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This paper introduces a new numerical scheme for a system that includes evolution equations describing a perfect plasticity model with a time-dependent yield surface. We demonstrate that the solution to the proposed scheme is stable under suitable norms. Moreover, the stability leads to the existence of an exact solution, and we also prove that the solution to the proposed scheme converges strongly to the exact solution under suitable norms.

The increasing reliance on numerical methods for controlling dynamical systems and training machine learning models underscores the need to devise algorithms that dependably and efficiently navigate complex optimization landscapes. Classical gradient descent methods offer strong theoretical guarantees for convex problems; however, they demand meticulous hyperparameter tuning for non-convex ones. The emerging paradigm of learning to optimize (L2O) automates the discovery of algorithms with optimized performance leveraging learning models and data - yet, it lacks a theoretical framework to analyze convergence of the learned algorithms. In this paper, we fill this gap by harnessing nonlinear system theory. Specifically, we propose an unconstrained parametrization of all convergent algorithms for smooth non-convex objective functions. Notably, our framework is directly compatible with automatic differentiation tools, ensuring convergence by design while learning to optimize.

Topology optimization is used to systematically design contact-aided thermo-mechanical regulators, i.e. components whose effective thermal conductivity is tunable by mechanical deformation and contact. The thermo-mechanical interactions are modeled using a fully coupled non-linear thermo-mechanical finite element framework. To obtain the intricate heat transfer response, the components leverage self-contact, which is modeled using a third medium contact method. The effective heat transfer properties of the regulators are tuned by solving a topology optimization problem using a traditional gradient based algorithm. Several designs of thermo-mechanical regulators in the form of switches, diodes and triodes are presented.

Random effect models for time-to-event data, also known as frailty models, provide a conceptually appealing way of quantifying association between survival times and of representing heterogeneities resulting from factors which may be difficult or impossible to measure. In the literature, the random effect is usually assumed to have a continuous distribution. However, in some areas of application, discrete frailty distributions may be more appropriate. The present paper is about the implementation and interpretation of the Addams family of discrete frailty distributions. We propose methods of estimation for this family of densities in the context of shared frailty models for the hazard rates for case I interval-censored data. Our optimization framework allows for stratification of random effect distributions by covariates. We highlight interpretational advantages of the Addams family of discrete frailty distributions and the K-point distribution as compared to other frailty distributions. A unique feature of the Addams family and the K-point distribution is that the support of the frailty distribution depends on its parameters. This feature is best exploited by imposing a model on the distributional parameters, resulting in a model with non-homogeneous covariate effects that can be analysed using standard measures such as the hazard ratio. Our methods are illustrated with applications to multivariate case I interval-censored infection data.

We propose a two-step procedure to model and predict high-dimensional functional time series, where the number of function-valued time series $p$ is large in relation to the length of time series $n$. Our first step performs an eigenanalysis of a positive definite matrix, which leads to a one-to-one linear transformation for the original high-dimensional functional time series, and the transformed curve series can be segmented into several groups such that any two subseries from any two different groups are uncorrelated both contemporaneously and serially. Consequently in our second step those groups are handled separately without the information loss on the overall linear dynamic structure. The second step is devoted to establishing a finite-dimensional dynamical structure for all the transformed functional time series within each group. Furthermore the finite-dimensional structure is represented by that of a vector time series. Modelling and forecasting for the original high-dimensional functional time series are realized via those for the vector time series in all the groups. We investigate the theoretical properties of our proposed methods, and illustrate the finite-sample performance through both extensive simulation and two real datasets.

Model-based clustering of moderate or large dimensional data is notoriously difficult. We propose a model for simultaneous dimensionality reduction and clustering by assuming a mixture model for a set of latent scores, which are then linked to the observations via a Gaussian latent factor model. This approach was recently investigated by Chandra et al. (2023). The authors use a factor-analytic representation and assume a mixture model for the latent factors. However, performance can deteriorate in the presence of model misspecification. Assuming a repulsive point process prior for the component-specific means of the mixture for the latent scores is shown to yield a more robust model that outperforms the standard mixture model for the latent factors in several simulated scenarios. The repulsive point process must be anisotropic to favor well-separated clusters of data, and its density should be tractable for efficient posterior inference. We address these issues by proposing a general construction for anisotropic determinantal point processes. We illustrate our model in simulations as well as a plant species co-occurrence dataset.

