Computational tools for characterizing electromagnetic scattering from objects with uncertain shapes are needed in various applications ranging from remote sensing at microwave frequencies to Raman spectroscopy at optical frequencies. Often, such computational tools use the Monte Carlo (MC) method to sample a parametric space describing geometric uncertainties. For each sample, which corresponds to a realization of the geometry, a deterministic electromagnetic solver computes the scattered fields. However, for an accurate statistical characterization the number of MC samples has to be large. In this work, to address this challenge, the continuation multilevel Monte Carlo (\CMLMC) method is used together with a surface integral equation solver.
The \CMLMC method optimally balances statistical errors due to sampling of
the parametric space, and numerical errors due to the discretization of the geometry using a hierarchy of discretizations, from coarse to fine.
The number of realizations of finer discretizations can be kept low, with most samples
computed on coarser discretizations to minimize computational cost.
Consequently, the total execution time is significantly reduced, in comparison to the standard MC scheme.
Computation of electromagnetic fields scattered from dielectric objects of un...Alexander Litvinenko
We develop fast and efficient stochastic methods for characterizing scattering
from objects of uncertain shapes. This is highly needed in the
fields of electromagnetics, optics, and photonics.
The continuation multilevel Monte Carlo (CMLMC) method is
used together with a surface integral equation solver. The
CMLMC method optimally balances statistical errors due to
sampling of the parametric space, and numerical errors due
to the discretization of the geometry using a hierarchy of
discretizations, from coarse to fine. The number of realizations
of finer discretizations can be kept low, with most samples
computed on coarser discretizations to minimize computational
work. Consequently, the total execution time is significantly
reduced, in comparison to the standard MC scheme.
Computation of electromagnetic fields scattered from dielectric objects of un...Alexander Litvinenko
Computational tools for characterizing electromagnetic scattering from objects with uncertain shapes are needed in various applications ranging from remote sensing at microwave frequencies to Raman spectroscopy at optical frequencies. Often, such computational tools use the Monte Carlo (MC) method to sample a parametric space describing geometric uncertainties. For each sample, which corresponds to a realization of the geometry, a deterministic electromagnetic solver computes the scattered fields. However, for an accurate statistical characterization the number of MC samples has to be large. In this work, to address this challenge, the continuation multilevel Monte Carlo (\CMLMC) method is used together with a surface integral equation solver.
The \CMLMC method optimally balances statistical errors due to sampling of
the parametric space, and numerical errors due to the discretization of the geometry using a hierarchy of discretizations, from coarse to fine.
The number of realizations of finer discretizations can be kept low, with most samples
computed on coarser discretizations to minimize computational cost.
Consequently, the total execution time is significantly reduced, in comparison to the standard MC scheme.
Computation of electromagnetic_fields_scattered_from_dielectric_objects_of_un...Alexander Litvinenko
Tools for electromagnetic scattering from objects with uncertain shapes are needed in various applications.
We develop numerical methods for predicting radar and scattering cross sections (RCS and SCS) of complex targets.
To reduce cost of Monte Carlo (MC) we offer modified multilevel MC (CMLMC) method.
Computation of Electromagnetic Fields Scattered from Dielectric Objects of Un...Alexander Litvinenko
We research how input uncertainties in the geometry shape propagate through the electromagnetic model to electro-magnetic fields. We use multi-level Monte Carlo methods.
Computation of electromagnetic fields scattered from dielectric objects of un...Alexander Litvinenko
We develop fast and efficient stochastic methods for characterizing scattering
from objects of uncertain shapes. This is highly needed in the
fields of electromagnetics, optics, and photonics.
The continuation multilevel Monte Carlo (CMLMC) method is
used together with a surface integral equation solver. The
CMLMC method optimally balances statistical errors due to
sampling of the parametric space, and numerical errors due
to the discretization of the geometry using a hierarchy of
discretizations, from coarse to fine. The number of realizations
of finer discretizations can be kept low, with most samples
computed on coarser discretizations to minimize computational
work. Consequently, the total execution time is significantly
reduced, in comparison to the standard MC scheme.
Computation of electromagnetic fields scattered from dielectric objects of un...Alexander Litvinenko
Computational tools for characterizing electromagnetic scattering from objects with uncertain shapes are needed in various applications ranging from remote sensing at microwave frequencies to Raman spectroscopy at optical frequencies. Often, such computational tools use the Monte Carlo (MC) method to sample a parametric space describing geometric uncertainties. For each sample, which corresponds to a realization of the geometry, a deterministic electromagnetic solver computes the scattered fields. However, for an accurate statistical characterization the number of MC samples has to be large. In this work, to address this challenge, the continuation multilevel Monte Carlo (\CMLMC) method is used together with a surface integral equation solver.
The \CMLMC method optimally balances statistical errors due to sampling of
the parametric space, and numerical errors due to the discretization of the geometry using a hierarchy of discretizations, from coarse to fine.
The number of realizations of finer discretizations can be kept low, with most samples
computed on coarser discretizations to minimize computational cost.
Consequently, the total execution time is significantly reduced, in comparison to the standard MC scheme.
Computation of electromagnetic_fields_scattered_from_dielectric_objects_of_un...Alexander Litvinenko
Tools for electromagnetic scattering from objects with uncertain shapes are needed in various applications.
We develop numerical methods for predicting radar and scattering cross sections (RCS and SCS) of complex targets.
To reduce cost of Monte Carlo (MC) we offer modified multilevel MC (CMLMC) method.
Computation of Electromagnetic Fields Scattered from Dielectric Objects of Un...Alexander Litvinenko
We research how input uncertainties in the geometry shape propagate through the electromagnetic model to electro-magnetic fields. We use multi-level Monte Carlo methods.
TWO DIMENSIONAL MODELING OF NONUNIFORMLY DOPED MESFET UNDER ILLUMINATIONVLSICS Design
A two dimensional numerical model of an optically gated GaAs MESFET with non uniform channel doping has been developed. This is done to characterize the device as a photo detector. First photo induced voltage (Vop) at the Schottky gate is calculated for estimating the channel profile. Then Poisson’s equation for the device is solved numerically under dark and illumination condition. The paper aims at developing the MESFET 2-D model under illumination using Monte Carlo Finite Difference method. The results discuss about the optical potential developed in the device, variation of channel potential under different biasing and illumination and also about electric fields along X and Y directions. The Cgs under different illumination is also calculated. It has been observed from the results that the characteristics of the device are strongly influenced by the incident optical illumination.
