Abstract : In recent decades, the concept of verified numerical computations and computer-assisted proofs is gaining increasing attention and importance.
These methods prove mathematically rigorous results using a combination of analytical arguments such as fixed-point theorems and numerical computations.
This minisymposium focuses on some general tools of accurate and verified numerical computations for the solution of linear and nonlinear systems and eigenvalue problems together with their applications to the proof of solvability and uniqueness for ordinary and partial differential equations.
New developments in that area will be presented.
[02902] High relative accuracy computing with the Cauchon algorithm
Format : Talk at Waseda University
Author(s) :
Juergen Garloff (University Konstanz)
Mohammad Adm (Palestine Polytechnic University Hebron)
Fatima Rasheed (Palestine Polytechnic University Hebron)
Abstract : We present the condensed form of the so-called Cauchon Algorithm and reformulate the computations in such a way that they can be performed without any subtraction of numbers of equal sign. This provides the basis for an algorithm needing O(n³) arithmetic operations for the computation of all eigenvalues of an n-by-n nonsingular totally nonnegative matrix, i.e., a matrix having all its minors nonnegative, with guaranteed high relative accuracy, independently of the condition number of the matrix.
[03658] Floating-point matrices with specified solutions for linear algebra problems
Format : Talk at Waseda University
Author(s) :
Katsuhisa Ozaki (Shibaura Institute of Technology)
Yuki Uchino (Shibaura Institute of Technology)
Takeshi Terao (Kyushu University)
Abstract : This research aims to rigorously verify the accuracy of the numerical results for numerical linear algebra problems. If an exact solution to a problem is known in advance, we can observe the relative error of the computed result. We proposed methods that generate a test problem based on an error-free transformation of floating-point numbers. We focus on liner systems, eigenvalue decomposition, singular value decomposition, and least squares problems with specified solutions.
[05571] Adaptive precision sparse matrix-product and application to Krylov solvers
Format : Online Talk on Zoom
Author(s) :
Stef Graillat (LIP6, Sorbonne Université)
Fabienne Jezequel (LIP6, Sorbonne Université)
Theo Mary (Sorbonne Université, CNRS, LIP6)
Romeo Molina (Sorbonne Université, CNRS)
Abstract : We introduce a mixed precision algorithm for computing sparse matrix-vector products and use
it to accelerate the solution of sparse linear systems by iterative methods. Our approach is based on the idea
of adapting the precision of each matrix element to their magnitude: we split the elements into buckets and use
progressively lower precisions for the buckets of progressively smaller elements. We carry out a rounding error
analysis of this algorithm that provides us with an explicit rule to decide which element goes into which bucket
and allows us to rigorously control the accuracy of the algorithm. We implement the algorithm on a multicore
computer and obtain significant speedups (up to a factor 7×) with respect to uniform precision algorithms, without
loss of accuracy, on a range of sparse matrices from real-life applications. We showcase the effectiveness of our
algorithm by plugging it into various Krylov solvers for sparse linear systems and observe that the convergence of
the solution is essentially unaffected by the use of adaptive precision.
[05577] Iterative refinement for an eigenpair subset of real symmetric matrices
Format : Talk at Waseda University
Author(s) :
Takeshi Terao (Kyushu University)
Toshiyuki Imamura (RIKEN Center for Computational Science)
Katsuhisa Ozaki (Shibaura Institute of Technology)
Abstract : Numerical computation for eigenvalue decomposition plays a crucial role in many scientific fields, and highly accurate eigenpairs are required in certain domains. A novel method was proposed in this study for the iterative refinement of the eigenpair of a real symmetric matrix, which is based on the Ogita-Aishima method and uses compact WY representation. The proposed method can refine the accuracy of a partial eigenpair without using a full eigenvector matrix.
[05276] Lower Bounds for Smallest Singular-Values of Asymptotic Diagonal Dominant Matrices
Format : Talk at Waseda University
Author(s) :
Shin'ichi Oishi (Waseda University)
Abstract : This article presents three classes of real square matrices. They are models of coefficient matrices of linearized Galerkin's equations of first order nonlinear delay differential equations with smooth nonlinearity. This paper shows results of computer experiments stating that the minimum singular values of these matrices are unchanged even if orders of matrices are increased. A theorem is presented based on the Schur complement. Through it, tight lower bounds are derived for the minimum singular values of such three matrices. It is proved that these lower bounds are unchanged even if orders of matrices are increased.
