## Uncertainty Quantification Group

Welcome to the Uncertainty Quantification Group, in the Institute of Applied Mathematics at METU.

Uncertainty quantification (UQ) is a modern inter-disciplinary science that cuts across traditional research groups and combines statistics, numerical analysis and computational applied mathematics. When we attempt to simulate complex real-world phenomena, e.g., fluid dynamics, climate science, chemically reacting systems, oil field research, the price of stock pricing, using mathematical and computer models, there is almost always uncertainty in our predictions. The objective is usually that of propagating quantitative information on the data through a computation to the solution. Our research focuses on advancing fundamental computational methodology for UQ in complex physical systems.

The aim of this research group is to answer the following core questions?

- How to quantify confidence in computational predictions?
- How to build or refine models of complex physical processes from indirect and limited observations?
- What information is needed to drive inference, design, and control?
- How to calibrate and validate our computational models?

In classical setting we solve partial differential equations (PDEs) for a given sets of input data such as material properties, domain geometries, boundary and intial conditions, forcing terms. However, in real-life applications, it is not possible to obtain all this information due to the limited information. If the inputs to the systems under consideration are uncertain, we require mathematical techniques that propagate this uncertainty to the output quantities of interest and methods for computing probabilities of events rather than specific solutions. Our current research includes investigating the numerical solution of partial differential equations (PDEs) with random input data, especially convection dominated problems, for a range of linear and non-linear flow problems.

**Team Members:**

- Pelin Çiloğlu
- Mustafa Kütük
- Eda Oktay
- Sıtkı Can Toraman
- M. Alp Üreten
- Hamdullah Yücel

**Projects:**

- Numerical Studies for Petrol and Gas Reservoir Problems (TUBITAK 1001: 119F022, August 01 2019 - August 01 2021)
- Numerical Studies of Korteweg-de Vries Equation with Random Input Data (Proje No: YÖP-705-2018-2820, May 29 2018 - May 29 2019)

**News:**

**September 10, 2019:**Mustafa Kütük gives a research talk, titled "Hyperspectral Target Detection by using Superpixels and Signature Based Methods" in BEYOND 2019: Computational Science and Engineering Conference**.****September 10, 2019:**Pelin Çiloğlu gives a research talk, titled "Discontinuous Galerkin Methods for Convection Diffusion Equations with Random Coefficients" in BEYOND 2019: Computational Science and Engineering Conference.**March 26, 2019:**P. Çiloğlu, H. Yücel, "Discontinuous Galerkin Methods for Convection--Diffusion Equations with Random Coefficients", Submitted.**October 21, 2018:**Pelin Çiloğlu gives a research talk, titled "Discontinuous Galerkin Methods for Unsteady Convection Diffusion Equation with Random Coefficients" in BEYOND: Workshop on Computational Science and Engineering**.****September 24, 2018:**Pelin Çiloğlu gives a research talk, titled "Unsteady Convection Diffusion Equation with Random Input Data" in the Chemnitz FE Symposium 2018.**September 03, 2018**: M. Alp Üreten successfully defends his Master thesis, titled "Numerical Studies of Korteweg-de Vries Equation with Random Input Data".

**Lecture Series: **

**Low-Rank Approximatian:**In this lecture series, we discuss theoretical foundations of low-rank approximation and applications in engineering, scientific computing, data analysis, ... where low-rank approximation plays a central role. We will follow the lecture notes of Danie Kressner.

**February 17, 2020 (15:30-16:30) : Hamdullah Yücel****February 20, 2020 (16:00-17:00) : Pelin Çiloğlu****March 02, 2020 (15:30-16:30): M. Alp Üreten****March 05, 2020 (15:30-16:30): Eda Oktay****March 09, 2020 (15:30-16:30): Pelin Çiloğlu****March 12, 2020 (14:00-15:30): Hamdullah Yücel**

**Group Meetings: ****Since February 06, 2019**, we regularly come together on Thursdays, 14:00 to discuss the revelant topics in the scope of UC Research Group on** in S212**. Each person give a talk related to his/her research interests. If you are interested, please feel free to join the meetings.

