Youngsoo Choi | Stanford University (original) (raw)

Papers by Youngsoo Choi

ArXiv, 2020

Traditional linear subspace reduced order models (LS-ROMs) are able to accelerate physical simula... more Traditional linear subspace reduced order models (LS-ROMs) are able to accelerate physical simulations, in which the intrinsic solution space falls into a subspace with a small dimension, i.e., the solution space has a small Kolmogorov n-width. However, for physical phenomena not of this type, such as advection-dominated flow phenomena, a low-dimensional linear subspace poorly approximates the solution. To address cases such as these, we have developed an efficient nonlinear manifold ROM (NM-ROM), which can better approximate high-fidelity model solutions with a smaller latent space dimension than the LS-ROMs. Our method takes advantage of the existing numerical methods that are used to solve the corresponding full order models (FOMs). The efficiency is achieved by developing a hyper-reduction technique in the context of the NM-ROM. Numerical results show that neural networks can learn a more efficient latent space representation on advection-dominated data from 2D Burgers' equa...

ArXiv, 2021

This work is the first to employ and adapt the image-to-image translation concept based on condit... more This work is the first to employ and adapt the image-to-image translation concept based on conditional generative adversarial networks (cGAN) towards learning a forward and an inverse solution operator of partial differential equations (PDEs). Even though the proposed framework could be applied as a surrogate model for the solution of any PDEs, here we focus on steady-state solutions of coupled hydro-mechanical processes in heterogeneous porous media. Strongly heterogeneous material properties, which translate to the heterogeneity of coefficients of the PDEs and discontinuous features in the solutions, require specialized techniques for the forward and inverse solution of these problems. Additionally, parametrization of the spatially heterogeneous coefficients is excessively difficult by using standard reduced order modeling techniques. In this work, we overcome these challenges by employing the image-to-image translation concept to learn the forward and inverse solution operators a...

ArXiv, 2018

Several reduced order models have been successfully developed for nonlinear dynamical systems. To... more Several reduced order models have been successfully developed for nonlinear dynamical systems. To achieve a considerable speedup, a hyper-reduction step is needed to reduce the computational complexity due to nonlinear terms. Many hyper-reduction techniques require the construction of nonlinear term basis, which introduces a computationally expensive offline phase. A novel way of constructing nonlinear term basis within the hyper-reduction process is introduced. In contrast to the traditional hyper-reduction techniques where the collection of nonlinear term snapshots is required, the SNS method completely avoids the use of the nonlinear term snapshots. Instead, it uses the solution snapshots that are used for building a solution basis. Furthermore, it avoids an extra data compression of nonlinear term snapshots. As a result, the SNS method provides a more efficient offline strategy than the traditional model order reduction techniques, such as the DEIM, GNAT, and ST-GNAT methods. Th...

ArXiv, 2019

Although design optimization has shown its great power of automatizing the whole design process a... more Although design optimization has shown its great power of automatizing the whole design process and providing an optimal design, using sophisticated computational models, its process can be formidable due to a computationally expensive large-scale linear system of equations to solve, associated with underlying physics models. We introduce a general reduced order model-based design optimization acceleration approach that is applicable not only to design optimization problems, but also to any PDE-constrained optimization problems. The acceleration is achieved by two techniques: i) allowing an inexact linear solve and ii) reducing the number of iterations in Krylov subspace iterative methods. The choice between two techniques are made, based on how close a current design point to an optimal point. The advantage of the acceleration approach is demonstrated in topology optimization examples, including both compliance minimization and stress-constrained problems, where it achieves a treme...

Traditional linear subspace reduced order models (LS-ROMs) are able to accelerate physical simula... more Traditional linear subspace reduced order models (LS-ROMs) are able to accelerate physical simulations, in which the intrinsic solution space falls into a subspace with a small dimension, i.e., the solution space has a small Kolmogorov n-width. However, for physical phenomena not of this type, e.g., any advection-dominated flow phenomena, such as in traffic flow, atmospheric flows, and air flow over vehicles, a low-dimensional linear subspace poorly approximates the solution. To address cases such as these, we have developed a fast and accurate physics-informed neural network ROM, namely nonlinear manifold ROM (NM-ROM), which can better approximate high-fidelity model solutions with a smaller latent space dimension than the LS-ROMs. Our method takes advantage of the existing numerical methods that are used to solve the corresponding full order models. The efficiency is achieved by developing a hyper-reduction technique in the context of the NM-ROM. Numerical results show that neural...

