I am a tenure-track Goenka Family Assistant Professor at the Department of Mathematics, Cornell University.

I obtained my Ph.D. in 2018 from the Department of Mathematics, UT Austin, supervised by Prof. Björn Engquist. I was a Courant Instructor at Courant Institute of Mathematical Sciences, NYU, from 09/2018 to 08/2021, and a Simons-Berkeley Research Fellow at Simons Institute for the Theory of Computing in Fall 2021. I was an Advanced Fellow at the Institute for Theoretical Studies, ETH Zürich from 01/2022 to 06/2023.

I am interested in Numerical Analysis, Inverse Problems, Nonconvex Optimization, Optimal Transport, and Machine Learning. Here is my Curriculum Vitae.

Office: Malott Hall 582, Tower Rd, Ithaca, NY 14853, USA

Email: yunan.yang@cornell.edu

is formalized curiosity. It is poking and prying with a purpose. — Zora Neale Hurston

The optimal mass transport problem seeks the most efficient way of transforming one mass distribution to the other relative to a given cost function. It was first brought up by Monge in 1781 and later expanded by Kantorovich. The topic of optimal transportation has been a prominent subject of study for several decades, owing to the sound theoretical foundation established over the past two centuries through mathematical analysis. One of my main research interests is exploring the potential of optimal transport in Applied Mathematics. My PhD thesis work was on this topic.

Here is a list of my work focusing on the application and extension on the topic of optimal transportation, especially on seismic inversion, e.g., Full-Waveform Inversion (FWI):

- Botvinick-Greenhouse, J.,
**Yang, Y.**, Maulik, R., 2023. Generative Modeling of Time-Dependent Densities via Optimal Transport and Projection Pursuit. submitted; Chaos; arXiv preprint arXiv:2304.09663. [pdf] - Han, R., Slepčev, D.,
**Yang, Y.**, 2023. HV Geometry for Signal Comparison. To Appear in Quarterly of Applied Mathematics. [pdf] **Yang, Y.**, Nurbekyan, L., Negrini, E., Martin, R. and Pasha, M., 2023. Optimal transport for parameter identification of chaotic dynamics via invariant measures. SIAM Journal on Applied Dynamical Systems, 22(1):269–310, 2023. [pdf]- Frederick, C. and
**Yang, Y.**, 2022. Snapshots of modern mathematics from Oberwolfach, Mathematisches Forschungsinstitut Oberwolfach, 2022-04. [pdf] - Engquist, B. and
**Yang, Y.**, 2022. Optimal Transport Based Seismic Inversion: Beyond Cycle Skipping. Communications on Pure and Applied Mathematics. [pdf] - Dunlop, M. and
**Yang, Y.**, 2021. Stability of Gibbs Posteriors from the Wasserstein Loss for Bayesian Full Waveform Inversion. SIAM/ASA Journal on Uncertainty Quantification, 9(4), pp.1499-1526. [pdf] - Engquist, B., Ren, K. and
**Yang, Y.**, 2020. The quadratic Wasserstein metric for inverse data matching. Inverse Problems, 36(5), p.055001. [pdf] - Engquist, B. and
**Yang, Y.**, 2019. Seismic imaging and optimal transport. Communications in Information and Systems, Vol. 19, No. 2 (2019), pp. 95-145 [pdf] - Engquist, B. and
**Yang, Y.**, 2019. Seismic inversion and the data normalization for optimal transport. Methods and Applications of Analysis, Vol. 26, No. 2 (2019), pp. 133-148. [pdf] **Yang, Y.**and Engquist, B., 2018. Analysis of optimal transport and related misfit functions in full-waveform inversion. Geophysics, 83(1), pp.A7-A12. [pdf]-
**Yang, Y.**, Engquist, B., Sun, J. and Hamfeldt, B.F., 2018. Application of optimal transport and the quadratic Wasserstein metric to full-waveform inversion. Geophysics, 83(1), pp.R43-R62. [pdf] - Engquist, B., Froese, B.D. and
**Yang, Y.**, 2016. Optimal transport for seismic full waveform inversion. Communications in Mathematical Science, 14(8):2309-2330, 2016. [pdf]

