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Solving multiscale diffusion problems is often computationally expensive due to the spatial and temporal discretization challenges arising from high-contrast coefficients. To address this issue, a partially explicit temporal splitting…
We reexamine Smale's alpha theory as a way to certify a numerical solution to an analytic system. For a given point and a system, Smale's alpha theory determines whether Newton's method applied to this point shows the quadratic convergence…
In the first part of this work, we develop a novel scheme for solving nonparametric regression problems. That is the approximation of possibly low regular and noised functions from the knowledge of their approximate values given at some…
Iterative optimization algorithms depend on access to information about the objective function. In a differentiable programming framework, this information, such as gradients, can be automatically derived from the computational graph. We…
Functional time series have become an integral part of both functional data and time series analysis. Important contributions to methodology, theory and application for the prediction of future trajectories and the estimation of functional…
Given a smooth function $f$, we develop a general approach to turn Monte Carlo samples with expectation $m$ into an unbiased estimate of $f(m)$. Specifically, we develop estimators that are based on randomly truncating the Taylor series…
In this paper we introduce a new class of diffeomorphic smoothers based on general spline smoothing techniques and on the use of some tools that have been recently developed in the context of image warping to compute smooth diffeomorphisms.…
Pseudospectral numerical schemes for solving the Dirac equation in general static curved space are derived using a pseudodifferential representation of the Dirac equation along with a simple Fourier-basis technique. Owing to the presence of…
Compressed Sensing suggests that the required number of samples for reconstructing a signal can be greatly reduced if it is sparse in a known discrete basis, yet many real-world signals are sparse in a continuous dictionary. One example is…
A conformable time-scale fractional calculus of order $\alpha \in ]0,1]$ is introduced. The basic tools for fractional differentiation and fractional integration are then developed. The Hilger time-scale calculus is obtained as a particular…
The non-stationary evolution of observable quantities in complex systems can frequently be described as a juxtaposition of quasi-stationary spells. Given that standard theoretical and data analysis approaches usually rely on the assumption…
The paper discusses the relationships between electrical and affine differential geometry quantities, establishing a link between frequency and time derivatives of voltage, through the utilization of affine geometric invariants. Based on…
Stochastic optimization naturally arises in machine learning. Efficient algorithms with provable guarantees, however, are still largely missing, when the objective function is nonconvex and the data points are dependent. This paper studies…
We introduce new method of optimization for finding free parameters of affine iterated function systems (IFS), which are used for fractal approximation. We provide the comparison of effectiveness of fractal and quadratic types of…
In order to completely eliminate, or greatly reduce the number of phase wraps in 2D wrapped phase map, Gdeisat et al. proposed an algorithm, which uses shifting the spectrum towards the origin. But the spectrum can be shifted only by an…
Being able to capture the characteristics of a time series with a feature vector is a very important task with a multitude of applications, such as classification, clustering or forecasting. Usually, the features are obtained from linear…
The Fast Fourier Transform (FFT) is a fundamental tool for signal analysis, widely used across various fields. However, traditional FFT methods encounter challenges in adjusting the frequency bin interval, which may impede accurate spectral…
Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately…
In many applications and physical phenomena, bivariate signals are polarized, i.e. they trace an elliptical trajectory over time when viewed in the 2D planes of their two components. The smooth evolution of this elliptical trajectory,…
We design a sublinear-time approximation algorithm for quadratic function minimization problems with a better error bound than the previous algorithm by Hayashi and Yoshida (NIPS'16). Our approximation algorithm can be modified to handle…