Computer Methods in Applied Mechanics and Engineering

Mathematics

A classical reduced order model (ROM) for dynamical problems typically involves only the spatial ... more A classical reduced order model (ROM) for dynamical problems typically involves only the spatial reduction of a given problem. Recently, a novel space–time ROM for linear dynamical problems has been developed [Choi et al., Space–tume reduced order model for large-scale linear dynamical systems with application to Boltzmann transport problems, Journal of Computational Physics, 2020], which further reduces the problem size by introducing a temporal reduction in addition to a spatial reduction without much loss in accuracy. The authors show an order of a thousand speed-up with a relative error of less than 10−5 for a large-scale Boltzmann transport problem. In this work, we present for the first time the derivation of the space–time least-squares Petrov–Galerkin (LSPG) projection for linear dynamical systems and its corresponding block structures. Utilizing these block structures, we demonstrate the ease of construction of the space–time ROM method with two model problems: 2D diffusion...

Computer Methods in Applied Mechanics and Engineering

Journal of Computational Physics

A novel domain-decomposition least-squares Petrov-Galerkin (DD-LSPG) model-reduction method appli... more A novel domain-decomposition least-squares Petrov-Galerkin (DD-LSPG) model-reduction method applicable to parameterized systems of nonlinear algebraic equations (e.g., arising from discretizing a parameterized partial-differential-equations problem) is proposed. In contrast with previous works, we adopt an algebraically non-overlapping decomposition strategy rather than a spatial-decomposition strategy, which facilitates application to different spatial-discretization schemes. Rather than constructing a low-dimensional subspace for the entire state space in a monolithic fashion, the methodology constructs separate subspaces for the different subdomains/components characterizing the original model. During the offline stage, the method constructs low-dimensional bases for the interior and interface of subdomains/components. During the online stage, the approach constructs an LSPG reduced-order model for each subdomain/component (equipped with hyper-reduction in the case of nonlinear operators), and enforces strong or weak compatibility on the 'ports' connecting them. We propose several different strategies for defining the ingredients characterizing the methodology: (i) four different ways to construct reduced bases on the interface/ports of subdomains, and (ii) different ways to enforce compatibility across connecting ports. In particular, we show that the appropriate compatibility-constraint strategy depends strongly on the basis choice. In addition, we derive a posteriori and a priori error bounds for the DD-LSPG solutions. Numerical results performed on nonlinear benchmark problems in heat transfer and fluid dynamics that employ both finite-element and finite-difference spatial discretizations demonstrate that the proposed method performs well in terms of both accuracy and (parallel) computational cost, with different choices of basis and compatibility constraints yielding different performance profiles.

New methods for solving certain types of PDE-constrained optimization problems are presented in t... more New methods for solving certain types of PDE-constrained optimization problems are presented in this thesis. The approach taken is to augment state-of-the-art PDE methods. The PDE variables and the optimization variables are solved for simultaneously. This impacts the PDE method by changing the core from solving a system of nonlinear equations to that of finding a nonlinear saddle point. This in turn alters the character of the linear equations that are solved at each iteration. In the problem we address, the objective function has to match a given target state. Both volume and boundary controls are considered in order to match the target. Regularization is added to the objective function to aid stability and to facilitate computing the solution of the linear systems that arise within the algorithm. Solving such linear systems has been the focus of much research, with many methods being proposed. How to do this simultaneously and efficiently for the specific systems that arise in the PDE-constrained optimization problems of interest is the main focus of our work. The new methods have been implemented by modifying the cutting-edge software AERO-S. Numerical results are presented for a variety of problems including a flapping wing, a robotic control problem, and a thermal control problem. v I am thankful to be thankful. There are many to whom I would like to express gratitude. First and foremost, I am thankful to God for keeping me from going in a wrong direction and disciplining me because He loves me so much. He has been my strength and protection. He also has provided me with many good people who are essential for my life to flourish.

