The goal here is to utilize a mathematical model of the transport phenomenon for Source id. This mathematical model is often a PDE that is defined over a domain of interest. We consider Advection-Diffusion and Acoustic PDEs as running examples but the developed techniques can be applied to other transport models as well. In particular, given an arbitrary domain that contains a set of unknown sources and a set of stationary sensors that can measure a quantity generated by the sources, we are interested in predicting the shape, location, and intensity of the sources based on a limited number of noisy measurements.

## Robotic Source Identification

Source Identification refers to the estimation of a source using a set of measurements of a quantity that is generated under the action of that source. The Source id problem has various applications ranging from environmental protection to human safety. Locating atmospheric, underground, or underwater pollutants, finding the source of a hazardous chemical leakage, demining, and fire detection are a few examples. In addition, Source id can be an important component in higher level tasks like search and rescue missions and crowd evacuation.

Due to its importance, Source id has been investigated extensively. Many approaches are heuristics that neglect the physics of the problem or model-based methods that are usually computationally expensive given the resources available to a mobile robot. In this project, we are interested in developing practical algorithms for a single mobile robot or a team of autonomous agents that can be implemented onboard to solve realistic Source id problems in real-time. To reach this goal, we need to address two connected points:

- Design of tractable model-based Source id algorithms that can solve realistic problems. To this end, we utilize the latest advancements in the field of numerical computing. Particularly, we use finite element method, model order reduction techniques, parallel computing, and state of the art optimization algorithms. This step includes formulating tractable Source id problems and employing efficient optimization schemes to solve them in real-time.
- Design of trajectories that the agents follow in the domain of interest to collect optimal measurements. We formulate this task as an optimization problem that is also solved online since it requires feedback from the Source id algorithm. The resulting optimization problem depends on the optimality measure and the mathematical model used to approximate the physics of the transport phenomenon.

An integrated part of this project is experimental validation. We are also interested in a stochastic view of the Source id problem and we are currently developing tractable methods to quantify the identification uncertainty. A short description of the work that has been done to address these points is given in the following.

### Model-based Source Identification

#### Sparse Source Identification

Here we follow a discretize-optimize approach for Source id that results in a convex formulation of the problem in the expense of high dimensionality. Particularly, we discretize the PDE using the finite element method to formulate a Source id optimization problem. Since the sought source vector is typically sparse, we propose a novel Reweighted l_{1} Regularization technique combined with Least Squares Debiasing to obtain a unique, sparse, reconstructed source vector. This method is shown to work well especially when the signal to noise ratio is large.

- "Model-based Sparse Source Identification“,
**R. Khodayi-mehr**, W. Aquino, and M. M. Zavlanos,*American Control Conference, Chicago, July 2015, pp.1818-1823.*

#### Nonlinear Source Identification

Although convex formulations of the Source id problem possess various nice properties, they are not directly suitable for implementation on autonomous robots. Here we develop an extremely low-dimensional formulation of the Source id problem at the expense of nonlinearity. Particularly, we follow an optimize-discretize approach to formulate an optimization problem in function space and utilize the adjoint method to calculate the gradient. To obtain a finite dimensional representation, we employ proper orthogonal decomposition and parametrize the source function by nonlinear tower functions. The above approximations give rise to a low dimensional nonlinear optimization problem that can be solved for the desired source function.

- "Nonlinear Reduced Order Source Identification",
**R. Khodayi-mehr**, W. Aquino, and M. M. Zavlanos,*American Control Conference, Boston, July 2016, pp. 6302-6307.*

#### Distributed Source Identification

The objective here is to propose a scalable decentralized Source id algorithm that a team of autonomous agents cooperatively implement to estimate the desired source function.

- "Distributed Reduced Order Source Identification",
**R. Khodayi-mehr**, W. Aquino, and M. M. Zavlanos,*American Control Conference, Seattle, May 2017*(under review).

### Stochastic Source Identification

The deterministic approaches to Source id problem only provide a point estimate of the unknown source function without any confidence measures for the solution. The purpose of this work is to develop tractable stochastic Source id algorithms that in addition to point estimates, also quantify the uncertainty of the solution.

### Active Source Identification

The second component for an efficient solution of the Source id problem is the collection of optimal measurements. From a robotics point of view, this is equivalent to optimal path planning for the autonomous agents. The solution to this problem depends on the particular formulation of the Source id problem and the value of the unknown source, creating a feedback loop between the two components, namely the Source id algorithm and the path planning problem.

- "Model-Based Active Source Identification in Complex Environments",
**R. Khodayi-mehr**, W. Aquino, and M. M. Zavlanos,*IEEE Transactions on Robotics*(under review).

## Future Research

To appear ...