arXiv subjects

Samuli Siltanen

Publications and source records attributed to Samuli Siltanen.

60 records · Page 4Linked to original sources

arXiv2012

Exceptional circles of radial potentials

A nonlinear scattering transform is studied for the two-dimensional Schrodinger equation at zero energy with a radial potential. First explicit examples are presented, both theoretically and computationally, of potentials with nontrivial singularities in the scattering transform. The singularities arise from non-uniqueness of the complex geometric optics solutions that define the scattering transform. The values of the complex spectral parameter at which the singularities appear are called exceptional points. The singularity formation is closely related to the fact that potentials of conductivity type are critical in the sense of Murata.

math.AP↗

arXiv2011

Reconstructing conductivities with boundary corrected D-bar method

The aim of electrical impedance tomography is to form an image of the conductivity distribution inside an unknown body using electric boundary measurements. The computation of the image from measurement data is a non-linear ill-posed inverse problem and calls for a special regularized algorithm. One such algorithm, the so-called D-bar method, is improved in this work by introducing new computational steps that remove the so far necessary requirement that the conductivity should be constant near the boundary. The numerical experiments presented suggest two conclusions. First, for most conductivities arising in medical imaging, it seems the previous approach of using a best possible constant near the boundary is sufficient. Second, for conductivities that have high contrast features at the boundary, the new approach produces reconstructions with smaller quantitative error and with better visual quality.

math.NA↗

arXiv2011

The Novikov-Veselov Equation and the Inverse Scattering Method, Part I: Analysis

The Novikov-Veselov (NV) equation is a (2+1)-dimensional nonlinear evolution equation that generalizes the (1+1)-dimensional Korteweg-deVries (KdV) equation. Solution of the NV equation using the inverse scattering method has been discussed in the literature, but only formally (or with smallness assumptions in case of nonzero energy) because of the possibility of exceptional points, or singularities in the scattering data. In this work, absence of exceptional points is proved at zero energy for evolutions with compactly supported, smooth and rotationally symmetric initial data of the conductivity type: $q_0=γ^{-1/2}Δγ^{1/2}$ with a strictly positive function $γ$. The inverse scattering evolution is shown to be well-defined, real-valued, and preserving conductivity-type. There is no smallness assumption on the initial data.

math.AP↗

arXiv2010

Determining electrical and heat transfer parameters using coupled boundary measurements

Let $Ω\subset\R^n$, $n\ge 3$, be a smooth bounded domain and consider a coupled system in $Ω$ consisting of a conductivity equation $\nabla \cdot γ(x) \nabla u(t,x)=0$ and an anisotropic heat equation $κ^{-1}(x)\partial_tψ(t,x)=\nabla\cdot (A(x)\nabla ψ(t,x))+(γ\nabla u(t,x))\cdot \nabla u(t,x), \quad t\ge 0$. It is shown that the coefficients $γ$, $κ$ and $A=(a_{jk})$ are uniquely determined from the knowledge of the boundary map $u|_{\partialΩ}\mapsto ν\cdot A\nabla ψ|_{\partialΩ}$, where $ν$ is the unit outer normal to $\partialΩ$. The coupled system models the following physical phenomenon. Given a fixed voltage distribution, maintained on the boundary $\partialΩ$, an electric current distribution appears inside $Ω$. The current in turn acts as a source of heat inside $Ω$, and the heat flows out of the body through the boundary. The boundary measurements above then correspond to the map taking a voltage distribution on the boundary to the resulting heat flow through the boundary. The presented mathematical results suggest a new hybrid diffuse imaging modality combining electrical prospecting and heat transfer-based probing.

math.AP↗

arXiv2009

Discretization-invariant Bayesian inversion and Besov space priors

Bayesian solution of an inverse problem for indirect measurement $M = AU + {\mathcal{E}}$ is considered, where $U$ is a function on a domain of $R^d$. Here $A$ is a smoothing linear operator and $ {\mathcal{E}}$ is Gaussian white noise. The data is a realization $m_k$ of the random variable $M_k = P_kA U+P_k {\mathcal{E}}$, where $P_k$ is a linear, finite dimensional operator related to measurement device. To allow computerized inversion, the unknown is discretized as $U_n=T_nU$, where $T_n$ is a finite dimensional projection, leading to the computational measurement model $M_{kn}=P_k A U_n + P_k {\mathcal{E}}$. Bayes formula gives then the posterior distribution $π_{kn}(u_n | m_{kn})\simπ_n(u_n) \exp(-{1/2}\|m_{kn} - P_kA u_n\|_2^2)$ in $R^d$, and the mean $U^{CM}_{kn}:=\int u_n π_{kn}(u_n | m_k) du_n$ is considered as the reconstruction of $U$. We discuss a systematic way of choosing prior distributions $\prior_n$ for all $n\geq n_0>0$ by achieving them as projections of a distribution in a infinite-dimensional limit case. Such choice of prior distributions is {\em discretization-invariant} in the sense that $\prior_n$ represent the same {\em a priori} information for all $n$ and that the mean $U^{CM}_{kn}$ converges to a limit estimate as $k,n\to\infty$. Gaussian smoothness priors and wavelet-based Besov space priors are shown to be discretization invariant. In particular, Bayesian inversion in dimension two with $B^1_{11}$ prior is related to penalizing the $\ell^1$ norm of the wavelet coefficients of $U$.

math.PR↗

arXiv2007

Time reversal methods in unknown medium and inverse problems

A novel method to solve inverse problems for the wave equation is introduced. The method is a combination of the boundary control method and an iterative time reversal scheme, leading to adaptive imaging of coefficient functions of the wave equation using focusing waves in unknown medium. The approach is computationally effective since the iteration lets the medium do most of the processing of the data. The iterative time reversal scheme also gives an algorithm for constructing boundary controls for which the corresponding final values are as close as possible to the final values of a given wave in a part of the domain, and as close as possible to zero elsewhere. The algorithm does not assume that the coefficients of the wave equation are known.

math.AP↗