This section contains an overview of my current research interests, and some of the class and research projects that I have worked on before, listed
in reverse chronological order.
This section is incomplete and is being updated!
Current/Ongoing
Thesis Work
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Computational aspects of waveform inversion and extended waveform inversion.
Joint work and advised by Prof. Biondo Biondi.
This work is broadly motivated by the computational challenges in extended waveform inversion, in particular the method of
"Tomographic full waveform inversion". In 2D, we have shown that a direct factorization of the Helmoltz equation is a viable approach towards
making this technique computationally efficient. This method does not work in 3D, and so what to do in this setting is the current objective of
my research.
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Numerical implementation of the inverse problem for the local geodesic X-ray transform on Riemannian manifolds with boundary.
Advised by Prof. András Vasy.
This work is based on this paper by András Vasy and Gunther Uhlmann.
I am working on a numerical implementation of the layer-stripping algorithm described in the paper.
Other
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Seismic image segmentation using topological data analysis.
Joint work with Bradley J. Nelson.
See here.
2020
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The index of invariace and its implications for a parameterized least squares problem.
[More info]
Joint work with Léopold Cambier.
We are working to generalize a result of Eric Hallman and Ming Gu that shows that minimizers of a certain least squares type objective function,
that generalizes MINRES, lie of low dimensional affine subspaces. We show that the result holds not just for Krylov subspaces, but on any affine
subspaces. The dimension of the affine subspace is bounded by a quantity called the "index of invariance", which we introduce in this work.
Some properties of this quantity are studied in the paper. We also provide some conditions when the bound is tight. Finally a new matrix manifold
is characterized.
Updates:
Aug 2020: The paper is now available on arXiv.
May 2020: We have a short presentation
in the ICME XPO 2020 guide.
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A graph-based formalism for surface codes and twists. [More info]
Joint work with Theodore J. Yoder.
We have constructed a new class of quantum stabilizer codes based on rotation systems that generalize many existing code families, such as
stellated codes, Kitaev's toric code etc. Since rotation systems fully encode how an undirected graph is 2-cell embedded on a closed 2-dimensional
manifold, our codes can be viewed as a class of topological quantum codes. Extremal cyclically anticommuting lists of Paulis play a key role in
this construction. Notions such as graph "checkerboardability" turn out to be important for deriving relationships between the number of encoded qubits,
and the number of qubits used in the code, along with the genus of the manifold. Checkerboardability also influences the logical operators that one obtains
from our constructions.
Updates:
Jan 2021: The first draft of the paper is now online on arXiv.
Aug 2020: Current status of the work is being presented at the FTQT Meeting.
June 2020: I presented updates of this work at the SESAAI 2020 Annual Meeting.
[Slides]
[Video]
May 2020: I presented updates of this work for the Schlumberger Innovation Fellowship Talk at
STIC. [Slides]
Mar 2020: Ted's
APS March meeting talk
was delivered online, due to COVID restrictions.
2019
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On sets of commuting and anticommuting Paulis. [More info]
Joint work with Ewout van den Berg.
The Pauli group is important in the theory of quantum computing. For example, commuting sets of Paulis
are central in the theory of quantum stabilizer codes, while anticommuting sets of Paulis are used to map
Fermionic systems to qubit algebra, such as using the Jordan-Wigner transform, which allows Fermionic systems to be simulated
using quantum computers. Anticommuting Paulis are also used in wireless communication, such as in the design
of space-time codes. In this work, we study structure and cardinality of maximal sets of commuting and anticommuting
Paulis in the setting of the abelian Pauli group. We provide necessary and sufficient conditions for
anticommuting sets to be maximal, and present an efficient algorithm for generating anticommuting sets
of maximum size. These results appear to be new, even though the Pauli group has been extremely well-studied before.
The current version of our paper is available on arXiv.
Updates:
Nov 2019: This work has been accepted for publication in the journal Research in the Mathematical Sciences.
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Efficient quadratic Ising Hamiltonian generation with qubit reduction. [More info]
Joint work with Marco Pistoia.
We start with an optimization problem with a polynomial objective function, and polynomial equality and inequality constraints, where the
variables are integer-valued. We tackle the problem of how to efficiently map it to a quantum computer, specifically focusing on decreasing
the qubit count needed to represent the problem. This is important for near-term applications in the NISQ era of quantum computing. The problem
is first transformed into an unconstrained higher order binary optimization problem, and then quadratization techniques are employed to reduce it
to an equivalent quadratic pseudo-boolean function (PBF). Finally roof duality and extended roof duality techniques are employed to automatically
fix some of the binary variables in the PBF, to obtain a new PBF with less number of binary variables. This technique leads to a reduction in the
number of required qubits in a problem-dependent fashion.
Updates:
Nov 2019: A US Patent has been filed by IBM Corporation.
