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Karen Willcox

Professor, Department of Aeronautics & Astronautics
Co-Director, Center for Computational Engineering
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139

E-mail: kwillcox@mit.edu
Office: Rm 33-412a (map)
Phone: (617) 253-3503



My research interests lie in computational simulation and optimization of engineering systems, with specific contributions in (1) model reduction of large-scale systems with applications in optimization, real-time simulation, control, and uncertainty quantification, and (2) multifidelity methods for design and uncertainty quantification.

Model reduction is a mathematical and computational field of study that aims to systematically generate cost-efficient representations of large-scale computational models, such as those resulting from discretization of partial differential equations. Model reduction targets the critical need for low-dimensional, efficient dynamical models that retain predictive fidelity of high-resolution simulations. Such models are often essential for multidisciplinary applications, design under uncertainty, and real-time control and optimization of dynamic processes. Despite sustained advances in computing power, computational models for such applications yield large systems that are computationally intractable to solve. For example, uncertainty quantification (UQ) applications require the physical system to be simulated repeatedly. As another example, a real-time simulation capability requires the development of accurate models that can be solved sufficiently rapidly to permit control decisions in real time. My group has contributed new model reduction methods for goal-oriented, structure-exploiting approaches to problems with high-dimensional parameters.

Multifidelity modeling refers to the situation where we have multiple sources of information that describe a system of interest. Most often these information sources are models of differing fidelities and costs, but they could also include historical data, expert opinions, experimental data, etc. Multifidelity methods seek to use all available models and data in concert, guided by rigorous quantification of uncertainty. My group has contributed new multifidelity modeling methods for solution of problems in optimization, UQ, and inverse problems.

Current Research Projects

DiaMonD: An Integrated Multifaceted Approach to Mathematics at the Interfaces of Data, Models, and Decisions is a U.S. Department of Energy Mathematical Multifaceted Integrated Capabilities Center (MMICC) involving researchers from Colorado State University, Florida State University, Los Alamos National Laboratory, Massachusetts Institute of Technology, Oak Ridge National Laboratory, University of Texas at Austin, and Stanford University. For more information see the DiaMonD center webpage.

M3: Managing multiple information sources of multi-physics systems is a U.S. Air Force Office of Scientific Research (AFOSR) Multidisciplinary Research Program of the University Research Initiative (MURI) involving researchers from Arizona State University, Cornell University, Massachusetts Institute of Technology, Santa Fe Institute, Texas A&M University, and University of Michigan. For more information see the M3 project webpage.

Dynamic data-driven methods for a self-aware aerospace vehicle
A self-aware aerospace vehicle can dynamically adapt the way it performs missions by gathering information about itself and its surroundings and responding intelligently. We consider the specific challenge of an unmanned aerial vehicle that can dynamically and autonomously sense its structural state and re-plan its mission according to its estimated current structural health. The challenge is to achieve each of these tasks in real time--executing online models and exploiting dynamic data streams--while also accounting for uncertainty. Our approach combines information from physics-based models, simulated offline to build a scenario library, together with dynamic sensor data in order to estimate current flight capability. Our physics-based models analyze the system at both the local panel level and the global vehicle level. This project is conducted in collaboration with Cory Kays and Jeff Chambers at Aurora Flight Sciences, and Doug Allaire at TAMU. It is funded by the AFOSR Dynamic Data-Driven Application Systems (DDDAS) program.

Multifidelity methods for uncertainty quantification
We have proposed a multifidelity approach to forward propagation of uncertainty that makes use of inexpensive, low-fidelity models to provide approximate information about the expensive, high-fidelity model. The multifidelity estimator is developed based on the control variate method to reduce the computational cost of achieving a specified mean square error in the statistic estimate. The method optimally allocates the computational load among multiple models based on their relative evaluation costs and the strength of the correlations between them. We have also developed this approach in the context of multifidelity optimization under uncertainty . In the optimization setting, we propose an information-reuse estimator that exploits the autocorrelation structure of the high-fidelity model in the design space to reduce the cost of repeatedly estimating statistics during the course of optimization (i.e., we can reuse information from past optimization iterations as a cheap but effective surrogate model). In design problems ranging from acoustic horn geometry design to aerostructural wing design to conceptual aircraft vehicle-level optimization, we see computational speedups in the range of 80 to 90%.

