Related papers: Model input and output dimension reduction using K…
The Karhunen-Lo\`{e}ve (KL) expansion is a popular method for approximating random fields by transforming an infinite-dimensional stochastic domain into a finite-dimensional parameter space. Its numerical approximation is of central…
This article provides a primer on the spectral representation of random fields via the Karhunen-Lo\`eve Expansion (KLE). The goal is to bridge the gap between the theoretical foundations of the KLE and its application in computational…
We present a distributed active subspace method for training surrogate models of complex physical processes with high-dimensional inputs and function valued outputs. Specifically, we represent the model output with a truncated…
We present a new physics-informed machine learning approach for the inversion of PDE models with heterogeneous parameters. In our approach, the space-dependent partially-observed parameters and states are approximated via Karhunen-Lo\`eve…
In many Bayesian inverse problems the goal is to recover a spatially varying random field. Such problems are often computationally challenging especially when the forward model is governed by complex partial differential equations (PDEs).…
The Karhunen-Lo\`eve Expansion (KLE) of a stochastic process is a well understood eigenfunction expansion used widely in time series analysis, stochastic PDEs, and signal processing. Karhunen-Lo\`eve expansions have also been proven to…
Coupled problems with various combinations of multiple physics, scales, and domains are found in numerous areas of science and engineering. A key challenge in the formulation and implementation of corresponding coupled numerical models is…
We use a conditional Karhunen-Lo\`eve (KL) model to quantify and reduce uncertainty in a stochastic partial differential equation (SPDE) problem with partially-known space-dependent coefficient, $Y(x)$. We assume that a small number of…
This paper deals with the study, from a probabilistic point of view, of logistic-type differential equations with uncertainties. We assume that the initial condition is a random variable and the diffusion coefficient is a stochastic…
This report examines numerical aspects of constructing Karhunen-Lo\`{e}ve expansions (KLEs) for second-order stochastic processes. The KLE relies on the spectral decomposition of the covariance operator via the Fredholm integral equation of…
Diffusion denoising models have become a popular approach for image generation, but they often suffer from slow convergence during training. In this paper, we identify that this slow convergence is partly due to the complexity of the…
Fluid antenna systems (FAS) achieve spatial diversity by dynamically switching among $N$ densely packed ports, but the resulting spatially correlated Rayleigh channels render exact outage analysis intractable. Existing block-correlation…
We propose a diffuse interface model to describe tumor as a multicomponent deformable porous medium. We include mechanical effects in the model by coupling the mass balance equations for the tumor species and the nutrient dynamics to a…
We propose a methodology for improving the accuracy of surrogate models of the observable response of physical systems as a function of the systems' spatially heterogeneous parameter fields with applications to uncertainty quantification…
Detection of abrupt spatial changes in physical properties representing unique geometric features such as buried objects, cavities, and fractures is an important problem in geophysics and many engineering disciplines. In this context,…
We consider elliptic diffusion problems with a random anisotropic diffusion coefficient, where, in a notable direction given by a random vector field, the diffusion strength differs from the diffusion strength perpendicular to this notable…
Karhunen-Loeve expansions (KLE) of stochastic processes are important tools in mathematics, the sciences, economics, and engineering. However, the KLE is primarily useful for those processes for which we can identify the necessary…
This paper makes two main contributions. First, we present a pedagogical review of the derivation of the three-term recurrence relation for Legendre polynomials, without relying on the classical Legendre differential equation, Rodrigues'…
We present a computational framework for dimension reduction and surrogate modeling to accelerate uncertainty quantification in computationally intensive models with high-dimensional inputs and function-valued outputs. Our driving…
In this work, we present and analyze a system of PDEs, which models tumor growth by considering chemotaxis, active transport, and random effects. The stochasticity of the system is modelled by random initial data and Wiener noises that…