Signals with discontinuities appear in many problems in the applied sciences ranging from mechanics, electrical engineering to biology and medicine. The concrete data acquired are typically discrete, indirect and noisy measurements of some quantities describing the signal under consideration. The task is to restore the signal and, in particular, the discontinuities. In this respect, classical m...

Surface tension in multi-phase fluid flow engenders pressure discontinuities on phase interfaces. In this work we present a finite element method to solve viscous incompressible flows problems, especially designed to cope with such a situation. Taking as a model the Stokes system we study a finite element solution method based on a classical Galerkin-least-squares formulation with an added pres...

We propose three novel methods for recovering edges in piecewise smooth functions from their possibly incomplete and noisy spectral information. The proposed methods utilize three different approaches: #1. The randomly-based sparse Inverse Fast Fourier Transform (sIFT); #2. The Total Variation-based (TV) compressed sensing; and #3. The modified zero crossing. The different approaches share a co...

The concentration method of edge detection was developed to compute the locations and values of jump discontinuities in a piecewise-analytic function from its first few Fourier series coefficients. The accuracy and characteristic features of the resulting jump approximation depend on Fourier space “filter” factors known as concentration factors. In this paper, we provide a flexible, iterative f...

In this paper, we consider the problem of mean-variance hedging in an incomplete market where the underlying assets are jump diffusion processes which are driven by Brownian motion and doubly stochastic Poisson processes. This problem is formulated as a stochastic control problem and closed form expressions for the optimal hedging policy are obtained using methods from stochastic control and th...

In this paper, we propose the use of partial differential equation (PDE) based novel PDE transforms for determining jump discontinuity locations in piecewise smooth data containting potential isolated jump discontinuities. The PDE transform method involves evolving coupled PDEs under different time scales to extract edge information. Multiscale edge detection is done by varying the parameters a...

Abstract Let f ∈ C(−π, π), where m > 0 is an integer. An algorithm is proposed for representing f as a convergent series which admits m times term-wise differentiation. This algorithm is illustrated by numerical examples. It can be used, for example, for acceleration of convergence of Fourier series. The algorithm is generalized to the case when f is piecewise-C(−π, π) function with known posit...

We extend a well-studied ODE model for collective behaviour by considering anisotropic interactions among individuals. Anisotropy is modeled by limited sensorial perception of individuals, that depends on their current direction of motion. Consequently, the firstorder model becomes implicit, and new analytical issues, such as non-uniqueness and jump discontinuities in velocities, are being rais...

A numerical method in which the Rankine-Hugoniot condition is enforced at the discrete level is developed. The simple format of central discretization in a finite volume method is used together with the jump condition to develop a simple and yet accurate numerical method free of Riemann solvers and complicated flux splittings. The steady discontinuities are captured accurately by this numerical...

We propose three novel methods for recovering edges in piecewise smooth functions from their possibly incomplete and noisy spectral information. The proposed methods utilize three different approaches: #1. the randomly-based sparse Inverse Fast Fourier Transform (sIFT); #2. the Total Variation-based (TV) compressive sensing; and #3. the modified zero crossing. The different approaches share a c...