Inverse problems andtheory of reproducing
kernels – theory(abstract)
Saburou Saitoh
Institute of Reproducing Kernels
saburou.saitoh@gmail.com
December 8, 2020

1Introduction

At least, until about 30 years ago, we had very diﬀicult inverse problems thatare important in many practical problems (fundamentals) as follows:1): Inverse source problem; that is in the Poisson equation△u = −ρ,from the observation of the potential u for the out side of the support ρ, look
for the source ρ.
2): The problem in the heat conduction; that is, from some heat u(x, t)observation at a time t, look for the initial heat u(x, 0).
3): Real inversion formulas for the Laplace transform.
These problems were indeed diﬀicult in both mathematics and numericalrealizations of the solutions and so, they are called ill-posed problems and very
famous diﬀicult problems.
We were able to solve these problems by using the theory of reproducing
kernels applying the Tikhonov regularization. However, for the real inversion
formula of Laplace transform, we needed the great power of computers by H.
Fujiwara.
For any practical numerical analysis, the important problem is on the discretization procedure for analytical inverse problem solutions. At this very
important point, we will see that the theory of reproducing kernels is very good1

mathematics. These global theories were published in the book [22] and our
method is applicable in some general problems in the viewpoint of practicalproblems. Here, we state their essential parts. Here, we will state its theoriticalparts and Professor T. Matsuura will give their numerical examples.

2Inversion Formulas

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Best Approximations, as a connection

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The Tikhonov Regularization

5Real and Numerical Inversions of the Laplace
Transform

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The Aveiro Discretization Method

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A Typical Example of the Aveiro Discretization Method With ODE

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87-101[3] L. P. Castro, H. Fujiwara, M. M. Rodrigues, S. Saitoh and V. K. Tuan,Aveiro Discretization Method in Mathematics: A New Discretization Principle, MATHEMATICS WITHOUT BOUNDARIES: SURVEYS IN PURE
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[22] S. Saitoh and Y. Sawano, Theory of Reproducing Kernels and Applications,
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