Related papers: Approximation by convolutions with probability den…
The method of maximum entropy has proven to be a rather powerful way to solve the inverse problem consisting of determining a probability density $f_S(s)$ on $[0,\infty)$ from the knowledge of the expected value of a few generalized…
Hypercomplex Fourier transforms are increasingly used in signal processing for the analysis of higher-dimensional signals such as color images. A main stumbling block for further applications, in particular concerning filter design in the…
We introduce another new type of combinations of Bernstein operators in this paper, which can be used to approximate the functions with inner singularities. The direct and inverse results of the weighted approximation of this new type…
This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…
Hessian operators arising in inverse problems governed by partial differential equations (PDEs) play a critical role in delivering efficient, dimension-independent convergence for both Newton solution of deterministic inverse problems, as…
We introduce a new procedure to select the optimal cutoff parameter for Fourier density estimators that leads to adaptive rate optimal estimators, up to a logarithmic factor. This adaptive procedure applies for different inverse problems.…
Inverse problems arise anywhere we have indirect measurement. As, in general they are ill-posed, to obtain satisfactory solutions for them needs prior knowledge. Classically, different regularization methods and Bayesian inference based…
We construct intrinsic on-and off-diagonal upper and lower estimates for the transition probability density of a L\'evy process in small time. By intrinsic we mean that such estimates reflect the structure of the characteristic exponent of…
Reciprocal transformations mix the role of the dependent and independent variables to achieve simpler versions or even linearized versions of nonlinear PDEs. These transformations help in the identification of a plethora of PDEs available…
The paper presents transfer functions for limited memory time-invariant linear integral predictors for continuous time processes such that the corresponding predicting kernels have bounded support. It is shown that processes with…
We present a framework which enables the analysis of dynamic inverse problems for wave phenomena that are modeled through second-order hyperbolic PDEs. This includes well-posedness and regularity results for the forward operator in an…
We introduce the concepts of the Fourier transform and convolution generated by an arbitrary restriction of the differentiation operator in the space $L_{2}(0,b).$ In contrast to the classical convolution, the introduced convolution…
The main object of this paper is to construct new Durrmeyer type operators which have better features than the classical one. Some results concerning the rate of convergence and asymptotic formulas of the new operator are given. Finally,…
In metric of spaces $L_{s}, \ 1< s\leq\infty$, we obtain exact order estimates of best approximations and approximations by Fourier sums of classes of convolutions the periodic functions that belong to unit ball of space $L_{1}$, with…
We provide a rigorous convergence proof demonstrating that the well-known semi-analytical Fourier cosine (COS) formula for the inverse Fourier transform of continuous probability distributions can be extended to discrete probability…
We use the method of similar operators to study a mixed problem for a differential equation with an involution and an operator-valued potential function. The differential operator defined by the equation is transformed into a similar…
The inverse of a large matrix can often be accurately approximated by a polynomial of degree significantly lower than the order of the matrix. The iteration polynomial generated by a run of the GMRES algorithm is a good candidate, and its…
Inverse problems involving systems of partial differential equations (PDEs) with many measurements or experiments can be very expensive to solve numerically. In a recent paper we examined dimensionality reduction methods, both stochastic…
This paper introduces a computationally efficient algorithm in system theory for solving inverse problems governed by linear partial differential equations (PDEs). We model solutions of linear PDEs using Gaussian processes with priors…
Diffusion models have emerged as powerful generative tools with applications in computer vision and scientific machine learning (SciML), where they have been used to solve large-scale probabilistic inverse problems. Traditionally, these…