Akitoshi Takayasu

Verified Computations for Hyperbolic 3-Manifolds

For a given cusped 3-manifold M admitting an ideal triangulation, we describe a method to rigorously prove that either M or a filling of M admits a complete hyperbolic structure via verified computer calculations. Central to our method is an implementation of interval arithmetic and Krawczyk’s test. These techniques represent an improvement over existing algorithms as they are faster while accounting for error accumulation in a more direct and user-friendly way.

Numerical validation of blow-up solutions of ordinary differential equations

This paper focuses on blow-up solutions of ordinary differential equations (ODEs). We present a method for validating blow-up solutions and their blow-up times, which is based on compactifications and the Lyapunov function validation method. The necessary criteria for this construction can be verified using interval arithmetic techniques. Some numerical examples are presented to demonstrate the applicability of our method.

Numerical verification for existence of a global-in-time solution to semilinear parabolic equations

Abstract This paper presents a method of numerical verification for the existence of a global-in-time solution to a class of semilinear parabolic equations. Such a method is based on two main theorems in this paper. One theorem gives a sufficient condition for proving the existence of a solution to the semilinear parabolic equations with the initial point t = t ′ ≥ 0 . If the sufficient condition does not hold, the other theorem is used for enclosing the solution for time t ∈ ( 0 , τ ] , τ > 0 in a neighborhood of a numerical solution. Numerical results of obtaining a global-in-time solution for a certain semilinear parabolic equation are also given.

A Method of Verified Computations for Solutions to Semilinear Parabolic Equations Using Semigroup Theory

This paper presents a numerical method for verifying the existence and local uniqueness of a solution for an initial-boundary value problem of semilinear parabolic equations. The main theorem of this paper provides a sufficient condition for a unique solution to be enclosed within a neighborhood of a numerical solution. In the formulation used in this paper, the initial-boundary value problem is transformed into a fixed-point form using an analytic semigroup. The sufficient condition is derived from Banachs fixed-point theorem. This paper also introduces a recursive scheme to extend a time interval in which the validity of the solution can be verified. As an application of this method, the existence of a global-in-time solution is demonstrated for a certain semilinear parabolic equation.

Verified Computations for Solutions to Semilinear Parabolic Equations Using the Evolution Operator

This article presents a theorem for guaranteeing existence of a solution for an initial-boundary value problem of semilinear parabolic equations. The sufficient condition of our main theorem is derived by a fixed-point formulation using the evolution operator. We note that the sufficient condition can be checked by verified numerical computations.

A Priori Inverse Operator Estimation for Guaranteed Error Estimate

A guaranteed error estimate procedure for linear or nonlinear two-point boundary value problems is established by authors. ‘Guaranteed’ error estimate is rigorous, i.e. it takes into account every error such as the discretization error and the rounding error when we compute an approximate solution. We can also prove the existence and the uniqueness of the exact solution. Namely, we can solve the problem with mathematically rigorous. In order to bound the guaranteed error, an inverse operator norm estimation is needed. In the previous work of authors, the inverse operator estimate needs the norm of the inverse matrix, which is a posteriori constant. So we need much time to compute the inverse operator norm estimation. In this paper, we propose a priori estimation concerning the inverse operator. The proposed estimate is given without the norm of the inverse matrix. By using a priori estimation, we can estimate the inverse operator as a priori constant. An extremely improvement of the computational speed is expected. On the other hand, it needs a condition to use a priori estimation. Finally, we present some numerical results.