scholarly journals Model Input and Output Dimension Reduction Using Karhunen–Loève Expansions With Application to Biotransport

2019 ◽  
Vol 5 (4) ◽  
Author(s):  
Alen Alexanderian ◽  
William Reese ◽  
Ralph C. Smith ◽  
Meilin Yu
Keyword(s):  
Dimension Reduction ◽  
Pressure Field ◽  
Input Parameter ◽  
Gaussian Random Field ◽  
Elliptic Partial Differential Equation ◽  
Heterogeneous Material ◽  
Correlation Lengths ◽  
Reduced Order ◽  
Output Dimension ◽  
Permeability Field

Abstract We consider biotransport in tumors with uncertain heterogeneous material properties. Specifically, we focus on the elliptic partial differential equation (PDE) modeling the pressure field inside the tumor. The permeability field is modeled as a log-Gaussian random field with a prespecified covariance function. We numerically explore dimension reduction of the input parameter and model output. Specifically, truncated Karhunen–Loève (KL) expansions are used to decompose the log-permeability field, as well as the resulting random pressure field. We find that although very high-dimensional representations are needed to accurately represent the permeability field, especially in presence of small correlation lengths, the pressure field is not sensitive to high-order KL terms of the input parameter. Moreover, we find that the pressure field itself can be represented accurately using a KL expansion with a small number of terms. These observations are used to guide a reduced-order modeling approach to accelerate computational studies of biotransport in tumors.

Efficient Uncertainty Quantification for Biotransport in Tumors With Uncertain Material Properties

2018 ◽  
Author(s):  
Alen Alexanderian ◽  
William Reese ◽  
Ralph C. Smith ◽  
Meilin Yu
Keyword(s):  
Porous Media ◽  
Random Field ◽  
Material Properties ◽  
Pressure Field ◽  
Input Parameter ◽  
Gaussian Random Field ◽  
Porous Media Flow ◽  
Expansion Technique ◽  
Low Dimensional ◽  
Permeability Field

We consider modeling of single phase fluid flow in heterogeneous porous media governed by elliptic partial differential equations (PDEs) with random field coefficients. Our target application is biotransport in tumors with uncertain heterogeneous material properties. We numerically explore dimension reduction of the input parameter and model output. In the present work, the permeability field is modeled as a log-Gaussian random field, and its covariance function is specified. Uncertainties in permeability are then propagated into the pressure field through the elliptic PDE governing porous media flow. The covariance matrix of pressure is constructed via Monte Carlo sampling. The truncated Karhunen–Loève (KL) expansion technique is used to decompose the log-permeability field, as well as the random pressure field resulting from random permeability. We find that although very high-dimensional representation is needed to recover the permeability field when the correlation length is small, the pressure field is not sensitive to high-oder KL terms of input parameter, and itself can be modeled using a low-dimensional model. Thus a low-rank representation of the pressure field in a low-dimensional parameter space is constructed using the truncated KL expansion technique.

scholarly journals Stochastic reconstruction of the microstructure of equilibrium form snow and computation of effective elastic properties

2010 ◽  
Vol 56 (197) ◽  
pp. 405-414 ◽  
Author(s):  
Hongyan Yuan ◽  
Jonah H. Lee ◽  
James E. Guilkey
Keyword(s):  
Mechanical Properties ◽  
Gaussian Random Field ◽  
Three Dimensional ◽  
Physical And Mechanical Properties ◽  
Heterogeneous Material ◽  
Reconstruction Technique ◽  
Structure Property ◽  
Stochastic Reconstruction ◽  
Snow Samples

AbstractThree-dimensional geometric descriptions of microstructure are indispensable to obtain the structure–property relationships of snow. Because snow is a random heterogeneous material, it is often helpful to construct stochastic geometric models that can be used to model physical and mechanical properties of snow. In the present study, the Gaussian random field-based stochastic reconstruction of the sieved and sintered dry-snow sample with grain size less than 1 mm is investigated. The one- and two-point correlation functions of the snow samples are used as input for the stochastic snow model. Several statistical descriptors not used as input to the stochastic reconstruction are computed for the real and reconstructed snow to assess the quality of the reconstructed images. For the snow samples and the reconstructed snow microstructure, we also estimate the mechanical properties and the size of the associated representative volume element using numerical simulations as additional assessment of the quality of the reconstructed images. The results indicate that the stochastic reconstruction technique used in this paper is reasonably accurate, robust and highly efficient in numerical computations for the high-density snow samples we consider.

scholarly journals Multilevel and quasi-Monte Carlo methods for uncertainty quantification in particle travel times through random heterogeneous porous media

2017 ◽  
Vol 4 (8) ◽  
pp. 170203 ◽  
Author(s):  
D. Crevillén-García ◽  
H. Power
Keyword(s):  
Porous Media ◽  
Monte Carlo ◽  
Uncertainty Quantification ◽  
Input Parameter ◽  
Computational Cost ◽  
Gaussian Random Field ◽  
Heterogeneous Porous Media ◽  
Quasi Monte Carlo ◽  
Average Travel Time ◽  
The Mean

In this study, we apply four Monte Carlo simulation methods, namely, Monte Carlo, quasi-Monte Carlo, multilevel Monte Carlo and multilevel quasi-Monte Carlo to the problem of uncertainty quantification in the estimation of the average travel time during the transport of particles through random heterogeneous porous media. We apply the four methodologies to a model problem where the only input parameter, the hydraulic conductivity, is modelled as a log-Gaussian random field by using direct Karhunen–Loéve decompositions. The random terms in such expansions represent the coefficients in the equations. Numerical calculations demonstrating the effectiveness of each of the methods are presented. A comparison of the computational cost incurred by each of the methods for three different tolerances is provided. The accuracy of the approaches is quantified via the mean square error.

Determination of interface width using EELS fine structure changes

1991 ◽  
Vol 49 ◽  
pp. 730-731
Author(s):  
Vinayak P. Dravid ◽  
M.R. Notis ◽  
C.E. Lyman
Keyword(s):  
Fine Structure ◽  
Materials Science ◽  
Input Parameter ◽  
Core Loss ◽  
Directionally Solidified ◽  
Interface Width ◽  
Structure Changes ◽  
Important Input ◽  
Directionally Solidified Eutectics

The concept of interfacial width is often invoked in many materials science phenomena which relate to the structure and properties of internal interfaces. The numerical value of interface width is an important input parameter in diffusion equations, sintering theories as well as in many electronic devices/processes. Most often, however, this value is guessed rather than determined or even estimated. In this paper we present a method of determining the effective structural and electronic- structural width of interphase interfaces using low- and core loss fine structure effects in EELS spectra.The specimens used in the study were directionally solidified eutectics (DSEs) in the system; NiO-ZrO2(CaO), NiO-Y2O3 and MnO-ZrO2(ss). EELS experiments were carried out using a VG HB-501 FE STEM and a Hitachi HF-2000 FE TEM.

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