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A multi-step extended maximum residual Kaczmarz method is presented for the solution of the large inconsistent linear system of equations by using the multi-step iterations technique. Theoretical analysis proves the proposed method is convergent and gives an upper bound on its convergence rate. Numerical experiments show that the proposed method is effective and outperforms the existing extended Kaczmarz methods in terms of the number of iteration steps and the computational costs.

Accurate delineation of key waveforms in an ECG is a critical initial step in extracting relevant features to support the diagnosis and treatment of heart conditions. Although deep learning based methods using a segmentation model to locate the P, QRS, and T waves have shown promising results, their ability to handle signals exhibiting arrhythmia remains unclear. This study builds on existing research by introducing a U-Net-like segmentation model for ECG delineation, with a particular focus on diverse arrhythmias. For this purpose, we curate an internal dataset containing waveform boundary annotations for various arrhythmia types to train and validate our model. Our key contributions include identifying segmentation model failures in different arrhythmia types, developing a robust model using a diverse training set, achieving comparable performance on benchmark datasets, and introducing a classification guided strategy to reduce false P wave predictions for specific arrhythmias. This study advances deep learning based ECG delineation in the context of arrhythmias and highlights its challenges.

Model-based sequential approaches to discrete "black-box" optimization, including Bayesian optimization techniques, often access the same points multiple times for a given objective function in interest, resulting in many steps to find the global optimum. Here, we numerically study the effect of a postprocessing method on Bayesian optimization that strictly prohibits duplicated samples in the dataset. We find the postprocessing method significantly reduces the number of sequential steps to find the global optimum, especially when the acquisition function is of maximum a posterior estimation. Our results provide a simple but general strategy to solve the slow convergence of Bayesian optimization for high-dimensional problems.

We consider the problem of minimizing the makespan on batch processing identical machines, subject to compatibility constraints, where two jobs are compatible if they can be processed simultaneously in a same batch. These constraints are modeled by an undirected graph $G$, in which compatible jobs are represented by adjacent vertices. We show that several subproblems are polynomial. We propose some exact polynomial algorithms to solve these subproblems. To solve the general case, we propose a mixed-integer linear programming (MILP) formulation alongside with heuristic approaches. Furthermore, computational experiments are carried out to measure the performance of the proposed methods.

In this work, we develop first-order (Hessian-free) and zero-order (derivative-free) implementations of the Cubically regularized Newton method for solving general non-convex optimization problems. For that, we employ finite difference approximations of the derivatives. We use a special adaptive search procedure in our algorithms, which simultaneously fits both the regularization constant and the parameters of the finite difference approximations. It makes our schemes free from the need to know the actual Lipschitz constants. Additionally, we equip our algorithms with the lazy Hessian update that reuse a previously computed Hessian approximation matrix for several iterations. Specifically, we prove the global complexity bound of $\mathcal{O}( n^{1/2} \epsilon^{-3/2})$ function and gradient evaluations for our new Hessian-free method, and a bound of $\mathcal{O}( n^{3/2} \epsilon^{-3/2} )$ function evaluations for the derivative-free method, where $n$ is the dimension of the problem and $\epsilon$ is the desired accuracy for the gradient norm. These complexity bounds significantly improve the previously known ones in terms of the joint dependence on $n$ and $\epsilon$, for the first-order and zeroth-order non-convex optimization.

This work considers Bayesian experimental design for the inverse boundary value problem of linear elasticity in a two-dimensional setting. The aim is to optimize the positions of compactly supported pressure activations on the boundary of the examined body in order to maximize the value of the resulting boundary deformations as data for the inverse problem of reconstructing the Lam\'e parameters inside the object. We resort to a linearized measurement model and adopt the framework of Bayesian experimental design, under the assumption that the prior and measurement noise distributions are mutually independent Gaussians. This enables the use of the standard Bayesian A-optimality criterion for deducing optimal positions for the pressure activations. The (second) derivatives of the boundary measurements with respect to the Lam\'e parameters and the positions of the boundary pressure activations are deduced to allow minimizing the corresponding objective function, i.e., the trace of the covariance matrix of the posterior distribution, by a gradient-based optimization algorithm. Two-dimensional numerical experiments are performed to demonstrate the functionality of our approach.

Electrical circuits are present in a variety of technologies, making their design an important part of computer aided engineering. The growing number of tunable parameters that affect the final design leads to a need for new approaches of quantifying their impact. Machine learning may play a key role in this regard, however current approaches often make suboptimal use of existing knowledge about the system at hand. In terms of circuits, their description via modified nodal analysis is well-understood. This particular formulation leads to systems of differential-algebraic equations (DAEs) which bring with them a number of peculiarities, e.g. hidden constraints that the solution needs to fulfill. We aim to use the recently introduced dissection concept for DAEs that can decouple a given system into ordinary differential equations, only depending on differential variables, and purely algebraic equations that describe the relations between differential and algebraic variables. The idea then is to only learn the differential variables and reconstruct the algebraic ones using the relations from the decoupling. This approach guarantees that the algebraic constraints are fulfilled up to the accuracy of the nonlinear system solver, which represents the main benefit highlighted in this article.

We present a new approach to understanding the relationship between loss curvature and input-output model behaviour in deep learning. Specifically, we use existing empirical analyses of the spectrum of deep network loss Hessians to ground an ansatz tying together the loss Hessian and the input-output Jacobian of a deep neural network over training samples throughout training. We then prove a series of theoretical results which quantify the degree to which the input-output Jacobian of a model approximates its Lipschitz norm over a data distribution, and deduce a novel generalisation bound in terms of the empirical Jacobian. We use our ansatz, together with our theoretical results, to give a new account of the recently observed progressive sharpening phenomenon, as well as the generalisation properties of flat minima. Experimental evidence is provided to validate our claims.

We present a computational design method that optimizes the reinforcement of dental prostheses and increases the durability and fracture resistance of dentures. Our approach optimally places reinforcement, which could be implemented by modern multi-material, three-dimensional printers. The study focuses on reducing deformation by identifying regions within the structure that require reinforcement (E-glass material). Our method is applied to a three-dimensional removable lower jaw dental prosthesis and aims to improve the living quality of denture patients and pretend fracture of dental reinforcement in clinical studies. To do this, we compare the deformation results of a non-reinforced denture and a reinforced denture that has two materials. The results indicate the maximum deformation is lower and node-based displacement distribution demonstrates that the average displacement distribution is much better in the reinforced denture.

Hawkes processes are often applied to model dependence and interaction phenomena in multivariate event data sets, such as neuronal spike trains, social interactions, and financial transactions. In the nonparametric setting, learning the temporal dependence structure of Hawkes processes is generally a computationally expensive task, all the more with Bayesian estimation methods. In particular, for generalised nonlinear Hawkes processes, Monte-Carlo Markov Chain methods applied to compute the doubly intractable posterior distribution are not scalable to high-dimensional processes in practice. Recently, efficient algorithms targeting a mean-field variational approximation of the posterior distribution have been proposed. In this work, we first unify existing variational Bayes approaches under a general nonparametric inference framework, and analyse the asymptotic properties of these methods under easily verifiable conditions on the prior, the variational class, and the nonlinear model. Secondly, we propose a novel sparsity-inducing procedure, and derive an adaptive mean-field variational algorithm for the popular sigmoid Hawkes processes. Our algorithm is parallelisable and therefore computationally efficient in high-dimensional setting. Through an extensive set of numerical simulations, we also demonstrate that our procedure is able to adapt to the dimensionality of the parameter of the Hawkes process, and is partially robust to some type of model mis-specification.