**March 19, 2020:**S. Can Toraman, " "**March 05, 2020:**M. Alp Üreten, " "**February 20, 2020:**Eda Oktay, " Iterative Solvers for the Stochastic Finite Element Method"**February 13, 2020:**Pelin Çiloğlu, "A Low-Rank Solver for the Navier-Stokes Equations with Uncertain Viscosity"**February 06, 2020:**Mustafa Kütük, "An Energy Approach to the Solution of Partial Differential Equation via Machine/Deep Learning"**December 20, 2019:**M. Alp Üreten, "Defect-Deferred Correction Methods for Time Dependent Problems**December 06, 2019:**Pelin Çiloğlu, "Low-Rank Solution of Unsteady Diffusion Equations with Stochastic Coefficients"**November 29, 2019:**Saliha Yıldırım, "Numerical Techniques for Advective Cahn-Hilliard Equations"**November 22, 2019:**Gizem Yıldız, "Numerical Techniques for Advective Allen-Cahn Equations"**November 15, 2019:**Sıtkı Can Toraman, "Kernel Principal Component Analysis for Stochastic Input Model Generation "**November 08, 2019:**Mustafa Kütük, "DEEPXDE: A Deep Learning Library for Solving Differential Equations"**November 01, 2019:**Eda Oktay, "Solvers and Precondtioners based on Gauss-Seidel and Jacobi Algorithms for Non-symmetric Stochastic Galerkin System of Equations"**October 25, 2019:**Pelin Çiloğlu, "Optimal Control of Elliptic PDEs with Random Coefficients"**October 18, 2019:**M. Alp Üreten, "Monte Carlo and Multilevel Monte Carlo Methods "**August 02, 2019:**Mustafa Kütük, "Simulator Free Solution of High dimensional Stochastic Elliptic Partial Differential Equations using Deep Neural Networks"**July 19, 2019:**Mustafa Kütük, "Deep Learning: An Introduction for Applied Mathematicians: Part II"**June 28, 2019:**Pelin Çiloğlu, "Ensemble Time-Stepping Algorithm for the Convection-Diffusion Equation with Random Diffusivity"**June 21, 2019:**Mustafa Kütük, "Deep Learning: An Introduction for Applied Mathematicians: Part I"**June 14, 2019:**M. Alp Üreten, "Further Analysis of Multilevel Monte Carlo Methods for Elliptic PDEs with Random Coefficients"**May 30, 2019:**Pelin Çiloğlu, "A Domain Decomposition Model Reduction Method for Linear Convection Diffusion Equations with Random Coefficients"**May 30, 2019:**Eda Oktay, " Graph Partitioning Techniques"**May 10, 2019:**Ertuğrul Umut Yıldırım, "Numerical Analysis of Stochastic Advection-Diffusion Equation via Karhunen-Loeve Expansion"**May 10, 2019:**İsa Eren Yıldırım, "First-Arrival Traveltime Tomography based on the Adjoint-State Method"**May 03, 2019:**M. Alp Üreten, "Finite Element Methods with MATLAB"**April 12, 2019:**Bülent Karasözen "Quantification of Uncertainty"**April 05, 2019:**Hamdullah Yücel, "PDE-constrained optimization"**March 22, 2019:**Mustafa Kütük, "Target Detection Algorithms*"***March 15, 2019**: M. Alp Üreten, "Implementation of Multilevel Monte Carlo Methods"**March 08, 2019:**Pelin Çiloğlu, "Preconditioning"**March 01, 2019:**M. Alp Üreten, "Introduction to Multilevel Monte Carlo Methods*"*

**Prospective Students:**

If you are interested in uncertainty quantification or for any other comments or questions, please feel free to contact me at yucelh[at]metu.edu.tr. We are always looking for motivated and talented research members.