Studying Inverse Problems is like solving a puzzle. You are given a partial clue about the ground truth (i.e., data and model), and you must figure out the answer following the clue. Sometimes, we are concerned about whether it is solvable or not (i.e., the problem's ill-/well-posedness). Sometimes, we are interested in developing a way to find the solution (computational inverse problem). Different inverse problems exhibit totally different properties, making the entire subject extremely fun to study! Here is a list of my related work on Inverse Problems (minus those solved using Optimal Transport):

- Einkemmer, L., Li, Q., Wang, L. and
**Yang, Y.**, 2023. Suppressing Instability in a Vlasov-Poisson System by an External Electric Field Through Constrained Optimization. arXiv preprint arXiv:2305.17994. [pdf] - Li, Q., Wang, L. and
**Yang, Y.**, 2023. Differential-Equation Constrained Optimization With Stochasticity. submitted; arXiv preprint arXiv:2305.04024. [pdf] - Mahankali, S. and
**Yang, Y.**, 2023. Norm-dependent convergence and stability of the inverse scattering series for diffuse and scalar waves. Inverse Problems, 39(5), p.054005. [pdf] - Zhu, B., Hu, J., Lou, Y. and
**Yang, Y.**, 2021. Implicit Regularization Effects of the Sobolev Norms in Image Processing. submitted; arXiv preprint arXiv:2109.06255. [pdf]

Many challenging inverse problems, optimal control, and optimal design tasks are eventually translated into optimization problems to be solved on the computer. Most of them are nonlinear, and the resulting optimization problems suffer from severe nonconvexity, for example, PDE-constrained optimization problems and machine learning training. My interests in designing optimization algorithms originated from dealing with these challenges. The main ideas are to design algorithms (1) for global optimization (i.e., the iterates converge to the global minimizer), (2) for faster convergence with the minimal computational cost, and (3) to incorporate the special property of forward model (e.g., the modeling PDE, or specific NN architecture) into the choice of optimization algorithms. Here is a list of my related work:

- Engquist, B., Ren, K. and
**Yang, Y.**, 2023. Adaptive State-Dependent Diffusion for Derivative-Free Optimization. submitted; arXiv preprint arXiv:2302.04370. [pdf] - Engquist, B., Ren, K. and
**Yang, Y.**, 2022. An Algebraically Converging Stochastic Gradient Descent Algorithm for Global Optimization. submitted; arXiv preprint arXiv:2204.05923. [pdf] - Nurbekyan, L., Lei, W. and
**Yang, Y.**, 2023. Efficient Natural Gradient Descent Methods for Large-Scale PDE-Based Optimization Problems. To appear in SIAM Journal on Scientific Computing. [pdf]

Machine Learning has immense potential and is a crucial subject in today's world. There are many similarities between Machine Learning concepts and other areas of Applied Mathematics, and understanding these connections is key to a comprehensive understanding of Machine Learning. Additionally, Machine Learning can be utilized to tackle persistent challenges in various domains of Applied Mathematics. Here is a list of my related work on understanding and utilizing Machine Learning:

- Molinaro, R.,
**Yang, Y.**, Engquist, B. and Mishra, S., 2023. Neural Inverse Operators for Solving PDE Inverse Problems. To appear in ICML 2023; arXiv preprint arXiv:2301.11167. [pdf] - Yu, A.,
**Yang, Y.**and Townsend, A., 2023. Tuning Frequency Bias in Neural Network Training with Nonuniform Data. The 11th International Conference on Learning Representations (ICLR) 2023. [pdf] - Engquist, B., Ren, K. and
**Yang, Y.**, 2022. A Generalized Weighted Optimization Method for Computational Learning and Inversion. The 10th International Conference on Learning Representations (ICLR) 2022. [pdf]

In a sequence of works, I investigate how to mathematically "differentiate" a Monte Carlo method, which is random in nature, with respect to parameters of interest. In particular, I studied such Monte Carlo methods designed to solve certain Partial Differential Equations (PDEs), including the Radiative Transport Equation (RTE) and the Boltzmann Equation. Here is a list of my related work:

