24 research outputs found
A class of well-posed parabolic final value problems
This paper focuses on parabolic final value problems, and well-posedness is
proved for a large class of these. The clarification is obtained from Hilbert
spaces that characterise data that give existence, uniqueness and stability of
the solutions. The data space is the graph normed domain of an unbounded
operator that maps final states to the corresponding initial states. It induces
a new compatibility condition, depending crucially on the fact that analytic
semigroups always are invertible in the class of closed operators. Lax--Milgram
operators in vector distribution spaces constitute the main framework. The
final value heat conduction problem on a smooth open set is also proved to be
well posed, and non-zero Dirichlet data are shown to require an extended
compatibility condition obtained by adding an improper Bochner integral.Comment: 16 pages. To appear in "Applied and numerical harmonic analysis"; a
reference update. Conference contribution, based on arXiv:1707.02136, with
some further development
Quantitative permeability imaging of plant tissues
A method for mapping tissue permeability based on time-dependent diffusion measurements is presented. A pulsed field gradient sequence to measure the diffusion encoding time dependence of the diffusion coefficients based on the detection of stimulated spin echoes to enable long diffusion times is combined with a turbo spin echo sequence for fast NMR imaging (MRI). A fitting function is suggested to describe the time dependence of the apparent diffusion constant in porous (bio-)materials, even if the time range of the apparent diffusion coefficient is limited due to relaxation of the magnetization. The method is demonstrated by characterizing anisotropic cell dimensions and permeability on a subpixel level of different tissues of a carrot (Daucus carota) taproot in the radial and axial directions
A Multiscale Guide to Brownian Motion
We revise the Levy's construction of Brownian motion as a simple though still rigorous approach to operate with various Gaussian processes. A Brownian path is explicitly constructed as a linear combination of wavelet-based "geometrical features" at multiple length scales with random weights. Such a wavelet representation gives a closed formula mapping of the unit interval onto the functional space of Brownian paths. This formula elucidates many classical results about Brownian motion (e.g., non-differentiability of its path), providing intuitive feeling for non-mathematicians. The illustrative character of the wavelet representation, along with the simple structure of the underlying probability space, is different from the usual presentation of most classical textbooks. Similar concepts are discussed for fractional Brownian motion, Ornstein-Uhlenbeck process, Gaussian free field, and fractional Gaussian fields. Wavelet representations and dyadic decompositions form the basis of many highly efficient numerical methods to simulate Gaussian processes and fields, including Brownian motion and other diffusive processes in confining domains