SIAM Journal on Scientific Computing

This work proposes a space-time least-squares Petrov-Galerkin (ST-LSPG) projection method for mod... more This work proposes a space-time least-squares Petrov-Galerkin (ST-LSPG) projection method for model reduction of nonlinear dynamical systems. In contrast to typical nonlinear model-reduction methods that first apply (Petrov-)Galerkin projection in the spatial dimension and subsequently apply time integration to numerically resolve the resulting low-dimensional dynamical system, the proposed method applies projection in space and time simultaneously. To accomplish this, the method first introduces a low-dimensional space-time trial subspace, which can be obtained by computing tensor decompositions of state-snapshot data. The method then computes discrete-optimal approximations in this space-time trial subspace by minimizing the residual arising after time discretization over all space and time in a weighted 2-norm. This norm can be defined to enable complexity reduction (i.e., hyper-reduction) in time, which leads to space-time collocation and space-time Gauss-Newton with Approximated Tensors (GNAT) variants of the ST-LSPG method. Advantages of the approach relative to typical spatial-projection-based nonlinear model reduction methods such as Galerkin projection and least-squares Petrov-Galerkin projection include a reduction of both the spatial and temporal dimensions of the dynamical system, and a priori error bounds that bound the solution error by the best space-time approximation error and whose stability constants exhibit slower growth in time. Numerical examples performed on model problems in fluid dynamics demonstrate the ability of the method to generate orders-of-magnitude computational savings relative to spatial-projection-based reduced-order models without sacrificing accuracy for a fixed spatio-temporal discretization.

An iterative, CFD-based approach for aeroelastic computations in the frequency domain is presente... more An iterative, CFD-based approach for aeroelastic computations in the frequency domain is presented. The method relies on a linearized formulation of the aeroelastic problem and a fixed-point iteration approach and enables the computation of the
eigenproperties of each of the wet aeroelastic eigenmodes. Numerical experiments on the aeroelastic analysis and design optimization of two wing configurations illustrate the capability of the method for the fast and accurate aeroelastic analysis of aircraft configurations and its advantage over classical time-domain approaches.

A novel methodology for accelerating the solution of PDE-constrained optimization is introduced. ... more A novel methodology for accelerating the solution of PDE-constrained optimization is introduced. It is based on an offline construction of database of local ROMs and an online interpolation within the database. The online flexibility of the ROM database approach makes it amenable to speedingup optimization-intensive applications such as robust optimization, multi-objectives optimization, and multi-start strategies for locating global optima. The accuracy of the ROM database model can be tuned in the offline phase where the database of local ROMs is constructed through a greedy procedure. In this work, a novel greedy algorithm based on saturation assumption is introduced to speed-up the ROM database construction procedure. The ROM database approach is applied to a
realistic wing design problems and leads to a large online speed-up.

New methods for solving certain types of PDE-constrained optimization problems are presented in t... more New methods for solving certain types of PDE-constrained optimization problems are presented in this thesis. The approach taken is to augment state-of-the-art PDE methods. The PDE variables and the optimization variables are solved for simultaneously. This impacts the PDE method by
changing the core from solving a system of nonlinear equations to that of finding a nonlinear saddle point. This in turn alters the character of the linear equations that are solved at each iteration.
In the problem we address, the objective function has to match a given target state. Both volume and boundary controls are considered in order to match the target. Regularization is added to the objective function to aid stability and to facilitate computing the solution of the linear systems that arise within the algorithm. Solving such linear systems has been the focus of much research, with many methods being proposed. How to do this simultaneously and efficiently for the specific systems that arise in the PDE-constrained optimization problems of interest is the main focus of our work.
The new methods have been implemented by modifying the cutting-edge software AERO-S. Numerical results are presented for a variety of problems including a flapping wing, a robotic control problem, and a thermal control problem.

Solving large-scale PDE-constrained optimization problems presents computational challenges due t... more Solving large-scale PDE-constrained optimization problems presents computational challenges due to the large dimensional set of underlying equations that have to be handled by the optimizer. Recently, projection-based nonlinear reduced-order models have been proposed to be used in place of high-dimensional models in a design optimization procedure. The dimensionality of the solution space is reduced using a reduced-order basis constructed by Proper Orthogonal Decomposition. In the case of nonlinear equations, however, this is not sufficient to ensure that the cost associated with the optimization procedure does not scale with the high dimension. To achieve that goal, an additional reduction step, hyper-reduction is applied. Then, solving the resulting reduced set of equations only requires a reduced dimensional domain and large speedups can be achieved. In the case of design optimization, it is shown in this paper that an additional approximation of the objective function is required. This is achieved by the construction of a surrogate objective using radial basis functions. The proposed method is illustrated with two applications: the shape optimization of a simplified nozzle inlet model and the design optimization of a chemical reaction.

