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There has been growing interest in high-order tensor methods for nonconvex optimization, with adaptive regularization, as they possess better/optimal worst-case evaluation complexity globally and faster convergence asymptotically. These…
Discrete tensor train decomposition is widely employed to mitigate the curse of dimensionality in solving high-dimensional PDEs through traditional methods. However, the direct application of the tensor train method typically requires…
The rapid pace of recent advancements in numerical computation, notably the rise of GPU and TPU hardware accelerators, have allowed tensor network (TN) algorithms to scale to even larger quantum simulation problems, and to be employed more…
Tensor networks represent the state-of-the-art in computational methods across many disciplines, including the classical simulation of quantum many-body systems and quantum circuits. Several applications of current interest give rise to…
This paper presents a novel learning-based trajectory planning framework for quadrotors that combines model-based optimization techniques with deep learning. Specifically, we formulate the trajectory optimization problem as a quadratic…
We investigate the use of piecewise linear systems, whose coefficient matrix is a piecewise constant function of the solution itself. Such systems arise, for example, from the numerical solution of linear complementarity problems and in the…
For linear time-invariant (LTI) systems, the design of an optimal controller is a commonly encountered problem in many applications. Among all the optimization approaches available, the linear quadratic regulator (LQR) methodology certainly…
The goal of this paper is to study approaches to bridge the gap between first-order and second-order type methods for composite convex programs. Our key observations are: i) Many well-known operator splitting methods, such as…
Explicit step-truncation tensor methods have recently proven successful in integrating initial value problems for high-dimensional partial differential equations (PDEs). However, the combination of non-linearity and stiffness may introduce…
This paper proposes an efficient numerical method based on second-order cone programming (SOCP) to solve dynamic optimal transport (DOT) problems with quadratic cost on staggered grid discretization. By properly reformulating discretized…
In pattern classification, polynomial classifiers are well-studied methods as they are capable of generating complex decision surfaces. Unfortunately, the use of multivariate polynomials is limited to kernels as in support vector machines,…
This paper concerns the inclusion of Newton's method into an adaptive finite element method (FEM) for the solution of nonlinear partial differential equations (PDEs). It features an adaptive choice of the damping parameter in the Newton…
This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give an algorithm with provable approximation guarantees for its…
Physics-based models often involve large systems of parametrized partial differential equations, where design parameters control various properties. However, high-fidelity simulations of such systems on large domains or with high grid…
In this study, we introduce a tensor-train (TT) finite difference WENO method for solving compressible Euler equations. In a step-by-step manner, the tensorization of the governing equations is demonstrated. We also introduce…
Transformers and linear state space models can be evaluated in parallel on modern hardware, but evaluating nonlinear RNNs appears to be an inherently sequential problem. Recently, however, Lim et al. '24 developed an approach called DEER,…
We propose several new nonsmooth Newton methods for solving convex composite optimization problems with polyhedral regularizers, while avoiding the computation of complicated second-order information on these functions. Under the…
Physics-Informed Neural Networks (PINNs) have emerged as a promising paradigm for solving partial differential equations (PDEs) by embedding physical laws into neural network training objectives. However, their deployment on…
Understanding how molecules arrange on surfaces is fundamental to surface chemistry and essential for the rational design of catalytic and functional materials. In particular, the energetically most stable configuration provides valuable…
This paper presents the Tensor Product Network (TPNet), a novel neural architecture for efficient and accurate function approximation and PDE solving. The core of the proposal involves constructing the solution explicitly as a linear…