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In this paper we study the problem of computing wavelet coefficients of compactly supported functions from their Fourier samples. For this, we use the recently introduced framework of generalized sampling. Our first result demonstrates that…
We discuss a Hilbert space method that allows to prove analytical well-posedness of a class of linear strongly damped wave equations. The main technical tool is a perturbation lemma for sesquilinear forms, which seems to be new. In most…
Wavelet based algorithms in numerical analysis are similar to other transform methods in that vectors and operators are expanded into a basis and the computations take place in this new system of coordinates. However, due to the recursive…
The diffuse domain method for partial differential equations on complicated geometries recently received strong attention in particular from practitioners, but many fundamental issues in the analysis are still widely open. In this paper we…
We propose a high-order spacetime wavelet method for the solution of nonlinear partial differential equations with a user-prescribed accuracy. The technique utilizes wavelet theory with a priori error estimates to discretize the problem in…
This paper is concerned with the problem of reconstructing an infinite-dimensional signal from a limited number of linear measurements. In particular, we show that for binary measurements (modelled with Walsh functions and Hadamard…
We consider state-dependent delay equations (SDDE) obtained by adding delays to a planar ordinary differential equation with a limit cycle. These situations appear in models of several physical processes, where small delay effects are…
We propose a general framework for obtaining probabilistic solutions to PDE-based inverse problems. Bayesian methods are attractive for uncertainty quantification but assume knowledge of the likelihood model or data generation process. This…
We consider delay differential equations with a polynomially distributed delay. We derive an equivalent system of delay differential equations, which includes just two discrete delays. The stability of the equivalent system and its…
We study Dirichlet-type problems for the simplest third-order linear dispersive PDE, often referred to as the Airy equation. Such problems have not been extensively studied, perhaps due to the complexity of the spectral structure of the…
In a recent paper a mathematical model for quantum measurement was presented. The phenomenon of wave particle duality, which is introduced in every beginning course of quantum theory, can be explained using this model. Although it is a…
Variational regularization of ill-posed inverse problems is based on minimizing the sum of a data fidelity term and a regularization term. The balance between them is tuned using a positive regularization parameter, whose automatic choice…
Using a capacity approach, and the theory of measure's perturbation of Dirichlet forms, we give the probabilistic representation of the General Robin boundary value problems on an arbitrary domain $\Omega$, involving smooth measures, which…
We lift the constraint of a diagonal representation of the Hamiltonian by searching for square integrable bases that support an infinite tridiagonal matrix representation of the wave operator. The class of solutions obtained as such…
A semilinear wave equation with slowly varying wave speed is considered in one to three space dimensions on a bounded interval, a rectangle or a box, respectively. It is shown that the action, which is the harmonic energy divided by the…
In dispersive media, dissipation appears in the Schr\"odinger representation of classical Maxwell equations as a sparse diagonal operator occupying an $r$-dimensional subspace. A first order Suzuki-Trotter approximation for the evolution…
An original approach to solving rather difficult probabilistic problems arising in studying the readout of random discrete fields and having no exact analytical solutions at the moment is proposed. Several algorithms for direct, iterative,…
The continuous extension of a discrete random variable is amongst the computational methods used for estimation of multivariate normal copula-based models with discrete margins. Its advantage is that the likelihood can be derived…
A method is introduced for studying large deviations in the context of statistical physics of disordered systems. The approach, based on an extension of the cavity method to atypical realizations of the quenched disorder, allows us to…
Recent progress concerning regularization of supersymmetric theories is reviewed. Dimensional reduction is reformulated in a mathematically consistent way, and an elegant and general method is presented that allows to study the…