Search results for "Different"

Refined instability estimates for some inverse problems

2022

Many inverse problems are known to be ill-posed. The ill-posedness can be manifested by an instability estimate of exponential type, first derived by Mandache [29]. In this work, based on Mandache's idea, we refine the instability estimates for two inverse problems, including the inverse inclusion problem and the inverse scattering problem. Our aim is to derive explicitly the dependence of the instability estimates on key parameters. The first result of this work is to show how the instability depends on the depth of the hidden inclusion and the conductivity of the background medium. This work can be regarded as a counterpart of the depth-dependent and conductivity-dependent stability estim…

Increasing stability in the linearized inverse Schrödinger potential problem with power type nonlinearities

2022

We consider increasing stability in the inverse Schr\"{o}dinger potential problem with power type nonlinearities at a large wavenumber. Two linearization approaches, with respect to small boundary data and small potential function, are proposed and their performance on the inverse Schr\"{o}dinger potential problem is investigated. It can be observed that higher order linearization for small boundary data can provide an increasing stability for an arbitrary power type nonlinearity term if the wavenumber is chosen large. Meanwhile, linearization with respect to the potential function leads to increasing stability for a quadratic nonlinearity term, which highlights the advantage of nonlinearit…

Generalized solutions of a system of differential equations of the first order and elliptic type with discontinuous coefficients

2009

On some partial data Calderón type problems with mixed boundary conditions

2021

In this article we consider the simultaneous recovery of bulk and boundary potentials in (degenerate) elliptic equations modelling (degenerate) conducting media with inaccessible boundaries. This connects local and nonlocal Calderón type problems. We prove two main results on these type of problems: On the one hand, we derive simultaneous bulk and boundary Runge approximation results. Building on these, we deduce uniqueness for localized bulk and boundary potentials. On the other hand, we construct a family of CGO solutions associated with the corresponding equations. These allow us to deduce uniqueness results for arbitrary bounded, not necessarily localized bulk and boundary potentials. T…

Shape optimization utilizing consistent sensitivities

2010

Navier-Stokes equations

2009

Systematic implementation of higher order Whitney forms in methods based on discrete exterior calculus

2022

AbstractWe present a systematic way to implement higher order Whitney forms in numerical methods based on discrete exterior calculus. Given a simplicial mesh, we first refine the mesh into smaller simplices which can be used to define higher order Whitney forms. Cochains on this refined mesh can then be interpolated using higher order Whitney forms. Hence, when the refined mesh is used with methods based on discrete exterior calculus, the solution can be expressed as a higher order Whitney form. We present algorithms for the three required steps: refining the mesh, solving the coefficients of the interpolant, and evaluating the interpolant at a given point. With our algorithms, the order of…

GPU-accelerated time integration of Gross-Pitaevskii equation with discrete exterior calculus

2022

The quantized vortices in superfluids are modeled by the Gross-Pitaevskii equation whose numerical time integration is instrumental in the physics studies of such systems. In this paper, we present a reliable numerical method and its efficient GPU-accelerated implementation for the time integration of the three-dimensional Gross-Pitaevskii equation. The method is based on discrete exterior calculus which allows us the usage of more versatile spatial discretization than traditional finite difference and spectral methods are applicable to. We discretize the problem using six different natural crystal structures and observe the correct choices of spatial tiling to decrease the truncation error…

On a numerical solution of the Maxwell equations by discrete exterior calculus

2014

p-Laplacen operaattorin ominaisarvo-ongelmasta

2016

Tämän tutkielman tarkoitus on tutustua epälineaarisiin ominaisarvo-ongelmiin p-Laplacen operaattorin ominaisarvo-ongelman kautta. p-Laplacen operaattori on Laplacen operaattorin eräs yleistys ja tarkastelun kohteena oleva ominaisarvo-ongelma on Dirichletin ominaisarvo-ongelman yleistys. Tutkielmassa kerrataan ensin tarvittavia taustatietoja Sobolevin avaruuksista ja funktionaalianalyysistä, ja keskitytään sitten itse ongelmaan. Päätulokset koskevat ensimmäistä ominaisarvoa, ja ne ovat ensimmäisen ominaisarvon olemassaolo, ensimmäisen ominaisarvon karakterisointi Rayleighin osamäärän avulla, ensimmäisen ominaisfunktion yksinkertaisuus, ja se, että ensimmäinen ominaisfunktio on ainoa ominaisf…