Multifidelity methods for design using derivative-free optimization
Multifidelity optimization approaches seek to bring higher-fidelity analyses earlier into the design process by using performance estimates from lower-fidelity models to accelerate convergence towards the optimum of a high-fidelity design problem. We have developed multifidelity optimization methods that combine Bayesian calibration and radial basis function approximations with trust region model management, to yield a provably-convergent algorithm. The gradient-free version of the approach does not require the gradient of the high-fidelity function---in this case, convergence is guaranteed using sensitivity information from the calibrated low-fidelity models. We have demonstrated the method for aerodynamic shape optimization, showing an 80% reduction in the number of high-fidelity analyses compared with a single-fidelity sequential quadratic programming formulation and a similar number of high-fidelity analyses compared with a multifidelity trust-region algorithm that estimates the high-fidelity gradient using finite differences.

International Design Center, Singapore University of Design and Technology
The Singapore University of Design and Technology is a newly established university that focuses on design and technology. I am leading the Design Computation Research Thrust of the International Design Center.

Past Research Projects

Model reduction for probabilistic analysis and design under uncertainty
Effective computational tools to support decision-making under uncertainty are becoming essential in the design and operation of aerospace systems. The accurate and efficient propagation of uncertainties in parameters through complex, high fidelity computational models is a significant challenge. Since analytical characterizations of uncertainties in the system outputs are typically not available, numerical methods must be used that require repeated evaluations of models at suitably sampled parameters. Model reduction is a promising technique to substantially reduce the computational cost involved in the propagation of uncertainty. While model reduction has been applied successfully to many different applications in deterministic simulation and optimization settings, the goal of this project is the development, analysis, and application of reduced-order models to support decision-making under uncertainty. We are applying our methods to applications in reacting flows and fluid-structure interaction. This project is being conducted in collaboration with Prof. M. Heinkenschloss and Prof. D. Sorensen of Rice University.

Multi-scale fusion of information for uncertainty quantification and management in large-scale simulations
This MURI project sponsored by AFOSR aims to develop an integrated methodology for uncertainty quantification. The MURI project includes five research areas: (1) Mathematical analysis of SPDEs and multiscale formulation, (2) Numerical solution of SPDEs, (3) Reduced-Order modeling, (4) Estimation/Inverse problems, and (5) Robust optimization and control. Our work in areas (3) and (5) is combining certified reduced models with robust optimization methods and is developing multifidelity approach for optimization under uncertainty. More information can be found at the MURI project website.

FAA Environmental Tools Suite
The increasing concern over environmental impacts of aircraft is reflected in an ongoing commitment by the FAA to develop a comprehensive suite of tools to support aviation environmental policy decision-making. Components of this toolkit include tools to assess the interdependencies among aviation-related noise and emissions, impacts on health and welfare, and industry and consumer costs, under different policy, technology, operations and market scenarios, as well as aircraft design tools that conduct an integrated analysis of noise, emissions and economics at the aircraft level for future and current aircraft. The scale and complexity of this problem are immense; for example, simulation of noise and environmental footprints over one year involves over two million flights with 350 aircraft types, analyzed with black-box models spanning airline economics, environmental economics, aircraft operations, aircraft performance and emissions, noise, local air quality, and global climate. Furthermore, while just simulating the system is a daunting task, uncertainty must be characterized, computed and communicated in a way tangible to the domestic and international policy decision-makers. Together with a large team of researchers from the PARTNER Center of Excellence, we are working to create methods to assess the uncertainties in the FAA Environmental Tools Suite. Surrogate models, building on concepts from model reduction, are a critical component of addressing this challenge.

Reduced-order models for unsteady aerodynamic applications, active flow control, and aeroelasticity
My group's contributions in model reduction have focused on development of rigorous methodology suitable for application to large-scale systems, with a focus on CFD. Classical reduction techniques are mathematically rigorous, but are limited in application to small systems; conversely, large-scale reduction methods have lacked mathematical rigor and robustness. Using dynamical systems concepts we have introduced new methodology to link large-scale reduction approaches to more rigorous classical techniques. For example, we proposed a balanced version of POD that includes system outputs for POD-based reduction of large-scale systems. Including output information is important for control applications and also leads to more efficient models. In collaboration with Prof. Alexandre Megretski, I introduced Fourier model reduction (FMR) for large-scale linear time-invariant systems. FMR uses a finite number of discrete-time Fourier coefficients, yielding a reduced model with a theoretical error bound. In comparison with standard POD, for smooth transfer functions, FMR is computationally more efficient, yields models with rigorous guarantees of accuracy/stability, and exploits both input and output information. These approaches have been applied to a range of problems including turbomachinery aeroelasticity, compressor mistuning, supersonic inlet flow dynamics, and active flow controller design.

