arXiv subjects

Mirza Karamehmedović

Publications and source records attributed to Mirza Karamehmedović.

At least 19 recordsLinked to original sources

arXiv2026

A variational mollification approach to circular deconvolution

We propose a variational mollification approach to circular deconvolution based on the reconstruction of a mollified target object rather than the exact solution itself. The method is formulated as a convex variational problem whose solution admits an explicit Fourier representation. We establish the consistency of the proposed reconstruction as the target resolution increases and derive convergence rates for deterministic data perturbations. For ordinary smooth kernels, the method achieves the classical order-optimal algebraic convergence rates under Besov-Nikolskii smoothness assumptions, while for supersmooth kernels it attains the corresponding order-optimal logarithmic rates. Numerical experiments on synthetic and wind-direction data illustrate the effectiveness of the proposed approach and confirm the theoretical predictions.

math.NA↗

arXiv2026

Programmable photonic nanojets via phase-only time-reversal: a numerical study

We present a phase-only time-reversal framework for steering photonic nanojets without mechanical motion or amplitude modulation. Time-reversed radiation by a synthetic source placed at the target PNJ location helps define a phase-only modulation on a control line, compatible with a spatial light modulator, that produces the desired PNJ. Full-wave finite-difference frequency-domain (FDFD) simulations demonstrate robust lateral and axial steering with subwavelength confinement and low sidelobes. A parametric study of microelement geometries shows that nanojet formation is largely insensitive to moderate boundary variations, with simple shapes providing competitive performance. Robustness to fabrication and alignment errors is confirmed via uncertainty analysis.

physics.optics↗

arXiv2026

Photonic nanojets as emergent free-space power flux funnels

A reduced local field model derived from full-wave electromagnetic simulations shows that photonic nanojet formation corresponds to an emergent mesoscopic funnel of propagating power flux sustained by an effective free-space transverse mode structure. This interpretation moves beyond purely geometric-optics or interference-based explanations by identifying a self-consistent redistribution of phase gradients and effective longitudinal wavenumber near the nanojet waist. The model quantitatively captures characteristic nanojet morphology, including the formation and local structure of the jet waist. It also yields a geometry-independent lower bound on the nanojet waist, linking transverse confinement to the effective axial wavenumber through an explicit trade-off. The model establishes a direct connection between full-wave Maxwell fields and a reduced free-space oscillator description, yielding new physical insight into nanojet confinement and suggesting design principles for nanojet-assisted imaging, lithography, and subwavelength field localization.

physics.optics↗

arXiv2026

Interpolating between Tikhonov regularization and spectral cutoff

Regularizing a linear ill-posed operator equation can be achieved by manipulating the spectrum of the operator's pseudo-inverse. Tikhonov regularization and spectral cutoff are well-known techniques within this category. This paper introduces an interpolating formula that defines a one-parameter family of regularizations, where Tikhonov and spectral cutoff methods are represented as limiting cases. By adjusting the interpolating parameter taking into account the specific operator equation under consideration, it is possible to mitigate the limitations associated with both Tikhonov and spectral cutoff regularizations. The proposed approach is demonstrated through numerical simulations in the fields of signal and image processing.

math.NA↗

arXiv2025

Multi-resolution deconvolution

We extend the classical deconvolution framework in Rn to the case with a pseudodifferential-like solution operator with a symbol depending on both the base and cotangent variable. Our framework enables deconvolution with spatially varying resolution while maintaining a set global stability, and it additionally allows rather general distributional convolution kernels. We provide consistency, convergence and stability results, as well as convergence rates. Finally, we include numerical examples supporting our results and demonstrating advantages of the generalized framework.

math.SP↗

arXiv2025

Rethinking photonic nanojets: a new definition and design paradigm

We propose a rigorous, physically interpretable, and quantifiable definition of the photonic nanojet (PNJ). This framework resolves longstanding ambiguities in measuring PNJ dimensions and leverages an optimal mass transport-based metric to quantify PNJ quality. Building on this metric, we develop a PNJ steering methodology that requires no opto-mechanical intervention, relying solely on phase-only illumination modulation.

physics.optics↗

arXiv2025

Localization Coefficients of Functions with Applications in Partial Differential Equations

