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Partition of unity methods (PUM) are of domain decomposition type and provide the opportunity for multiscale and multiphysics numerical modeling. Different physical models can exist within a PUM scheme for handling problems with zones of linear elasticity and zones where fractures occur. Here, the peridynamic (PD) model is used in regions of fracture and smooth PUM is used in the surrounding linear elastic media. The method is a so-called global-local enrichment strategy. The elastic fields of the undamaged media provide appropriate boundary data for the localized PD simulations. The first steps for a combined PD/PUM simulator are presented. In part I of this series, we show that the local PD approximation can be utilized to enrich the global PUM approximation to capture the true material response with high accuracy efficiently. Test problems are provided demonstrating the validity and potential of this numerical approach.

The dynamic iteration method with a restricted additive Schwarz splitting is investigated to co-simulate linear differential algebraic equations system coming from RLC electrical circuit with linear components. We show the pure linear convergence or divergence of the method with respect to the linear operator belonging to the restricted additive Schwarz interface. It allows us to accelerate it toward the true solution with the Aitken's technique for accelerating convergence. This provides a dynamic iteration method less sensitive to the splitting. Numerical examples with convergent and divergent splitting show the efficiency of the proposed approach. We also test it on a linear RLC circuit combining different types of circuit modeling (Transient Stability model and Electro-Magnetic Transient model) with overlapping partitions. Finally, some results for a weakly nonlinear differential algebraic equations system are also provided.

We consider the evolution of curve networks in two dimensions (2d) and surface clusters in three dimensions (3d). The motion of the interfaces is described by surface diffusion, with boundary conditions at the triple junction points/lines, where three interfaces meet, and at the boundary points/lines, where an interface meets a fixed planar boundary. We propose a parametric finite element method based on a suitable variational formulation. The constructed method is semi-implicit and can be shown to satisfy the volume conservation of each enclosed bubble and the unconditional energy-stability, thus preserving the two fundamental geometric structures of the flow. Besides, the method has very good properties with respect to the distribution of mesh points, thus no mesh smoothing or regularization technique is required. A generalization of the introduced scheme to the case of anisotropic surface energies and non-neutral external boundaries is also considered. Numerical results are presented for the evolution of two-dimensional curve networks and three-dimensional surface clusters in the cases of both isotropic and anisotropic surface energies.

We introduce a geometric stencil selection algorithm for Laplacian in 3D that significantly improves octant-based selection considered earlier. The goal of the algorithm is to choose a small subset from a set of irregular points surrounding a given point that admits an accurate numerical differentiation formula. The subset serves as an influence set for the numerical approximation of the Laplacian in meshless finite difference methods using either polynomial or kernel-based techniques. Numerical experiments demonstrate a competitive performance of this method in comparison to the finite element method and to other selection methods for solving the Dirichlet problems for the Poisson equation on several STL models. Discretization nodes for these domains are obtained either by 3D triangulations or from Cartesian grids or Halton quasi-random sequences.

We report on a general purpose method for the scalar Stefan problem inspired by the standard boundary updating method used in several existence proofs. By suitably modifying it we can solve numerically any kind of Stefan problem. We present a theoretical justification of the method and several computational results.

This paper proposes to test the number of common factors in high-dimensional factor models by bootstrap. We provide asymptotic distributions for the eigenvalues of bootstrapped sample covariance matrix under mild conditions. The spiked eigenvalues converge weakly to Gaussian limits after proper scaling and centralization. The limiting distribution of the largest non-spiked eigenvalue is mainly determined by order statistics of bootstrap resampling weights, and follows extreme value distribution. We propose two testing schemes based on the disparate behavior of the spiked and non-spiked eigenvalues. The testing procedures can perform reliably with weak factors, cross-sectionally and serially correlated errors. Our technical proofs contribute to random matrix theory with convexly decaying density and unbounded support, or with general elliptical distributions.

