Computational Mathematics Research Digest · Issue 1
Weekly computational mathematics digest
Published · Curated by Hassan Mohammad
Coverage: substantive postings from 2–12 August 2026, ranked by relevance and potential research value.
1. Optimal Near-Optimality Bounds for the Lanczos Method for Matrix Functions
Tyler Chen and David Persson · arXiv · 7 August 2026 · Primary source ↗
Why it matters For Hermitian positive-definite A, the authors show that Lanczos approximations to f(A)b are within an explicit condition-number-dependent factor of the best approximation in the same Krylov space. The result covers Stieltjes-related functions and the constant is proved optimal.
Read first Section 1.2, especially Theorems 1–2 and Figure 2, followed by Remark 3 for the sharpened Euclidean-norm interpretation for conjugate gradients.
2. Dynamic Proximal Point Method for Unconstrained Minimization
Enrico Bertolazzi, Alberto De Marchi and Davide Stocco · arXiv · 4 August 2026 · Primary source ↗
Why it matters The method combines adaptively scaled proximal subproblems with an inner Newton–line-search solver, including a reduced Newton system, merit function, diagonal scaling, stopping rules, and implementation-oriented pseudocode.
Read first Algorithms 1–2 and Sections 2.0.1 and 3.1 for the outer regularization and inner solver, then Section 4’s tests on 100 benchmark problems.
3. Alternating Levenberg–Marquardt Training of Physics-Informed Neural Networks with Fourier-Enhanced Features
Yulun Wu, Matthieu Barreau, Miguel Aguiar and Karl H. Johansson · arXiv · 6 August 2026 · Primary source ↗
Why it matters The framework separates basis learning from coefficient fitting, handling the latter through Levenberg–Marquardt and exposing more exploitable least-squares structure than ordinary end-to-end PINN training. It includes convergence results and reports errors up to two orders of magnitude below selected baselines.
Read first Sections 3.3–3.4 and Theorem 1 for the alternating LM framework and convergence conditions, followed by Sections 4.3–4.5 on the high-frequency heat, cavity-flow, and Burgers problems.
4. Rake–Compress Riccati Recursions for Parallel Scenario-Tree Model Predictive Control
Why it matters The exact tree-contraction solver has O(N) work and storage but O(log N) parallel span, independently of tree height or balance. It reconstructs Riccati coefficients, primal variables, and multipliers, and is accompanied by JAX implementations.
Read first Sections IV–V for the contraction construction, Theorem 3 for equivalence with the KKT system, and Section VII for parallel complexity.
Alberto Bemporad and Tatiana Tatarenko · arXiv · 7 August 2026 · Primary source ↗
Why it matters The principal structural result converts a monotone linear-quadratic variational GNE problem into one convex quadratic program. Regularized and accelerated variants obtain O(1/k²) approximate-equilibrium convergence, while an invertible game Jacobian permits a smaller dual-space QP.
Read first The QP-equivalence theorem and lower-dimensional reduction, followed by the game-theoretic MPC experiment and comparisons with extragradient-type methods.
Selection and commentary are editorial. Dates and bibliographic details link to the cited primary sources.