CMAI Research

Optimization & Control

Optimization problem with constraints given by partial (ordinary) differential equations (PDEs / ODEs) can be written as

min J(u,z),
subject to: e(u,z) = 0,
over (u,z) ∈ U_ad x Z_ad.

Here, J: U x Z → ℜ stands for the objective functional depending on the state variables u ∈ U and the control variables z ∈ Z. The equation corresponds to the PDE (linear/nonlinear), and U_ad ⊂ U, Z_ad ⊂ Z refer to the sets of admissible states and control variables, respectively. The variable z could be an optimal control (optimal control problems) or a shape parameter (shape optimization problems).

The numerical solution to PDE constrained optimization problems involves a series of theoretical and practical challenges:

Solving a PDE constrained optimization problem not only require a solution to state equations but adjoint equations as well.
The PDEs may not equalities, but could be complementarity problems or variational inequalities. Also, these problems may contain uncertainty due to unknown boundary conditions, coefficients, etc.
The structural interaction between optimization algorithms and the underlying PDE and the impact of the discretization processes have to be taken into account.
The numerical approaches typically lead to large-scale nonlinear programming problems. With regard to algorithmic complexity, their numerical solution requires the use of efficient iterative schemes such as multilevel techniques.
Significant savings both in terms of memory and computational time maybe needed via model reduction techniques.

[1]	H. Antil, Drew Kouri, Martin Lacasse, and Denis Ridzal. Frontiers in PDE-constrained Optimization. The IMA Volumes in Mathematics and its Applications, Springer 2018.

Shape Optimization

Magnetic Drug Targeting

Moving the drug from left to right with the help of three magnets located outside the domain.

COVID Simulation

Distribution of virus after sneezing in a typical hospital room

Electrowetting

The goal is to move the droplet placed on top of electrodes, acting as control, to a desired location.

Big Data and Reduced Order Models

Consider a linear or nonlinear dynamical system and quantity of interest given by y. We are interested in deriving a reduced order model, which is cheaper and faster to evaluate.

Full 3D turbulence model and vorticity formation .

Gravitational Waves

Imaging

Event or Neuromorphic cameras are novel biologically inspired sensors that record data based on the change in light intensity at each pixel asynchronously. They have a temporal resolution of microseconds. This is useful for scenes with fast moving objects that can cause motion blur in traditional cameras, which record the average light intensity over an exposure time for each pixel synchronously. In our work, we consider bilevel optimization based variational framework for neuromorphic imaging.

Image Denoising

Denoised image using total variation regularization

Denoised image using variable order fractional model

Deep Learning

Our group’s work on deep neural networks (DNNs) focuses on writing DNNs as constrained optimization problems. In particular, we have introduced DNNs with memory, which helps overcome the vanishing gradient challenge. We have also explored reducing the computational complexity of DNNs by introducing a bias ordering. These proposed DNNs have been shown to be excellent surrogates to parameterized (nonlinear) partial differential equations (PDEs), Bayesian inverse problems, and data assimilation problems, with multiple advantages over the traditional approaches. We have applied these DNNs to chemically reacting flow problems. The latter requires solving a system of stiff ODEs and fluid flow equations. These are highly challenging problems, for instance, for combustion the number of reactions can be significant (over 100). Due to the large CPU requirements of chemical reactions (over 99% of total CPU time), a large number of flow and combustion problems are presently beyond the capabilities of even the largest supercomputers.

Nonlinear & Nonlocal PDEs

It is becoming clear that the assumptions leading to classical PDE based models are not always accurate or realistic. Fractional models have emerged as a replacement because: they can capture sharp transitions across interfaces and they can capture nonlocal effects. Such problems occur in imaging science and phase field models. The fractional Laplacian has received a tremendous amount of attention partly due to the fact that it is the generator of the Le ́vy process. Surprisingly, the probability community has been using these concepts for several decades. A not so well-known fact is that fractional operators natural arise in the celebrated Haldane-Shastry and Calogero-Moser quantum spin chains models and they are also emerging as a natural candidate for classical Oseen-Frank energy for a nonlocal Liquid crystal model. The nonlocal models are also playing a significant role in geophysical electromagnetics and machine learning. Using the Nonlocal models, it is also possible to introduce novel optimal control concepts, such as exterior optimal control.