadrature rules: The first is best suited for the parameter range $σ\in (0,1)$ and consists of a Riemann-Maruyama approximation on a randomly shifted grid. The second quadrature rule considered in this paper applies to the case of a deterministic integrand of fractional Sobolev regularity with $σ\in (1,2)$. In both cases the order of convergence is equal to $σ$ with respect to the $L^p$-norm. As an application, we consider the stochastic integration of a Poisson process, which has discontinuous sample paths. The theoretical results are accompanied by numerical experiments.

math.PR↗

arXiv2018

On a randomized backward Euler method for nonlinear evolution equations with time-irregular coefficients

In this paper we introduce a randomized version of the backward Euler method, that is applicable to stiff ordinary differential equations and nonlinear evolution equations with time-irregular coefficients. In the finite-dimensional case, we consider Carathéodory type functions satisfying a one-sided Lipschitz condition. After investigating the well-posedness and the stability properties of the randomized scheme, we prove the convergence to the exact solution with a rate of $0.5$ in the root-mean-square norm assuming only that the coefficient function is square integrable with respect to the temporal parameter. These results are then extended to the numerical solution of infinite-dimensional evolution equations under monotonicity and Lipschitz conditions. Here we consider a combination of the randomized backward Euler scheme with a Galerkin finite element method. We obtain error estimates that correspond to the regularity of the exact solution. The practicability of the randomized scheme is also illustrated through several numerical experiments.

math.NA↗

arXiv2018

Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data

For initial value problems associated with operator-valued Riccati differential equations posed in the space of Hilbert--Schmidt operators existence of solutions is studied. An existence result known for algebraic Riccati equations is generalized and used to obtain the existence of a solution to the approximation of the problem via a backward Euler scheme. Weak and strong convergence of the sequence of approximate solutions is established permitting a large class of right-hand sides and initial data.

math.AP↗

arXiv2017

Convergence analysis of domain decomposition based time integrators for degenerate parabolic equations

Domain decomposition based time integrators allow the usage of parallel and distributed hardware, making them well-suited for the temporal discretization of parabolic systems, in general, and degenerate parabolic problems, in particular. The latter is due to the degenerate equations' finite speed of propagation. In this study, a rigours convergence analysis is given for such integrators without assuming any restrictive regularity on the solutions or the domains. The analysis is conducted by first deriving a new variational framework for the domain decomposition, which is applicable to the two standard degenerate examples. That is, the $p$-Laplace and the porous medium type vector fields. Secondly, the decomposed vector fields are restricted to the underlying pivot space and the time integration of the parabolic problem can then be interpreted as an operators splitting applied to a dissipative evolution equation. The convergence results then follow by employing elements of the approximation theory for nonlinear semigroups.

math.NA↗