[05029] Error estimation for the FEM solution with a few bad elements
Format : Talk at Waseda University
Author(s) :
Kenta Kobayashi (Hitotsubashi University)
Abstract : In conventional error analysis for the finite element method, even one bad element results in poor error estimation. However, numerical results suggest that a few bad elements do not lead to an increase in error. We have provided theoretical proof for this fact, together with the error estimation, that under certain conditions, the presence of a few bad elements does not worsen the error of the finite element method.
[03173] Verified Numerical Computations for multiple solutions of the Henon equation
Format : Talk at Waseda University
Author(s) :
Taisei Asai (Waseda University)
Kazuaki Tanaka (Waseda University)
Shin'ichi Oishi (Waseda University)
Abstract : In this talk, we describe a numerical verification of the Henon equation in which some asymmetric solutions arise due to the nonlinear term.
The existence of multiple solutions is verified on various domains, and the relationship between the domain and the symmetry of the solution will be discussed.
[05290] Rigorous solution-enclosures of elliptic boundary value problems between piecewise linear functions
Format : Talk at Waseda University
Author(s) :
Kazuaki Tanaka (Waseda University)
Kaname Matsue (Kyushu University)
Hiroyuki Ochiai (Kyushu University)
Abstract : Sub- and super-solutions are useful for obtaining stable solutions of elliptic boundary value problems. However, their conventional definition requires smoothness, which makes it difficult to construct sub- and super-solutions using piecewise linear functions. To overcome this issue, we propose a definition that uses a variational inequality and constrained test functions. We show that the generalized sub- and super-solutions enclose the desired solutions, and that even a simple difference method can construct sub- and super-solutions that enclose the true solutions.
[04361] Verified computation for shape derivative of the Laplacian eigenvalues
Format : Talk at Waseda University
Author(s) :
Ryoki Endo (Niigata University)
Xuefeng Liu (Niigata University)
Abstract : The shape derivative of Laplacian eigenvalues with respect to domain deformations was theoretically investigated by Hadamard in the early 20th century. However, the rigorous computation of these derivatives is not an easy task, since there exists the singularity for repeated eigenvalues. In this study, we propose a verified computation method for the shape derivative of Laplacian eigenvalues using guaranteed computation of both eigenvalues and eigenfunctions.
[04568] Constructive error estimates for a full-discretized periodic solution of heat equation
Format : Talk at Waseda University
Author(s) :
Takuma Kimura (Saga University)
Teruya Minamoto (Saga University)
Mitsuhiro T. Nakao (Waseda University)
Abstract : In this talk, we consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation with time-periodic condition;
$\frac{\partial u}{\partial t}-\nu\Delta u=f(x,t)$ in $\Omega \times J$, $u(x,t)=0$ on $\partial\Omega \times J$, $u(x,0)=u(x,T)$ on $\Omega$.
Our numerical scheme is based on the finite element semi-discretization in space direction combined with the Fourier expansion in time.
Several numerical examples will be shown to illustrate the theoretical results.
[03685] A Numerical verification method for a self-similar solution to the linear elliptic differential equation
Format : Talk at Waseda University
Author(s) :
Kouta Sekine (Chiba Institute of Technology)
Taiyou Fuse (Chiba Institute of Technology)
Abstract : In this talk, we propose a numerical verification method for self-similar solutions of linear partial differential equations on \({\mathbb R}\) using the Galerkin approximation with extended Hermite polynomials. In particular, we derive a Gaussian quadrature method for extended Hermite polynomials to errors in numerical quadrature over infinite interval. Also, we also introduce the projection error constant for obtaining the discretisation error of the Hermite-Galerkin approximation.
[04321] A computer-assisted proof for a nonlinear differential equation involved with self-similar blowup in wave equations
Format : Talk at Waseda University
Author(s) :
Yoshitaka Watanabe (Kyushu University)
Kaori Nagatou (Karlsruhe Institute of Technology)
Michael Plum (Karlsruhe Institute of Technology)
Birgit Schörkhuber (University of Innsbruck)
Mitsuhiro T. Nakao (Waseda University)
Abstract : An existence proof with its specific shape of a non-trivial solution of a nonlinear ordinary differential equation involved with self-similar blowup in three-dimensional wave equations is presented. The proof is computer-assisted based on a fixed-point and Newton-type formulation, and the result takes into account the effects of rounding errors of floating-point arithmetic in computer.