A distributed optimal control problem with the constraint of a linear elliptic partial differenti... more A distributed optimal control problem with the constraint of a linear elliptic partial differential equation is considered. A necessary optimality condition for this problem forms a saddle point system, the efficient and accurate solution of which is crucial. A new factorization of the Schur complement for such a system is proposed and its characteristics discussed. The factorization introduces two complex factors that are complex conjugate to each other. The proposed solution methodology involves the application of a parallel linear domain decomposition solver---FETI-DPH---for the solution of the subproblems with the complex factors. Numerical properties of FETI-DPH in this context are demonstrated, including numerical and parallel scalability and regularization dependence. The new factorization can be used to solve Schur complement systems arising in both range-space and full-space formulations. In both cases, numerical results indicate that the complex factorization is promising.

Risk Analysis, 2010

We superimpose a radiation fallout model onto a traffic flow model to assess the evacuation versu... more We superimpose a radiation fallout model onto a traffic flow model to assess the evacuation versus shelter-in-place decisions after the daytime ground-level detonation of a 10-kt improvised nuclear device in Washington, DC. In our model, ≈80k people are killed by the prompt effects of blast, burn, and radiation. Of the ≈360k survivors without access to a vehicle, 42.6k would die if they immediately self-evacuated on foot. Sheltering above ground would save several thousand of these lives and sheltering in a basement (or near the middle of a large building) would save of them. Among survivors of the prompt effects with access to a vehicle, the number of deaths depends on the fraction of people who shelter in a basement rather than self-evacuate in their vehicle: 23.1k people die if 90% shelter in a basement and 54.6k die if 10% shelter. Sheltering above ground saves approximately half as many lives as sheltering in a basement. The details related to delayed (i.e., organized) evacuation, search and rescue, decontamination, and situational awareness (via, e.g., telecommunications) have very little impact on the number of casualties. Although antibiotics and transfusion support have the potential to save ≈10k lives (and the number of lives saved from medical care increases with the fraction of people who shelter in basements), the logistical challenge appears to be well beyond current response capabilities. Taken together, our results suggest that the government should initiate an aggressive outreach program to educate citizens and the private sector about the importance of sheltering in place in a basement for at least 12 hours after a terrorist nuclear detonation.

ArXiv, 2020

Traditional linear subspace reduced order models (LS-ROMs) are able to accelerate physical simula... more Traditional linear subspace reduced order models (LS-ROMs) are able to accelerate physical simulations, in which the intrinsic solution space falls into a subspace with a small dimension, i.e., the solution space has a small Kolmogorov n-width. However, for physical phenomena not of this type, such as advection-dominated flow phenomena, a low-dimensional linear subspace poorly approximates the solution. To address cases such as these, we have developed an efficient nonlinear manifold ROM (NM-ROM), which can better approximate high-fidelity model solutions with a smaller latent space dimension than the LS-ROMs. Our method takes advantage of the existing numerical methods that are used to solve the corresponding full order models (FOMs). The efficiency is achieved by developing a hyper-reduction technique in the context of the NM-ROM. Numerical results show that neural networks can learn a more efficient latent space representation on advection-dominated data from 2D Burgers' equa...

ArXiv, 2021

This work is the first to employ and adapt the image-to-image translation concept based on condit... more This work is the first to employ and adapt the image-to-image translation concept based on conditional generative adversarial networks (cGAN) towards learning a forward and an inverse solution operator of partial differential equations (PDEs). Even though the proposed framework could be applied as a surrogate model for the solution of any PDEs, here we focus on steady-state solutions of coupled hydro-mechanical processes in heterogeneous porous media. Strongly heterogeneous material properties, which translate to the heterogeneity of coefficients of the PDEs and discontinuous features in the solutions, require specialized techniques for the forward and inverse solution of these problems. Additionally, parametrization of the spatially heterogeneous coefficients is excessively difficult by using standard reduced order modeling techniques. In this work, we overcome these challenges by employing the image-to-image translation concept to learn the forward and inverse solution operators a...

