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We propose finite-volume schemes for general continuity equations which preserve positivity and global bounds that arise from saturation effects in the mobility function. In the particular case of gradient flows, the schemes dissipate the free energy at the fully discrete level. Moreover, these schemes are generalised to coupled systems of non-linear continuity equations, such as multispecies models in mathematical physics or biology, preserving the bounds and the dissipation of the energy whenever applicable. These results are illustrated through extensive numerical simulations which explore known behaviours in biology and showcase new phenomena not yet described by the literature.

We devise 3-field and 4-field finite element approximations of a system describing the steady state of an incompressible heat-conducting fluid with implicit non-Newtonian rheology. We prove that the sequence of numerical approximations converges to a weak solution of the problem. We develop a block preconditioner based on augmented Lagrangian stabilisation for a discretisation based on the Scott-Vogelius finite element pair for the velocity and pressure. The preconditioner involves a specialised multigrid algorithm that makes use of a space-decomposition that captures the kernel of the divergence and non-standard intergrid transfer operators. The preconditioner exhibits robust convergence behaviour when applied to the Navier-Stokes and power-law systems, including temperature-dependent viscosity, heat conductivity and viscous dissipation.

We study a matrix recovery problem with unknown correspondence: given the observation matrix $M_o=[A,\tilde P B]$, where $\tilde P$ is an unknown permutation matrix, we aim to recover the underlying matrix $M=[A,B]$. Such problem commonly arises in many applications where heterogeneous data are utilized and the correspondence among them are unknown, e.g., due to privacy concerns. We show that it is possible to recover $M$ via solving a nuclear norm minimization problem under a proper low-rank condition on $M$, with provable non-asymptotic error bound for the recovery of $M$. We propose an algorithm, $\text{M}^3\text{O}$ (Matrix recovery via Min-Max Optimization) which recasts this combinatorial problem as a continuous minimax optimization problem and solves it by proximal gradient with a Max-Oracle. $\text{M}^3\text{O}$ can also be applied to a more general scenario where we have missing entries in $M_o$ and multiple groups of data with distinct unknown correspondence. Experiments on simulated data, the MovieLens 100K dataset and Yale B database show that $\text{M}^3\text{O}$ achieves state-of-the-art performance over several baselines and can recover the ground-truth correspondence with high accuracy.

This work formulates a new approach to reduced modeling of parameterized, time-dependent partial differential equations (PDEs). The method employs Operator Inference, a scientific machine learning framework combining data-driven learning and physics-based modeling. The parametric structure of the governing equations is embedded directly into the reduced-order model, and parameterized reduced-order operators are learned via a data-driven linear regression problem. The result is a reduced-order model that can be solved rapidly to map parameter values to approximate PDE solutions. Such parameterized reduced-order models may be used as physics-based surrogates for uncertainty quantification and inverse problems that require many forward solves of parametric PDEs. Numerical issues such as well-posedness and the need for appropriate regularization in the learning problem are considered, and an algorithm for hyperparameter selection is presented. The method is illustrated for a parametric heat equation and demonstrated for the FitzHugh-Nagumo neuron model.

The matrix normal model, the family of Gaussian matrix-variate distributions whose covariance matrix is the Kronecker product of two lower dimensional factors, is frequently used to model matrix-variate data. The tensor normal model generalizes this family to Kronecker products of three or more factors. We study the estimation of the Kronecker factors of the covariance matrix in the matrix and tensor models. We show nonasymptotic bounds for the error achieved by the maximum likelihood estimator (MLE) in several natural metrics. In contrast to existing bounds, our results do not rely on the factors being well-conditioned or sparse. For the matrix normal model, all our bounds are minimax optimal up to logarithmic factors, and for the tensor normal model our bound for the largest factor and overall covariance matrix are minimax optimal up to constant factors provided there are enough samples for any estimator to obtain constant Frobenius error. In the same regimes as our sample complexity bounds, we show that an iterative procedure to compute the MLE known as the flip-flop algorithm converges linearly with high probability. Our main tool is geodesic strong convexity in the geometry on positive-definite matrices induced by the Fisher information metric. This strong convexity is determined by the expansion of certain random quantum channels. We also provide numerical evidence that combining the flip-flop algorithm with a simple shrinkage estimator can improve performance in the undersampled regime.

