For the last several years my
research has focused on theory and
applications of numerical methods for partial differential equations
(PDEs). My current research efforts continue in this direction, but
with particular emphasis on model reduction of parametrized PDEs via
the Reduced Basis method.
This methodology enables orders of magnitude reduction in computational
effort compared to classical numerical discretizations such as the
finite element method, while at the same time providing rigorous
guarantees of fidelity via a posteriori error analysis. My research on the Reduced Basis method is in collaboration with Prof. Tony Patera and Dr. Phuong Huynh.
We have recently employed this technology to implement "Supercomputing on a Smartphone." For example, we developed a finite element model of a bridge in Les Montes, El Salvador (built by Bridges to Prosperity); the model has 1.5 million degrees of freedom and simulating this bridge takes minutes on a supercomputer.
We applied the Reduced Basis method to this model and as a result were able to simulate the footbridge in real-time on an Android smartphone. Moreover, our smartphone simulations are endowed with rigorous error bounds with respect to the high-fidelity (and expensive!) finite element model.
We have also recently begun collaborating with software engineers from National Instruments in order to develop a Reduced Basis plug-in for LabVIEW. We currently have a working prototype which we plan to use in the real-time measurement and control contexts in which LabVIEW is widely deployed.
Below are screenshots of a 4-core CPU thermal simulation model that we developed to demonstrate the capabilities of the plug-in. A demo video is also available here.
A major emphasis of my current research is the development of a component-based approach to model reduction, which is detailed in the paper "A Static Condensation Reduced Basis Element Method: Approximation and A Posteriori Error Analysis" (see Publications). The classical approach to model reduction is to develop a monolothic reduced order model for an entire parametrized system. As an alternative to this monolothic approach, we propose the idea of generating stand-alone parametrized Reduced Basis Components which are stored in a Library. These components can then be connected together in order to assemble a system model in a "bottom-up" fashion, and the resulting reduced order model is, again, orders of magnitude less computationally espensive than the corresponding finite element model.
This component based approach has a number of advantages significant advantages compared to the standard Reduced Basis method. In particular, it is much more flexible as one can easily make major topological changes to an assembled model by adding, removing or replacing components. The component-based approach simplifies the Offline stage since each component building block are typically far less complex (in terms of geometry and parametrization) than full system models. Also, it enables development of reduced order models with rigorous error bounds for problems with O(100) parameters, which is out of reach with classical state-space model reduction methods.
Polymeric Fluids
I have worked for some time on numerical methods for simulating polymeric fluids. These types of fluids are inherently multiscale in nature and are governed by a coupled Navier-Stokes Fokker-Planck system of PDEs. This system is very challenging from a computational point of view because the Fokker-Planck equation is posed on a high-dimensional domain.
In my Ph.D. thesis I developed theory and analysis of numerical methods (specifically, finite element and spectral methods) for simulating dilute polymeric fluids. I analyzed and implemented a novel alternating-direction scheme for this problem which was amenable to efficient parallelization and was able to perform the first fully-3D simulations of this type of system.
More recently, I have also developed Reduced Basis algorithms for the Fokker-Planck equation in order to reduce the computational effort that is required to model these types of multiscale fluids. See my Publications page for references to papers on this work.
Reduced Basis Models on a Smartphone
We have recently employed this technology to implement "Supercomputing on a Smartphone." For example, we developed a finite element model of a bridge in Les Montes, El Salvador (built by Bridges to Prosperity); the model has 1.5 million degrees of freedom and simulating this bridge takes minutes on a supercomputer.
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| Bridge in Les Montes, El Salvador |
Finite element mesh |
We applied the Reduced Basis method to this model and as a result were able to simulate the footbridge in real-time on an Android smartphone. Moreover, our smartphone simulations are endowed with rigorous error bounds with respect to the high-fidelity (and expensive!) finite element model.
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| Set simulation parameters |
Reduced Basis solutions (displacements) computed on smartphone for two sets of bridge parameters |
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Reduced Basis Models in LabVIEW
We have also recently begun collaborating with software engineers from National Instruments in order to develop a Reduced Basis plug-in for LabVIEW. We currently have a working prototype which we plan to use in the real-time measurement and control contexts in which LabVIEW is widely deployed.
Below are screenshots of a 4-core CPU thermal simulation model that we developed to demonstrate the capabilities of the plug-in. A demo video is also available here.
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| Geometry of CPU chip model (separated and assembled view) |
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| Reduced Basis solutions (CPU temperature profiles) in LabVIEW for two different sets of parameters |
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Reduced Basis Components
A major emphasis of my current research is the development of a component-based approach to model reduction, which is detailed in the paper "A Static Condensation Reduced Basis Element Method: Approximation and A Posteriori Error Analysis" (see Publications). The classical approach to model reduction is to develop a monolothic reduced order model for an entire parametrized system. As an alternative to this monolothic approach, we propose the idea of generating stand-alone parametrized Reduced Basis Components which are stored in a Library. These components can then be connected together in order to assemble a system model in a "bottom-up" fashion, and the resulting reduced order model is, again, orders of magnitude less computationally espensive than the corresponding finite element model.
This component based approach has a number of advantages significant advantages compared to the standard Reduced Basis method. In particular, it is much more flexible as one can easily make major topological changes to an assembled model by adding, removing or replacing components. The component-based approach simplifies the Offline stage since each component building block are typically far less complex (in terms of geometry and parametrization) than full system models. Also, it enables development of reduced order models with rigorous error bounds for problems with O(100) parameters, which is out of reach with classical state-space model reduction methods.
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| Assembly and simulation of a component based Reduced Basis bridge mode | |
Polymeric Fluids
I have worked for some time on numerical methods for simulating polymeric fluids. These types of fluids are inherently multiscale in nature and are governed by a coupled Navier-Stokes Fokker-Planck system of PDEs. This system is very challenging from a computational point of view because the Fokker-Planck equation is posed on a high-dimensional domain.
In my Ph.D. thesis I developed theory and analysis of numerical methods (specifically, finite element and spectral methods) for simulating dilute polymeric fluids. I analyzed and implemented a novel alternating-direction scheme for this problem which was amenable to efficient parallelization and was able to perform the first fully-3D simulations of this type of system.
More recently, I have also developed Reduced Basis algorithms for the Fokker-Planck equation in order to reduce the computational effort that is required to model these types of multiscale fluids. See my Publications page for references to papers on this work.
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| Six components of the polymeric extra stress tensor governed by the Fokker-Planck equation | ||