We introduce a general and efficient method to recover piecewise constant coefficients occurring in elliptic partial differential equations as well as the interface where these coefficients have jump discontinuities. For this purpose, we use an output least squares approach with level set and augmented Lagrangian methods. Our formulation incorporates the inherent nature of the piecewise constan...

The sequence {B2}, {B4}, {B6}, ... is dense in the unit interval [0, 1], but it is not uniformly distributed [4]. Certain rational numbers appear with positive probability: 1/6 is most likely with probability 0.151..., 29/30 is next with probability 0.064... [5]. In fact, the limiting distribution F is piecewise linear with countably many jump discontinuities: F increases only when jumping (see...

Edge detection is important in a variety of applications. While there are many algorithms available for detecting edges from pixelated images or equispaced Fourier data, much less attention has been given to determining edges from nonuniform Fourier data. There are applications in sensing (e.g. MRI) where the data is given in this way, however. This paper introduces a method for determining the...

We prove a hydrodynamic limit for the totally asymmetric simple exclusion process with spatially inhomogeneous jump rates given by a speed function that may admit discontinuities. The limiting density profiles are described with a variational formula. This formula enables us to compute explicit density profiles even though we have no information about the invariant distributions of the process....

Techniques of interval extensions and interval Newton methods have been developed for verified solution of nonlinear systems of equations and for global optimization. In most of the literature to date, such interval extensions and interval Newton methods are applicable when the functions are given by smooth expressions, without conditional branches. In fact, however, many practical problems, in...

We investigate an optimal stopping time problem which arises from pricing Russian options (i.e. perpetual look-back options) on a stock whose price fluctuations are modelled by adjoining a hidden Markov process to the classical Black–Scholes geometric Brownian motion model. By extending the technique of smooth fit to allow jump discontinuities, we obtain an explicit closed-form solution. It giv...

We compute the asymptotics of the eigenvalues of the classical Sturm–Liouville problem with a piecewise smooth coefficient q. This means that q and/or its derivatives can have jump discontinuities. The boundary conditions are arbitrary. Our results extend the classical work of H. Hochstadt (see [Comm. Pure Appl. Math., 14 (1961), pp. 749–764]) and some related formulas discovered by G. Borg (se...

Weconstruct finite difference discretizations of the acousticwave equation in complicated geometries and heterogeneous media. Particular emphasis is placed on the accurate treatment of interfaces at which the underlying media parameters have jump discontinuities. Discontinuousmedia is treated by subdividing the domain into blockswith continuousmedia. The equation on each block is then discretiz...

We investigate the asymptotic behavior of minimizers to sequences of elliptic variational problems posed on thin three-dimensional domains. These domains arise as thin neighborhoods of artibrary graphs that contain severe constrictions near the graph nodes. We characterize an appropriate limit of minimizers as a function of one variable defined on the graph that necessarily minimizes a one-dime...