A numerical method for an inverse optimization problem through the generalized method of lines

Fabio Silva Botelho

arxiv: 1907.02503 · v1 · pith:ODZDWHYVnew · submitted 2019-07-04 · 🧮 math.OC

A numerical method for an inverse optimization problem through the generalized method of lines

Fabio Silva Botelho This is my paper

Pith reviewed 2026-05-25 09:03 UTC · model grok-4.3

classification 🧮 math.OC

keywords inverse optimizationshape optimizationLaplace equationgeneralized method of linesmixed boundary conditionsnumerical methodboundary value problem

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The pith

The generalized method of lines computes the optimal internal boundary shape that satisfies mixed boundary conditions for the Laplace equation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper applies the generalized method of lines to an inverse optimization problem for the Laplace equation on a domain with internal and external boundaries. Dirichlet conditions are imposed on both boundaries while a non-homogeneous Neumann condition is also required on the external boundary, and the internal boundary shape must be adjusted until all conditions hold simultaneously. The method reduces the problem to a solvable system along lines in the domain. A sympathetic reader would care because it supplies a direct numerical route to shape determination without separate optimization procedures.

Core claim

The generalized method of lines serves as a tool to compute a solution for the inverse optimization problem of finding the optimal shape of the internal boundary such that the Laplace equation holds together with the prescribed Dirichlet boundary conditions on the internal and external boundaries and the non-homogeneous Neumann boundary condition on the external boundary.

What carries the argument

The generalized method of lines, which discretizes the domain into lines and reduces the elliptic boundary-value problem to a system of ordinary differential equations solved for the unknown boundary geometry.

If this is right

The optimal internal boundary geometry is obtained by solving the algebraic system that arises after discretization along the lines.
The mixed Dirichlet and Neumann conditions are enforced simultaneously within the same numerical scheme.
The procedure yields a numerical approximation to the shape that makes the entire boundary-value problem consistent.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same line-based reduction could be tested on the Poisson equation or other linear elliptic operators to check broader applicability.
Comparison against manufactured solutions on circular or elliptical domains would provide a direct accuracy check.
The approach may simplify design iterations in applications where an internal boundary must be tuned to meet flux or potential specifications.

Load-bearing premise

The generalized method of lines extends directly to the inverse shape-optimization setting for the Laplace equation with the stated mixed boundary conditions and produces a convergent numerical solution without additional domain restrictions or regularization.

What would settle it

A concrete test case with a known exact optimal internal boundary where the computed shape from the method deviates from the exact shape by more than the expected discretization error.

read the original abstract

This article develops a solution for an inverse problem through the generalized method of lines. We consider a Laplace equation on a domain with internal and external boundaries with standard Dirichlet boundary conditions. Also, we specify a third non-homogeneous Newmann type boundary condition for the external boundary, and consider the problem of finding the optimal shape for the internal boundary such that all the prescribed boundary conditions are satisfied. The novelty here presented is the application of the generalized method of lines as a tool to compute a solution for such an inverse optimization problem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Applies generalized method of lines to an inverse shape problem for the Laplace equation but supplies no convergence or stability analysis.

read the letter

The paper's core move is to recast the inverse problem of recovering an optimal inner boundary for the Laplace equation (Dirichlet on both boundaries plus non-homogeneous Neumann on the exterior) as a shape optimization task and then attack it with the generalized method of lines. That application is presented as the novelty. The setup itself is stated cleanly: find the interior boundary shape that makes the given boundary data consistent. If the method of lines extension works, it could be a practical tool for this narrow class of inverse problems in potential theory. The text does lay out the boundary conditions and the optimization goal without obvious internal contradictions. That is the positive part. The main weakness is the absence of any convergence analysis, consistency proof, or numerical study for the discretized shape-optimization step. The inverse problem is known to be mildly ill-posed, yet the manuscript offers no a priori estimates, no study of how the solution behaves as the number of lines and collocation points grows, and no checks against regularization or domain restrictions. Without those, the claim that the method directly yields a convergent numerical solution remains unsupported. The work is aimed at specialists already using the method of lines for elliptic problems who might want to see one more inverse-problem application. A reader looking for a fully supported numerical method or a new theoretical guarantee will not find it here. It is coherent on its own terms and shows straightforward engagement with the literature on the method, so it clears the bar for serious refereeing even though the central numerical claim needs substantial additional evidence.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a numerical approach to an inverse shape-optimization problem for the Laplace equation on an annular-type domain. Dirichlet data are prescribed on both the unknown interior boundary and the fixed exterior boundary; an additional non-homogeneous Neumann condition is imposed on the exterior boundary. The goal is to recover the interior boundary shape that renders the over-determined boundary-value problem consistent. The proposed solver is the generalized method of lines, whose novelty is asserted to lie in its direct application to this inverse setting.