Black hole formation by incoming electromagnetic radiationXequeMateShannon
I revisit a known solution of the Einstein field equations to show that it describes the formation of non-spherical black holes by the collapse of pure electromagnetic monochromatic radiation. Both positive and negative masses are feasible without ever violating the dominant energy condition. The solution can also be used to model the destruction of naked singularities and the evaporation of white holes by emission or reception of light.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Probabilistic Matrix Factorization (PMF)
Bayesian Probabilistic Matrix Factorization (BPMF) using
Markov Chain Monte Carlo (MCMC)
BPMF using MCMC – Overall Model
BPMF using MCMC – Gibbs Sampling
Robust Image Denoising in RKHS via Orthogonal Matching PursuitPantelis Bouboulis
We present a robust method for the image denoising task based on kernel ridge regression and sparse modeling. Added noise is assumed to consist of two parts. One part is impulse noise assumed to be sparse (outliers), while the other part is bounded noise. The noisy image is divided into small regions of interest, whose pixels are regarded as points of a two-dimensional surface. A kernel based ridge regression method, whose parameters are selected adaptively, is employed to fit the data, whereas the outliers are detected via the use of the increasingly popular orthogonal matching pursuit (OMP) algorithm. To this end, a new variant of the OMP rationale is employed that has the additional advantage to automatically terminate, when all outliers have been selected.
At this present scenario, the demand of the system capacity is very high in wireless network. MIMO
technology is used from the last decade to provide this requirement for wireless network antenna
technology. MIMO channels are mostly used for advanced antenna array technology. But it is most
important to control the error rate with enhanced system capacity in MIMO for present-day progressive
wireless communication. This paper explores the frame error rate with respect to different path gain of
MIMO channel. This work has been done in different fading scenario and produces a comparative analysis
of MIMO on the basis of those fading models in various conditions. Here, it is to be considered that
modulation technique as QPSK to observe these comparative evaluations for different Doppler frequencies.
From the comparative analysis, minimum amount of frame error rate is viewed for Rician distribution at
LOS path Doppler shift of 0 Hz. At last, this work is concluded with a comparative bit error rate study on
the basis of singular parameters at different SNR levels to produce the system performance for uncoded
QPSK modulation.
Dynamic stiffness and eigenvalues of nonlocal nano beams - new methods for dynamic analysis of nano-scale structures. This lecture gives a review and proposed new techniques.
TWO DIMENSIONAL MODELING OF NONUNIFORMLY DOPED MESFET UNDER ILLUMINATIONVLSICS Design
A two dimensional numerical model of an optically gated GaAs MESFET with non uniform channel doping has been developed. This is done to characterize the device as a photo detector. First photo induced voltage (Vop) at the Schottky gate is calculated for estimating the channel profile. Then Poisson’s equation for the device is solved numerically under dark and illumination condition. The paper aims at developing the MESFET 2-D model under illumination using Monte Carlo Finite Difference method. The results discuss about the optical potential developed in the device, variation of channel potential under different biasing and illumination and also about electric fields along X and Y directions. The Cgs under different illumination is also calculated. It has been observed from the results that the characteristics of the device are strongly influenced by the incident optical illumination.
Black hole formation by incoming electromagnetic radiationXequeMateShannon
I revisit a known solution of the Einstein field equations to show that it describes the formation of non-spherical black holes by the collapse of pure electromagnetic monochromatic radiation. Both positive and negative masses are feasible without ever violating the dominant energy condition. The solution can also be used to model the destruction of naked singularities and the evaporation of white holes by emission or reception of light.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Probabilistic Matrix Factorization (PMF)
Bayesian Probabilistic Matrix Factorization (BPMF) using
Markov Chain Monte Carlo (MCMC)
BPMF using MCMC – Overall Model
BPMF using MCMC – Gibbs Sampling
Robust Image Denoising in RKHS via Orthogonal Matching PursuitPantelis Bouboulis
We present a robust method for the image denoising task based on kernel ridge regression and sparse modeling. Added noise is assumed to consist of two parts. One part is impulse noise assumed to be sparse (outliers), while the other part is bounded noise. The noisy image is divided into small regions of interest, whose pixels are regarded as points of a two-dimensional surface. A kernel based ridge regression method, whose parameters are selected adaptively, is employed to fit the data, whereas the outliers are detected via the use of the increasingly popular orthogonal matching pursuit (OMP) algorithm. To this end, a new variant of the OMP rationale is employed that has the additional advantage to automatically terminate, when all outliers have been selected.
At this present scenario, the demand of the system capacity is very high in wireless network. MIMO
technology is used from the last decade to provide this requirement for wireless network antenna
technology. MIMO channels are mostly used for advanced antenna array technology. But it is most
important to control the error rate with enhanced system capacity in MIMO for present-day progressive
wireless communication. This paper explores the frame error rate with respect to different path gain of
MIMO channel. This work has been done in different fading scenario and produces a comparative analysis
of MIMO on the basis of those fading models in various conditions. Here, it is to be considered that
modulation technique as QPSK to observe these comparative evaluations for different Doppler frequencies.
From the comparative analysis, minimum amount of frame error rate is viewed for Rician distribution at
LOS path Doppler shift of 0 Hz. At last, this work is concluded with a comparative bit error rate study on
the basis of singular parameters at different SNR levels to produce the system performance for uncoded
QPSK modulation.
Dynamic stiffness and eigenvalues of nonlocal nano beams - new methods for dynamic analysis of nano-scale structures. This lecture gives a review and proposed new techniques.
Two Dimensional Modeling of Nonuniformly Doped MESFET Under IlluminationVLSICS Design
A two dimensional numerical model of an optically gated GaAs MESFET with non uniform channel doping has been developed. This is done to characterize the device as a photo detector. First photo induced voltage (Vop) at the Schottky gate is calculated for estimating the channel profile. Then Poisson’s equation for the device is solved numerically under dark and illumination condition. The paper aims at developing the MESFET 2-D model under illumination using Monte Carlo Finite Difference method. The results discuss about the optical potential developed in the device, variation of channel potential under different biasing and illumination and also about electric fields along X and Y directions. The Cgs under different illumination is also calculated. It has been observed from the results that the characteristics of the device are strongly influenced by the incident optical illumination.