Many combinatorial optimization problems can be formulated as the search for a subgraph that satisfies certain properties and minimizes the total weight. We assume here that the vertices correspond to points in a metric space and can take any position in given uncertainty sets. Then, the cost function to be minimized is the sum of the distances for the worst positions of the vertices in their uncertainty sets. We propose two types of polynomial-time approximation algorithms. The first one relies on solving a deterministic counterpart of the problem where the uncertain distances are replaced with maximum pairwise distances. We study in details the resulting approximation ratio, which depends on the structure of the feasible subgraphs and whether the metric space is Ptolemaic or not. The second algorithm is a fully-polynomial time approximation scheme for the special case of $s-t$ paths.

Symplectic integrators are widely implemented numerical integrators for Hamiltonian mechanics, which preserve the Hamiltonian structure (symplecticity) of the system. Although the symplectic integrator does not conserve the energy of the system, it is well known that there exists a conserving modified Hamiltonian, called the shadow Hamiltonian. For the Nambu mechanics, which is a kind of generalized Hamiltonian mechanics, we can also construct structure-preserving integrators by the same procedure used to construct the symplectic integrators. In the structure-preserving integrator, however, the existence of shadow Hamiltonians is nontrivial. This is because the Nambu mechanics is driven by multiple Hamiltonians and it is nontrivial whether the time evolution by the integrator can be cast into the Nambu mechanical time evolution driven by multiple shadow Hamiltonians. In this paper we present a general procedure to calculate the shadow Hamiltonians of structure-preserving integrators for Nambu mechanics, and give an example where the shadow Hamiltonians exist. This is the first attempt to determine the concrete forms of the shadow Hamiltonians for a Nambu mechanical system. We show that the fundamental identity, which corresponds to the Jacobi identity in Hamiltonian mechanics, plays an important role in calculating the shadow Hamiltonians using the Baker-Campbell-Hausdorff formula. It turns out that the resulting shadow Hamiltonians have indefinite forms depending on how the fundamental identities are used. This is not a technical artifact, because the exact shadow Hamiltonians obtained independently have the same indefiniteness.

Approximation of solutions to partial differential equations (PDE) is an important problem in computational science and engineering. Using neural networks as an ansatz for the solution has proven a challenge in terms of training time and approximation accuracy. In this contribution, we discuss how sampling the hidden weights and biases of the ansatz network from data-agnostic and data-dependent probability distributions allows us to progress on both challenges. In most examples, the random sampling schemes outperform iterative, gradient-based optimization of physics-informed neural networks regarding training time and accuracy by several orders of magnitude. For time-dependent PDE, we construct neural basis functions only in the spatial domain and then solve the associated ordinary differential equation with classical methods from scientific computing over a long time horizon. This alleviates one of the greatest challenges for neural PDE solvers because it does not require us to parameterize the solution in time. For second-order elliptic PDE in Barron spaces, we prove the existence of sampled networks with $L^2$ convergence to the solution. We demonstrate our approach on several time-dependent and static PDEs. We also illustrate how sampled networks can effectively solve inverse problems in this setting. Benefits compared to common numerical schemes include spectral convergence and mesh-free construction of basis functions.

In this paper, to address the optimization problem on a compact matrix manifold, we introduce a novel algorithmic framework called the Transformed Gradient Projection (TGP) algorithm, using the projection onto this compact matrix manifold. Compared with the existing algorithms, the key innovation in our approach lies in the utilization of a new class of search directions and various stepsizes, including the Armijo, nonmonotone Armijo, and fixed stepsizes, to guide the selection of the next iterate. Our framework offers flexibility by encompassing the classical gradient projection algorithms as special cases, and intersecting the retraction-based line-search algorithms. Notably, our focus is on the Stiefel or Grassmann manifold, revealing that many existing algorithms in the literature can be seen as specific instances within our proposed framework, and this algorithmic framework also induces several new special cases. Then, we conduct a thorough exploration of the convergence properties of these algorithms, considering various search directions and stepsizes. To achieve this, we extensively analyze the geometric properties of the projection onto compact matrix manifolds, allowing us to extend classical inequalities related to retractions from the literature. Building upon these insights, we establish the weak convergence, convergence rate, and global convergence of TGP algorithms under three distinct stepsizes. In cases where the compact matrix manifold is the Stiefel or Grassmann manifold, our convergence results either encompass or surpass those found in the literature. Finally, through a series of numerical experiments, we observe that the TGP algorithms, owing to their increased flexibility in choosing search directions, outperform classical gradient projection and retraction-based line-search algorithms in several scenarios.