arXiv subjects

Anna Peruso

Publications and source records attributed to Anna Peruso.

5 recordsLinked to original sources

arXiv2026

Fully discrete least-squares splitting scheme for the Monge-Amp\`ere equation: finite element analysis and convergence

The least-squares splitting algorithm for the Monge-Amp\`ere equation has been used successfully in computations for several years, but a convergence theory for fully discrete splitting schemes of this type has remained unavailable. In this work, we introduce and analyze a finite element framework for smooth solutions of the Dirichlet Monge-Amp\`ere equation in two dimensions. The proposed schemes combine a discrete Hessian reconstruction with a local projection onto the determinant constraint. Under a discrete Miranda-Talenti estimate and standard approximation properties of the Hessian reconstruction, we prove local convergence of the iterative scheme and optimal-order convergence of its limit to the exact solution in an $H^2$-type norm. We verify the estimates for conforming $C^1$ schemes, including the Argyris element, and for $C^0$-interior penalty and DG schemes of degree at least three; quadratic $C^0$-interior penalty and DG schemes are also covered when $|u|_{H^3(\Omega)}$ is sufficiently small. To the best of our knowledge, these discretizations have not previously been proposed or analyzed for least-squares splitting methods. Numerical experiments confirm the theoretical convergence rates.

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arXiv2026

An adaptive Deep Ritz framework for second-order fully nonlinear partial differential equations

As an alternative to PINNs, a Deep Ritz framework is proposed to solve fully nonlinear PDEs. A least-squares algorithm is advocated to decouple the nonlinearities from the variational features of several fully nonlinear PDEs. A splitting method allows to iteratively solve local nonlinear problems and linear variational problems at each iteration. While existing nonlinear solvers are applied to solve for nonlinearities, we propose a novel coupling with a Deep Ritz neural network approach that is well-suited to the variational flavor of the linear variational problems. An adaptive sampling strategy for the selection of collocation points is incorporated to increase the efficiency of the algorithm without sacrificing its accuracy. Numerical experiments are presented to solve the Dirichlet problem for several fully nonlinear equations, starting with the prototypical Monge-Amp\`ere equation, showing the flexibility of the approach. Numerical results are compared with results obtained using a full PINNs approach. Finally, numerical experiments are extended to address the optimal transport Monge-Amp\`ere problem with transport boundary conditions.

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arXiv2026

Convergence of a least-squares splitting method for the Monge-Amp\`ere equation

We study the theoretical convergence of the nonlinear least-squares splitting method for the Monge-Amp\`ere equation in which each iteration decouples the pointwise nonlinearity from the differential operator and consists of a local nonlinear update followed by the solution of two sequential Poisson-type elliptic problems. While the method performs well in computations, a rigorous convergence theory has remained unavailable. We observe that the iteration admits a reformulation as an alternating-projection scheme on Sobolev spaces $H^m$, $m\ge 0$. At a solution, the G\^ateaux differentials of the projection maps are the linear projections onto the corresponding tangent spaces. We prove that these tangent spaces are transverse, and hence the linearization of the alternating-projection map is a contraction by classical Hilbert-space theory for alternating projections. Building on this geometric characterization, we prove linear convergence in $H^2$ of the splitting method on the two-dimensional torus $\mathbb{T}^2$ for initial data sufficiently close to a solution $u\in H^4$. To the best of our knowledge, this yields the first rigorous convergence result for this splitting method in the periodic setting and provides a functional-analytic explanation for its observed numerical robustness.

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arXiv2025

Error estimates and adaptivity for a least-squares method applied to the Monge-Amp\`ere equation

We introduce novel a posteriori error indicators for a nonlinear least-squares solver for smooth solutions of the Monge--Amp\`ere equation on convex polygonal domains in $\mathbb{R}^2$. At each iteration, our iterative scheme decouples the problem into (i) a pointwise nonlinear minimization problem and (ii) a linear biharmonic variational problem. For the latter, we derive an equivalence to a biharmonic problem with Navier boundary conditions and solve it via mixed piecewise-linear finite elements. Reformulating this as a coupled second-order system, we derive a priori and a posteriori $\mathbb{P}^1$ finite element error estimators and we design a robust adaptive mesh refinement strategy. Numerical tests confirm that errors in different norms scale appropriately. Finally, we demonstrate the effectiveness of our a posteriori indicators in guiding mesh refinement.

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arXiv2025

Convex Physics Informed Neural Networks for the Monge-Amp\`ere Optimal Transport Problem

Optimal transportation of raw material from suppliers to customers is an issue arising in logistics that is addressed here with a continuous model relying on optimal transport theory. A physics informed neuralnetwork method is advocated here for the solution of the corresponding generalized Monge-Amp`ere equation. Convex neural networks are advocated to enforce the convexity of the solution to the Monge-Amp\`ere equation and obtain a suitable approximation of the optimal transport map. A particular focus is set on the enforcement of transport boundary conditions in the loss function. Numerical experiments illustrate the solution to the optimal transport problem in several configurations, and sensitivity analyses are performed.

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