arXiv subjects

César Herrera

Publications and source records attributed to César Herrera.

9 recordsLinked to original sources

arXiv2026

Efficient Shallow Ritz Method: 1D Semi-Linear Diffusion-Reaction and Allen-Cahn Equations

An efficient shallow Ritz method was developed in [8, 9] for one-dimensional linear diffusion and diffusion-reaction problems. In this paper, the method is extended to a nonlinear diffusion-reaction equation with a cubic reaction term: (1) a basic error estimate of the shallow Ritz discretization is established and (2) a reduced block Newton (rBN) method for solving the resulting non-convex optimization problem is developed. Again, the computational cost of the rBN method is $\mathcal{O}(n)$ for one-dimensional problems, where $n$ is the number of neurons. Combining a convex splitting scheme for time discretization with the efficient shallow Ritz method, we develop a time-stepping neural network method for solving the Allen-Cahn equation. The unconditional energy stability of the time discretization and an error estimate in the $L^2$ norm are established. Numerical experiments demonstrate that the neural network approach yields accurate solutions for singularly perturbed nonlinear diffusion-reaction equations and the Allen-Cahn equation. The method effectively resolves interior and boundary layers without oscillations and significantly outperforms the finite element method on fixed uniform meshes.

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arXiv2026

A two-level preconditioner for discontinuous virtual element methods with irregular subdomains

We introduce and analyze a two-level overlapping additive Schwarz preconditioner for discontinuous Galerkin discretizations of virtual element spaces on general polygonal meshes in two dimensions, for a second-order elliptic problem whose coefficient is constant on each subdomain and may have large jumps across the interface. The discretization is a weighted symmetric interior penalty formulation: the averages carry the coefficient and the penalty is scaled by its harmonic mean across each edge, which keeps the method consistent and makes the constants of the analysis independent of the coefficient. The coarse space is obtained from a virtual, continuous partition of unity with a single degree of freedom per subdomain vertex, and is therefore applicable to the irregular subdomains produced by a graph partitioner; the discrete harmonic extension used to build it may be replaced by a cheaper projection onto polynomials of low degree. We prove that the condition number of the preconditioned system is bounded by $C(1+\log(H/h))(1+H/δ)$, where $H$, $h$ and $δ$ denote the subdomain diameter, the mesh size and the overlap, and where $C$ is independent of the number of subdomains and of the contrast of the coefficient, provided the latter is quasi-monotone. This is the same bound known for the conforming case and the analysis allows for hanging nodes. We also show, by an explicit example, that the quasi-monotonicity assumption cannot be dispensed with. Numerical experiments with structured triangular, hexagonal and Voronoi meshes, and with subdomains obtained both by uniform partitioning and by METIS, confirm the theory.

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arXiv2026

Performance Evaluation of Stabilized Corrections for Mixed Precision Runge--Kutta Methods

Mixed precision Runge--Kutta methods reduce the cost of the expensive implicit solves in diagonally implicit Runge--Kutta (DIRK) schemes by evaluating them in low precision, while retaining the accuracy of the scheme for larger time steps. The accuracy lost to the low-precision perturbation can be recovered through inexpensive explicit corrections; however, these corrections have an adverse impact on stability. Recently proposed stabilized corrections remedy this by applying a stabilization matrix to the correction step, but their runtime cost has not previously been quantified. In this work, we present a numerical study of the runtime performance of these stabilized corrections. Using spectral semi-discretizations of two nonlinear partial differential equations, the inviscid Burgers' equation and the porous medium equation, we compare uncorrected mixed precision DIRK methods against explicitly corrected and stabilized variants across half, single, double, and quadruple precision pairings, for SDIRK methods of orders two through four. We report convergence, runtime, and speedups, and show that the stabilized corrections improve the accuracy of the mixed precision schemes while preserving substantial runtime savings. All experiments were performed on an Intel Xeon Platinum 8480+ CPU with Julia version 1.11.4.

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arXiv2026

Structure-Guided Gauss-Newton Method: Linear Advection-Reaction Equation

The least-squares neural network (LSNN) method introduced in [5] for linear advection-reaction equations is capable of accurately approximating discontinuous solutions without a priori knowledge of the interface location. However, the resulting discretization is a non-convex optimization problem that is computationally intensive and complex. In this paper, we propose a structure-guided Gauss-Newton (SgGN) method that alternates between the linear (output) and the nonlinear (hidden layer) parameters. At each outer iteration, the linear parameters are computed by a linear solver, and the nonlinear parameters are updated by a modified Gauss-Newton (GN) method that explicitly removes the singularities of the GN matrix. Numerical experiments for all test problems presented in [5] show that the SgGN method is superior to the Adam optimizer [13], the commonly used first-order optimization algorithm, not only in computational cost but, more importantly, in accuracy

