Reseach
Uncertainty quantification in multiscale/multiphysics systems
Experimental evidence suggests that the performance of most devices as well as the dynamics of many physical phenomena is significantly affected by the underlying uncertainties associated with variations in properties and fluctuations in operating conditions. To accurately predict the performance of physical systems, it becomes essential for one to include the effects of input uncertainties into the model system and understand how they propagate and alter the final solution. The presence of uncertainties can be modeled in the system through reformulation of the governing equations as stochastic ordinary/partial differential equations (SODEs/SPDEs). In recent years, there has been increasing interest in analyzing and quantifying the effects of random inputs and uncertainties in the solution of SPDEs that describe complex physical systems.
Project 1: A stabilized stochastic finite element second-order projection method for modeling natural convection in random porous media
Bayesian Inference and Inverse Problems
Inverse problems arise frequently in diverse engineering applications, i.e. heat conduction, geophysics, fluid mechanics and solid mechanics. In a typical inverse problem, one is interested in identifying the initial, boundary and/or material properties given sensor measurements of the dependent variable inside the domain. A typical example is the one of estimating permeability from measurements of flow data. The inverse problem is often ill-posed in the sense that the solution may not exist or may not be unique. The majority of the deterministic approaches restate the problem as a least-squares minimization problem and lead to estimates of unknowns without rigorously considering system uncertainties and without providing quantification of uncertainty in the inverse problem . Several methods have been proposed and Bayesian inference approach attracts much more interests.
The Bayesian inference approach provides a systematic means of taking system variabilities and parameter fluctuations into account. This framework formulates a complete probabilistic description of the unknown parameters and system uncertainties given measurement data. The Bayesian approach incorporates the known information regarding the unknown parameters into a prior distribution model that is then combined with the likelihood to formulate the posterior probability density function (PPDF). The PPDF serves as the solution of the inverse problem and various statistics can be estimated from the samples of this distribution, such as mean, marginal distribution and quantiles. This methodology has been used with great success to solve a variety of problems.
Stochastic Multiscale Modeling
Observation of the structure of different man-made and naturally occurring materials leads to the conclusion that the concept of the homogenous continuum, which underlies the classical theories of material behavior (e.g. elasticity theory, electrodynamics), is often too idealized and does not reflect the complexity and heterogeneity inherent in real materials. Examples of such media include various types of composite materials, porous and cracked solids, polycrystals, soils, rocks, concrets as well as a variety of biological media. Although the nature of heterogeneity is different in each of the examples above, they all share a common feature - the existence of a microstructure, i.e., an underlying heterogeneous material structure at a scale that is small compared to the characteristic dimension (macroscale) of the specimen. There has been increasing interest in reliably modelling and predicting the thermal and mechanical behavior of such media.
This poses a significant computational challenge with several interesting physical, computational and mathematical issues awaiting resolution. A detailed study is necessary regarding the nature of uncertainties (in particular of the underlying medium topology) and their interactions across length scales in such systems. The key is to develop a mathematically consistent multiscale approach to address stochastic transport processes in such media. This natural confluence of multiscale systems and uncertainty is still a fairly new and exciting topic and there exist many avenues for significant mathematical and physical insights into this fundamental problem.
Stochastic Reduced-order Modeling
Stochastic analysis of random heterogeneous media provides useful information only if realistic input models of the material property variation are used. These input models are often constructed from a set of experimental samples of the underlying random field. This leads to the problem of probabilistic model identification or stochastic reduced-order modeling, where the purpose is to find a parametric representation of the random field through only limited experimental data.
Project 6: Kernel principal component analysis of stochastic input model generation