Sparse data formats and efficient numerical methods for uncertainties in nume...Alexander Litvinenko
Description of methodologies and overview of numerical methods, which we used for modeling and quantification of uncertainties in numerical aerodynamics
Ill-posedness formulation of the emission source localization in the radio- d...Ahmed Ammar Rebai PhD
To contact the authors : tarek.salhi@gmail.com and ahmed.rebai2@gmail.com
In the field of radio detection in astroparticle physics, many studies have shown the strong dependence of the solution of the radio-transient sources localization problem (the radio-shower time of arrival on antennas) such solutions are purely numerical artifacts. Based on a detailed analysis of some already published results of radio-detection experiments like : CODALEMA 3 in France, AERA in Argentina and TREND in China, we demonstrate the ill-posed character of this problem in the sens of Hadamard. Two approaches have been used as the existence of solutions degeneration and the bad conditioning of the mathematical formulation problem. A comparison between experimental results and simulations have been made, to highlight the mathematical studies. Many properties of the non-linear least square function are discussed such as the configuration of the set of solutions and the bias.
This poster was created in LaTeX on a Dell Inspiron laptop with a Linux Fedora Core 4 operating system. The background image and the animation snapshots are dxf meshes of elastic waveform solutions, rendered on a Windows machine using 3D Studio Max.
A CPW-fed Rectangular Patch Antenna for WLAN/WiMAX ApplicationsIDES Editor
This paper presents a CPW fed Rectangular
shaped patch antenna for the frequency 3.42GHz which
falls in WiMAX and 5.25GHz for WLAN applications.
The measured -10dB impedance bandwidth is about
650MHz (2.98GHz-3.63GHz) for WiMAX and 833MHz
(4.95GHz-5.78GHz) for WLAN applications. The effect of
slot width, rectangular patch height, and substrate
dielectric constant have been evaluated. The results of
antenna are simulated by using Zeeland’s MOM based
IE3D tool. Two dimensional radiation patterns with
elevation and azimuth angles, VSWR<2, Return loss of
-24dB and -18dB for WiMAX and WLAN applications,
antenna efficiency about 90%, gain above 3.5dB are
obtained. The compact aperture area of the antenna is
46.2 X 41.66 mm2.
Finite-difference modeling, accuracy, and boundary conditions- Arthur Weglein...Arthur Weglein
This short report gives a brief review on the finite difference modeling method used in MOSRP
and its boundary conditions as a preparation for the Green’s theorem RTM. The first
part gives the finite difference formulae we used and the second part describes the implemented
boundary conditions. The last part, using two examples, points out some impacts of the accuracy
of source fields on the results of modeling.
Response Surface in Tensor Train format for Uncertainty QuantificationAlexander Litvinenko
We apply low-rank Tensor Train format to solve PDEs with uncertain coefficients. First, we approximate uncertain permeability coefficient in TT format, then the operator and then apply iterations to solve stochastic Galerkin system.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Numerical Investigation of Multilayer Fractal FSSIJMER
Numerical investigations are presented for a multilayer frequency selective surface with Koch
fractal (levels 1 and 2) conducting patch elements. The structure investigated is obtained using two FSS
screens separated by an air gap layer. For the proposed investigation were used three different values an
air gap height. The results obtained using the numerical method were compared with other technique and
using the commercial software Ansoft DesignerTM. A good agreement was observed in terms of the
bandwidth.
Similar to Computation of electromagnetic fields scattered from dielectric objects of uncertain shapes using MLMC algorithm (20)
Poster to be presented at Stochastic Numerics and Statistical Learning: Theory and Applications Workshop 2024, Kaust, Saudi Arabia, https://cemse.kaust.edu.sa/stochnum/events/event/snsl-workshop-2024.
In this work we have considered a setting that mimics the Henry problem \cite{Simpson2003,Simpson04_Henry}, modeling seawater intrusion into a 2D coastal aquifer. The pure water recharge from the ``land side'' resists the salinisation of the aquifer due to the influx of saline water through the ``sea side'', thereby achieving some equilibrium in the salt concentration. In our setting, following \cite{GRILLO2010}, we consider a fracture on the sea side that significantly increases the permeability of the porous medium.
The flow and transport essentially depend on the geological parameters of the porous medium, including the fracture. We investigated the effects of various uncertainties on saltwater intrusion. We assumed uncertainties in the fracture width, the porosity of the bulk medium, its permeability and the pure water recharge from the land side. The porosity and permeability were modeled by random fields, the recharge by a random but periodic intensity and the thickness by a random variable. We calculated the mean and variance of the salt mass fraction, which is also uncertain.
The main question we investigated in this work was how well the MLMC method can be used to compute statistics of different QoIs. We found that the answer depends on the choice of the QoI. First, not every QoI requires a hierarchy of meshes and MLMC. Second, MLMC requires stable convergence rates for $\EXP{g_{\ell} - g_{\ell-1}}$ and $\Var{g_{\ell} - g_{\ell-1}}$. These rates should be independent of $\ell$. If these convergence rates vary for different $\ell$, then it will be hard to estimate $L$ and $m_{\ell}$, and MLMC will either not work or be suboptimal. We were not able to get stable convergence rates for all levels $\ell=1,\ldots,5$ when the QoI was an integral as in \eqref{eq:integral_box}. We found that for $\ell=1,\ldots 4$ and $\ell=5$ the rate $\alpha$ was different. Further investigation is needed to find the reason for this. Another difficulty is the dependence on time, i.e. the number of levels $L$ and the number of sums $m_{\ell}$ depend on $t$. At the beginning the variability is small, then it increases, and after the process of mixing salt and fresh water has stopped, the variance decreases again.
The number of random samples required at each level was estimated by calculating the decay of the variances and the computational cost for each level. These estimates depend on the minimisation function in the MLMC algorithm.
To achieve the efficiency of the MLMC approach presented in this work, it is essential that the complexity of the numerical solution of each random realisation is proportional to the number of grid vertices on the grid levels.
We investigated the applicability and efficiency of the MLMC approach to the Henry-like problem with uncertain porosity, permeability and recharge. These uncertain parameters were modelled by random fields with three independent random variables. Permeability is a function of porosity. Both functions are time-dependent, have multi-scale behaviour and are defined for two layers. The numerical solution for each random realisation was obtained using the well-known ug4 parallel multigrid solver. The number of random samples required at each level was estimated by calculating the decay of the variances and the computational cost for each level.
