Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2209.11474

Generating Optimally Focal and Intense Current Patterns in tES via Metaheuristic L1-L1 Search: Interior-Point vs. Simplex Algorithms

F. Galaz Prieto·M. Samavaki·S. Pursiainen
View original source ↗Read source document ↗Export metadata ↓

Abstract

This numerical simulation study investigates solving the L1-norm fitted and regularized (L1-L1) linear programming (LP) problem to find a well-localized volumetric current density in transcranial electrical stimulation (tES), where a current pattern is attached through contact electrodes attached to the skin to create a stimulus in a targeted brain region. We consider a metaheuristic optimization process where the problem parameters are selected so that the final solution found is optimal with respect to given metacriteria, e.g., the intensity and focality of the volumetric current density in the brain. We focus on the effect of the LP algorithm on the solution. We examine interior-point and simplex algorithms, which constitute two major alternative ways to solve an LP task; the interior-point algorithms are based on determining a feasible solution set to allow finding an optimizer via Newton's method, while the simplex methods take steps along the edges of a polytope, subdividing the set of candidate solutions into simplicial polygons. Interior-point methods stand out among them for their complexity and predictability in terms of convergence properties at the same time. To find the current pattern, we apply five alternative optimization toolboxes: Matlab, MOSEK, Gurobi, SDPT3, and SeDuMi. We suggest that the mutual differences between these vary based on the placement of the target as well as the resolution of the two-parameter lattice. In the numerical experiments, we investigate maximizing the focality and intensity of the L1-L1 optimized stimulation current and, in the latter regard, examine its relationship to the reciprocal current pattern, maximizing the focused current density.

Explore related subjects

math.OC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

F. Galaz Prieto, M. Samavaki, S. Pursiainen. 2022-09-23. Generating Optimally Focal and Intense Current Patterns in tES via Metaheuristic L1-L1 Search: Interior-Point vs. Simplex Algorithms. https://arxiv.org/abs/2209.11474

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Subpath-Based Column Generation for Electric Vehicle Routing Problems

Motivated by widespread electrification targets, this paper studies an Electric Vehicle Routing Problem with Time Windows and Nonlinear Charging (EVRPTWNL) that jointly optimizes routing-scheduling decisions and charging decisions given vehicle capacities, time windows and battery capacities. We develop a column generation scheme with a subpath-based label-setting algorithm that decomposes the pricing problem into two phases: (i) generating subpaths between charging stations, and (ii) combining subpaths into paths while optimizing charging decisions in between. We formalize a domination framework to establish the convergence and exactness of the algorithm, and prove that the methodology can solve a range of EVRP variants (e.g., with vehicle capacities, time windows, and nonlinear charging) and relaxation-tightening strategies (e.g., ng-relaxations and subset-row cuts). Computational results show improvements over path-based benchmarks in both computational time and solution quality, especially when time windows become wider, when vehicles can perform multiple tasks on a single charge and when vehicles still need to recharge several times across the planning horizon. Ultimately, the methodology can scale to otherwise intractable instances with up to 100 customers, thereby enhancing fleet management capabilities across electrified logistics areas.

math.OC↗
arXiv2026

Polynomial Scaling is Possible For Neural Operator Approximations of Structured Families of BSDEs

Neural operator (NO) architectures learn nonlinear maps between infinite-dimensional function spaces and are widely used to accelerate simulation and enable data-driven model discovery. While universality results ensure expressivity, they do not address \emph{complexity}: for broad operator classes described only through regularity (e.g.\ uniform continuity or $C^r$-regularity), information-theoretic lower bounds imply that minimax-optimal NO approximation rates scale \emph{exponentially} in the reciprocal accuracy $1/\varepsilon$. This has shifted the focus of NO theory toward identifying additional problem-specific structure, beyond regularity, under which suitably tailored NO architectures can leverage to unlock polynomial scaling in $1/\varepsilon$. We exhibit the first polynomial-scaling regime for NO approximations of solution operators in stochastic analysis; by identifying structured families of \emph{non-Markovian} BSDEs with randomized terminal condition parameterized by the Sobolev-regular terminal condition and by Sobolev-regular additive nonlinear perturbations of the generator. We prove that their solution operator can be approximated (uniformly over the family) by a tailored NO whose number of trainable parameters grows \emph{polynomially} in $1/\varepsilon$. We unlock this polynomial scaling regime by \emph{informing the NO's inductive bias} by factoring out the singular part of the associated semilinear elliptic PDE Green's function and by incorporating the Doléans--Dade exponential of the BSDE's common non-Markovian factor into the NO's decoding layers. As a byproduct, we extend polynomial-scaling guarantees from families of linear elliptic PDEs on regular domains to the semilinear setting.

math.OC↗
arXiv2026

The Competive Spectral Radius of Families of Nonexpansive Mappings

We consider a new class of repeated zero-sum games in which the payoff is the escape rate of a switched dynamical system, where at every stage, the transition is given by a nonexpansive operator depending on the actions of both players. This generalizes to the two-player (and non-linear) case the notion of joint spectral radius of a family of matrices. We show that the value of this game does exist, and we characterize it in terms of an infinite dimensional non-linear eigenproblem. This provides a two-player analogue of Mañe's lemma from ergodic control. This also extends to the two-player case results of Kohlberg and Neyman (1981), Karlsson (2001), and Vigeral and the second author (2012), concerning the asymptotic behavior of nonexpansive mappings. We discuss two special cases of this game: order preserving and positively homogeneous self-maps of a cone equipped with Funk's and Thompson's metrics, and translations of a finite dimensional normed space.

math.OC↗