With the field of rigid-body robotics having matured in the last fifty years, routing, planning, and manipulation of deformable objects have emerged in recent years as a more untouched research area in many fields ranging from surgical robotics to industrial assembly and construction. Routing approaches for deformable objects which rely on learned implicit spatial representations (e.g., Learning-from-Demonstration methods) make them vulnerable to changes in the environment and the specific setup. On the other hand, algorithms that entirely separate the spatial representation of the deformable object from the routing and manipulation, often using a representation approach independent of planning, result in slow planning in high dimensional space. This paper proposes a novel approach to spatial representation combined with route planning that allows efficient routing of deformable one-dimensional objects (e.g., wires, cables, ropes, threads). The spatial representation is based on the geometrical decomposition of the space into convex subspaces, which allows an efficient coding of the configuration. Having such a configuration, the routing problem can be solved using a dynamic programming matching method with a quadratic time and space complexity. The proposed method couples the routing and efficient configuration for improved planning time. Our tests and experiments show the method correctly computing the next manipulation action in sub-millisecond time and accomplishing various routing and manipulation tasks.

We consider increasingly complex models of matrix denoising and dictionary learning in the Bayes-optimal setting, in the challenging regime where the matrices to infer have a rank growing linearly with the system size. This is in contrast with most existing literature concerned with the low-rank (i.e., constant-rank) regime. We first consider a class of rotationally invariant matrix denoising problems whose mutual information and minimum mean-square error are computable using techniques from random matrix theory. Next, we analyze the more challenging models of dictionary learning. To do so we introduce a novel combination of the replica method from statistical mechanics together with random matrix theory, coined spectral replica method. This allows us to derive variational formulas for the mutual information between hidden representations and the noisy data of the dictionary learning problem, as well as for the overlaps quantifying the optimal reconstruction error. The proposed method reduces the number of degrees of freedom from $\Theta(N^2)$ matrix entries to $\Theta(N)$ eigenvalues (or singular values), and yields Coulomb gas representations of the mutual information which are reminiscent of matrix models in physics. The main ingredients are a combination of large deviation results for random matrices together with a new replica symmetric decoupling ansatz at the level of the probability distributions of eigenvalues (or singular values) of certain overlap matrices and the use of HarishChandra-Itzykson-Zuber spherical integrals.

Gaussian Graphical Models (GGMs) are widely used for exploratory data analysis in various fields such as genomics, ecology, psychometry. In a high-dimensional setting, when the number of variables exceeds the number of observations by several orders of magnitude, the estimation of GGM is a difficult and unstable optimization problem. Clustering of variables or variable selection is often performed prior to GGM estimation. We propose a new method allowing to simultaneously infer a hierarchical clustering structure and the graphs describing the structure of independence at each level of the hierarchy. This method is based on solving a convex optimization problem combining a graphical lasso penalty with a fused type lasso penalty. Results on real and synthetic data are presented.

In electricity markets, retailers or brokers want to maximize profits by allocating tariff profiles to end consumers. One of the objectives of such demand response management is to incentivize the consumers to adjust their consumption so that the overall electricity procurement in the wholesale markets is minimized, e.g. it is desirable that consumers consume less during peak hours when cost of procurement for brokers from wholesale markets are high. We consider a greedy solution to maximize the overall profit for brokers by optimal tariff profile allocation. This in-turn requires forecasting electricity consumption for each user for all tariff profiles. This forecasting problem is challenging compared to standard forecasting problems due to following reasons: i. the number of possible combinations of hourly tariffs is high and retailers may not have considered all combinations in the past resulting in a biased set of tariff profiles tried in the past, ii. the profiles allocated in the past to each user is typically based on certain policy. These reasons violate the standard i.i.d. assumptions, as there is a need to evaluate new tariff profiles on existing customers and historical data is biased by the policies used in the past for tariff allocation. In this work, we consider several scenarios for forecasting and optimization under these conditions. We leverage the underlying structure of how consumers respond to variable tariff rates by comparing tariffs across hours and shifting loads, and propose suitable inductive biases in the design of deep neural network based architectures for forecasting under such scenarios. More specifically, we leverage attention mechanisms and permutation equivariant networks that allow desirable processing of tariff profiles to learn tariff representations that are insensitive to the biases in the data and still representative of the task.