ArXiv, 2018

Several reduced order models have been successfully developed for nonlinear dynamical systems. To... more Several reduced order models have been successfully developed for nonlinear dynamical systems. To achieve a considerable speedup, a hyper-reduction step is needed to reduce the computational complexity due to nonlinear terms. Many hyper-reduction techniques require the construction of nonlinear term basis, which introduces a computationally expensive offline phase. A novel way of constructing nonlinear term basis within the hyper-reduction process is introduced. In contrast to the traditional hyper-reduction techniques where the collection of nonlinear term snapshots is required, the SNS method completely avoids the use of the nonlinear term snapshots. Instead, it uses the solution snapshots that are used for building a solution basis. Furthermore, it avoids an extra data compression of nonlinear term snapshots. As a result, the SNS method provides a more efficient offline strategy than the traditional model order reduction techniques, such as the DEIM, GNAT, and ST-GNAT methods. Th...

ArXiv, 2019

Although design optimization has shown its great power of automatizing the whole design process a... more Although design optimization has shown its great power of automatizing the whole design process and providing an optimal design, using sophisticated computational models, its process can be formidable due to a computationally expensive large-scale linear system of equations to solve, associated with underlying physics models. We introduce a general reduced order model-based design optimization acceleration approach that is applicable not only to design optimization problems, but also to any PDE-constrained optimization problems. The acceleration is achieved by two techniques: i) allowing an inexact linear solve and ii) reducing the number of iterations in Krylov subspace iterative methods. The choice between two techniques are made, based on how close a current design point to an optimal point. The advantage of the acceleration approach is demonstrated in topology optimization examples, including both compliance minimization and stress-constrained problems, where it achieves a treme...

Traditional linear subspace reduced order models (LS-ROMs) are able to accelerate physical simula... more Traditional linear subspace reduced order models (LS-ROMs) are able to accelerate physical simulations, in which the intrinsic solution space falls into a subspace with a small dimension, i.e., the solution space has a small Kolmogorov n-width. However, for physical phenomena not of this type, e.g., any advection-dominated flow phenomena, such as in traffic flow, atmospheric flows, and air flow over vehicles, a low-dimensional linear subspace poorly approximates the solution. To address cases such as these, we have developed a fast and accurate physics-informed neural network ROM, namely nonlinear manifold ROM (NM-ROM), which can better approximate high-fidelity model solutions with a smaller latent space dimension than the LS-ROMs. Our method takes advantage of the existing numerical methods that are used to solve the corresponding full order models. The efficiency is achieved by developing a hyper-reduction technique in the context of the NM-ROM. Numerical results show that neural...

Computer Methods in Applied Mechanics and Engineering

Mathematics

A classical reduced order model (ROM) for dynamical problems typically involves only the spatial ... more A classical reduced order model (ROM) for dynamical problems typically involves only the spatial reduction of a given problem. Recently, a novel space–time ROM for linear dynamical problems has been developed [Choi et al., Space–tume reduced order model for large-scale linear dynamical systems with application to Boltzmann transport problems, Journal of Computational Physics, 2020], which further reduces the problem size by introducing a temporal reduction in addition to a spatial reduction without much loss in accuracy. The authors show an order of a thousand speed-up with a relative error of less than 10−5 for a large-scale Boltzmann transport problem. In this work, we present for the first time the derivation of the space–time least-squares Petrov–Galerkin (LSPG) projection for linear dynamical systems and its corresponding block structures. Utilizing these block structures, we demonstrate the ease of construction of the space–time ROM method with two model problems: 2D diffusion...

Computer Methods in Applied Mechanics and Engineering

Journal of Computational Physics

A novel domain-decomposition least-squares Petrov-Galerkin (DD-LSPG) model-reduction method appli... more A novel domain-decomposition least-squares Petrov-Galerkin (DD-LSPG) model-reduction method applicable to parameterized systems of nonlinear algebraic equations (e.g., arising from discretizing a parameterized partial-differential-equations problem) is proposed. In contrast with previous works, we adopt an algebraically non-overlapping decomposition strategy rather than a spatial-decomposition strategy, which facilitates application to different spatial-discretization schemes. Rather than constructing a low-dimensional subspace for the entire state space in a monolithic fashion, the methodology constructs separate subspaces for the different subdomains/components characterizing the original model. During the offline stage, the method constructs low-dimensional bases for the interior and interface of subdomains/components. During the online stage, the approach constructs an LSPG reduced-order model for each subdomain/component (equipped with hyper-reduction in the case of nonlinear operators), and enforces strong or weak compatibility on the 'ports' connecting them. We propose several different strategies for defining the ingredients characterizing the methodology: (i) four different ways to construct reduced bases on the interface/ports of subdomains, and (ii) different ways to enforce compatibility across connecting ports. In particular, we show that the appropriate compatibility-constraint strategy depends strongly on the basis choice. In addition, we derive a posteriori and a priori error bounds for the DD-LSPG solutions. Numerical results performed on nonlinear benchmark problems in heat transfer and fluid dynamics that employ both finite-element and finite-difference spatial discretizations demonstrate that the proposed method performs well in terms of both accuracy and (parallel) computational cost, with different choices of basis and compatibility constraints yielding different performance profiles.

