arXiv subjects

Martin S. Andersen

Publications and source records attributed to Martin S. Andersen.

At least 19 recordsLinked to original sources

arXiv2026

Informativity and Identifiability for Identification of Networks of Dynamical Systems

In this paper, we show how informativity and identifiability for networks of dynamical systems can be investigated using Gr\"obner bases. We provide a sufficient condition for informativity in terms of positive definiteness of the spectrum of external signals and full generic rank of the transfer function relating the external signals to the inputs of the predictor. Moreover, we show how generic local network identifiability can be investigated by computing the dimension of the fiber associated with the closed loop transfer function from external measurable signals to the measured outputs.

eess.SY↗

arXiv2025

Maximum Likelihood Estimation of Dynamic Sub-Networks with Missing Data

Maximum likelihood estimation is effective for identifying dynamical systems, but applying it to large networks becomes computationally prohibitive. This paper introduces a maximum likelihood estimation method that enables identification of sub-networks within complex interconnected systems without estimating the entire network. The key insight is that under specific topological conditions, a sub-network's parameters can be estimated using only local measurements: signals within the target sub-network and those in the directly connected to the so-called separator sub-network. This approach significantly reduces computational complexity while enhancing privacy by eliminating the need to share sensitive internal data across organizational boundaries. We establish theoretical conditions for network separability, derive the probability density function for the sub-network, and demonstrate the method's effectiveness through numerical examples.

eess.SY↗

arXiv2025

Numerically Efficient and Stable Algorithms for Kernel-Based Regularized System Identification Using Givens-Vector Representation

Numerically efficient and stable algorithms are essential for kernel-based regularized system identification. The state of art algorithms exploit the semiseparable structure of the kernel and are based on the generator representation of the kernel matrix. However, as will be shown from both the theory and the practice, the algorithms based on the generator representation are sometimes numerically unstable, which limits their application in practice. This paper aims to address this issue by deriving and exploiting an alternative Givens-vector representation of some widely used kernel matrices. Based on the Givens-vector representation, we derive algorithms that yield more accurate results than existing algorithms without sacrificing efficiency. We demonstrate their usage for the kernel-based regularized system identification. Monte Carlo simulations show that the proposed algorithms admit the same order of computational complexity as the state-of-the-art ones based on generator representation, but without issues with numerical stability.

math.NA↗

arXiv2025

Maximum Likelihood Estimation for System Identification of Networks of Dynamical Systems

This paper investigates maximum likelihood estimation for direct system identification in networks of dynamical systems. We establish that the proposed approach is both consistent and efficient. In addition, it is more generally applicable than existing methods, since it can be employed even when measurements are unavailable for all network nodes, provided that network identifiability is satisfied. Finally, we demonstrate that the maximum likelihood problem can be formulated without relying on a predictor, which is key to achieving computationally efficient numerical solutions.

eess.SY↗

arXiv2025

Connecting Kaporin's condition number and the Bregman log determinant divergence

This paper presents some theoretical results relating the Bregman log determinant matrix divergence to Kaporin's condition number. These can be viewed as nearness measures between a preconditioner and a given matrix, and we show under which conditions these two functions coincide. We also give examples of constraint sets over which it is equivalent to minimise these two objectives. We focus on preconditioners that are the sum of a positive definite and low-rank matrix, which were developed in a previous work. These were constructed as minimisers of the aforementioned divergence, and we show that they are only a constant scaling from also minimising Kaporin's condition number. We highlight connections to information geometry and comment on future directions.

math.NA↗

arXiv2024

A New Matrix Truncation Method for Improving Approximate Factorisation Preconditioners

In this experimental work, we present a general framework based on the Bregman log determinant divergence for preconditioning Hermitian positive definite linear systems. We explore this divergence as a measure of discrepancy between a preconditioner and a matrix. Given an approximate factorisation of a given matrix, the proposed framework informs the construction of a low-rank approximation of the typically indefinite factorisation error. The resulting preconditioner is therefore a sum of a Hermitian positive definite matrix given by an approximate factorisation plus a low-rank matrix. Notably, the low-rank term is not generally obtained as a truncated singular value decomposition (TSVD). This framework leads to a new truncation where principal directions are not based on the magnitude of the singular values, and we prove that such truncations are minimisers of the aforementioned divergence. We present several numerical examples showing that the proposed preconditioner can reduce the number of PCG iterations compared to a preconditioner constructed using a TSVD for the same rank. We also propose a heuristic to approximate the proposed preconditioner in the case where exact truncations cannot be computed explicitly (e.g. in a large-scale setting) and demonstrate its effectiveness over TSVD-based approaches.

math.NA↗

arXiv2024

Direct System Identification of Dynamical Networks with Partial Measurements: a Maximum Likelihood Approach