The MLMC method was used to compute the expected value and variance of several QoIs, such as the solution at a few preselected points $(t,\bx)$, the solution integrated over a small subdomain, and the time evolution of the freshwater integral. We have found that some QoIs require only 2-3 mesh levels and samples from finer meshes would not significantly improve the result. Other QoIs require more grid levels.
1. Investigated efficiency of MLMC for Henry problem with
uncertain porosity, permeability, and recharge.
2. Uncertainties are modeled by random fields.
3. MLMC could be much faster than MC, 3200 times faster !
4. The time dependence is challenging.
Remarks:
1. Check if MLMC is needed.
2. The optimal number of samples depends on the point (t;x)
3. An advanced MLMC may give better estimates of L and m`.
Density Driven Groundwater Flow with Uncertain Porosity and PermeabilityAlexander Litvinenko
In this work, we solved the density driven groundwater flow problem with uncertain porosity and permeability. An accurate solution of this time-dependent and non-linear problem is impossible because of the presence of natural uncertainties in the reservoir such as porosity and permeability.
Therefore, we estimated the mean value and the variance of the solution, as well as the propagation of uncertainties from the random input parameters to the solution.
We started by defining the Elder-like problem. Then we described the multi-variate polynomial approximation (\gPC) approach and used it to estimate the required statistics of the mass fraction.
Utilizing the \gPC method allowed us
to reduce the computational cost compared to the classical quasi Monte Carlo method.
\gPC assumes that the output function $\sol(t,\bx,\thetab)$ is square-integrable and smooth w.r.t uncertain input variables $\btheta$.
Many factors, such as non-linearity, multiple solutions, multiple stationary states, time dependence and complicated solvers, make the investigation of the convergence of the \gPC method a non-trivial task.
We used an easy-to-implement, but only sub-optimal \gPC technique to quantify the uncertainty. For example, it is known that by increasing the degree of global polynomials (Hermite, Langange and similar), Runge's phenomenon appears. Here, probably local polynomials, splines or their mixtures would be better. Additionally, we used an easy-to-parallelise quadrature rule, which was also only suboptimal. For instance, adaptive choice of sparse grid (or collocation) points \cite{ConradMarzouk13,nobile-sg-mc-2015,Sudret_sparsePCE,CONSTANTINE12,crestaux2009polynomial} would be better, but we were limited by the usage of parallel methods. Adaptive quadrature rules are not (so well) parallelisable. In conclusion, we can report that: a) we developed a highly parallel method to quantify uncertainty in the Elder-like problem; b) with the \gPC of degree 4 we can achieve similar results as with the \QMC method.
In the numerical section we considered two different aquifers - a solid parallelepiped and a solid elliptic cylinder. One of our goals was to see how the domain geometry influences the formation, the number and the shape of fingers.
Since the considered problem is nonlinear,
a high variance in the porosity may result in totally different solutions; for instance, the number of fingers, their intensity and shape, the propagation time, and the velocity may vary considerably.
The number of cells in the presented experiments varied from $241{,}152$ to $15{,}433{,}728$ for the cylindrical domain and from $524{,}288$ to $4{,}194{,}304$ for the parallelepiped. The maximal number of parallel processing units was $600\times 32$, where $600$ is the number of parallel nodes and $32$ is the number of computing cores on each node. The total computing time varied from 2 hours for the coarse mesh to 24 hours for the finest mesh.
Saltwater intrusion occurs when sea levels rise and saltwater moves onto the land. Usually, this occurs during storms, high tides, droughts, or when saltwater penetrates freshwater aquifers and raises the groundwater table. Since groundwater is an essential nutrition and irrigation resource, its salinization may lead to catastrophic consequences. Many acres of farmland may be lost because they can become too wet or salty to grow crops. Therefore, accurate modeling of different scenarios of saline flow is essential to help farmers and researchers develop strategies to improve the soil quality and decrease saltwater intrusion effects.
Saline flow is density-driven and described by a system of time-dependent nonlinear partial differential equations (PDEs). It features convection dominance and can demonstrate very complicated behavior.
As a specific model, we consider a Henry-like problem with uncertain permeability and porosity.
These parameters may strongly affect the flow and transport of salt.
We consider a class of density-driven flow problems. We are particularly interested in the problem of the salinization of coastal aquifers. We consider the Henry saltwater intrusion problem with uncertain porosity, permeability, and recharge parameters as a test case.
The reason for the presence of uncertainties is the lack of knowledge, inaccurate measurements,
and inability to measure parameters at each spatial or time location. This problem is nonlinear and time-dependent. The solution is the salt mass fraction, which is uncertain and changes in time. Uncertainties in porosity, permeability, recharge, and mass fraction are modeled using random fields. This work investigates the applicability of the well-known multilevel Monte Carlo (MLMC) method for such problems. The MLMC method can reduce the total computational and storage costs. Moreover, the MLMC method runs multiple scenarios on different spatial and time meshes and then estimates the mean value of the mass fraction.
The parallelization is performed in both the physical space and stochastic space. To solve every deterministic scenario, we run the parallel multigrid solver ug4 in a black-box fashion.
We use the solution obtained from the quasi-Monte Carlo method as a reference solution.
We investigated the applicability and efficiency of the MLMC approach for the Henry-like problem with uncertain porosity, permeability, and recharge. These uncertain parameters were modeled by random fields with three independent random variables. The numerical solution for each random realization was obtained using the well-known ug4 parallel multigrid solver. The number of required random samples on each level was estimated by computing the decay of the variances and computational costs for each level. We also computed the expected value and variance of the mass fraction in the whole domain, the evolution of the pdfs, the solutions at a few preselected points $(t,\bx)$, and the time evolution of the freshwater integral value. We have found that some QoIs require only 2-3 of the coarsest mesh levels, and samples from finer meshes would not significantly improve the result. Note that a different type of porosity may lead to a different conclusion.
The results show that the MLMC method is faster than the QMC method at the finest mesh. Thus, sampling at different mesh levels makes sense and helps to reduce the overall computational cost.