New methods for solving certain types of PDE-constrained optimization problems are presented in t... more New methods for solving certain types of PDE-constrained optimization problems are presented in this thesis. The approach taken is to augment state-of-the-art PDE methods. The PDE variables and the optimization variables are solved for simultaneously. This impacts the PDE method by changing the core from solving a system of nonlinear equations to that of finding a nonlinear saddle point. This in turn alters the character of the linear equations that are solved at each iteration. In the problem we address, the objective function has to match a given target state. Both volume and boundary controls are considered in order to match the target. Regularization is added to the objective function to aid stability and to facilitate computing the solution of the linear systems that arise within the algorithm. Solving such linear systems has been the focus of much research, with many methods being proposed. How to do this simultaneously and efficiently for the specific systems that arise in the PDE-constrained optimization problems of interest is the main focus of our work. The new methods have been implemented by modifying the cutting-edge software AERO-S. Numerical results are presented for a variety of problems including a flapping wing, a robotic control problem, and a thermal control problem. v I am thankful to be thankful. There are many to whom I would like to express gratitude. First and foremost, I am thankful to God for keeping me from going in a wrong direction and disciplining me because He loves me so much. He has been my strength and protection. He also has provided me with many good people who are essential for my life to flourish.

SIAM Journal on Scientific Computing

This work proposes a space-time least-squares Petrov-Galerkin (ST-LSPG) projection method for mod... more This work proposes a space-time least-squares Petrov-Galerkin (ST-LSPG) projection method for model reduction of nonlinear dynamical systems. In contrast to typical nonlinear model-reduction methods that first apply (Petrov-)Galerkin projection in the spatial dimension and subsequently apply time integration to numerically resolve the resulting low-dimensional dynamical system, the proposed method applies projection in space and time simultaneously. To accomplish this, the method first introduces a low-dimensional space-time trial subspace, which can be obtained by computing tensor decompositions of state-snapshot data. The method then computes discrete-optimal approximations in this space-time trial subspace by minimizing the residual arising after time discretization over all space and time in a weighted 2-norm. This norm can be defined to enable complexity reduction (i.e., hyper-reduction) in time, which leads to space-time collocation and space-time Gauss-Newton with Approximated Tensors (GNAT) variants of the ST-LSPG method. Advantages of the approach relative to typical spatial-projection-based nonlinear model reduction methods such as Galerkin projection and least-squares Petrov-Galerkin projection include a reduction of both the spatial and temporal dimensions of the dynamical system, and a priori error bounds that bound the solution error by the best space-time approximation error and whose stability constants exhibit slower growth in time. Numerical examples performed on model problems in fluid dynamics demonstrate the ability of the method to generate orders-of-magnitude computational savings relative to spatial-projection-based reduced-order models without sacrificing accuracy for a fixed spatio-temporal discretization.

An iterative, CFD-based approach for aeroelastic computations in the frequency domain is presente... more An iterative, CFD-based approach for aeroelastic computations in the frequency domain is presented. The method relies on a linearized formulation of the aeroelastic problem and a fixed-point iteration approach and enables the computation of the
eigenproperties of each of the wet aeroelastic eigenmodes. Numerical experiments on the aeroelastic analysis and design optimization of two wing configurations illustrate the capability of the method for the fast and accurate aeroelastic analysis of aircraft configurations and its advantage over classical time-domain approaches.

A novel methodology for accelerating the solution of PDE-constrained optimization is introduced. ... more A novel methodology for accelerating the solution of PDE-constrained optimization is introduced. It is based on an offline construction of database of local ROMs and an online interpolation within the database. The online flexibility of the ROM database approach makes it amenable to speedingup optimization-intensive applications such as robust optimization, multi-objectives optimization, and multi-start strategies for locating global optima. The accuracy of the ROM database model can be tuned in the offline phase where the database of local ROMs is constructed through a greedy procedure. In this work, a novel greedy algorithm based on saturation assumption is introduced to speed-up the ROM database construction procedure. The ROM database approach is applied to a
realistic wing design problems and leads to a large online speed-up.