This paper introduces a novel direct approach to system identification of dynamic networks with missing data based on maximum likelihood estimation. Dynamic networks generally present a singular probability density function, which poses a challenge in the estimation of their parameters. By leveraging knowledge about the network's interconnections, we show that it is possible to transform the problem into a more tractable form by applying linear transformations. This results in a nonsingular probability density function, enabling the application of maximum likelihood estimation techniques. Our preliminary numerical results suggest that when combined with global optimization algorithms or a suitable initialization strategy, we are able to obtain a good estimate of the dynamics of the internal systems.

eess.SY↗

arXiv2023

Preconditioner Design via the Bregman Divergence

We study a preconditioner for a Hermitian positive definite linear system, which is obtained as the solution of a matrix nearness problem based on the Bregman log determinant divergence. The preconditioner is of the form of a Hermitian positive definite matrix plus a low-rank matrix. For this choice of structure, the generalised eigenvalues of the preconditioned matrix are easily calculated, and we show under which conditions the preconditioner minimises the $\ell_2$ condition number of the preconditioned matrix. We develop practical numerical approximations of the preconditioner based on the randomised singular value decomposition (SVD) and the Nyström approximation and provide corresponding approximation results. Furthermore, we prove that the Nyström approximation is in fact also a matrix approximation in a range-restricted Bregman divergence and establish several connections between this divergence and matrix nearness problems in different measures. Numerical examples are provided to support the theoretical results.

math.NA↗

arXiv2023

AdaSub: Stochastic Optimization Using Second-Order Information in Low-Dimensional Subspaces

We introduce AdaSub, a stochastic optimization algorithm that computes a search direction based on second-order information in a low-dimensional subspace that is defined adaptively based on available current and past information. Compared to first-order methods, second-order methods exhibit better convergence characteristics, but the need to compute the Hessian matrix at each iteration results in excessive computational expenses, making them impractical. To address this issue, our approach enables the management of computational expenses and algorithm efficiency by enabling the selection of the subspace dimension for the search. Our code is freely available on GitHub, and our preliminary numerical results demonstrate that AdaSub surpasses popular stochastic optimizers in terms of time and number of iterations required to reach a given accuracy.

math.OC↗

arXiv2022

A Bayesian Approach to CT Reconstruction with Uncertain Geometry

Computed tomography is a method for synthesizing volumetric or cross-sectional images of an object from a collection of projections. Popular reconstruction methods for computed tomography are based on idealized models and assumptions that may not be valid in practice. One such assumption is that the exact projection geometry is known. The projection geometry describes the relative location of the radiation source, object, and detector for each projection. However, in practice, the geometric parameters used to describe the position and orientation of the radiation source, object, and detector are estimated quantities with uncertainty. A failure to accurately estimate the geometry may lead to reconstructions with severe misalignment artifacts that significantly decrease their scientific or diagnostic value. We propose a novel reconstruction method that jointly estimates the reconstruction and the projection geometry. The reconstruction method is based on a Bayesian approach that yields a point estimate for the reconstruction and geometric parameters and, in addition, provides valuable information regarding their uncertainty. This is achieved by approximately sampling from the joint posterior distribution of the reconstruction and projection geometry using a hierarchical Gibbs sampler. Using real tomographic data, we demonstrate that the proposed reconstruction method significantly reduces misalignment artifacts. Compared with two commonly used alignment methods, our proposed method achieves comparable or better results under challenging conditions.

math.NA↗

arXiv2021

Regularization by Denoising Sub-sampled Newton Method for Spectral CT Multi-Material Decomposition

Spectral Computed Tomography (CT) is an emerging technology that enables to estimate the concentration of basis materials within a scanned object by exploiting different photon energy spectra. In this work, we aim at efficiently solving a model-based maximum-a-posterior problem to reconstruct multi-materials images with application to spectral CT. In particular, we propose to solve a regularized optimization problem based on a plug-in image-denoising function using a randomized second order method. By approximating the Newton step using a sketching of the Hessian of the likelihood function, it is possible to reduce the complexity while retaining the complex prior structure given by the data-driven regularizer. We exploit a non-uniform block sub-sampling of the Hessian with inexact but efficient Conjugate gradient updates that require only Jacobian-vector products for denoising term. Finally, we show numerical and experimental results for spectral CT materials decomposition.

math.OC↗

arXiv2019

On the Robustness and Scalability of Semidefinite Relaxation for Optimal Power Flow Problems

Semidefinite relaxation techniques have shown great promise for nonconvex optimal power flow problems. However, a number of independent numerical experiments have led to concerns about scalability and robustness of existing SDP solvers. To address these concerns, we investigate some numerical aspects of the problem and compare different state-of-the-art solvers. Our results demonstrate that semidefinite relaxations of large problem instances with on the order of 10,000 buses can be solved reliably and to reasonable accuracy within minutes. Furthermore, the semidefinite relaxation of a test case with 25,000 buses can be solved reliably within half an hour; the largest test case with 82,000 buses is solved within eight hours. We also compare the lower bound obtained via semidefinite relaxation to locally optimal solutions obtained with nonlinear optimization methods and calculate the optimality gap.

