Related papers: Fastest quotient iteration with variational princi…
A type of adaptive finite element method for the eigenvalue problems is proposed based on the multilevel correction scheme. In this method, adaptive finite element method to solve eigenvalue problems involves solving associated boundary…
Generalized eigenvalue problems (GEPs) find applications in various fields of science and engineering. For example, principal component analysis, Fisher's discriminant analysis, and canonical correlation analysis are specific instances of…
We present an efficient method for preparing the initial state required by the eigenvalue approximation quantum algorithm of Abrams and Lloyd. Our method can be applied when solving continuous Hermitian eigenproblems, e.g., the Schroedinger…
Bilevel optimization aims to optimize an outer objective function that depends on the solution to an inner optimization problem. It is routinely used in Machine Learning, notably for hyperparameter tuning. The conventional method to compute…
For a singular oscillator, the Schrodinger equation is obtained an equation of eigenvalues, and the dependence of energy on the self-adjoint extension parameter is established. It is shown that the self-adjoint extension violates the…
The ParaDiag family of algorithms solves differential equations by using preconditioners that can be inverted in parallel through diagonalization. In the context of optimal control of linear parabolic PDEs, the state-of-the-art ParaDiag…
In this paper, we study an adaptive finite element method for multiple eigenvalue problems of a class of second order elliptic equations. By using some eigenspace approximation technology and its crucial property which is also presented in…
Successive quadratic approximations, or second-order proximal methods, are useful for minimizing functions that are a sum of a smooth part and a convex, possibly nonsmooth part that promotes regularization. Most analyses of iteration…
The paper suggests a preconditioning type method for fast solving of elliptic equations with oscillating quasiperiodic coefficients $A_\epsilon$ specified by the small parameter $\epsilon>0$. We use an iteration method generated by an…
Quantization is a fundamental optimization for many machine-learning use cases, including compressing gradients, model weights and activations, and datasets. The most accurate form of quantization is \emph{adaptive}, where the error is…
The method of self-similar factor approximants is shown to be very convenient for solving different evolution equations and boundary-value problems typical of physical applications. The method is general and simple, being a straightforward…
Dynamic optimization is currently limited by sensitivity computations that require information from full forward and adjoint wave fields. Since the forward and adjoint solutions are computed in opposing time directions, the forward solution…
For large-scale eigenvalue problems requiring many mutually orthogonal eigenvectors, traditional numerical methods suffer substantial computational and communication costs with limited parallel scalability, primarily due to explicit…
We introduce a novel eigenvalue algorithm for near-diagonal matrices inspired by Rayleigh-Schr\"odinger perturbation theory and termed Iterative Perturbative Theory (IPT). Contrary to standard eigenvalue algorithms, which are either…
We discuss the close connection between eigenvalue computation and optimization using the Newton method and subspace methods. From the connection we derive a new class of Newton updates. The new update formulation is similar to the…
In this article we study the estimation of bifurcation coefficients in nonlinear branching problems by means of Rayleigh-Ritz approximation to the eigenvectors of the corresponding linearized problem. It is essential that the approximations…
We describe a strategy for solving nonlinear eigenproblems numerically. Our approach is based on the approximation of a vector-valued function, defined as solution of a non-homogeneous version of the eigenproblem. This approximation step is…
Derivative-based algorithms are ubiquitous in statistics, machine learning, and applied mathematics. Automatic differentiation offers an algorithmic way to efficiently evaluate these derivatives from computer programs that execute relevant…
We propose, analyze, and test a proximal-gradient method for solving regularized optimization problems with general constraints. The method employs a decomposition strategy to compute trial steps and uses a merit function to determine step…
It is shown how to compute quotients efficiently in non-commutative univariate polynomial rings. This extends earlier work where efficient generic quotients were studied with a primary focus on commutative domains. Fast algorithms are given…