New methods for solving certain types of PDE-constrained optimization problems are presented in t... more New methods for solving certain types of PDE-constrained optimization problems are presented in this thesis. The approach taken is to augment state-of-the-art PDE methods. The PDE variables and the optimization variables are solved for simultaneously. This impacts the PDE method by
changing the core from solving a system of nonlinear equations to that of finding a nonlinear saddle point. This in turn alters the character of the linear equations that are solved at each iteration.
In the problem we address, the objective function has to match a given target state. Both volume and boundary controls are considered in order to match the target. Regularization is added to the objective function to aid stability and to facilitate computing the solution of the linear systems that arise within the algorithm. Solving such linear systems has been the focus of much research, with many methods being proposed. How to do this simultaneously and efficiently for the specific systems that arise in the PDE-constrained optimization problems of interest is the main focus of our work.
The new methods have been implemented by modifying the cutting-edge software AERO-S. Numerical results are presented for a variety of problems including a flapping wing, a robotic control problem, and a thermal control problem.

Solving large-scale PDE-constrained optimization problems presents computational challenges due t... more Solving large-scale PDE-constrained optimization problems presents computational challenges due to the large dimensional set of underlying equations that have to be handled by the optimizer. Recently, projection-based nonlinear reduced-order models have been proposed to be used in place of high-dimensional models in a design optimization procedure. The dimensionality of the solution space is reduced using a reduced-order basis constructed by Proper Orthogonal Decomposition. In the case of nonlinear equations, however, this is not sufficient to ensure that the cost associated with the optimization procedure does not scale with the high dimension. To achieve that goal, an additional reduction step, hyper-reduction is applied. Then, solving the resulting reduced set of equations only requires a reduced dimensional domain and large speedups can be achieved. In the case of design optimization, it is shown in this paper that an additional approximation of the objective function is required. This is achieved by the construction of a surrogate objective using radial basis functions. The proposed method is illustrated with two applications: the shape optimization of a simplified nozzle inlet model and the design optimization of a chemical reaction.

A distributed optimal control problem with the constraint of a linear elliptic partial differenti... more A distributed optimal control problem with the constraint of a linear elliptic partial differential equation is considered. A necessary optimality condition for this problem forms a saddle point system, the efficient and accurate solution of which is crucial. A new factorization of the Schur complement for such a system is proposed and its characteristics discussed. The factorization introduces two complex factors that are complex conjugate to each other. The proposed solution methodology involves the application of a parallel linear domain decomposition solver---FETI-DPH---for the solution of the subproblems with the complex factors. Numerical properties of FETI-DPH in this context are demonstrated, including numerical and parallel scalability and regularization dependence. The new factorization can be used to solve Schur complement systems arising in both range-space and full-space formulations. In both cases, numerical results indicate that the complex factorization is promising.

Risk Analysis, 2010

We superimpose a radiation fallout model onto a traffic flow model to assess the evacuation versu... more We superimpose a radiation fallout model onto a traffic flow model to assess the evacuation versus shelter-in-place decisions after the daytime ground-level detonation of a 10-kt improvised nuclear device in Washington, DC. In our model, ≈80k people are killed by the prompt effects of blast, burn, and radiation. Of the ≈360k survivors without access to a vehicle, 42.6k would die if they immediately self-evacuated on foot. Sheltering above ground would save several thousand of these lives and sheltering in a basement (or near the middle of a large building) would save of them. Among survivors of the prompt effects with access to a vehicle, the number of deaths depends on the fraction of people who shelter in a basement rather than self-evacuate in their vehicle: 23.1k people die if 90% shelter in a basement and 54.6k die if 10% shelter. Sheltering above ground saves approximately half as many lives as sheltering in a basement. The details related to delayed (i.e., organized) evacuation, search and rescue, decontamination, and situational awareness (via, e.g., telecommunications) have very little impact on the number of casualties. Although antibiotics and transfusion support have the potential to save ≈10k lives (and the number of lives saved from medical care increases with the fraction of people who shelter in basements), the logistical challenge appears to be well beyond current response capabilities. Taken together, our results suggest that the government should initiate an aggressive outreach program to educate citizens and the private sector about the importance of sheltering in place in a basement for at least 12 hours after a terrorist nuclear detonation.