Here the interest is mainly to compute characterisations like the entropy,
the Kullback-Leibler divergence, more general $f$-divergences, or other such characteristics based on
the probability density. The density is often not available directly,
and it is a computational challenge to just represent it in a numerically
feasible fashion in case the dimension is even moderately large. It
is an even stronger numerical challenge to then actually compute said characteristics
in the high-dimensional case.
The task considered here was the numerical computation of characterising statistics of
high-dimensional pdfs, as well as their divergences and distances,
where the pdf in the numerical implementation was assumed discretised on some regular grid.
We have demonstrated that high-dimensional pdfs,
pcfs, and some functions of them
can be approximated and represented in a low-rank tensor data format.
Utilisation of low-rank tensor techniques helps to reduce the computational complexity
and the storage cost from exponential $\C{O}(n^d)$ to linear in the dimension $d$, e.g.\
$O(d n r^2 )$ for the TT format. Here $n$ is the number of discretisation
points in one direction, $r<<n$ is the maximal tensor rank, and $d$ the problem dimension.
Talk presented on this workshop "Workshop: Imaging With Uncertainty Quantification (IUQ), September 2022",
https://people.compute.dtu.dk/pcha/CUQI/IUQworkshop.html
We consider a weakly supervised classification problem. It
is a classification problem where the target variable can be unknown
or uncertain for some subset of samples. This problem appears when
the labeling is impossible, time-consuming, or expensive. Noisy measurements
and lack of data may prevent accurate labeling. Our task
is to build an optimal classification function. For this, we construct and
minimize a specific objective function, which includes the fitting error on
labeled data and a smoothness term. Next, we use covariance and radial AQ1
basis functions to define the degree of similarity between points. The further
process involves the repeated solution of an extensive linear system
with the graph Laplacian operator. To speed up this solution process,
we introduce low-rank approximation techniques. We call the resulting
algorithm WSC-LR. Then we use the WSC-LR algorithm for analysis
CT brain scans to recognize ischemic stroke disease. We also compare
WSC-LR with other well-known machine learning algorithms.
Computing f-Divergences and Distances of High-Dimensional Probability Density...Alexander Litvinenko
Poster presented on Stochastic Numerics and Statistical Learning: Theory and Applications Workshop in KAUST, Saudi Arabia.
The task considered here was the numerical computation of characterising statistics of
high-dimensional pdfs, as well as their divergences and distances,
where the pdf in the numerical implementation was assumed discretised on some regular grid.
Even for moderate dimension $d$, the full storage and computation with such objects become very quickly infeasible.
We have demonstrated that high-dimensional pdfs,
pcfs, and some functions of them
can be approximated and represented in a low-rank tensor data format.
Utilisation of low-rank tensor techniques helps to reduce the computational complexity
and the storage cost from exponential $\C{O}(n^d)$ to linear in the dimension $d$, e.g.
O(d n r^2) for the TT format. Here $n$ is the number of discretisation
points in one direction, r<n is the maximal tensor rank, and d the problem dimension.
The particular data format is rather unimportant,
any of the well-known tensor formats (CP, Tucker, hierarchical Tucker, tensor-train (TT)) can be used,
and we used the TT data format. Much of the presentation and in fact the central train
of discussion and thought is actually independent of the actual representation.
In the beginning it was motivated through three possible ways how one may
arrive at such a representation of the pdf. One was if the pdf was given in some approximate
analytical form, e.g. like a function tensor product of lower-dimensional pdfs with a
product measure, or from an analogous representation of the pcf and subsequent use of the
Fourier transform, or from a low-rank functional representation of a high-dimensional
RV, again via its pcf.
The theoretical underpinnings of the relation between pdfs and pcfs as well as their
properties were recalled in Section: Theory, as they are important to be preserved in the
discrete approximation. This also introduced the concepts of the convolution and of
the point-wise multiplication Hadamard algebra, concepts which become especially important if
one wants to characterise sums of independent RVs or mixture models,
a topic we did not touch on for the sake of brevity but which follows very naturally from
the developments here. Especially the Hadamard algebra is also
important for the algorithms to compute various point-wise functions in the sparse formats.
Computing f-Divergences and Distances of\\ High-Dimensional Probability Densi...Alexander Litvinenko
Talk presented on SIAM IS 2022 conference.
Very often, in the course of uncertainty quantification tasks or
data analysis, one has to deal with high-dimensional random variables (RVs)
(with values in $\Rd$). Just like any other RV,
a high-dimensional RV can be described by its probability density (\pdf) and/or
by the corresponding probability characteristic functions (\pcf),
or a more general representation as
a function of other, known, random variables.
Here the interest is mainly to compute characterisations like the entropy, the Kullback-Leibler, or more general
$f$-divergences. These are all computed from the \pdf, which is often not available directly,
and it is a computational challenge to even represent it in a numerically
feasible fashion in case the dimension $d$ is even moderately large. It
is an even stronger numerical challenge to then actually compute said characterisations
in the high-dimensional case.
In this regard, in order to achieve a computationally feasible task, we propose
to approximate density by a low-rank tensor.
Low rank tensor approximation of probability density and characteristic funct...Alexander Litvinenko
Very often one has to deal with high-dimensional random variables (RVs). A high-dimensional RV can be described by its probability density (\pdf) and/or by the corresponding probability characteristic functions (\pcf), or by a function representation. Here the interest is mainly to compute characterisations like the entropy, or
relations between two distributions, like their Kullback-Leibler divergence, or more general measures such as $f$-divergences,
among others. These are all computed from the \pdf, which is often not available directly, and it is a computational challenge to even represent it in a numerically feasible fashion in case the dimension $d$ is even moderately large. It is an even stronger numerical challenge to then actually compute said characterisations in the high-dimensional case.
In this regard, in order to achieve a computationally feasible task, we propose to represent the density by a high order tensor product, and approximate this in a low-rank format.
Identification of unknown parameters and prediction of missing values. Compar...Alexander Litvinenko
H-matrix approximation of large Mat\'{e}rn covariance matrices, Gaussian log-likelihoods.
Identifying unknown parameters and making predictions
Comparison with machine learning methods.
kNN is easy to implement and shows promising results.
Identification of unknown parameters and prediction with hierarchical matrice...Alexander Litvinenko
We compare four numerical methods for the prediction of missing values in four different datasets.
These methods are 1) the hierarchical maximum likelihood estimation (H-MLE), and three machine learning (ML) methods, which include 2) k-nearest neighbors (kNN), 3) random forest, and 4) Deep Neural Network (DNN).
