Abstract : The minisymposium discusses recent development on inverse problems for partial differential equations on manifolds and inverse problems on graphs. Inverse problems typically study the reconstruction of system parameters and geometric or combinatorial structures from indirect measurements. They naturally appear in various imaging problems such as in geophysics, medical imaging, network tomography, material science and non-destructive testing. Many inverse problems are highly sensitive to noise, and understanding this unstable nature is important to applications. Inverse problems on manifolds and graphs in general exhibit different nature, and this minisymposium seeks new connections between them.
Organizer(s) : Matti Lassas, Jinpeng Lu, Lauri Oksanen
Abstract : Many areas of interest within the Earth are anisotropic, meaning that the speed of sound is different in different directions. It turns out that pressure waves are far better behaved than shear waves, but fortunately the different polarizations are coupled together through algebraic geometry. I will explain the surprising power of algebraic geometry in the study of anisotropic inverse problems.
Abstract : The pure Yang-Mills theory is only able to describe the behavior of massless gauge bosons. But experiments show massive gauge bosons do exist. According to the Higgs mechanism of mass generation, the mass of gauge bosons is acquired through the interactions with Higgs bosons. Therefore, the combined Yang-Mills-Higgs Lagrangian together with its Euler-Lagrange equation is of great scientific significance. We show that one can detect the coupled Yang-Mills-Higgs fields from active local measurements of Yang-Mills-Higgs equations.
[03095] Inverse problems for the graph Laplacian
Format : Talk at Waseda University
Author(s) :
Jinpeng Lu (University of Helsinki)
Abstract : We study the discrete version of Gel'fand's inverse spectral problem, of determining the graph structure of a finite weighted graph from the spectral data of its graph Laplacian. We prove that the problem is uniquely solvable under a novel Two-Points Condition. We also consider an inverse problem for random walks on finite graphs and its unique solvability under this condition. This is a joint work with E. Blåsten, H. Isozaki and M. Lassas.
[03169] Quantum computing algorithms for inverse problems on graphs
Format : Talk at Waseda University
Author(s) :
Joonas Ilmavirta (University of Jyväskylä)
Matti Lassas (University of Helsinki)
Jinpeng Lu (University of Helsinki)
Lauri Oksanen (University of Helsinki)
Lauri Ylinen (University of Helsinki)
Abstract : We consider a quantum algorithm for an inverse travel time problem on a graph. This problem is a discrete version of the inverse travel time problem encountered in seismic and medical imaging and the boundary rigidity problem studied in Riemannian geometry. We also consider the computational complexity of the inverse problem, and show that the quantum algorithm has a quadratic improvement in computational cost when compared to the standard classical algorithm.
Abstract : We will discuss recent density results for products of harmonic functions, and applications of these results to inverse problems such as the linearized Calderon problem and the Calderon problem for nonlinear partial differential equations. The emphasis will be on the geometric setting that corresponds to anisotropic conductivity coefficients given by a Riemannian metric.
[03644] Rellich type theorem for lattice Hamiltonians
Format : Talk at Waseda University
Author(s) :
Hiroshi Isozaki (University of Tsukuba)
Abstract :
We announce results for a Rellich type theorem on a locally perturbed periodic
lattice containing square, triangular and hexagonal lattices.
This is a joint work with K. Ando and H. Morioka.
[04868] Continuum limits of discrete Schr\"odinger operators on lattices
Format : Talk at Waseda University
Author(s) :
Yukihide Tadano (Tokyo University of Science)
Abstract : We consider continuum limit problems of discrete Schr\”odinger operators defined on lattices.
We show that the corresponding Schr\”odinger operators on the Euclidean space are obtained as the above continuum limits in the generalized norm resolvent sense.
This talk is based on joint work with Shu Nakamura.