math.OC↗

arXiv2016

Technical Note: Proximal Ordered Subsets Algorithms for TV Constrained Optimization in CT Image Reconstruction

This article is intended to supplement our 2015 paper in Medical Physics titled "Noise properties of CT images reconstructed by use of constrained total-variation, data-discrepancy minimization", in which ordered subsets methods were employed to perform total-variation constrained data-discrepancy minimization for image reconstruction in X-ray computed tomography. Here we provide details regarding implementation of the ordered subsets algorithms and suggestions for selection of algorithm parameters. Detailed pseudo-code is included for every algorithm implemented in the original manuscript.

physics.med-ph↗

arXiv2015

Distributed Primal-dual Interior-point Methods for Solving Loosely Coupled Problems Using Message Passing

In this paper, we propose a distributed algorithm for solving loosely coupled problems with chordal sparsity which relies on primal-dual interior-point methods. We achieve this by distributing the computations at each iteration, using message-passing. In comparison to already existing distributed algorithms for solving such problems, this algorithm requires far less number of iterations to converge to a solution with high accuracy. Furthermore, it is possible to compute an upper-bound for the number of required iterations which, unlike already existing methods, only depends on the coupling structure in the problem. We illustrate the performance of our proposed method using a set of numerical examples.

math.OC↗

arXiv2015

Distributed Semidefinite Programming with Application to Large-scale System Analysis

Distributed algorithms for solving coupled semidefinite programs (SDPs) commonly require many iterations to converge. They also put high computational demand on the computational agents. In this paper we show that in case the coupled problem has an inherent tree structure, it is possible to devise an efficient distributed algorithm for solving such problems. This algorithm can potentially enjoy the same efficiency as centralized solvers that exploit sparsity. The proposed algorithm relies on predictor-corrector primal-dual interior-point methods, where we use a message-passing algorithm to compute the search directions distributedly. Message-passing here is closely related to dynamic programming over trees. This allows us to compute the exact search directions in a finite number of steps. Furthermore this number can be computed a priori and only depends on the coupling structure of the problem. We use the proposed algorithm for analyzing robustness of large-scale uncertain systems distributedly. We test the performance of this algorithm using numerical examples.

math.OC↗

arXiv2015

Tomographic Image Reconstruction using Training images

We describe and examine an algorithm for tomographic image reconstruction where prior knowledge about the solution is available in the form of training images. We first construct a nonnegative dictionary based on prototype elements from the training images; this problem is formulated as a regularized non-negative matrix factorization. Incorporating the dictionary as a prior in a convex reconstruction problem, we then find an approximate solution with a sparse representation in the dictionary. The dictionary is applied to non-overlapping patches of the image, which reduces the computational complexity compared to other algorithms. Computational experiments clarify the choice and interplay of the model parameters and the regularization parameters, and we show that in few-projection low-dose settings our algorithm is competitive with total variation regularization and tends to include more texture and more correct edges.

cs.CV↗

arXiv2014

Distributed Robustness Analysis of Interconnected Uncertain Systems Using Chordal Decomposition

Large-scale interconnected uncertain systems commonly have large state and uncertainty dimensions. Aside from the heavy computational cost of solving centralized robust stability analysis techniques, privacy requirements in the network can also introduce further issues. In this paper, we utilize IQC analysis for analyzing large-scale interconnected uncertain systems and we evade these issues by describing a decomposition scheme that is based on the interconnection structure of the system. This scheme is based on the so-called chordal decomposition and does not add any conservativeness to the analysis approach. The decomposed problem can be solved using distributed computational algorithms without the need for a centralized computational unit. We further discuss the merits of the proposed analysis approach using a numerical experiment.

math.OC↗

arXiv2013

Distributed Interior-point Method for Loosely Coupled Problems

In this paper, we put forth distributed algorithms for solving loosely coupled unconstrained and constrained optimization problems. Such problems are usually solved using algorithms that are based on a combination of decomposition and first order methods. These algorithms are commonly very slow and require many iterations to converge. In order to alleviate this issue, we propose algorithms that combine the Newton and interior-point methods with proximal splitting methods for solving such problems. Particularly, the algorithm for solving unconstrained loosely coupled problems, is based on Newton's method and utilizes proximal splitting to distribute the computations for calculating the Newton step at each iteration. A combination of this algorithm and the interior-point method is then used to introduce a distributed algorithm for solving constrained loosely coupled problems. We also provide guidelines on how to implement the proposed methods efficiently and briefly discuss the properties of the resulting solutions.

math.OC↗