AESIM2026: Inverse problems and applications

Venue: Azerbaijan University, Baku, Azerbaijan

Dates: 16/04/2026-25/04/2026 (postponed to November 1-14, 2026.)

Inscription Deadline: 10 October 2026.

For Registration, please click here

           1. Coordinators: 

Abdeljalil Nachaoui, Laboratoire de Mathéatiques-Jean Leray, Nantes Université, France Phone +33251125937 E-mail : Abdeljalil.Nachaoui@univ-nantes.fr

Fatima M. Aboud, Department of mathematics, College of Sciences, University of Diyala. Phone +9647700397254 E-mail : Fatima.Aboud@sciences.uodiyala.edu.iq (female)

 

         2. Scientific committee 

  1. Mr. Abdeljalil Nachaoui, Laboratoire de Mathématiques Jean Leray, Nantes Université, France, Abdeljalil.Nachaoui@univ-nantes.fr
  2. Mr. Yusif Gasimov, Azerbaijan University, Baku, Azerbaijan
  3. Mr. Amine Laghrib, Equipe EMI, Université Sultan Moulay Slimane, Béni-Mellal, Morocco
  4. Mr. Tamaz Tadumdaze, Institute of Applied Mathematics, Tbilisi State University, Tbilisi,Georgia
  5. Mrs. Ashrafova, Yegana R., Baku State University, Baku, Azerbaijan

 

            3. Organizing committee 

  • The person in charge: Mrs. F. Aboud, Department of mathematics, College of Sciences, University of Diyala, Iraq, (female) (Fatima.Aboud@sciences.uodiyala.edu.iq).
  • Local coordinator: Mr. Yusif Gasimov, Azerbaijan University, Baku, Azerbaijan
  • Mrs. Latifa Agamalieva, Azerbaijan University
  • Mr. Rakib Efendiev, Baku Engineering University  
  • Mrs. Khayala Seyfullayeva, Sumgayit State University
  • Mrs.Tunzale Huseynova, Azerbaijan State Pedagogical University
  • Mrs. Jamila Asadova, Azerbaijan University of Architecture and Construction

 

         4. Scientific content

  1.     Description of the project

This school aims to introduce students and non-experts in the field of inverse problems in order to build momentum around this topic that covers many areas of research ranging from modeling to simulation through mathematical analysis of partial differential equations and ordinary and delay differential equations as well as the development of new solution algorithms. The scientific program of this school will include lectures, seminars and slots reserved for participants who wish to present their work. The objective of this last possibility is to know the participating researchers and their areas of current or initial training, to encourage those who cannot afford to go exhibit in international conferences.

Presentations, courses and seminars will be prepared as courses for students and not as seminars for colleagues. That is, they will prepare an objective of initiation, and stimulus to research activity, and therefore start with reminders and detailed behavior of many examples of motivation and application.

Topics to be covered: Introduction to Inverse Problems, Inverse Source Problems, Parameter Identification Problems, Inverse Problems for Delay Differential Equations (DDEs), Identification of Boundary Conditions, Boundary Reconstruction Problems, Solution methods and applications.

  • Course 1: Identification of Boundary Conditions

Mrs. Fatima ABOUD, Department of Mathematics, College of Science, University of Diyala, Iraq (female) (Fatima.Aboud@uodiyala.edu.iq)

               https://www.scopus.com/authid/detail.uri?authorId=34978302800

              ORCID: 0009-0007-9254-998X

https://www.scopus.com/authid/detail.uri?authorId=6701831303

ORCID: 0000-0003-0463-3633

  • Course 3: Boundary Reconstruction Problems

Yusif Gasimov, Azerbaijan University, Baku, Azerbaijan yusif.gasimov@au.edu.az

https://www.scopus.com/authid/detail.uri?authorId=24171373800

ORCID: 0000-0001-9875-1280

 https://www.scopus.com/authid/detail.uri?authorId=42261115900

ORCID: 0000-0002-0698-4253            

  • Course 6: Solution Methods for Inverse Problems

    Amine Laghrib, Equipe EMI, Université Sultan Moulay Slimane, Béni-Mellal, Morocco, laghrib.amine@gmail.com

http://scopus.com/authid/detail.uri?authorId=56485972100

ORCID: 0000-0003-4851-3617

 