From the ML methods, the best results (for considered datasets) were obtained by the kNN method with three (or seven) neighbors.
On one dataset, the MLE method showed a smaller error than the kNN method, whereas, on another, the kNN method was better.
The MLE method requires a lot of linear algebra computations and works fine on almost all datasets. Its result can be improved by taking a smaller threshold and more accurate hierarchical matrix arithmetics. To our surprise, the well-known kNN method produces similar results as H-MLE and worked much faster.
1. Motivation: why do we need low-rank tensors
2. Tensors of the second order (matrices)
3. CP, Tucker and tensor train tensor formats
4. Many classical kernels have (or can be approximated in ) low-rank tensor format
5. Post processing: Computation of mean, variance, level sets, frequency
Propagation of Uncertainties in Density Driven Groundwater FlowAlexander Litvinenko
Major Goal: estimate risks of the pollution in a subsurface flow.
How?: we solve density-driven groundwater flow with uncertain porosity and permeability.
We set up density-driven groundwater flow problem,
review stochastic modeling and stochastic methods, use UG4 framework (https://gcsc.uni-frankfurt.de/simulation-and-modelling/ug4),
model uncertainty in porosity and permeability,
2D and 3D numerical experiments.
Simulation of propagation of uncertainties in density-driven groundwater flowAlexander Litvinenko
Consider stochastic modelling of the density-driven subsurface flow in 3D. This talk was presented by Dmitry Logashenko on the IMG conference in Kunming, China, August 2019.
Large data sets result large dense matrices, say with 2.000.000 rows and columns. How to work with such large matrices? How to approximate them? How to compute log-likelihood? determination? inverse? All answers are in this work.
In this paper, we solve a semi-supervised regression
problem. Due to the luck of knowledge about the
data structure and the presence of random noise, the considered data model is uncertain. We propose a method which combines graph Laplacian regularization and cluster ensemble methodologies. The co-association matrix of the ensemble is calculated on both labeled and unlabeled data; this matrix is used as a similarity matrix in the regularization framework to derive the predicted outputs. We use the low-rank decomposition of the co-association matrix to significantly speedup calculations and reduce memory. Two clustering problem examples are presented.
Full version is here https://arxiv.org/abs/1901.03919
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
For more technical information, visit our website https://intellaparts.com
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
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Computation of electromagnetic fields scattered from dielectric objects of uncertain shapes using MLMC algorithm
1. Computation of Electromagnetic Fields Scattered From Dielectric Objects of
Uncertain Shapes Using MLMC
A. Litvinenko1
, A. C. Yucel3
, H. Bagci2
, J. Oppelstrup4
, E. Michielssen5
, R. Tempone1,2
1
RWTH Aachen, 2
KAUST, 3
Nanyang Technological University in Singapore, 4
KTH Royal Institute of Technology, 5
University of Michigan
GAMM 2020 in Kassel, Germany
2. Motivation: Comput. tools for electromagnetic scattering from
objects with uncertain shapes are needed in various applications.
Goal: Develop numerical methods for predicting radar and scattering
cross sections (RCS and SCS) of complex targets.
How: To reduce comp. cost we use CMLMC method (advanced
version of Multi-Level Monte Carlo).
CMLMC optimally balances statistical and discretization errors. It
requires very few samples on fine meshes and more on coarse.
taken from wiki, reddit.com, EMCoS 1/33
3. Plan:
1. Scattering problem setup
2. Deterministic solver
3. Generation of random shapes
4. Shape transformation
5. QoI on perturbed shape
6. Continuation Multi Level Monte Carlo (CMLMC)
7. Results (time, work vs. TOL, weak and strong convergences)
8. Conclusion
9. Best practices
10. Appendix (details of CMLMC)
2/33
5. Deterministic solver
Electromagnetic scattering from dielectric objects is analyzed by
using the Poggio-Miller-Chan-Harrington-Wu-Tsai surface integral
equation (PMCHWT-SIE) solver.
The PMCHWT-SIE is discretized using the method of moments
(MoM) and the iterative solution of the resulting matrix system is
accelerated using a (parallelized) fast multipole method (FMM) - fast
Fourier transform (FFT) scheme (FMM-FFT).
4/33
6. Generation of random shapes
Perturbed shape v(ϑm, ϕm) is defined as
v(ϑm, ϕm) ≈ ˜v(ϑm, ϕm) +
K
k=1
akκk(ϑm, ϕm). (1)
where ϑm and ϕm are angular coordinates of node m,
˜v(ϑm, ϕm) = 1 m is unperturbed radial coordinate on the unit sphere.
κk(ϑ, ϕ) obtained from spherical harmonics by re-scaling their
arguments, κ1(ϑ, ϕ) = cos(α1ϑ), κ2(ϑ, ϕ) = sin(α2ϑ) sin(α3ϕ),
where α1, α2, α3 > 0.
-1
1
-0.5
1
0
0.5
0
1
0
-1 -1
1
0
-1-1
-0.5
0
0.5
1
0.5
0
-0.5
-1
1
1
0
-1
1
0
-1
1
0.5
0
-0.5
-1
-1.5
5/33
7. Mesh transformation
Mesh P0, which is now after the application of (1), is also rotated
and scaled using the simple transformation
xm
ym
zm
:=L(lx, ly, lz)Rx(ϕx)Ry(ϕy)Rz(ϕz)
xm
ym
zm
. (2)
Node (xm, ym, zm) before and after transformation.
matrices Rx(ϕx), Ry(ϕy), and Rz(ϕz) perform rotations around x, y,
and z axes by angles ϕx, ϕy, and ϕz,
matrix L(lx, ly, lz) implements scaling along x, y, and z axes by lx, ly,
and lz, respectively.
6/33
8. Random rotation, stretching and expanding
rotations around axes x, y, and z by angles ϕx, ϕy, and ϕz:
Rx(ϕx) =
1 0 0
0 cos ϕx − sin ϕx
0 sin ϕx cos ϕx
Ry(ϕy) =
cos ϕy 0 sin ϕy
0 1 0
− sin ϕy 0 cos ϕy
Rz(ϕz) =
cos ϕz − sin ϕz 0
sin ϕz cos ϕz 0
0 0 1
.