- Li, Q., Wang, L., and
**Yang, Y.**, 2022. Monte Carlo Gradient in Optimization Constrained by Radiative Transport Equation. submitted; arXiv preprint arXiv:2209.12114. [pdf] **Yang, Y.**, Silantyev, D. and Caflisch, R., 2023. Adjoint DSMC for Nonlinear Spatially-Homogeneous Boltzmann Equation With a General Collision Model. Journal of Computational Physics, p.112247. [pdf]- Caflisch, R., Silantyev, D. and
**Yang, Y.**, 2021. Adjoint DSMC for nonlinear Boltzmann equation constrained optimization. Journal of Computational Physics, 439, p.110404. [pdf]

Working with dynamical systems is always a humbling experience as the complexity of the trajectory behavior is so rich and case-dependent. However, there are many aspects to explore, especially from a data-driven perspective. Here is list of my related work on data-driven approaches for learning dynamical systems:

- Botvinick-Greenhouse, J.,
**Yang, Y.**, Maulik, R., 2023. Generative Modeling of Time-Dependent Densities via Optimal Transport and Projection Pursuit. submitted; Chaos; arXiv preprint arXiv:2304.09663. [pdf] - Botvinick-Greenhouse, J., Martin, R. and
**Yang, Y.**, 2023. Learning Dynamical Systems From Invariant Measures. submitted; arXiv preprint arXiv:2301.05193. [pdf] **Yang, Y.**, Nurbekyan, L., Negrini, E., Martin, R. and Pasha, M., 2023. Optimal transport for parameter identification of chaotic dynamics via invariant measures. SIAM Journal on Applied Dynamical Systems, 22(1):269–310, 2023. [pdf]

- Einkemmer, L., Li, Q., Wang, L. and
**Yang, Y.**, 2023. Suppressing Instability in a Vlasov-Poisson System by an External Electric Field Through Constrained Optimization. arXiv preprint arXiv:2305.17994. [pdf] - Li, Q., Wang, L. and
**Yang, Y.**, 2023. Differential-Equation Constrained Optimization With Stochasticity. submitted; arXiv preprint arXiv:2305.04024. [pdf] - Botvinick-Greenhouse, J.,
**Yang, Y.**, Maulik, R., 2023. Generative Modeling of Time-Dependent Densities via Optimal Transport and Projection Pursuit. submitted; Chaos; arXiv preprint arXiv:2304.09663. [pdf] - Engquist, B., Ren, K. and
**Yang, Y.**, 2023. Adaptive State-Dependent Diffusion for Derivative-Free Optimization. submitted; arXiv preprint arXiv:2302.04370. [pdf] - Li, Q., Wang, L., and
**Yang, Y.**, 2022. Monte Carlo Gradient in Optimization Constrained by Radiative Transport Equation. submitted; arXiv preprint arXiv:2209.12114. [pdf] - Engquist, B., Ren, K. and
**Yang, Y.**, 2022. An Algebraically Converging Stochastic Gradient Descent Algorithm for Global Optimization. submitted; arXiv preprint arXiv:2204.05923. [pdf] - Zhu, B., Hu, J., Lou, Y. and
**Yang, Y.**, 2021. Implicit Regularization Effects of the Sobolev Norms in Image Processing. submitted; arXiv preprint arXiv:2109.06255. [pdf] **Yang, Y**., 2019. Analysis and Application of Optimal Transport For Challenging Seismic Inverse Problems. arXiv preprint arXiv:1902.01226. [pdf]