2018
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A recurrent neural-network based deep-learning learning approach for seismic velocity estimation in time. [More info]
Joint work with Gabriel Fabien-Ouellet.
We design a recurrent neural-network for the task of estimating the interval velocity and the root-mean-square velocity in time, starting
from CMP gathers. An important question that this work addresses is whether semblance calculation is strictly necessary or not for
seismic velocity analysis, as is done in any conventional processing workflow. Our work shows that it is not, and in fact our opinion is
that starting from the data itself as inputs for a ML based velocity estimation workflow, is much better than starting with semblance
panels because semblance panels have strictly less information content than the gathers themselves. Our work assumes a horizontally
layered medium, which is a standard assumption for seismic velocity analysis in time.
Updates:
Nov 2019: This work has now been accepted for publication in the journal Geophysics.
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Seismic image segmentation using topological data analysis. [More info]
Joint work with Bradley J. Nelson.
We propose a supervised classification scheme for automatic segmentation of a seismic image. The image is broken up into tiles, and then
persistent barcodes are derived from a sub- or super-level set filtration of the image tile. Polynomial features are then derived from this
barcode representation. These barcodes track how homology changes as a function of the filtration. We show that these polynomial features can be
used to classify the seismic images. The classification accuracy is comparable in performance to state-of-the-art CNNs for the same task.
The TDA based method has the advantage that there is no need to augment the training data set based on rotations, as the topological features
are rotation invariant. We test our method on the LANDMASS datasets. We are currently working on a draft of the full paper, to be put on arXiv soon.
Updates:
Jun 2019: I presented this work at the 81st EAGE Conference & Exhibition in London.
May 2019: I and Brad presented a poster on this work at ICME Xpo 2019.
May 2019: I presented this work at the SEP annual meeting in Monterrey.
Mar 2019: Brad presented this work in the SEP seminar.
[pdf]
Mar 2019: Our conference paper on the current status of this work is accepted for publication in the
81st EAGE Conference & Exhibition in London.
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Snell tomography for net-to-gross estimation using quantum annealing. [More info]
Joint work with Stewart A. Levin.
In this work, we explore how to exploit the potential power of quantum annealers in an unusual tomographic challenge of
estimating material percentages such as net-to-gross ratios using sparse offset-traveltime information.
By restricting the constituent materials in an isotropic horizontally-stratified medium to a specific set with well-known
acoustic properties, transmission tomography can provide the relative fraction of each material within the set of layers
over which the ray traverses. In this paper, this problem is formulated as a quadratic unconstrained binary optimization
problem suitable for solving using a quantum annealer.
Updates:
Oct 2018: I presented this paper at the 88th SEG Annual International Meeting in Anaheim.
Jun 2018: Our conference paper on this work is accepted for publication in the
88th SEG Annual International Meeting in Anaheim.
2017
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Information directed reinforcement learning. [More info]
Joint work with Andrea Zanette.
This was a project done as part of the CS234 class Reinforcement learning,
taught by Emma Brunskill, at Stanford University.
Efficient exploration is recognized as a key difficulty in reinforcement learning. We consider an episodic undiscounted MDP
where the goal is to minimize the sum of regrets over different episodes. Classical methods are either based on optimism in
the face of uncertainty or on probability matching. In this project we explore an approach that aims at quantifying the cost
of exploration while remaining computationally tractable.
2016
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Automated Aircraft Touchdown. [More info]
Joint work with Amy Shoemaker and Sagar Vare.
This was a project done as part of the CS238 class Decision making under uncertainty,
taught by Mykel J. Kochenderfer, at Stanford University.
We tackle the problem of making an aircraft land safely in an automated manner in difficult landing conditions. In a simplified
version of the problem, we model the aircraft's decisions to be from a finite set that includes sideways and vertical movements.
A simulator was designed to perform experiments in various conditions including stormy weather and
starting from off-trajectory starting points. We experiment with a wide range of techniques to learn the optimal flight
policy including reinforcement learning, Monte Carlo based dynamic programming and direct policy search; and through our
experiments we are able to analyze the efficiency of these strategies with respect to the problem at hand.
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Enclosing an ellipse with N rectangles so as to minimize the area between the ellipse and the rectangles. [More info]
This is an interesting class project done as part of the CME304 Numerical Optimization class at Stanford University,
taught by Walter Murray, where the goal is to find a cover for an ellipse with
N rectangles, such that the area outside the ellipse is minimized. In this project, I first formulate an equivalent problem that
reduces to finding a cover for a circle with the same number of rectangles. The problem is then solved using a Modified-Newton
Hessian based approach, and the performance is compared against Steepest Descent. It is found that Modified Newton based approach
is much more efficient, although each iteration is significantly more expensive compared to Steepest Descent. I also compare the
performance of using different line search algorithms like Goldstein vs Strong-Wolfe conditions, and conclude that the Strong-Wolfe
conditions provide much better results.