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We prove that the Sinkhorn algorithm converges at a rate of $O(k^{-1} \log k)$ in $\ell_1$-norm marginal error, in the asymptotically scalable case. This almost closes the gap between the lower bound $\Omega(k^{-1})$ (Qu et al., 2025) and…
SDE driven by an $\alpha $-stable process, $\alpha \in \lbrack 1,2),$ with Lipshitz continuous coefficient and $\beta $-H\"older drift is considered. The existence and uniqueness of a strong solution is proved when $\beta >1-\alpha /2$ by…
Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\Omega)$, and optimizing functionals arising from some…
This paper considers the approximation of spatial convolution with a given radial integral kernel. Previous studies have demonstrated that approximating spatial convolution using a system of partial differential equations (PDEs) can…
We derive optimal rates of convergence in the supremum norm for estimating the H\"older-smooth mean function of a stochastic process which is repeatedly and discretely observed with additional errors at fixed, multivariate, synchronous…
We present a new method based on functional tensor decomposition and dynamic tensor approximation to compute the solution of a high-dimensional time-dependent nonlinear partial differential equation (PDE). The idea of dynamic approximation…
We study function estimation in the empirical Bayes setting for Poisson and normal means. Specifically, given observations $Y_i\sim f(\cdot; \theta_i)$ with latent parameters $\theta_i\sim \pi$, the goal is to estimate…
Our main interest in this paper is to study some approximation problems for classes of functions with mixed smoothness. We use technique, based on a combination of results from hyperbolic cross approximation, which were obtained in 1980s --…
Using geometric considerations, we provide a clear derivation of the integral representation for the error function, known as the Craig formula. We calculate the corresponding power series expansion and prove the convergence. The same…
It is known that a subharmonic function of finite order $\rho$ can be approximated by the logarithm of the modulus of an entire function at the point $z$ outside an exceptional set up to $C\log|z|$. In this article we prove that if such an…
We consider the asymptotic evaluation of the integral transform $\int_0^\infty f(x) \, \sin^n(\lambda x)/x^n \,\text{d} x$ of an exponential type function $f(x)$ of type $\tau>0$, for large values of the parameter $\lambda$, where $n$ is a…
This paper proposes a robust approximation method for solving chance constrained optimization (CCO) of polynomials. Assume the CCO is defined with an individual chance constraint that is affine in the decision variables. We construct a…
This work deals with the numerical solution of systems of oscillatory second-order differential equations which often arise from the semi-discretization in space of partial differential equations. Since these differential equations exhibit…
Stochastic optimization algorithms with variance reduction have proven successful for minimizing large finite sums of functions. Unfortunately, these techniques are unable to deal with stochastic perturbations of input data, induced for…
In this paper, we derive a strong convergence rate of spatial finite difference approximations for both focusing and defocusing stochastic cubic Schr\"odinger equations driven by a multiplicative $Q$-Wiener process. Beyond the uniform…
We pose the problem of the optimal approximation of a given nonnegative signal $y_t$ with the scalar autoconvolution $(x*x)_t$ of a nonnegative signal $x_t$, where $x_t$ and $y_t$ are signals of equal length. The $\mathcal{I}$-divergence…
We derive a three-term recurrence relation for computing the polynomial of best approximation in the uniform norm to $x^{-1}$ on a finite interval with positive endpoints. As application, we consider two-level methods for scalar elliptic…
In this article, we describe a function fitting method that has potential applications in machine learning and also prove relevant theorems. The described function fitting method is a convex minimization problem and can be solved using a…
The (1+1)-evolution strategy (ES) with success-based step-size adaptation is analyzed on a general convex quadratic function and its monotone transformation, that is, $f(x) = g((x - x^*)^\mathrm{T} H (x - x^*))$, where…
It is known that, if $S(z)$ is a non-constant, singular inner function defined on the unit disk, then $\min_{|z|\le r}|S(z)|\to0$ as $r\to1^-$. We show that the convergence may be arbitrarily slow.