- Botvinick-Greenhouse, J., Martin, R. and
**Yang, Y.**, 2023. Learning dynamics on invariant measures using PDE-constrained optimization. Chaos: An Interdisciplinary Journal of Nonlinear Science, 33(6). [pdf] **Yang, Y.**, Silantyev, D. and Caflisch, R., 2023. Adjoint DSMC for Nonlinear Spatially-Homogeneous Boltzmann Equation With a General Collision Model. To appear in Journal of Computational Physics. [pdf]- Nurbekyan, L., Lei, W. and
**Yang, Y.**, 2023. Efficient Natural Gradient Descent Methods for Large-Scale PDE-Based Optimization Problems. To appear in SIAM Journal on Scientific Computing. [pdf] - Han, R., Slepčev, D.,
**Yang, Y.**, 2023. HV Geometry for Signal Comparison. To Appear in Quarterly of Applied Mathematics. [pdf] - Mahankali, S. and
**Yang, Y.**, 2023. Norm-dependent convergence and stability of the inverse scattering series for diffuse and scalar waves. Inverse Problems, 39(5), p.054005. [pdf] **Yang, Y.**, Nurbekyan, L., Negrini, E., Martin, R. and Pasha, M., 2023. Optimal transport for parameter identification of chaotic dynamics via invariant measures. SIAM Journal on Applied Dynamical Systems, 22(1):269–310, 2023. [pdf]- Frederick, C. and
**Yang, Y.**, 2022. Snapshots of modern mathematics from Oberwolfach, Mathematisches Forschungsinstitut Oberwolfach, 2022-04. [pdf] - Engquist, B. and
**Yang, Y.**, 2022. Optimal Transport Based Seismic Inversion: Beyond Cycle Skipping. Communications on Pure and Applied Mathematics. [pdf] **Yang, Y.**, Townsend, A. and Appelö, D., 2022. Anderson acceleration based on the $H^{-s}$ Sobolev norm for contractive and noncontractive fixed-point operators. Journal of Computational and Applied Mathematics, 403, p.113844. [pdf]- Caflisch, R., Silantyev, D. and
**Yang, Y.**, 2021. Adjoint DSMC for nonlinear Boltzmann equation constrained optimization. Journal of Computational Physics, 439, p.110404. [pdf] - Dunlop, M. and
**Yang, Y.**, 2021. Stability of Gibbs Posteriors from the Wasserstein Loss for Bayesian Full Waveform Inversion. SIAM/ASA Journal on Uncertainty Quantification, 9(4), pp.1499-1526. [pdf] **Yang, Y.**, 2020. Anderson Acceleration for Seismic Inversion. Geophysics, 86(1), pp.R99-R108. [pdf]- Engquist, B., Ren, K. and
**Yang, Y.**, 2020. The quadratic Wasserstein metric for inverse data matching. Inverse Problems, 36(5), p.055001. [pdf] - Engquist, B. and
**Yang, Y.**, 2019. Seismic imaging and optimal transport. Communications in Information and Systems, Vol. 19, No. 2 (2019), pp. 95-145 [pdf] - Engquist, B. and
**Yang, Y.**, 2019. Seismic inversion and the data normalization for optimal transport. Methods and Applications of Analysis, Vol. 26, No. 2 (2019), pp. 133-148. [pdf] **Yang, Y.**and Engquist, B., 2018. Analysis of optimal transport and related misfit functions in full-waveform inversion. Geophysics, 83(1), pp.A7-A12. [pdf]-
**Yang, Y.**, Engquist, B., Sun, J. and Hamfeldt, B.F., 2018. Application of optimal transport and the quadratic Wasserstein metric to full-waveform inversion. Geophysics, 83(1), pp.R43-R62. [pdf] - Engquist, B., Froese, B.D. and
**Yang, Y.**, 2016. Optimal transport for seismic full waveform inversion. Communications in Mathematical Science, 14(8):2309-2330, 2016. [pdf] - Molinaro, R.,
**Yang, Y.**, Engquist, B. and Mishra, S., 2023. Neural Inverse Operators for Solving PDE Inverse Problems. To appear in ICML 2023; arXiv preprint arXiv:2301.11167. [pdf] - Yu, A.,
**Yang, Y.**and Townsend, A., 2023. Tuning Frequency Bias in Neural Network Training with Nonuniform Data. The 11th International Conference on Learning Representations (ICLR) 2023. [pdf] - Engquist, B., Ren, K. and
**Yang, Y.**, 2022. A Generalized Weighted Optimization Method for Computational Learning and Inversion. The 10th International Conference on Learning Representations (ICLR) 2022. [pdf] -
**Yang, Y.**2020. Anderson acceleration for seismic inversion. In SEG Technical Program Expanded Abstracts 2020 (pp. 880-884). Society of Exploration Geophysicists. [pdf] - Dunlop, M. and
**Yang, Y.**, 2020. New likelihood functions and level-set prior for Bayesian full-waveform inversion. In SEG Technical Program Expanded Abstracts 2020 (pp. 825-829). Society of Exploration Geophysicists. [pdf] -
**Yang, Y.**and Engquist, B., 2019. Improving optimal transport based FWI through data normalization. In SEG Technical Program Expanded Abstracts 2019 (pp. 1315-1319). Society of Exploration Geophysicists. [pdf] -
**Yang, Y**. and Engquist, B., 2018. Model recovery below reflectors by optimal-transport FWI. In SEG Technical Program Expanded Abstracts 2018 (pp. 1178-1182). Society of Exploration Geophysicists. [pdf] - J. Ramos-Martínez, L. Qiu, J. Kirkebø, A.A. Valenciano, and
**Y. Yang**, 2018. Long-wavelength FWI updates beyond cycle skipping. In SEG Technical Program Expanded Abstracts 2018 (pp. 1168-1172). Society of Exploration Geophysicists. [pdf] **Yang, Y.**and Engquist, B., 2017. Analysis of optimal transport and related misfit functions in fullwaveform inversion. In SEG Technical Program Expanded Abstracts 2017 (pp. 1291-1296). Society of Exploration Geophysicists. [pdf]- Qiu, L., Ramos-Martínez, J., Valenciano, A.,
**Yang, Y.**and Engquist, B., 2017. Full-waveform inversion with an exponentially encoded optimal-transport norm. In SEG Technical Program Expanded Abstracts 2017 (pp. 1286-1290). Society of Exploration Geophysicists [pdf] -
**Yang, Y.**, Engquist, B. and Sun, J., 2016. Convexity of the quadratic Wasserstein metric as a misfit function for full-waveform inversion. In SEG Technical Program Expanded Abstracts 2016 (pp. 1385-1389). Society of Exploration Geophysicists. [pdf] **Yang, Y.**and Engquist, B., 2017. Analysis of optimal transport related misfit functions in seismic imaging. In: Nielsen F., Barbaresco F. (eds) Geometric Science of Information. GSI 2017. Lecture Notes in Computer Science, vol 10589. Springer, Cham. [pdf]