We identify shortcomings in two popular measures of localization of functions: the $L^p-L^q$ participation ratio and the mass concentration comparison. We then introduce a novel localization measure for functions on bounded subsets of $\mathbf{R}^d$, $d=1,2,3,\dots$, based on a Wasserstein metric. For efficient computation, we prove the equality of this measure with a suitable Sobolev norm in dimension one. We demonstrate our approach by numerical experiments in one and two dimensions. Finally, we discuss and mitigate challenges arising from boundary effects.

math.AP↗

arXiv2024

Estimates on the stability constant for the truncated Fourier transform

In this paper we are interested in the inverse problem of recovering a compact supported function from its truncated Fourier transform. We derive new Lipschitz stability estimates for the inversion in terms of the truncation parameter. The obtained results show that the Lipschitz constant is of order one when the truncation parameter is larger than the spatial frequency of the function, and it grows exponentially when the truncation parameter tends to zero. Finally, we present some numerical examples of reconstruction of a compactly supported function from its noisy truncated Fourier transform. The numerical illustrations validate our theoretical results.

math.NA↗

arXiv2023

On the series solutions of integral equations in scattering

We study the validity of the Neumann or Born series approach in solving the Helmholtz equation and coefficient identification in related inverse scattering problems. Precisely, we derive a sufficient and necessary condition under which the series is strongly convergent. We also investigate the rate of convergence of the series. The obtained condition is optimal and it can be much weaker than the traditional requirement for the convergence of the series. Our approach makes use of reduction space techniques proposed by Suzuki \cite{Suzuki-1976}. Furthermore we propose an interpolation method that allows the use of the Neumann series in all cases. Finally, we provide several numerical tests with different medium functions and frequency values to validate our theoretical results.

math.NA↗

arXiv2023

Localization and the landscape function for regular Sturm-Liouville operators

We consider the localization in the eigenfunctions of regular Sturm-Liouville operators. After deriving non-asymptotic and asymptotic lower and upper bounds on the localization coefficient of the eigenfunctions, we characterize the landscape function in terms of the first eigenfunction. Several numerical experiments are provided to illustrate the obtained theoretical results.

math.CA↗

arXiv2022

Optimal design of photonic nanojets under uncertainty

Photonic nanojets (PNJs) have promising applications as optical probes in super-resolution optical microscopy, Raman microscopy, as well as fluorescence microscopy. In this work, we consider optimal design of PNJs using a heterogeneous lens refractive index with a fixed lens geometry and uniform plane wave illumination. In particular, we consider the presence of manufacturing error of heterogeneous lens, and propose a computational framework of Optimization Under Uncertainty (OUU) for robust optimal design of PNJ. We formulate a risk-averse stochastic optimization problem with the objective to minimize both the mean and the variance of a target function, which is constrained by the Helmholtz equation that governs the 2D transverse electric (2D TE) electromagnetic field in a neighborhood of the lens. The design variable is taken as a spatially-varying field variable, where we use a finite element method for its discretization, impose a total variation penalty to promote its sparsity, and employ an adjoint-based BFGS method to solve the resulting high-dimensional optimization problem. We demonstrate that our proposed OUU computational framework can achieve more robust optimal design than a deterministic optimization scheme to significantly mitigate the impact of manufacturing uncertainty.

cs.CE↗

arXiv2022

Localization of moving sources: uniqueness, stability, and Bayesian inference

We consider the subsonic moving point source problem for the scalar wave equation in $\pmb{R}^3$, proving a regularity result for the direct problem, and uniqueness and stability results for the inverse problem. We then present and investigate numerically a Bayesian framework for the inference of the source trajectory and intensity from wave field measurements. The framework employs Gaussian process priors, the pre-conditioned Crank-Nicholson scheme with Markov Chain Monte Carlo sampling, and conditioning on functionals to include prior information on the source trajectory.

math.AP↗

arXiv2021

On the transfer of information in multiplier equations

We derive spectral width estimates for traces of tempered solutions of a large class of multiplier equations in $\mathbf{R}^n$. The estimates are uniform for solutions up to a given order. In the process, we find a rather explicit expression for a tempered fundamental solution of a multiplier. We successfully verify our spectral width estimates against numerical results in several scenarios involving the inhomogeneous Helmholtz equation in $\mathbf{R}^n$ with $n=1,\dots,9$. Our main result is directly applicable in the stability analysis of solutions of inverse source problems.

