1,650 Works

The Hagedorn-Hermite Correspondence

Tomoki Ohsawa
I will explain the correspondence between the semiclassical wave packets of Hagedorn and the Hermite functions by looking into the relationship between their ladder operators. The correspondence provides simple derivations of some fundamental properties of the Hagedorn wave packets-such as its completeness, transformation properties, and their generating functions-by linking them to the corresponding properties of the Hermite functions.

Continuous and discontinuous approaches to moving mesh finite elements

Matthew Hubbard
The moving mesh finite element method of Baines, Hubbard and Jimack [1] determines mesh velocities by building the underlying PDE into a monitor conservation principle and recovering an approximation in a manner which preserves the original monitor distribution (so an equidistributed mesh remains so as time progresses). The first part of this talk will outline this method and show some examples which illustrate its ability to track interfaces accurately for implicit moving boundary problems [2]....

Inverse Scattering Problems for the Time Dependent Wave Equation

Fioralba Cakoni
In this presentation we will discuss recent progress in non-iterative methods in the time domain. The use of time dependent data is a remedy for the large spacial aperture that these method need to obtain a reasonable reconstructions. Fist we consider the linear sampling method for solving inverse scattering problem for inhomogeneous media. A fundamental tool for the justification of this method is the solvability of the time domain interior transmission problem that relies on...

Detection of conductivity inclusions in a semilinear elliptic problem via a phase field approach

Marco Verani
Abstract: we tackle the reconstruction of discontinuous coefficients in a semilinear elliptic equation from the knowledge of the solution on the boundary of the planar bounded domain. The problem is motivated by an application in cardiac electrophysiology. We formulate a constraint minimization problem involving a quadratic mismatch functional enhanced with a phase field term which penalizes the perimeter. After computing the optimality conditions of the phase-field optimization problem and introducing a discrete finite element formulation,...

Reduced order models for spectral domain inversion: Embedding into the continuous problem and generation of internal data

Shari Moskow
We generate reduced order Galerkin models for inversion of the Schr\"odinger equation given boundary data in the spectral domain for one and two dimensional problems. We show that in one dimension, after Lanczos orthogonalization, the Galerkin system is precisely the same as the three point staggered finite difference system on the corresponding spectrally matched grid. The orthogonalized basis functions depend only very weakly on the medium, and thus by embedding into the continuous problem, the...

Reconstruction via Bayesian hierarchical models: convexity, sparsity and model reduction

Daniela Calvetti
The reconstruction of sparse signals from indirect, noisy data is a challenging inverse problem. In the Bayesian framework, the sparsity belief can be encoded via hierarchical prior models. In this talk we discuss the convexity - or lack thereof - of the functional associated to different models, and we show that Krylov subspace methods for the computation of the MAP solution implicitly perform an effective and efficient model reduction.

Understanding angular momentum in General Relativity

Anna Sakovich
Conserved quantities â mass, linear momentum, center of mass, and angular momentum â are important in various areas of physics as in many cases they provide an essential characterization of a physical system. The concepts of mass and linear momentum are by now well-established in General Relativity, and recently there has been a lot of progress in understanding the center of mass of isolated systems. However, we believe that the somewhat more mysterious notion of...

PCM-TV-TFV: A Novel Two-Stage Framework for Image Reconstruction from Fourier Data

Weihong Guo
We propose in this paper a novel two-stage projection correction modeling (PCM) framework for image reconstruction from (nonuniform) Fourier measurements. PCM consists of a projection stage (P-stage) motivated by the multiscale Galerkin method and a correction stage (C-stage) with an edge guided regularity fusing together the advantages of total variation and total fractional variation. The P-stage allows for continuous modeling of the underlying image of interest. The given measurements are projected onto a space in...

Conservative architectures for deep neural networks

Eldad Haber
In this talk we discuss architectures for deep neural networks that preserve the energy of the propagated signal. We show that such networks can have significant computational advantages for some key problems in computer vision

Combining learned and model based approaches for inverse problems

Simon Arridge
Deep Learning (DL) has become a pervasive approach in many machine learning tasks and in particular in image processing problems such as denoising, deblurring, inpainting and segmentation. The application of DL within inverse problems is less well explored because it is not trivial to include Physics based knowledge of the forward operator into what is usually a purely data-driven framework. In addition some inverse problems are at a scale much larger than image or video...

New results on a variational inequality formulation of Lavrentiev regularization for nonlinear monotone ill-posed problems

Robert Plato

Combining the Runge approximation and the Whitney embedding theorem in hybrid imaging

Giovanni S. Alberti
The reconstruction in quantitative coupled physics imaging often requires that the solutions of certain PDEs satisfy some non-zero constraints, such as the absence of critical points or nodal points. After a brief review of several methods used to construct such solutions, I will focus on a recent approach that combines the Runge approximation and the Whitney embedding theorem.

The impact of conditional stability estimates on variational regularization and the distinguished case of oversmoothing penalties

Bernd Hofmann
Conditional stability estimates require additional regularization for obtaining stable approximate solutions if the validity area of such estimates is not completely known. The focus of this talk is on the Tikhonov regularization under conditional stability estimates for non-linear ill-posed problems in Hilbert scales, where the case that the penalty is oversmoothing plays a prominent role. This oversmoothing problem has been studied early for linear forward operators, most notably in the seminal paper by Natterer 1984....