- @ UT Austin
- 2017 M387C Numerical Analysis: Algebra and Approximation
- 2016 Fall M427J Differential Equations with Linear Algebra
- 2016 Spring M387D Numerical Analysis: Differential Equations
- 2014 Fall M408C Differential and Integral Calculus

- @ NYU
- 2020 Fall MATH-UA.140 Linear Algebra
- 2020 Summer MATH-UA.252 Numerical Analysis
- 2020 Spring MATH-UA.122 Calculus II
- 2019 Spring & Fall MATH-UA.120 Discrete Math
- 2018 Fall MATH-UA.211 Math For Economics

- @ Cornell
- 2023 Fall MATH 4250 Numerical Analysis and Differential Equations

- Undergraduate Students
- Stuti Saria (NYU, Econ & Math Major)
- Srinath Mahankali (Stuyvesant High School, now an undergraduate student at MIT)
- Carol Long (NYU, then a PhD student at Harvard SEAS)
- Han Liu (NYU, Math major, then a M.S. student at Rice Univ)
- Baoye Chen (NYU, then obtain M.S. degree at Columbia, currently working at Deutsche Bank at NYC)
- Linglai Chen (NYU, now a M.S. student at Harvard)
- Bowen Zhu (NYU, now a M.S. student at Harvard)
- Wanzhou Lei (NYU, now a M.S. student at Harvard)
- Liu Zhang (Yale-NUS, now a Ph.D. student at Princeton)
- Aparna Gupte (MIT, CS major)

- Master Students
- Nicolai Gerber (University of Bonn)
- Guerda Kenne (AIMS Cameroon), now a data scientist at Cameroon
- Brenda Anague (AIMS Cameroon), now a PhD student at Quantum Leap Africa at AIMS Rwanda

- Ph.D. Students
- Jonah Botvinick-Greenhouse (Cornell CAM)
- Esther Gallmeier (Cornell CAM)

Yunan Yang, May 22, 2023