Gappy POD for flow reconstruction, flow sensing, and nonlinear model reduction
The gappy POD is a modification of the basic POD method that handles incomplete or "gappy" data sets. An incomplete data vector can be reconstructed by representing it as a linear combination of known POD basis vectors. The modal content is determined by solving a small linear system. Further, if the snapshots themselves are damaged or incomplete, an iterative method can be used to derive the POD basis vectors. This method was developed by Everson and Sirovich in the context of reconstruction of images, such as human faces, from partial data. The gappy POD is relevant for flow problems where incomplete data is available. For example, in experiments, data may only be available on the airfoil surface. Our research has shown that the gappy POD can be used to reconstruct steady and unsteady flowfield data from limited surface pressure measurements. We have also used the gappy POD to develop the Missing Point Estimation approach for efficient reduction of nonlinear systems.

Model reduction for optimal design and inverse problem applications
We have developed enabling methodology to advance model reduction from simulation to optimization applications. For large-scale optimal design, optimal control, and inverse problem applications, a key challenge is deriving reduced models to capture variation over a parametric input space, which, for many optimization applications, is of high dimension. We have proposed a new model-constrained optimization methodology, building on the work of Patera et al. This methodology provides a systematic approach to sample high-dimensional input spaces by solving a sequence of model-constrained optimization problems.

For example, one challenge is to solve an inverse contaminant transport problem on a grid of millions of cells, with limited measurements, in order to determine the probable upstream source of a contaminant release, and the potential downstream impact areas-all within a few minutes to allow for emergency response. Using our model-constrained optimization formulation, we developed a new methodology to sample the high-dimensional space of possible initial conditions to determine a reduced model for this state estimation inverse problem setting. Under certain assumptions, our method provides an explicit solution to the sampling problem in the form of an eigenvalue problem.

In the case of general nonlinear parametric dependence, our method has a computational cost that scales linearly with the dimension of the parametric input space (for a reduced model of fixed size). For a heat transfer optimal design application with 18,000 states, we demonstrated the approach for parametric input spaces up to dimension 21. Sampling such high-dimensional input spaces with statistically-based sampling methods is, in general, a computationally prohibitive proposition; our model-constrained sampling yields reduced models with errors three to four orders of magnitude lower than those obtained using standard methods, such as Latin hypercube and log-random sampling. We also show how our approach enables probabilistic analysis, where the adaptive sampling method is used to derive a reduced basis that is effective over the joint probability density of the input parameters. We demonstrate probabilistic analysis of a CFD model, estimating through Monte Carlo simulation the effects of variations in blade geometry on the forced response of a subsonic compressor blade row.

Large-scale optimization for Bayesian inference in complex systems
The SAGUARO (Scalable Algorithms for Groundwater Uncertainty Analysis and Robust Optimization) Project focuses on the development of scalable numerical algorithms for large-scale Bayesian inversion in complex systems. Specifically, we are developing stochastic spectral approximations together with methods to exploit prior information and approximations of the Hessian operator, thereby capitalizing on advances in large-scale simulation-based optimization and inversion methods. Our application target is a challenging testbed problem in subsurface flow and transport. Collaborators on the SAGUARO project include Prof. G. Biros (GA-Tech), Prof.C. Dawson (UT-Austin), Prof.O. Ghattas (UT-Austin),and Prof. Y. Marzouk (MIT).

The MIT component of the SAGUARO Project addresses the intractability of conventional sampling methods for large-scale statistical inverse problems by devising reduced-order models that are faithful to the full-order model over a wide range of parameter values; sampling then employs the reduced model rather than the full model, resulting in very large computational savings. Results indicate little effect on the computed posterior distribution.

Bayesian-based multifidelity multidisciplinary vehicle design optimization
This project is developing new methods for multidisciplinary design that are based on estimation theory, with a focus on management of multifidelity models in MDO. The method employs maximum entropy characterizations of model uncertainties that can be established via expert opinion or historical data, and incorporates global sensitivity analysis to rigorously apportion variation in performance parameters associated with critical design constraints to individual disciplines. This provides a means of determining, with confidence, when low, medium, and high fidelity models need to be incorporated into the design process. This project was conducted in collaboration with Aurora Flight Sciences.

Integrated performance/cost optimization for aircraft design
As the aerospace industry moves from the era of "Higher, Faster, Farther" to the challenge of "Leaner, Meaner, Greener" the definition of best design has evolved considerably. The balance between performance and cost in commercial aircraft design is increasingly important, at both the conceptual and the preliminary design level. This research developed quantitative methods for including programmatic decisions and financial models in the conceptual and preliminary design phase, and implemented them in an integrated cost/performance design tool. The work is directed towards new aircraft concepts, such as the Blended-Wing-Body. Using a real options valuation approach combined with an MDO framework, we demonstrated the importance of emphasizing long-term cash flows over development costs for a commercial aircraft program and introduced new methodology to quantify the impact of technical and financial uncertainty and to combine technical and programmatic decisions.