Regularization of backwards diffusion by fractional time derivatives

Barbara Kaltenbacher
The backwards heat equation is one of the classical inverse problems, related to a wide range of applications and exponentially ill-posed. One of the first and maybe most intuitive approaches to its stable numerical solution was that of quasireversibility, whereby the parabolic operator is replaced by a differential operator for which the backwards problem in time is well posed. After a short overview of approaches in this vein, we will dwell on a new one...

Total variation based Lavrentiev regularisation

Markus Grasmair
In this talk we will discuss a non-linear variant of Lavrentiev regularisation, where the sub-differential of the total variation replaces the identity operator as regularisation term. The advantage of this approach over Tikhonov based total variation regularisation is that it avoids the evaluation of the adjoint operator on the data. As a consequence, it can be used, for instance, for the solution of Volterra integral equations of the first kind, where the adjoint would require...

Regularization of Inverse Problems via Time Discrete Geodesics in Image Spaces

Gabriele Steidl
This talk addresses the solution of inverse problems in imaging given an additional reference image. We combine a modification of the discrete geodesic path model of Berkels, Effland and Rumpf with a variational model, actually the L 2 -T V model, for image restoration. We prove that the space continuous model has a minimizer and propose a minimization procedure which alternates over the involved sequences of deformations and images. The minimization with respect to the...

A convex analysis approach to iterative regularization methods

Antonio Leitao
We address two well known iterative regularization methods for ill-posed problems (Landweber and iterated-Tikhonov methods) and discuss how to improve the performance of these classical methods by using convex analysis tools. The talk is based on two recent articles (2018): Range-relaxed criteria for choosing the Lagrange multipliers in nonstationary iterated Tikhonov method (with R.Boiger, B.F.Svaiter), and On a family of gradient type projection methods for nonlinear ill-posed problems (with B.F.Svaiter)

Discrete processes and their continuous limits

Uri Ascher
The possibility that a discrete process can be closely approximated by a continuous one, with the latter involving a differential system, is fascinating. Important theoretical insights, as well as significant computational efficiency gains may lie in store. A great success story in this regard are the Navier-Stokes equations, which model many phenomena in fluid flow rather well. Recent years saw many attempts to formulate more such continuous limits, and thus harvest theoretical and practical advantages,...

Stable determination of polygonal and polyhedral interfaces from boundary measurements

Elisa Francini
We present some Lipschitz stability estimates for the Hausdorff distance of polygonal or polyhedral inclusions in terms of the Dirichlet-to-Neumann map based on a series of papers in collaboration with Elena Beretta (New York University AbuDhabi) and Sergio Vessella (Università di Firenze).

Detecting presence of emission sources with low SNR. "Analysis" vs deep learning

Peter Kuchment
The talk will discuss the homeland security problem of detecting presence of emission sources at high noise conditions. (Semi-)analytic and deep learning techniques will be compared. This is a joint work with W. Baines and J. Ragusa.

Quantitative PAT-OCT Elastography for Biomechanical Parameter Imaging

Ekaterina Sherina
Diseases like cancer or arteriosclerosis often cause changes of tissue stiffness in the micrometer scale. Our work aims at developing a non-invasive method to quantitatively image these biomechanical changes and study the potential of the method for medical diagnostics. We focus on quantitative elastography combined with photoacoustic (PAT) and optical coherence tomography (OCT). The problem we deal with consists in estimating elastic material parameters from internal displacement data, which are evaluated from OCT-PAT recordered successive...

Infinite-dimensional inverse problems with finite measurements

Matteo Santacesaria
In this talk I will discuss how ideas from applied harmonic analysis, in particular sampling theory and compressed sensing, may be applied to inverse problems for partial differential equations. The focus will be on inverse boundary value problems for the conductivity and the Schrodinger equations, but the approach is very general and allows to handle many other classes of inverse problems. I will discuss uniqueness, stability and reconstruction, both in the linearized and in the...

A multiscale approach for inverse problems

Luca Rondi
We extend the hierarchical decomposition of an image as a sum of constituents of different scales, introduced by Tadmor, Nezzar and Vese in 2004, to a general setting. We develop a theory for multiscale decompositions which, besides extending the one of Tadmor, Nezzar and Vese to arbitrary L^2 functions, is applicable to nonlinear inverse problems, as well as to other imaging problems. As a significant example, we present applications to the inverse conductivity problem. This...

Deftly Divulging Delicate Data

Scott Baker
Open scholarship, which encompasses open science, open access, open data, open education, and all other forms of openness in the scholarly and research environment, is transforming how knowledge is created and shared. The 3rd annual Open Scholarship in Practice (OSiP) day was held at UBC on October 25, 2019 to explore innovative areas in open scholarship, and included a full day of hands-on workshops for faculty, staff, and students to learn how to incorporate Open...

Panel: Looking in

Edward Doolittle
Panelists discuss what they see and hope for in university mathematics instruction, from their particular â outsider-insiderâ perspectives. Ed Doolittle, Kari Marken, Rina Zazkis

Registration Year

  • 2019

Resource Types

  • Audiovisual