  1.    Description of each course

Course Title: Inverse Source Problems

Description: This chapter delves into the fundamental class of inverse source problems, where the objective is to reconstruct an unknown internal source term from measurements taken on the boundary of a domain. We begin by establishing the general mathematical formulation of such problems, highlighting their inherent ill-posedness. The theoretical framework is then explored through canonical examples, including source identification in the Helmholtz equation for acoustic imaging and the Poisson equation for electrostatic applications. A critical discussion on uniqueness and non-uniqueness results will reveal the fundamental limitations of source reconstruction. Finally, the chapter concludes by examining cutting-edge applications in biomedical engineering, specifically the localization of neural activity from EEG and MEG data.

Course Title: Parameter Identification Problems

Description: This chapter addresses parameter identification problems, which involve reconstructing unknown coefficients within a governing partial differential equation from observed data. We will formulate the general framework for recovering spatially distributed parameters, such as conductivity or diffusivity that define the material properties of a system. A central focus is the challenging problem of Electrical Impedance Tomography (EIT), where internal conductivity is inferred from surface voltage measurements. The scope extends to identifying elastic moduli in seismic imaging and other physical domains. The chapter also critically examines the significant challenges and specialized techniques required for recovering discontinuous parameters, a common feature in real-world scenarios with distinct material interfaces.

Course Title: Inverse Problems for Delay Differential Equations (DDEs)

Description: This chapter explores the unique challenges of inverse problems governed by Delay Differential Equations (DDEs), where the system's evolution depends on its past states. We begin by formulating key inverse problems for these infinite-dimensional systems. The focus will be on several core types: identifying unknown parameters, including discrete and distributed delays; reconstructing the unknown initial history function from final-time data; and solving inverse source problems. We will discuss the severe ill-posedness introduced by these tasks and analyze uniqueness and stability considerations. The chapter concludes with an overview of relevant numerical methods and their applications in modeling physiological systems, epidemic spread, and engineering control.

Course Title: Identification of Boundary Conditions

Description: This chapter is dedicated to the identification of unknown boundary conditions, a classic and practically significant class of inverse problems. We separate the analysis into two main categories. First, for elliptic systems, we examine the severely ill-posed Cauchy problem, which involves reconstructing missing data on an inaccessible boundary. Second, for time-dependent problems, we focus on the Inverse Heat Conduction Problem (IHCP) to estimate a transient boundary heat flux. The chapter covers foundational theory and established solution techniques, such as Beck's sequential method, highlighting the inherent instability of these problems. Practical applications in non-destructive testing, corrosion detection, and thermal protection of aerospace systems are discussed to ground the theory in real-world contexts.

Course Title: Boundary Reconstruction Problems

Description: This chapter investigates boundary reconstruction problems, a fundamental class of geometric inverse problems where the goal is to determine the shape of a domain or an inclusion within it from boundary measurement data. We will explore the mathematical formulation of such problems, focusing on two key paradigms: the inverse obstacle scattering problem, which aims to identify an object from scattered wave fields, and geometric inverse problems in Impedance Tomography. A central theme is the high nonlinearity introduced by the unknown geometry. The chapter will also cover powerful computational frameworks for solving these problems, with a particular emphasis on level set methods, which provide a robust approach for tracking the evolution of a boundary during reconstruction.

Course Title: Solution Methods and Applications

Description: This culminating chapter provides a comprehensive overview of the computational frameworks essential for solving inverse problems. We transition from theoretical formulation to practical implementation, exploring three major classes of solution methods. The discussion covers deterministic approaches, including variational methods like Tikhonov regularization and iterative techniques such as Landweber and conjugate gradient. Furthermore, the chapter introduces the probabilistic perspective of Bayesian inference, detailing how prior knowledge is formally incorporated. Finally, these methodologies are explicitly connected to their real-world impact, with concrete examples drawn from medical imaging, geophysics, remote sensing, and non-destructive evaluation, demonstrating the complete pipeline from mathematical theory to application.

 

  1.    Tentative schedule

To cover these courses, we will need 58 hours, which includes theoretical part and practical part. The participants will have the possibility to speak about their interesting domain of research; in addition there will be 10 hours of communications.