Significance. A convergent, stable discretization for this mildly ill-posed inverse problem would be of interest to the shape-optimization and inverse-PDE communities. The manuscript, however, supplies neither an a-priori error analysis nor any numerical convergence study, so the practical utility of the claimed method cannot yet be assessed.

major comments (2)

[Abstract and method description] No convergence or stability analysis is presented for the shape-optimization step under the mixed boundary conditions. The central claim that the generalized method of lines “computes a solution” therefore rests on an unverified assertion that the discretized system remains well-posed and converges to the true optimal boundary as the number of lines and collocation points increases.
[Problem formulation] The inverse problem is known to be mildly ill-posed; the manuscript contains no regularization strategy or domain-restriction argument. Without such safeguards, it is unclear whether the method can produce a stable reconstruction when the data are perturbed or when the interior boundary is allowed to vary freely.

minor comments (2)

[Abstract] The abstract states that “standard Dirichlet boundary conditions” are used on both boundaries, yet later refers to a “third non-homogeneous Neumann type boundary condition.” Clarify the precise boundary-condition set in the introduction.
[Method section] Notation for the interior boundary parametrization and the line-coordinate transformation is introduced without an accompanying figure or explicit mapping; readers cannot reconstruct the discretization from the text alone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and constructive feedback on our manuscript. We address each major comment below and outline the revisions we will make to strengthen the presentation of the numerical method.

read point-by-point responses

Referee: [Abstract and method description] No convergence or stability analysis is presented for the shape-optimization step under the mixed boundary conditions. The central claim that the generalized method of lines “computes a solution” therefore rests on an unverified assertion that the discretized system remains well-posed and converges to the true optimal boundary as the number of lines and collocation points increases.

Authors: We acknowledge that the manuscript does not contain an a-priori error analysis or a dedicated numerical convergence study. The generalized method of lines is applied directly to the inverse problem by discretizing along lines and solving the resulting optimization for the interior boundary coefficients. In the revised manuscript we will add a new section containing systematic numerical experiments that vary the number of lines and collocation points, reporting the observed convergence of the recovered boundary to a reference shape for the test cases already presented. We will also include a brief discussion of the algebraic well-posedness of the discretized system under the mixed boundary conditions. revision: yes
Referee: [Problem formulation] The inverse problem is known to be mildly ill-posed; the manuscript contains no regularization strategy or domain-restriction argument. Without such safeguards, it is unclear whether the method can produce a stable reconstruction when the data are perturbed or when the interior boundary is allowed to vary freely.

Authors: We agree that the underlying inverse shape problem is mildly ill-posed. The current formulation restricts the admissible interior boundaries to a finite-dimensional parametric family generated by the method-of-lines discretization; this constitutes an implicit domain restriction. In the revised version we will add an explicit statement of this restriction together with a short discussion of its regularizing effect. We will also report additional numerical tests with perturbed Neumann data to illustrate the practical stability observed within the chosen parametrization. revision: yes

Circularity Check

0 steps flagged

No circularity; method application is independent of inputs

full rationale

The paper presents the generalized method of lines as an existing numerical tool applied to an inverse shape-optimization problem for the Laplace equation under mixed boundary conditions. No equations, parameter fits, or self-citations are shown that reduce any claimed solution or convergence result to the problem data by construction. The derivation chain consists of discretizing the PDE via the method of lines and optimizing the interior boundary shape; this is a standard forward application of a pre-existing scheme rather than a self-definitional or fitted-input reduction. The absence of any quoted step that equates output to input by definition confirms the result is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are described.

pith-pipeline@v0.9.0 · 5602 in / 1072 out tokens · 40481 ms · 2026-05-25T09:03:06.603413+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

J(r(θ)) = ∥∇²u∥²_{2,Ω} + K∥∇u·n − w∥²_{2,∂Ω₁}

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