math.AP↗

arXiv2020

Low-dimensional offshore wave input for extreme event quantification

In offshore engineering design, nonlinear wave models are often used to propagate stochastic waves from an input boundary to the location of an offshore structure. Each wave realization is typically characterized by a high-dimensional input time series, and a reliable determination of the extreme events is associated with substantial computational effort. As the sea depth decreases, extreme events become more difficult to evaluate. We here construct a low-dimensional characterization of the candidate input time series to circumvent the search for extreme wave events in a high-dimensional input probability space. Each wave input is represented by a unique low-dimensional set of parameters for which standard surrogate approximations, such as Gaussian processes, can estimate the short-term exceedance probability efficiently and accurately. We demonstrate the advantages of the new approach with a simple shallow-water wave model based on the Korteweg-de Vries equation for which we can provide an accurate reference solution based on the simple Monte Carlo method. We furthermore apply the method to a fully nonlinear wave model for wave propagation over a sloping seabed. The results demonstrate that the Gaussian process can learn accurately the tail of the heavy-tailed distribution of the maximum wave crest elevation based on only $1.7\%$ of the required Monte Carlo evaluations.

physics.data-an↗

arXiv2020

Estimation of Failure Probabilities via Local Subset Approximations

We here consider the subset simulation method which approaches a failure event using a decreasing sequence of nested intermediate failure events. The method resembles importance sampling, which actively explores a probability space by conditioning the next evaluation on the previous evaluations using a Markov chain Monte Carlo (MCMC) algorithm. A Markov chain typically requires many steps to estimate the target distribution, which is impractical with expensive numerical models. Therefore, we propose to approximate each step of a Markov chain locally with Gaussian process (GP) regression. Benchmark examples of reliability analysis show that local approximations significantly improve overall efficiency of subset simulation. They reduce the number of expensive limit-state evaluations by over $80\%$. However, GP regression becomes computationally impractical with increasing dimension. Therefore, to make our use of a GP feasible, we employ the partial least squares (PLS) regression, a gradient-free reduction method, locally to explore and utilize a low-dimensional subspace within a Markov chain. Numerical experiments illustrate a significant computational gain with maintained sufficient accuracy.

stat.CO↗

arXiv2019

Active-Subspace Analysis of Exceedance Probability for Shallow-Water Waves

We model shallow-water waves using a one-dimensional Korteweg-de Vries equation with the wave generation parameterized by random wave amplitudes for a predefined sea state. These wave amplitudes define the high-dimensional stochastic input vector for which we estimate the short-term wave crest exceedance probability at a reference point. For this high-dimensional and complex problem, most reliability methods fail, while Monte Carlo methods become impractical due to the slow convergence rate. Therefore, first within offshore applications, we employ the dimensionality reduction method called \textit{Active-Subspace Analysis}. This method identifies a low-dimensional subspace of the input space that is most significant to the input-output variability. We exploit this to efficiently train a Gaussian process that models the maximum 10-minute crest elevation at the reference point, and to thereby efficiently estimate the short-term wave crest exceedance probability. The active low-dimensional subspace for the Korteweg-de Vries model also exposes the expected incident wave groups associated with extreme waves and loads. Our results show the advantages and the effectiveness of the active-subspace analysis against the Monte Carlo implementation for offshore applications.

physics.flu-dyn↗

arXiv2018

Explicit tight bounds on the stably recoverable information for the inverse source problem

For the inverse source problem with the two-dimensional Helmholtz equation, the singular values of the 'source-to-near field' forward operator reveal a sharp frequency cut-off in the stably recoverable information on the source. We prove and numerically validate an explicit, tight lower bound for the spectral location of this cut-off. We also conjecture and support numerically a tight upper bound for the cut-off. The bounds are expressed in terms of zeros of Bessel functions of the first and second kind.

math.AP↗

arXiv2018

Stable source reconstruction from a finite number of measurements in the Multi-frequency Inverse Source Problem

We consider the multi-frequency inverse source problem for the scalar Helmholtz equation in the plane. The goal is to reconstruct the source term in the equation from measurements of the solution on a surface outside the support of the source. We study the problem in a certain finite dimensional setting: From measurements made at a finite set of frequencies we uniquely determine and reconstruct sources in a subspace spanned by finitely many Fourier-Bessel functions. Further, we obtain a constructive criterion for identifying a minimal set of measurement frequencies sufficient for reconstruction, and under an additional, mild assumption, the reconstruction method is shown to be stable. Our analysis is based on a singular value decomposition of the source-to-measurement forward operators and the distribution of positive zeros of the Bessel functions of the first kind. The reconstruction method is implemented numerically and our theoretical findings are supported by numerical experiments.

math.NA↗