Given a fixed finite metric space $(V,\mu)$, the {\em minimum $0$-extension problem}, denoted as ${\tt 0\mbox{-}Ext}[\mu]$, is equivalent to the following optimization problem: minimize function of the form $\min\limits_{x\in V^n} \sum_i f_i(x_i) + \sum_{ij}c_{ij}\mu(x_i,x_j)$ where $c_{ij},c_{vi}$ are given nonnegative costs and $f_i:V\rightarrow \mathbb R$ are functions given by $f_i(x_i)=\sum_{v\in V}c_{vi}\mu(x_i,v)$. The computational complexity of ${\tt 0\mbox{-}Ext}[\mu]$ has been recently established by Karzanov and by Hirai: if metric $\mu$ is {\em orientable modular} then ${\tt 0\mbox{-}Ext}[\mu]$ can be solved in polynomial time, otherwise ${\tt 0\mbox{-}Ext}[\mu]$ is NP-hard. To prove the tractability part, Hirai developed a theory of discrete convex functions on orientable modular graphs generalizing several known classes of functions in discrete convex analysis, such as $L^\natural$-convex functions. We consider a more general version of the problem in which unary functions $f_i(x_i)$ can additionally have terms of the form $c_{uv;i}\mu(x_i,\{u,v\})$ for $\{u,v\}\in F$, where set $F\subseteq\binom{V}{2}$ is fixed. We extend the complexity classification above by providing an explicit condition on $(\mu,F)$ for the problem to be tractable. In order to prove the tractability part, we generalize Hirai's theory and define a larger class of discrete convex functions. It covers, in particular, another well-known class of functions, namely submodular functions on an integer lattice. Finally, we improve the complexity of Hirai's algorithm for solving ${\tt 0\mbox{-}Ext}[\mu]$ on orientable modular graphs.

In this work we consider a class of non-linear eigenvalue problems that admit a spectrum similar to that of a Hamiltonian matrix, in the sense that the spectrum is symmetric with respect to both the real and imaginary axis. More precisely, we present a method to iteratively approximate the eigenvalues of such non-linear eigenvalue problems closest to a given purely real or imaginary shift, while preserving the symmetries of the spectrum. To this end the presented method exploits the equivalence between the considered non-linear eigenvalue problem and the eigenvalue problem associated with a linear but infinite-dimensional operator. To compute the eigenvalues closest to the given shift, we apply a specifically chosen shift-invert transformation to this linear operator and compute the eigenvalues with the largest modulus of the new shifted and inverted operator using an (infinite) Arnoldi procedure. The advantage of the chosen shift-invert transformation is that the spectrum of the transformed operator has a "real skew-Hamiltonian"-like structure. Furthermore, it is proven that the Krylov space constructed by applying this operator, satisfies an orthogonality property in terms of a specifically chosen bilinear form. By taking this property into account in the orthogonalization process, it is ensured that even in the presence of rounding errors, the obtained approximation for, e.g., a simple, purely imaginary eigenvalue is simple and purely imaginary. The presented work can thus be seen as an extension of [V. Mehrmann and D. Watkins, "Structure-Preserving Methods for Computing Eigenpairs of Large Sparse Skew-Hamiltonian/Hamiltonian Pencils", SIAM J. Sci. Comput. (22.6), 2001], to the considered class of non-linear eigenvalue problems. Although the presented method is initially defined on function spaces, it can be implemented using finite dimensional linear algebra operations.

We consider a mesh-based approach for training a neural network to produce field predictions of solutions to parametric partial differential equations (PDEs). This approach contrasts current approaches for "neural PDE solvers" that employ collocation-based methods to make point-wise predictions of solutions to PDEs. This approach has the advantage of naturally enforcing different boundary conditions as well as ease of invoking well-developed PDE theory -- including analysis of numerical stability and convergence -- to obtain capacity bounds for our proposed neural networks in discretized domains. We explore our mesh-based strategy, called NeuFENet, using a weighted Galerkin loss function based on the Finite Element Method (FEM) on a parametric elliptic PDE. The weighted Galerkin loss (FEM loss) is similar to an energy functional that produces improved solutions, satisfies a priori mesh convergence, and can model Dirichlet and Neumann boundary conditions. We prove theoretically, and illustrate with experiments, convergence results analogous to mesh convergence analysis deployed in finite element solutions to PDEs. These results suggest that a mesh-based neural network approach serves as a promising approach for solving parametric PDEs with theoretical bounds.

This paper studies the $\tau$-coherence of a (n x p)-observation matrix in a Gaussian framework. The $\tau$-coherence is defined as the largest magnitude outside a diagonal bandwith of size $\tau$ of the empirical correlation coefficients associated to our observations. Using the Chen-Stein method we derive the limiting law of the normalized coherence and show the convergence towards a Gumbel distribution. We generalize here the results of Cai and Jiang [CJ11a]. We assume that the covariance matrix of the model is bandwise. Moreover, we provide numerical considerations highlighting issues from the high dimension hypotheses. We numerically illustrate the asymptotic behaviour of the coherence with Monte-Carlo experiment using a HPC splitting strategy for high dimensional correlation matrices.

This article derives closed-form parametric formulas for the Minkowski sums of convex bodies in d-dimensional Euclidean space with boundaries that are smooth and have all positive sectional curvatures at every point. Under these conditions, there is a unique relationship between the position of each boundary point and the surface normal. The main results are presented as two theorems. The first theorem directly parameterizes the Minkowski sums using the unit normal vector at each surface point. Although simple to express mathematically, such a parameterization is not always practical to obtain computationally. Therefore, the second theorem derives a more useful parametric closed-form expression using the gradient that is not normalized. In the special case of two ellipsoids, the proposed expressions are identical to those derived previously using geometric interpretations. In order to examine the results, numerical validations and comparisons of the Minkowski sums between two superquadric bodies are conducted. Applications to generate configuration space obstacles in motion planning problems and to improve optimization-based collision detection algorithms are introduced and demonstrated.