Asymptotic Behavior of Solutions of Integral Equations with Homogeneous Kernels

The multidimensional integral equation of second kind with a homogeneous of degree (−n) kernel is considered. The special class of continuous functions with a given asymptotic behavior in the neighborhood of zero is defined. It is proved that, if the free term of the integral equation belongs to this class and the equation itself is solvable, then its solution also belongs to this class. To solve this problem, a special research technique is used. The above-mentioned technique is based on the decomposition of both the solution and the free term in spherical harmonics.

ON POINTWISE APPROXIMATION OF FUNCTIONS OF SEVERAL VARIABLES BY SINGULAR INTEGRALS

We prove a theorem on weighted pointwise convergence of \ multidimensional integral operators with radial kernels to generating function of several variables, which are in general non-integrable in $n$-dimensional Euclidean space $E_{n}$ in the sense of Lebesgue$.$ Main result holds at almost every point of $E_{n}.$

Designed quadrature to approximate integrals in maximum simulated likelihood estimation

Maximum simulated likelihood estimation of mixed multinomial logit models requires evaluation of a multidimensional integral. Quasi-Monte Carlo (QMC) methods such as Halton sequences and modified Latin hypercube sampling are workhorse methods for integral approximation. Earlier studies explored the potential of sparse grid quadrature (SGQ), but SGQ suffers from negative weights. As an alternative to QMC and SGQ, we looked into the recently developed designed quadrature (DQ) method. DQ requires fewer nodes to get the same level of accuracy as of QMC and SGQ, is as easy to implement, ensures positivity of weights, and can be created on any general polynomial space. We benchmarked DQ against QMC in a Monte Carlo and an empirical study. DQ outperformed QMC in all considered scenarios, is practice-ready and has potential to become the workhorse method for integral approximation.

Up and down deconvolution in complex geological scenarios

The redatuming approach, often referred to as up-down deconvolution, is well-known and applied to remove water-layer and source-signature effects in seabed seismic surveys. The upgoing wavefield can be expressed as the multidimensional convolution of the downgoing wavefield with the earth’s reflectivity. Consequently, deconvolving the downgoing wavefield from the upgoing wavefield, gives us the earth’s reflectivity response. The deconvolution process requires solving a multidimensional integral equation but, in a laterally invariant medium, after that wavefields are decomposed into plane-wave components, deconvolution can be enormously simplified if performed as a spectral division in the Fourier or Radon domain. It has been experimentally observed that deconvolution carried out one plane-wave component at a time gives good results, even in the presence of complex subsurface structures, provided that the seabed is relatively flat. When this geological condition is not satisfied, the same problem can be formulated in terms of interferometric redatuming using multidimensional deconvolution, where the integral equation solution is achieved by introducing the point-spread function concept. We present a methodology based on numerical simulations to determine when the integral equations associated with the problem of up-down deconvolution can be solved under the assumption of shift-invariant wavefields and when it requires multidimensional deconvolution. In the latter case, we propose a regularized inverse procedure that mitigates the numerical problems due to the typically ill-posed nature of the inversion and that, combined with an interpolation strategy for the downgoing, enables the application of multidimensional deconvolution within the range of sampling scenarios considered so far. We apply this methodology to synthetic data, and we discuss on the potential to extend up-down deconvolution to a broader range of geological conditions.

Optimal Sensor Placement for Reliable Virtual Sensing Using Modal Expansion and Information Theory

A framework for optimal sensor placement (OSP) for virtual sensing using the modal expansion technique and taking into account uncertainties is presented based on information and utility theory. The framework is developed to handle virtual sensing under output-only vibration measurements. The OSP maximizes a utility function that quantifies the expected information gained from the data for reducing the uncertainty of quantities of interest (QoI) predicted at the virtual sensing locations. The utility function is extended to make the OSP design robust to uncertainties in structural model and modeling error parameters, resulting in a multidimensional integral of the expected information gain over all possible values of the uncertain parameters and weighted by their assigned probability distributions. Approximate methods are used to compute the multidimensional integral and solve the optimization problem that arises. The Gaussian nature of the response QoI is exploited to derive useful and informative analytical expressions for the utility function. A thorough study of the effect of model, prediction and measurement errors and their uncertainties, as well as the prior uncertainties in the modal coordinates on the selection of the optimal sensor configuration is presented, highlighting the importance of accounting for robustness to errors and other uncertainties.