arXiv subjects

Martin Redmann

Publications and source records attributed to Martin Redmann.

27 records · Page 2Linked to original sources

arXiv2019

An $L^2_T$-error bound for time-limited balanced truncation

Model order reduction (MOR) is often applied to spatially-discretized partial differential equations to reduce their order and hence decrease computational complexity. A reduced system can be obtained, e.g., by time-limited balanced truncation, a method that aims to construct an accurate reduced order model on a given finite time interval $[0, T]$. This particular balancing related MOR technique is studied in this paper. An $L^2_T$-error bound based on the truncated time-limited singular values is proved and is the main result of this paper. The derived error bound converges (as $T\rightarrow \infty$) to the well-known $\mathcal H_\infty$-error bound of unrestricted balanced truncation, a scheme that is used to construct a good reduced system on the entire time line. The techniques within the proofs of this paper can also be applied to unrestricted balanced truncation so that a relatively short time domain proof of the $\mathcal H_\infty$-error bound is found here.

math.OC↗

arXiv2019

Dynamic programming for optimal stopping via pseudo-regression

We introduce new variants of classical regression-based algorithms for optimal stopping problems based on computation of regression coefficients by Monte Carlo approximation of the corresponding $L^2$ inner products instead of the least-squares error functional. Coupled with new proposals for simulation of the underlying samples, we call the approach "pseudo regression". A detailed convergence analysis is provided and it is shown that the approach asymptotically leads to less computational cost for a pre-specified error tolerance, hence to lower complexity. The method is justified by numerical examples.

q-fin.CP↗

arXiv2019

A new type of singular perturbation approximation for stochastic bilinear systems

Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretized (stochastic) partial differential equations and hence reduce computational complexity. A particular class of MOR techniques is balancing related methods which rely on simultaneously diagonalizing the system Gramians. This has been extensively studied for deterministic linear systems. The balancing procedure has already been extended to bilinear equations [1], an important subclass of nonlinear systems. The choice of Gramians in [1] is referred to be the standard approach. In [18], a balancing related MOR scheme for bilinear systems called singular perturbation approximation (SPA) has been described that relies on the standard choice of Gramians. However, no error bound for this method could be proved. In this paper, we extend the setting used in [18] by considering a stochastic system with bilinear drift and linear diffusion term. Moreover, we propose a modified reduced order model and choose a different reachability Gramian. Based on this new approach, an $L^2$-error bound is proved for SPA which is the main result of this paper. This bound is new even for deterministic bilinear systems.

math.OC↗

arXiv2018

Energy estimates and model order reduction for stochastic bilinear systems

In this paper, we investigate a large-scale stochastic system with bilinear drift and linear diffusion term. Such high dimensional systems appear for example when discretizing a stochastic partial differential equations in space. We study a particular model order reduction technique called balanced truncation (BT) to reduce the order of spatially-discretized systems and hence reduce computational complexity. We introduce suitable Gramians to the system and prove energy estimates that can be used to identify states which contribute only very little to the system dynamics. When BT is applied the reduced system is obtained by removing these states from the original system. The main contribution of this paper is an $L^2$-error bound for BT for stochastic bilinear systems. This result is new even for deterministic bilinear equations. In order to achieve it, we develop a new technique which is not available in the literature so far.

math.OC↗

arXiv2018

Solving linear parabolic rough partial differential equations

We study linear rough partial differential equations in the setting of [Friz and Hairer, Springer, 2014, Chapter 12]. More precisely, we consider a linear parabolic partial differential equation driven by a deterministic rough path $\mathbf{W}$ of Hölder regularity $α$ with $1/3 < α \le 1/2$. Based on a stochastic representation of the solution of the rough partial differential equation, we propose a regression Monte Carlo algorithm for spatio-temporal approximation of the solution. We provide a full convergence analysis of the proposed approximation method which essentially relies on the new bounds for the higher order derivatives of the solution in space. Finally, a comprehensive simulation study showing the applicability of the proposed algorithm is presented.

math.PR↗

arXiv2017

Towards Time-Limited $\mathcal H_2$-Optimal Model Order Reduction

In order to solve partial differential equations numerically and accurately, a high order spatial discretization is usually needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretized systems and hence reduce computational complexity. A particular class of MOR techniques are $\mathcal H_2$-optimal methods such as the iterative rational Krylov subspace algorithm (IRKA) and related schemes. However, these methods are used to obtain good approximations on a infinite time-horizon. Thus, in this work, our main goal is to discuss MOR schemes for time-limited linear systems. For this, we propose an alternative time-limited $\mathcal H_2$-norm and show its connection with the time-limited Gramians. We then provide first-order optimality conditions for an optimal reduced order model (ROM) with respect to the time-limited $\mathcal H_2$-norm. Based on these optimality conditions, we propose an iterative scheme, which, upon convergence, aims at satisfying these conditions approximately. Then, we analyze how far away the obtained ROM due to the proposed algorithm is from satisfying the optimality conditions. We test the efficiency of the proposed iterative scheme using various numerical examples and illustrate that the newly proposed iterative method can lead to a better reduced-order compared to the unrestricted IRKA in the finite time interval of interest.

math.OC↗

arXiv2017

An $\mathcal H_2$-Type Error Bound for Time-Limited Balanced Truncation

When solving partial differential equations numerically, usually a high order spatial discretization is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretized systems and hence reduce computational complexity. A particular MOR technique to obtain a reduced order model (ROM) is balanced truncation (BT). However, if one aims at finding a good ROM on a certain finite time interval only, time-limited BT (TLBT) can be a more accurate alternative. So far, no error bound on TLBT has been proved. In this paper, we close this gap in the theory by providing an $\mathcal H_2$ error bound for TLBT with two different representations. The performance of the error bound is then shown in several numerical experiments.

math.OC↗

arXiv2017

Type II balanced truncation for deterministic bilinear control systems

When solving partial differential equations numerically, usually a high order spatial discretisation is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretised systems and hence reduce computational complexity. A particular MOR technique to obtain a reduced order model (ROM) is balanced truncation (BT), a method which has been extensively studied for deterministic linear systems. As so-called type I BT it has already been extended to bilinear equations, an important subclass of nonlinear systems. We provide an alternative generalisation of the linear setting to bilinear systems which is called type II BT. The Gramians that we propose in this context contain information about the control. It turns out that the new approach delivers energy bounds which are not just valid in a small neighbourhood of zero. Furthermore, we provide an $H_\infty$-error bound which so far is not known when applying type I BT to bilinear systems.

math.OC↗

arXiv2016

Balanced truncation and singular perturbation approximation model order reduction for stochastically controlled linear systems

When solving linear stochastic differential equations numerically, usually a high order spatial discretisation is used. Balanced truncation (BT) and singular perturbation approximation (SPA) are well-known projection techniques in the deterministic framework which reduce the order of a control system and hence reduce computational complexity. This work considers both methods when the control is replaced by a noise term. We provide theoretical tools such as stochastic concepts for reachability and observability, which are necessary for balancing related model order reduction of linear stochastic differential equations with additive Lévy noise. Moreover, we derive error bounds for both BT and SPA and provide numerical results for a specific example which support the theory.

math.NA↗