Solving High-dimensional Parametric Elliptic Equation Using Tensor Neural Network

In this paper, we introduce a tensor neural network based machine learning method for solving the elliptic partial differential equations with random coefficients in a bounded physical domain. With the help of tensor product structure, we can transform the high-dimensional integrations of tensor neural network functions to one-dimensional integrations which can be computed with the classical quadrature schemes with high accuracy. The complexity of its calculation can be reduced from the exponential scale to a polynomial scale. The corresponding machine learning method is designed for solving high-dimensional parametric elliptic equations. Some numerical examples are provided to validate the accuracy and efficiency of the proposed algorithms.Comment: 22 pages, 25 figures. arXiv admin note: substantial text overlap with arXiv:2311.0273

Complexity of Resolution of Parametric Systems of Polynomial Equations and Inequations

Consider a system of n polynomial equations and r polynomial inequations in n indeterminates of degree bounded by d with coefficients in a polynomial ring of s parameters with rational coefficients of bit-size at most $\sigma$. From the real viewpoint, solving such a system often means describing some semi-algebraic sets in the parameter space over which the number of real solutions of the considered parametric system is constant. Following the works of Lazard and Rouillier, this can be done by the computation of a discriminant variety. In this report we focus on the case where for a generic specialization of the parameters the system of equations generates a radical zero-dimensional ideal, which is usual in the applications. In this case, we provide a deterministic method computing the minimal discriminant variety reducing the problem to a problem of elimination. Moreover, we prove that the degree of the computed minimal discriminant variety is bounded by $D:=(n+r)d^{(n+1)}$ and that the complexity of our method is $\sigma^{\mathcal{O}(1)} D^{\mathcal{O}(n+s)}$ bit-operations on a deterministic Turing machine

A fast and well-conditioned spectral method

A novel spectral method is developed for the direct solution of linear ordinary differential equations with variable coefficients. The method leads to matrices which are almost banded, and a numerical solver is presented that takes $O(m^{2}n)$ operations, where $m$ is the number of Chebyshev points needed to resolve the coefficients of the differential operator and $n$ is the number of Chebyshev points needed to resolve the solution to the differential equation. We prove stability of the method by relating it to a diagonally preconditioned system which has a bounded condition number, in a suitable norm. For Dirichlet boundary conditions, this reduces to stability in the standard 2-norm

The automatic solution of partial differential equations using a global spectral method

A spectral method for solving linear partial differential equations (PDEs) with variable coefficients and general boundary conditions defined on rectangular domains is described, based on separable representations of partial differential operators and the one-dimensional ultraspherical spectral method. If a partial differential operator is of splitting rank $2$, such as the operator associated with Poisson or Helmholtz, the corresponding PDE is solved via a generalized Sylvester matrix equation, and a bivariate polynomial approximation of the solution of degree $(n_x,n_y)$ is computed in $\mathcal{O}((n_x n_y)^{3/2})$ operations. Partial differential operators of splitting rank $\geq 3$ are solved via a linear system involving a block-banded matrix in $\mathcal{O}(\min(n_x^{3} n_y,n_x n_y^{3}))$ operations. Numerical examples demonstrate the applicability of our 2D spectral method to a broad class of PDEs, which includes elliptic and dispersive time-evolution equations. The resulting PDE solver is written in MATLAB and is publicly available as part of CHEBFUN. It can resolve solutions requiring over a million degrees of freedom in under $60$ seconds. An experimental implementation in the Julia language can currently perform the same solve in $10$ seconds.Comment: 22 page

A clever elimination strategy for efficient minimal solvers

We present a new insight into the systematic generation of minimal solvers in computer vision, which leads to smaller and faster solvers. Many minimal problem formulations are coupled sets of linear and polynomial equations where image measurements enter the linear equations only. We show that it is useful to solve such systems by first eliminating all the unknowns that do not appear in the linear equations and then extending solutions to the rest of unknowns. This can be generalized to fully non-linear systems by linearization via lifting. We demonstrate that this approach leads to more efficient solvers in three problems of partially calibrated relative camera pose computation with unknown focal length and/or radial distortion. Our approach also generates new interesting constraints on the fundamental matrices of partially calibrated cameras, which were not known before.Comment: 13 pages, 7 figure

The complexity and geometry of numerically solving polynomial systems

These pages contain a short overview on the state of the art of efficient numerical analysis methods that solve systems of multivariate polynomial equations. We focus on the work of Steve Smale who initiated this research framework, and on the collaboration between Stephen Smale and Michael Shub, which set the foundations of this approach to polynomial system--solving, culminating in the more recent advances of Carlos Beltran, Luis Miguel Pardo, Peter Buergisser and Felipe Cucker