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Lassi Päivärinta

Publications and source records attributed to Lassi Päivärinta.

17 recordsLinked to original sources
arXiv2024

Inverse Problems for Screens

We study the inverse scattering from a screen with using only one incoming time--harmonic plane wave but with measurements of the scattered wave done at all directions. Especially we focus on the 2D--case i.e. (inverse) scattering from an open bounded smooth curve. Besides the inverse scattering problem we also study the inverse electrostatic problem. We then show that one Cauchy--data of any continuous and bounded function vanishing on the screen and harmonic outside it, determines the screen uniquely.

math.AP↗
arXiv2023

The Fourier, Hilbert and Mellin transforms on a half-line

We are interested in the singular behaviour at the origin of solutions to the equation $\mathscr{H} ρ= e$ on a half-axis, where $\mathscr H$ is the one-sided Hilbert transform, $ρ$ an unknown solution and $e$ a known function. This is a simpler model problem on the path to understanding wave field singularities caused by curve-shaped scatterers in a planar domain. We prove that $ρ$ has a singularity of the form $\mathscr M[e](1/2) / \sqrt{t}$ where $\mathscr M$ is the Mellin transform. To do this we use specially built function spaces $\mathscr M'(a,b)$ by Zemanian, and these allow us to precisely investigate the relationship between the Mellin and Hilbert transforms. Fourier comes into play in the sense that the Mellin transform is simpy the Fourier transform on the locally compact Abelian multiplicative group of the half-line, and as a more familiar operator it guides our investigation.

math.AP↗
arXiv2020

Unique determination of the shape of a scattering screen from a passive measurement

We consider the problem of fixed frequency acoustic scattering from a sound-soft flat screen. More precisely the obstacle is restricted to a two-dimensional plane and interacting with a arbitrary incident wave, it scatters acoustic waves to three-dimensional space. The model is particularly relevant in the study and design of reflecting sonars and antennas, cases where one cannot assume that the incident wave is a plane wave. Our main result is that given the plane where the screen is located, the far-field pattern produced by any single arbitrary incident wave determines the exact shape of the screen, as long as it is not antisymmetric with respect to the plane. This holds even for screens whose shape is an arbitrary simply connected smooth domain. This is in contrast to earlier work where the incident wave had to be a plane wave, or more recent work where only polygonal scatterers are determined.

math.AP↗
arXiv2020

Fractional Brownian motion and asymptotic Bayesian estimation

In this paper, we study the recovery of the Hurst parameter from a given discrete sample of fractional Brownian motion with statistical inverse theory. In particular, we show that in the limit the posteriori distribution of the parameter given the sample determines the parameter uniquely. In order to obtain this result, we first prove various strong laws of large numbers related to the problem at hand and then employ these limit theorems to verify directly the limiting behaviour of posteriori distributions without making additional technical or simplifying assumptions that are commonly used.

math.PR↗
arXiv2018

Detecting anisotropic inclusions through EIT

We study the evolution equation $\partial_{t}u=-Λ_{t}u$ where $Λ_ {t}$ is the Dirichlet-Neumann operator of a decreasing family of Riemannian manifolds with boundary $Σ_{t}$. We derive a lower bound for the solution of such an equation, and apply it to a quantitative density estimate for the restriction of harmonic functions on $\mathcal{M}=Σ_{0}$ to the boundaries of $\partialΣ_{t}$. Consequently we are able to derive a lower bound for the difference of the Dirichlet-Neumann maps in terms of the difference of a background metrics $g$ and an inclusion metric $g+χ_Σ(h-g)$ on a manifold $\mathcal{M}$.

math.AP↗
arXiv2017

Enhancing D-bar reconstructions for electrical impedance tomography with conformal maps

We present a few ways of using conformal maps in the reconstruction of two-dimensional conductivities in electrical impedance tomography. First, by utilizing the Riemann mapping theorem, we can transform any simply connected domain of interest to the unit disk where the D-bar method can be implemented most efficiently. In particular, this applies to the open upper half-plane. Second, in the unit disk we may choose a region of interest that is magnified using a suitable Möbius transform. To facilitate the efficient use of conformal maps, we introduce input current patterns that are named conformally transformed truncated Fourier basis; in practice, their use corresponds to positioning the available electrodes close to the region of interest. These ideas are numerically tested using simulated continuum data in bounded domains and simulated point electrode data in the half-plane. The connections to practical electrode measurements are also discussed.

math.NA↗
arXiv2014

Inverse acoustic scattering problem in half-space with anisotropic random impedance