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arXiv2026

Smooth perturbations of diagonally implicit Runge--Kutta methods

A mixed accuracy framework for Runge--Kutta methods presented in [Grant, JSC 2022] has been shown to speed up the computation in diagonally implicit Runge--Kutta (DIRK) methods by using less expensive low accuracy approaches for the implicit stages. This theory included both smooth and nonsmooth perturbations, and subsequent work focused primarily on the case of nonsmooth perturbations that arise from mixed precision simulations. In this work the focus is on smooth perturbations that arise from using less accurate models or under-resolved iterative solvers to simplify the implicit computations. We develop an accuracy and stability analysis based on the framework in [Grant, JSC 2022] to design methods that strategically replace the original operator by a lower accuracy operator to reduce computational cost while mitigating the effect of the perturbations. In particular, we focus on designing novel methods that are high order for smooth perturbations that satisfy additional local consistency conditions. Finally, we verify the performance of the novel perturbed DIRK methods designed in this work and numerically study the impact of different types of smooth perturbations on the accuracy and stability of the methods.

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arXiv2026

Stable corrections for perturbed diagonally implicit Runge--Kutta methods

A mixed accuracy framework for Runge--Kutta methods presented in Grant [JSC 2022] and applied to diagonally implicit Runge--Kutta (DIRK) methods can significantly speed up the computation by replacing the implicit solver by less expensive low accuracy approaches such as lower precision computation of the implicit solve, under-resolved iterative solvers, or simpler, less accurate models for the implicit stages. Understanding the effect of the perturbation errors introduced by the low accuracy computations enables the design of stable and accurate mixed accuracy DIRK methods where the errors from the low-accuracy computation are damped out by multiplication by \dt at multiple points in the simulation, resulting in a more accurate simulation than if low-accuracy was used for all computation. To improve upon this, explicit corrections were previously proposed and analyzed for accuracy, and their performance was tested in related work. Explicit corrections work well when the time-step is sufficiently small, but may introduce instabilities when the time-step is larger. In this work, the stability of the mixed accuracy approach is carefully studied, and used to design novel stabilized correction approaches.

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arXiv2026

Convergence Analysis of Block Newton Methods for 1D Shallow Neural Network Approximation

This paper analyzes local convergence of the block Newton (BN) method introduced in [5, 6] for one-dimensional shallow neural network approximation to functions and diffusion-reaction problems. The BN method consists of the 2x2 block nonlinear Gauss-Seidel, linear Gauss-Seidel, or Jacobi method for outer iteration and the Newton method for inner iteration. The blocks are corresponding to the linear and the nonlinear parameters. Under some reasonable assumptions, we establish local convergence of the BN methods as well as the reduced BN (rBN) method for one-dimensional diffusion-reaction problems and least-squares function approximation. Unlike common optimization methods, the rBN allows for the reduction of the number of parameters during the optimization process when some neurons contribute little to the approximation or are at nearly optimal locations.

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arXiv2025

Efficient Shallow Ritz Method For 1D Diffusion Problems

This paper studies the shallow Ritz method for solving the one-dimensional diffusion problem. It is shown that the shallow Ritz method improves the order of approximation dramatically for non-smooth problems. To realize this optimal or nearly optimal order of the shallow Ritz approximation, we develop a damped block Newton (dBN) method that alternates between updates of the linear and non-linear parameters. Per each iteration, the linear and the non-linear parameters are updated by exact inversion and one step of a modified, damped Newton method applied to a reduced non-linear system, respectively. The computational cost of each dBN iteration is $O(n)$. Starting with the non-linear parameters as a uniform partition of the interval, numerical experiments show that the dBN is capable of efficiently moving mesh points to nearly optimal locations. To improve efficiency of the dBN further, we propose an adaptive damped block Newton (AdBN) method by combining the dBN with the adaptive neuron enhancement (ANE) method [26].

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arXiv2025

Efficient Shallow Ritz Method For 1D Diffusion-Reaction Problems

This paper studies the shallow Ritz method for solving one-dimensional diffusion-reaction problems. The method is capable of improving the order of approximation for non-smooth problems. By following a similar approach to the one presented in [9], we present a damped block Newton (dBN) method to achieve nearly optimal order of approximation. The dBN method optimizes the Ritz functional by alternating between the linear and non-linear parameters of the shallow ReLU neural network (NN). For diffusion-reaction problems, new difficulties arise: (1) for the linear parameters, the mass matrix is dense and even more ill-conditioned than the stiffness matrix, and (2) for the non-linear parameters, the Hessian matrix is dense and may be singular. This paper addresses these challenges, resulting in a dBN method with computational cost of ${\cal O}(n)$. The ideas presented for diffusion-reaction problems can also be applied to least-squares approximation problems. For both applications, starting with the non-linear parameters as a uniform partition, numerical experiments show that the dBN method moves the mesh points to nearly optimal locations.

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