arXiv subjects

Bonan Sun

Publications and source records attributed to Bonan Sun.

4 recordsLinked to original sources

arXiv2026

A practical randomized trust-region method to escape saddle points in high dimension

Without randomization, escaping the saddle points of $f \colon \mathbb{R}^d \to \mathbb{R}$ requires at least $\Omega(d)$ pieces of information about $f$ (values, gradients, Hessian-vector products). With randomization, this can be reduced to a polylogarithmic dependence in $d$. The prototypical algorithm to that effect is perturbed gradient descent (PGD): through sustained jitter, it reliably escapes strict saddle points. However, it also never settles: there is no convergence. What is more, PGD requires precise tuning based on Lipschitz constants and a preset target accuracy. To improve on this, we modify the time-tested trust-region method with truncated conjugate gradients (TR-tCG). Specifically, we randomize the initialization of tCG (the subproblem solver), and we prove that tCG automatically amplifies the randomization near saddles (to escape) and absorbs it near local minimizers (to converge). Saddle escape happens over several iterations. Accordingly, our analysis is multi-step, with several novelties. The proposed algorithm is practical: it essentially tracks the good behavior of TR-tCG, with three minute modifications and a single new hyperparameter (the noise scale $\sigma$). We provide an implementation and numerical experiments.

math.OC↗

arXiv2025

Efficient Krylov methods for linear response in plane-wave electronic structure calculations

We propose a novel algorithm based on inexact GMRES methods for linear response calculations in density functional theory. Such calculations require iteratively solving a nested linear problem $\mathcal{E} \delta\rho = b$ to obtain the variation of the electron density $\delta \rho$. Notably each application of the dielectric operator $\mathcal{E}$ in turn requires the iterative solution of multiple linear systems, the Sternheimer equations. We develop computable bounds to estimate the accuracy of the density variation given the tolerances to which the Sternheimer equations have been solved. Based on this result we suggest reliable strategies for adaptively selecting the convergence tolerances of the Sternheimer equations, such that each application of $\mathcal{E}$ is no more accurate than needed. Experiments on challenging materials systems of practical relevance demonstrate our strategies to achieve superlinear convergence as well as a reduction of computational time by about 40% while preserving the accuracy of the returned response solution. Our algorithm seamlessly combines with standard preconditioning approaches known from the context of self-consistent field problems making it a promising framework for efficient response solvers based on Krylov subspace techniques.

math.NA↗

arXiv2025

A hybrid Chebyshev-Tucker tensor format for approximation of multivariate functions

We introduce and analyze a mesh-free two-level hybrid Chebyshev-Tucker tensor representation for approximating multivariate functions, which combines tensor-product Chebyshev interpolation with the low-rank Tucker decomposition of the tensor of Chebyshev coefficients. This construction allows to avoid the expensive rank-structured grid-based approximation of function-related tensors on large spatial grids, while benefiting from the Tucker decomposition of the moderate-sized core tensor of Chebyshev coefficients. Thus, we can compute the nearly optimal Tucker decomposition of the 3D function with controllable accuracy $\varepsilon >0$ without discretizing the function on a full fine grid in the domain, but only using its values at a small set of Chebyshev nodes computed either from the explicit analytic expression of the target function or from its data-sparse representation in a rank-structured tensor format with moderate rank parameter. Finally, we can represent the function in the algebraic Tucker format with optimal $\varepsilon$-rank on an arbitrarily large 3D tensor grid in the computational domain by discretizing the Chebyshev polynomials on that grid. The rank parameters of the nonlinear Tucker-ALS decomposition of the coefficient tensor can be much smaller than the polynomial degrees of the initial Chebyshev linear interpolation in the function independent polynomial basis set. It is shown that our techniques can be gainfully applied to the long-range part of the singular electrostatic potential of multi-particle systems represented on a fine grid in the range-separated (RS) tensor format. We provide error and complexity estimates and demonstrate the computational efficiency of the proposed techniques on challenging examples, including the collective electrostatic potential for large bio-molecular systems and lattice-type compounds.

math.NA↗

arXiv2022

Approximation in the extended functional tensor train format

This work proposes the extended functional tensor train (EFTT) format for compressing and working with multivariate functions on tensor product domains. Our compression algorithm combines tensorized Chebyshev interpolation with a low-rank approximation algorithm that is entirely based on function evaluations. Compared to existing methods based on the functional tensor train format, the adaptivity of our approach often results in reducing the required storage, sometimes considerably, while achieving the same accuracy. In particular, we reduce the number of function evaluations required to achieve a prescribed accuracy by up to over 96% compared to the algorithm from [Gorodetsky, Karaman and Marzouk, Comput. Methods Appl. Mech. Eng., 347 (2019)].

math.NA↗