We study an inverse acoustic scattering problem in half-space with a probabilistic impedance boundary value condition. The Robin coefficient (surface impedance) is assumed to be a Gaussian random function $λ= λ(x)$ with a pseudodifferential operator describing the covariance. We measure the amplitude of the backscattered field averaged over the frequency band and assume that the data is generated by a single realization of $λ$. Our main result is to show that under certain conditions the principal symbol of the covariance operator of $λ$ is uniquely determined. Most importantly, no approximations are needed and we can solve the full non-linear inverse problem. We concentrate on anisotropic models for the principal symbol, which leads to the analysis of a novel anisotropic spherical Radon transform and its invertibility.

math.AP↗
arXiv2014

Nonlinear Fourier analysis for discontinuous conductivities: computational results

Two reconstruction methods of Electrical Impedance Tomography (EIT) are numerically compared for nonsmooth conductivities in the plane based on the use of complex geometrical optics (CGO) solutions to D-bar equations involving the global uniqueness proofs for Calderón problem exposed in [Nachman; Annals of Mathematics 143, 1996] and [Astala and Päivärinta; Annals of Mathematics 163, 2006]: the Astala-Päivärinta theory-based "low-pass transport matrix method" implemented in [Astala et al.; Inverse Problems and Imaging 5, 2011] and the "shortcut method" which considers ingredients of both theories. The latter method is formally similar to the Nachman theory-based regularized EIT reconstruction algorithm studied in [Knudsen, Lassas, Mueller and Siltanen; Inverse Problems and Imaging 3, 2009] and several references from there. New numerical results are presented using parallel computation with size parameters larger than ever, leading mainly to two conclusions as follows. First, both methods can approximate piecewise constant conductivities better and better as the cutoff frequency increases, and there seems to be a Gibbs-like phenomenon producing ringing artifacts. Second, the transport matrix method loses accuracy away from a (freely chosen) pivot point located outside of the object to be studied, whereas the shortcut method produces reconstructions with more uniform quality.

math.AP↗
arXiv2014

The inverse Robin boundary value problem in a half-space

We study the inverse Robin problem for the Schrödinger equation in a half-space. The potential is assumed to be compactly supported. We first solve the direct problem for dimensions two and three. We then show that the Robin-to-Robin map uniquely determines the potential q.

math.AP↗
arXiv2014

Strictly convex corners scatter

We prove the absence of non-scattering energies for potentials in the plane having a corner of angle smaller than $π$. This extends the earlier result of Blåsten, Päivärinta and Sylvester who considered rectangular corners. In three dimensions, we prove a similar result for any potential with a circular conic corner whose opening angle is outside a countable subset of $(0,π)$.

math.AP↗
arXiv2013

Do corners always scatter?

We study time harmonic scattering for the Helmholtz equation in Rn. We show that certain penetrable scatterers with rectangular corners scatter every incident wave nontrivially. Even though these scatterers have interior transmission eigenvalues, the relative scattering (a.k.a. far field) operator has a trivial kernel and cokernel at every real wavenumber.

math.AP↗
arXiv2010

Transmission eigenvalues for elliptic operators

A reduction of the transmission eigenvalue problem for multiplicative sign-definite perturbations of elliptic operators with constant coefficients to an eigenvalue problem for a non-selfadjoint compact operator is given. Sufficient conditions for the existence of transmission eigenvalues and completeness of generalized eigenstates for the transmission eigenvalue problem are derived. In the trace class case, the generic existence of transmission eigenvalues is established.

math-ph↗
arXiv2010

Transmission eigenvalues for operators with constant coefficients

In this paper we study the interior transmission problem and transmission eigenvalues for multiplicative perturbations of linear partial differential operator of order $\ge 2$ with constant real coefficients. Under suitable growth conditions on the symbol of the operator and the perturbation, we show the discreteness of the set of transmission eigenvalues and derive sufficient conditions on the existence of transmission eigenvalues. We apply these techniques to the case of the biharmonic operator and the Dirac system. In the hypoelliptic case we present a connection to scattering theory.

math-ph↗
arXiv2009

Inverse scattering for the magnetic Schroedinger operator

We show that fixed energy scattering measurements for the magnetic Schroedinger operator uniquely determine the magnetic field and electric potential in dimensions $n \geq 3$. The magnetic potential, its first derivatives, and the electric potential are assumed to be exponentially decaying. This improves an earlier result of Eskin and Ralston which considered potentials with many derivatives. The proof is close to arguments in inverse boundary problems, and is based on constructing complex geometrical optics solutions to the Schroedinger equation via a pseudodifferential conjugation argument.

math.AP↗