Matrix L(lx, ly, lz) implements scaling along axes x, y, z by factors lx,
ly, and lz:
¯L(lx, ly, lz) =
1/lx 0 0
0 1/ly 0
0 0 1/lz
.
7/33
9. Input random vector
RVs used in generating the coarsest perturbed mesh P0 are:
1. perturbation weights ak, k = 1, . . . , K,
2. rotation angles ϕx, ϕy, and ϕz,
3. scaling factors lx, ly, and lz.
Thus, random input parameter vector:
ξ = (a1, . . . , aK , ϕx, ϕy, ϕz, lx, ly, lz) ∈ RK+6
(3)
defines the perturbed shape.
8/33
10. Mesh refinement
Mesh P1 ( = 1) is generated by refining each triangle of the
perturbed P0 into four (by halving all three edges and connecting
mid-points).
Mesh at level = 2, P2, is generated in the same way from P1.
All meshes P at all levels = 1, . . . , L are nested discretizations of
P0.
(!!!) No uncertainties are added on meshes P , > 0;
the uncertainty is introduced only at level = 0.
9/33
11. Refinement of the perturbed shape
4 nested meshes with {320, 1280, 5120, 20480} triangular elements.
10/33
13. Electric (left) and magnetic (right) surface current densities
Amplitudes of (a) J(r) and (b) M(r) induced on the unit sphere
under excitation by an ˆx-polarized plane wave propagating in −ˆz
direction at 300 MHz. Amplitudes of (c) J(r) and (d) M(r) induced
on the perturbed shape under excitation by the same plane wave. For
all figures, amplitudes are normalized to 1 and plotted in dB scale.
12/33
14. RCS of unit sphere and perturbed shape
3 /4 /2 /4 0 /4 /2 3 /4
(rad)
-10
-5
0
5
10
15
20
25
rcs
(dB)
Sphere
Perturbed surface
3 /4 /2 /4 0 /4 /2 3 /4
(rad)
-10
-5
0
5
10
15
20
25
rcs
(dB)
Sphere
Perturbed surface
RCS is computed on
(a) xz and (b) yz planes
under excitation by an ˆx-polarized
plane wave propagating in −ˆz di-
rection at 300 MHz.
(a) ϕ = 0 and ϕ = π rad in the
first and second halves of the hori-
zontal axis, respectively.
(b)ϕ = π/2 rad and ϕ = 3π/2 rad
in the first and second halves of the
horizontal axis.
13/33
15. Multilevel Monte Carlo Algorithm
Aim: to approximate the mean E (g(u)) of QoI g(u) to a given
accuracy ε := TOL, where u = u(ω) and ω - random perturbations
in the domain.
Idea: Balance discretization and statistical errors. Very few samples
on a fine grid and more on coarse (denote by M ).
Assume: have a hierarchy of L + 1 meshes {h }L
=0, h := h0β−
for
each realization of random domain.
14/33
16. CMLMC numerical tests
The QoI is the SCS over a user-defined solid angle
Ω = [1/6, 11/36]π rad × [5/12, 19/36]π rad (i.e., a measure of
far-field scattered power in a cone).
RVs are:
a1, a2 ∼ U[−0.14, 0.14] m,
ϕx, ϕy, ϕz ∼ U[0.2, 3] rad,
lx, ly, lz ∼ U[0.9, 1.1];
U[a, b] is the uniform distribution between a and b.
CMLMC runs for TOL ranging from 0.2 to 0.008.
At TOL ≈ 0.008, CMLMC requires L = 5 meshes with
{320, 1280, 5120, 20480, 81920} triangles.
15/33
17. Average time vs. TOL
10−3
10−2
10−1
100
TOL
104
105
106
107
108
AverageTime(s)
TOL−2
CMLMC
MC Estimate
16/33
18. Work estimate vs. TOL
10−3
10−2
10−1
100
TOL
101
102
103
104
105
106
Workestimate
TOL−2
CMLMC
MC Estimate
17/33
19. Time required to compute G vs. .
0 1 2 3 4
101
102
103
104
105
Time(s)
22
G
18/33
20. E = E (G ) vs. (weak convergence)
0 1 2 3 4
10−6
10−5
10−4
10−3
10−2
10−1
100
E
2−3
G
assumed weak convergence curve 2−3
(q1 = 3).
19/33
21. V = Var[G ] vs. (strong convergence)
0 1 2 3 4
10−7
10−6
10−5
10−4
10−3
10−2
10−1
V
2−5
G
assumed strong convergence curve 2−5
(q2 = 5).
20/33
23. Prob. density functions of g − g −1
-1 0 1 2 3 4
0
0.2
0.4
0.6
0.8
1
g1-g0
-0.2 -0.1 0 0.1 0.2
0
20
40
60
80
100
120
g2-g1
g3-g2
(a) = 1 and (b) = {2, 3}.
22/33
24. Best practices for applying CMLMC method to CEM problems
Download CMLMC:
https://github.com/StochasticNumerics/mimclib.git (or use
MLMC from M. Giles)
Implement interface to couple CMLMC and your deterministic
solver
Generate a hierarchy of meshes (mimimum 3), nested are better
Generate 5-7 random shapes on first 3 meshes
Estimate the strong and weak convergence rates, q1, q2, (later they
will be corrected by CMLMC algorithm)
Run CMLMC solver and check visually automatically generated
plots
23/33
25. Conclusion
Used CMLMC method to characterize EM wave scattering from
dielectric objects with uncertain shapes.
Researched how uncertainties in the shape propagate to the
solution.
Demonstrated that the CMLMC algorithm can be 10 times faster
than MC.
To increase the efficiency further, each of the simulations is carried
out using the FMM-FFT accelerated PMCHWT-SIE solver.
Confirmed that the known advantages of the CMLMC algorithm
can be observed when it is applied to EM wave scattering:
non-intrusiveness, dimension independence, better convergence
rates compared to the classical MC method, and higher immunity
to irregularity w.r.t. uncertain parameters, than, for example,
sparse grid methods.
24/33
26. Conclusion
WARING: Some random perturbations may affect the convergence
rates in CMLMC. With difficult-to-predict convergence rates, it is
hard for the CMLMC algorithm to estimate the computational cost
W , the number of levels L, the number of samples on each level M ,
the computation time, and the parameter θ, and the variance in QoI.
All these may result in a sub-optimal performance.
