arXiv subjects

Neil Dizon

Publications and source records attributed to Neil Dizon.

6 recordsLinked to original sources

arXiv2024

Online optimisation for dynamic electrical impedance tomography

Online optimisation studies the convergence of optimisation methods as the data embedded in the problem changes. Based on this idea, we propose a primal dual online method for nonlinear time-discrete inverse problems. We analyse the method through regret theory and demonstrate its performance in real-time monitoring of moving bodies in a fluid with Electrical Impedance Tomography (EIT). To do so, we also prove the second-order differentiability of the Complete Electrode Model (CEM) solution operator on $L^\infty$.

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arXiv2024

Differential estimates for fast first-order multilevel nonconvex optimisation

With a view on bilevel and PDE-constrained optimisation, we develop iterative estimates $\widetilde{F'}(x^k)$ of $F'(x^k)$ for composite functions $F :=J \circ S$, where $S$ is the solution mapping of the inner optimisation problem or PDE. The idea is to form a single-loop method by interweaving updates of the iterate $x^k$ by an outer optimisation method, with updates of the estimate by single steps of standard optimisation methods and linear system solvers. When the inner methods satisfy simple tracking inequalities, the differential estimates can almost directly be employed in standard convergence proofs for general forward-backward type methods. We adapt those proofs to a general inexact setting in normed spaces, that, besides our differential estimates, also covers mismatched adjoints and unreachable optimality conditions in measure spaces. As a side product of these efforts, we provide improved convergence results for nonconvex Primal-Dual Proximal Splitting (PDPS). We numerically evaluate our methods on Electrical Impedance Tomography (EIT) and minimal surface control.

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arXiv2024

Prediction techniques for dynamic imaging with online primal-dual methods

Online optimisation facilitates the solution of dynamic inverse problems, such as image stabilisation, fluid flow monitoring, and dynamic medical imaging. In this paper, we improve upon previous work on predictive online primal-dual methods on two fronts. Firstly, we provide a more concise analysis that symmetrises previously unsymmetric regret bounds, and relaxes previous restrictive conditions on the dual predictor. Secondly, based on the latter, we develop several improved dual predictors. We numerically demonstrate their efficacy in image stabilisation and dynamic positron emission tomography.

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arXiv2020

Constraint reduction reformulations for projection algorithms with applications to wavelet construction

We introduce a reformulation technique that converts a many-set feasibility problem into an equivalent two-set problem. This technique involves reformulating the original feasibility problem by replacing a pair of its constraint sets with their intersection, before applying Pierra's classical product space reformulation. The step of combining the two constraint sets reduces the dimension of the product spaces. We refer to this as the constraint reduction reformulation and use it to obtain constraint-reduced variants of well-known projection algorithms such as the Douglas--Rachford algorithm and the method of alternating projections, among others. We prove global convergence of constraint-reduced algorithms in the presence of convexity and local convergence in a nonconvex setting. In order to analyse convergence of the constraint-reduced Douglas--Rachford method, we generalize a classical result which guarantees that the composition of two projectors onto subspaces is a projector onto their intersection. Finally, we apply the constraint-reduced versions of Douglas--Rachford and alternating projections to solve the wavelet feasibility problems, and then compare their performance with their usual product variants.

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arXiv2020

Centering Projection Methods for Wavelet Feasibility Problems

We revisit the feasibility approach to the construction of compactly supported smooth orthogonal wavelets on the line. We highlight its flexibility and illustrate how symmetry and cardinality properties are easily embedded in the design criteria. We solve the resulting wavelet feasibility problems using recently introduced centering methods, and we compare performance. Solutions admit real-valued compactly supported smooth orthogonal scaling functions and wavelets with near symmetry and near cardinality properties.

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arXiv2019

Circumcentering Reflection Methods for Nonconvex Feasibility Problems

Recently, circumcentering reflection method (CRM) has been introduced for solving the feasibility problem of finding a point in the intersection of closed constraint sets. It is closely related with Douglas--Rachford method (DR). We prove local convergence of CRM in the same prototypical settings of most theoretical analysis of regular nonconvex DR, whose consideration is made natural by the geometry of the phase retrieval problem. For the purpose, we show that CRM is related to the method of subgradient projections. For many cases when DR is known to converge to a feasible point, we establish that CRM locally provides a better convergence rate. As a root finder, we show that CRM has local convergence whenever Newton--Raphson method does, has quadratic rate whenever Newton--Raphson method does, and exhibits superlinear convergence in many cases when Newton--Raphson method fails to converge at all. We also obtain explicit regions of convergence. As an interesting aside, we demonstrate local convergence of CRM to feasible points in cases when DR converges to fixed points that are not feasible. We demonstrate an extension in higher dimensions, and use it to obtain convergence rate guarantees for sphere and subspace feasibility problems. Armed with these guarantees, we experimentally discover that CRM is highly sensitive to compounding numerical error that may cause it to achieve worse rates than those guaranteed by theory. We then introduce a numerical modification that enables CRM to achieve the theoretically guaranteed rates. Any future works that study CRM for product space formulations of feasibility problems should take note of this sensitivity and account for it in numerical implementations.

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