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We consider an inverse problem involving the reconstruction of the solution to a nonlinear partial differential equation (PDE) with unknown boundary conditions. Instead of direct boundary data, we are provided with a large dataset of…
A template-based generic programming approach was presented in a previous paper that separates the development effort of programming a physical model from that of computing additional quantities, such as derivatives, needed for embedded…
We study the global structure of the set of radial solutions of a nonlinear Dirichlet problem involving the p-Laplacian with p>2, in the unit ball of $R^N$, $N \ges 1$. We show that all non-trivial radial solutions lie on smooth curves of…
Manifolds discovered by machine learning models provide a compact representation of the underlying data. Geodesics on these manifolds define locally length-minimising curves and provide a notion of distance, which are key for reduced-order…
Matrix computations, especially iterative PDE solving (and the sparse matrix vector multiplication subproblem within) using conjugate gradient algorithm, and LU/Cholesky decomposition for solving system of linear equations, form the kernel…
Boundary integral methods for the solution of boundary value PDEs are an alternative to `interior' methods, such as finite difference and finite element methods. They are attractive on domains with corners, particularly when the solution…
This paper aims to study time periodic solutions for 3D inviscid quasi-geostrophic model. We show the existence of non trivial rotating patches by suitable perturbation of stationary solutions given by generic revolution shapes around the…
These are lecture notes for the mini-course \textit{PDE and hypersurfaces with prescribed mean curvature} held in Federal University of S\~ao Carlos at the Workshop on Submanifold Theory and Geometric Analysis, August 05 -- 09, 2019. The…
Pseudospectral approximation provides a means to approximate the dynamics of delay differential equations (DDE) by ordinary differential equations (ODE). This article develops a computer-aided algorithm to determine the distance between the…
Recent investigations on the bifurcations in switching circuits have shown that many atypical bifurcations can occur in piecewise smooth maps which can not be classified among the generic cases like saddle-node, pitchfork or Hopf…
Standard Vision Transformers flatten 2D images into 1D sequences, disrupting the natural spatial topology. While Rotary Positional Embedding (RoPE) excels in 1D, it inherits this limitation, often treating spatially distant patches (e.g.,…
We propose a hybrid image-space/object-space solution to the classical hidden surface removal problem: Given n disjoint triangles in Real^3 and p sample points (``pixels'') in the xy-plane, determine the first triangle directly behind each…
In this paper we consider the spatial semi-discretization of conservative PDEs. Such finite dimensional approximations of infinite dimensional dynamical systems can be described as flows in suitable matrix spaces, which in turn leads to the…
We analyze the point decomposition problem (PDP) in binary elliptic curves. It is known that PDP in an elliptic curve group can be reduced to solving a particular system of multivariate non-linear system of equations derived from the so…
A novel and scalable geometric multi-level algorithm is presented for the numerical solution of elliptic partial differential equations, specially designed to run with high occupancy of streaming processors inside Graphics Processing…
Immersed boundary methods are high-order accurate computational tools used to model geometrically complex problems in computational mechanics. While traditional finite element methods require the construction of high-quality boundary-fitted…
We present a discrete theory for modeling developable surfaces as quadrilateral meshes satisfying simple angle constraints. The basis of our model is a lesser known characterization of developable surfaces as manifolds that can be…
Solving partial differential equations (PDEs) by numerical methods meet computational cost challenge for getting the accurate solution since fine grids and small time steps are required. Machine learning can accelerate this process, but…
In this paper we propose and analyze three parallel hybrid extragradient methods for finding a common element of the set of solutions of equilibrium problems involving pseudomonotone bifunctions and the set of fixed points of nonexpansive…
In this paper, we introduce the concept of Developmental Partial Differential Equation (DPDE), which consists of a Partial Differential Equation (PDE) on a time-varying manifold with complete coupling between the PDE and the manifold's…