arXiv subjects

Boou Jiang

Publications and source records attributed to Boou Jiang.

3 recordsLinked to original sources

arXiv2025

A polynomial dimension-dependence analysis of Bramble--Pasciak--Xu preconditioners

We investigate the dimension dependence of Bramble--Pasciak--Xu (BPX) preconditioners for high-dimensional partial differential equations and establish that the condition numbers of BPX-preconditioned systems grow only polynomially with the spatial dimension. Our analysis requires a careful derivation of the dimension dependence of several fundamental tools in the theory of finite element methods, including elliptic regularity, the Bramble--Hilbert lemma, trace inequalities, and inverse inequalities. We further analyze an averaged Scott--Zhang-type quasi-interpolation operator, and show that its associated constants scale polynomially with the dimension. Building on these ingredients, we prove a multilevel norm equivalence theorem and derive a BPX preconditioner with explicit polynomial bounds on its dimensional dependence. The analysis is motivated in part by recent tensor and quantum finite element methods, where dimension-explicit conditioning estimates for BPX preconditioners play an important role.

math.NA↗

arXiv2025

Randomized subspace correction methods for convex optimization

This paper introduces an abstract framework for randomized subspace correction methods for convex optimization, which unifies and generalizes a broad class of existing algorithms, including domain decomposition, multigrid, and block coordinate descent methods. We provide a convergence rate analysis ranging from minimal assumptions to more practical settings, such as sharpness and strong convexity. While most existing studies on block coordinate descent methods focus on nonoverlapping decompositions and smooth or strongly convex problems, our framework extends to more general settings involving arbitrary space decompositions, inexact local solvers, and problems with weaker smoothness or convexity assumptions. The proposed framework is broadly applicable to convex optimization problems arising in areas such as nonlinear partial differential equations, imaging, and data science.

math.OC↗

arXiv2025

Subspace correction as a general framework for convex optimization algorithms

This paper shows that subspace correction methods provide a common algorithmic and theoretical foundation for several classes of convex optimization algorithms, including operator splitting, alternating projection, and multiplier methods. The underlying principle is to decompose a problem into smaller subproblems and combine their solutions, a strategy that appears throughout iterative algorithms. The main tool is an iterate-level formalism, which we call dualization, that abstracts classical primal--dual correspondences and relates a subspace correction method for a dual problem to an algorithm for the corresponding primal problem via a primal--dual consistency relation. At the algorithmic level, dualizing successive subspace correction yields the Peaceman--Rachford and Douglas--Rachford splitting methods; the von Neumann and Dykstra alternating projection algorithms arise as special cases. Dualizing parallel subspace correction yields a parallel splitting method. Multiplier methods, including the alternating direction method of multipliers (ADMM), are connected to subspace correction through the same mechanism together with equivalent block formulations. In particular, a multi-block ADMM-type algorithm is obtained by dualizing a splitting method derived from successive subspace correction. At the theoretical level, the primal--dual consistency relation transfers convergence estimates from subspace correction to the derived algorithms under suitable assumptions. Thus, these operator splitting, alternating projection, and multiplier algorithms can be derived from subspace correction at both the algorithmic and theoretical levels.

math.OC↗