Search results for "Inverse problem"

On $L^p$ resolvent estimates for Laplace-Beltrami operators on compact manifolds

2011

Abstract. In this article we prove Lp estimates for resolvents of Laplace–Beltrami operators on compact Riemannian manifolds, generalizing results of Kenig, Ruiz and Sogge (1987) in the Euclidean case and Shen (2001) for the torus. We follow Sogge (1988) and construct Hadamard's parametrix, then use classical boundedness results on integral operators with oscillatory kernels related to the Carleson and Sjölin condition. Our initial motivation was to obtain Lp Carleman estimates with limiting Carleman weights generalizing those of Jerison and Kenig (1985); we illustrate the pertinence of Lp resolvent estimates by showing the relation with Carleman estimates. Such estimates are useful in the …

The Poisson embedding approach to the Calderón problem

2020

We introduce a new approach to the anisotropic Calder\'on problem, based on a map called Poisson embedding that identifies the points of a Riemannian manifold with distributions on its boundary. We give a new uniqueness result for a large class of Calder\'on type inverse problems for quasilinear equations in the real analytic case. The approach also leads to a new proof of the result by Lassas and Uhlmann (2001) solving the Calder\'on problem on real analytic Riemannian manifolds. The proof uses the Poisson embedding to determine the harmonic functions in the manifold up to a harmonic morphism. The method also involves various Runge approximation results for linear elliptic equations.

Virtual darwinian drug design: QSAR inverse problem, virtual combinatorial chemistry, and computational screening.

2001

The generation of diversity and its further selection by an external system is a common mechanism for the evolution of the living species and for the current drug design methods. This assumption allows us to label the methods based on generation and selection of molecular diversity as "Darwinian" ones, and to distinguish them from the structure-based, structure-modulation approaches. An example of a Darwinian method is the inverse QSAR. It consists of the computational generation of candidate chemical structures and their selection according to a previously established QSAR model. New trends in the field of combinatorial chemical syntheses comprise the concepts of virtual combinatorial synt…

HIDDEN CHARM MOLECULES IN A FINITE VOLUME

2013

In the present paper we address the interaction of charmed mesons in hidden charm channels in a finite box. We use the interaction from a recent model based on heavy quark spin symmetry that predicts molecules of hidden charm in the infinite volume. The energy levels in the box are generated within this model, and several methods for the analysis of these levels ("inverse problem") are investigated.

Compartmental analysis of dynamic nuclear medicine data: Models and identifiability

2016

Compartmental models based on tracer mass balance are extensively used in clinical and pre-clinical nuclear medicine in order to obtain quantitative information on tracer metabolism in the biological tissue. This paper is the first of a series of two that deal with the problem of tracer coefficient estimation via compartmental modelling in an inverse problem framework. Specifically, here we discuss the identifiability problem for a general n-dimension compartmental system and provide uniqueness results in the case of two-compartment and three-compartment compartmental models. The second paper will utilize this framework in order to show how non-linear regularization schemes can be applied t…

On the broken ray transform

2014

The fixed angle scattering problem and wave equation inverse problems with two measurements

2019

We consider two formally determined inverse problems for the wave equation in more than one space dimension. Motivated by the fixed angle inverse scattering problem, we show that a compactly supported potential is uniquely determined by the far field pattern generated by plane waves coming from exactly two opposite directions. This implies that a reflection symmetric potential is uniquely determined by its fixed angle scattering data. We also prove a Lipschitz stability estimate for an associated problem. Motivated by the point source inverse problem in geophysics, we show that a compactly supported potential is uniquely determined from boundary measurements of the waves generated by exactl…

A sampling method for detecting buried objects using electromagnetic scattering

2005

We consider a simple (but fully three-dimensional) mathematical model for the electromagnetic exploration of buried, perfect electrically conducting objects within the soil underground. Moving an electric device parallel to the ground at constant height in order to generate a magnetic field, we measure the induced magnetic field within the device, and factor the underlying mathematics into a product of three operations which correspond to the primary excitation, some kind of reflection on the surface of the buried object(s) and the corresponding secondary excitation, respectively. Using this factorization we are able to give a justification of the so-called sampling method from inverse scat…

Non-invasive imaging of biological microstructures by VHF waves

2014

A meshfree approach for brain activity source modeling

2015

Weak electrical currents in the brain flow as a consequence of acquisition, processing and transmission of information by neurons, giving raise to electric and magnetic fields, which are representable by means of quasi-stationary approximation of the Maxwell’s equations. Measurements of electric scalar potential differences at the scalp and magnetic fields near the head constitute the input data for, respectively, electroencephalography (EEG) and magnetoencepharography (MEG), which allow for reconstructing the cerebral electrical currents and thus investigating the neuronal activity in the human brain in a non-invasive way. This is a typical erectromagnetic inverse problem, since measuremen…