25/33
27. Acknowledgements
SRI UQ at KAUST and Alexander von Humboldt
foundation.
Results are published:
1. A. Litvinenko, A. C. Yucel, H. Bagci, J. Oppelstrup,
E. Michielssen, R. Tempone,
Computation of Electromagnetic Fields Scattered From Objects With
Uncertain Shapes Using Multilevel Monte Carlo Method,
IEEE J. on Multiscale and Multiphysics Comput. Techniques,
pp 37-50, 2019.
2. A. Litvinenko, A. C. Yucel, H. Bagci, J. Oppelstrup,
E. Michielssen, R. Tempone,
Computation of Electromagnetic Fields Scattered From Objects With
Uncertain Shapes Using Multilevel Monte Carlo Method,
arXiv:1809.00362, 2018
26/33
29. CMLMC repeating
Let {P }L
=0 be sequences of meshes with h = h0β−
, β > 1. Let
g (ξ) represent the approximation to g(ξ) computed using mesh P .
E[gL] =
L
=0
E[G ] (4)
where G is defined as
G =
g0 if = 0
g − g −1 if > 0
. (5)
Note that g and g −1 are computed using the same input random
parameter ξ.
28/33
30. CMLMC repeating
E[G ] ≈
∼
G = M−1 M
m=1 G ,m,
E[g − g ] ≈ QW hq1
(6a)
Var[g − g −1] ≈ QShq2
−1 (6b)
for QW = 0, QS > 0, q1 > 0, and 0 < q2 ≤ 2q1.
QoI A = L
=0
∼
G .
Let the average cost of generating one sample of G (cost of one
deterministic simulation for one random realization) be
W ∝ h−dγ
= h−dγ
0 β dγ
(7)
29/33
31. CMLMC repeating
The total CMLMC computational cost is
W =
L
=0
M W . (8)
The estimator A satisfies a tolerance with a prescribed failure
probability 0 < ν ≤ 1, i.e.,
P[|E[g] − A| ≤ TOL] ≥ 1 − ν (9)
while minimizing W . The total error is split into bias and statistical
error,
|E[g] − A| ≤ |E[g − A]|
Bias
+ |E[A] − A|
Statistical error
30/33
32. CMLMC repeating
Let θ ∈ (0, 1) be a splitting parameter, so that
TOL = (1 − θ)TOL
Bias tolerance
+ θTOL
Statistical error tolerance
. (10)
The CMLMC algorithm bounds the bias, B = |E[g − A]|, and the
statistical error as
B = |E[g − A]| ≤ (1 − θ)TOL (11)
|E[A] − A| ≤ θTOL (12)
where the latter bound holds with probability 1 − ν.
To satisfy condition in (12) we require:
Var[A] ≤
θTOL
Cν
2
(13)
for some given confidence parameter, Cν, such that Φ(Cν) = 1 − ν
2,
Φ is the cdf of a standard normal random variable. 31/33
33. CMLMC repeating
By construction of the MLMC estimator, E[A] = E[gL], and by
independence, Var[A] = L
=0 V M−1
, where V = Var[G ]. Given L,
TOL, and 0 < θ < 1, and by minimizing W subject to the statistical
constraint (13) for {M }L
=0 gives the following optimal number of
samples per level (apply ceiling function to M if necessary):
M =
Cν
θTOL
2
V
W
L
=0
V W . (14)
Summing the optimal numbers of samples over all levels yields the
following expression for the total optimal computational cost in terms
of TOL:
W (TOL, L) =
Cν
θTOL
2 L
=0
V W
2
. (15)
32/33
34. Literature
1. Collier, N., Haji-Ali, A., Nobile, F. et al. A continuation multilevel Monte Carlo algorithm. Bit Numer Math 55, 399–432 (2015).
https://doi.org/10.1007/s10543-014-0511-3
2. D Liu, A Litvinenko, C Schillings, V Schulz, Quantification of Airfoil Geometry-Induced Aerodynamic Uncertainties—Comparison of Approaches, SIAM/ASA
Journal on Uncertainty Quantification 5 (1), 334-352, 2017
3. Litvinenko A., Matthies H.G., El-Moselhy T.A. (2013) Sampling and Low-Rank Tensor Approximation of the Response Surface. In: Dick J., Kuo F., Peters
G., Sloan I. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2012. Springer Proceedings in Mathematics & Statistics, vol 65. Springer, Berlin, Heidelber
4. A. Litvinenko, Application of hierarchical matrices for solving multiscale problems, Dissertation, Leipzig University, Germany,
http://publications.rwth-aachen.de/record/754296/files/754296.pdf
5. A Litvinenko, R Kriemann, MG Genton, Y Sun, DE Keyes, HLIBCov: Parallel hierarchical matrix approximation of large covariance matrices and likelihoods
with applications in parameter identification MethodsX 7, 100600, 2020
6. A Litvinenko, D Logashenko, R Tempone, G Wittum, D Keyes, Solution of the 3D density-driven groundwater flow problem with uncertain porosity and
permeability. Int J Geomath 11, 10 (2020). https://doi.org/10.1007/s13137-020-0147-1
7. A Litvinenko, Y Sun, MG Genton, DE Keyes, Likelihood approximation with hierarchical matrices for large spatial datasets, Computational Statistics & Data
Analysis 137, 115-132, 2019
8. S. Dolgov, A. Litvinenko, D.Liu, KRIGING IN TENSOR TRAIN DATA FORMAT Conf. Proceedings, 3rd International Conference on Uncertainty
Quantification in Computational Sciences and Engineering, https://files.eccomasproceedia.org/papers/e-books/uncecomp_2019.pdf pp 309-329,
2019
9. A. Litvinenko, D. Keyes, V. Khoromskaia, B.N. Khoromskij, H. G. Matthies, Tucker tensor analysis of Mat´ern functions in spatial statistics, J.
Computational Methods in Applied Mathematics, Vol. 19, Issue 1, pp 101-122, 2019, De Gruyter
10. HG Matthies, E Zander, BV Rosic, A Litvinenko, Parameter estimation via conditional expectation: a Bayesian inversion, Advanced modeling and simulation
in engineering sciences 3 (1), 1-21, 2016
11. M Espig, W Hackbusch, A Litvinenko, HG Matthies, E Zander, Post-Processing of High-Dimensional Data, arXiv:1906.05669, 2019
33/33