 

Chapter 1: Introduction to Inverse Problems (Aboud/Gasimov/Nachaoui) (9 hours)

  1.  A Brief Review of Functional Analysis
  • Properties of Elliptic and Parabolic Partial Differential Equations
  • Numerical Methods for PDEs (Finite Elements, Finite Differences)
  • Introduction to Optimization and Numerical Linear Algebra

1.2. What is an Inverse Problem? Contrast with Forward Problems 

  • Key Concepts: Ill-Posedness, Existence, Uniqueness, and Stability
  • The Morozov Discrepancy Principle
  • Classification of Inverse Problems
  • Motivating Examples from Science and Engineering

              1.3. Mathematical Foundations of Ill-Posed Problems 

  • Compact Operators and Singular Value Decomposition (SVD)
  • The Picard Condition and its Meaning
  • Regularization Theory: Filtering the SVD
  • Tikhonov Regularization

Chapter 2: Inverse Source Problems 

(Nachaoui) (9 hours)

  • Problem Formulation: Recovering a Source from Boundary Measurements
  • The Helmholtz Equation and Acoustic Source Identification
  • The Poisson Equation and Inverse Source Problems in Electrostatics
  • Uniqueness and Non-Uniqueness Results
  • Applications in Bioelectric Source Localization (EEG/MEG)

                 Chapter 3: Parameter Identification Problems 

                 (Ashrafova) (6 hours)

  • Recovering Coefficients in Partial Differential Equations
  • Identification of Distributed Parameters (e.g., conductivity, diffusivity)
  • The Inverse Problem of Electrical Impedance Tomography (EIT)
  • Coefficient Identification in Elasticity and Seismic Imaging
  • Challenges in Recovering Discontinuous Parameters

Chapter 4: Inverse Problems for Delay Differential Equations (DDEs) (Shavadze/Nachaoui) (7.5 hours)

  • Introduction to Delay Differential Equations: Theory and Applications
  • Formulating Inverse Problems in DDE Systems
  • Parameter Identification in DDEs
  • Estimating Discrete and Distributed Delay Terms
  • Recovering System Parameters (e.g., growth rates, coefficients)
  • Inverse Initial History Problems
  • Recovering the Pre-History Function from Terminal-Time Data
  • Uniqueness and Stability Considerations
  • Inverse Source Problems for DDEs
  • Numerical Methods for Inverse DDE Problems
  • Applications in Physiology, Epidemiology, and Engineering Control

Chapter 5: Identification of Boundary Conditions 

(Aboud/Nachaoui) (9 hours)

5.1 Elliptic Problems 

  • Recovering Unknown Cauchy Data on an Inaccessible Boundary
  • The Cauchy Problem for the Laplace and Poisson Equations
  • Application to Corrosion Detection and Non-Destructive Testing

5.2. Time-Dependent Inverse Problems

  • Recovering Time-Dependent Boundary Heat Flux
  • The Inverse Heat Conduction Problem (IHCP)
  • Beck's Sequential Function Specification Method
  • Applications in Thermal Protection Systems

Chapter 6: Boundary Reconstruction Problems 

(Gasimov /Nachaoui) (7.5 hours)

  • Recovering the Shape of a Domain from Boundary Measurements
  • The Inverse Obstacle Scattering Problem
  • Geometric Inverse Problems in Impedance Tomography
  • Level Set Methods for Evolving Boundaries

Chapter 7: Solution methods and applications

(Laghrib /Nachaoui) (10 hours)

7.1. Solution Methods for Inverse Problems

7.1.1. Variational Methods

  • Output Least-Squares Formulation
  • Tikhonov Functional and its Minimization

7.2. Iterative Regularization Methods

  • Landweber Iteration
  • The alternating iterative method
  • The Conjugate Gradient Method for Ill-Posed Problems

7.3. Bayesian Inference for Inverse Problems

  • Modeling Unknowns as Random Variables
  • Prior, Likelihood, and Posterior Distributions
  • Maximum a Posteriori (MAP) Estimation

                7.4. Applications 

  • Medical Imaging: X-Ray CT, MRI, and Optical Tomography
  • Geophysics: Seismic Inversion and Ground-Penetrating Radar
  • Remote Sensing and Image Deblurring

  • Material Science and Non-Destructive Evaluation

     

     

 

    

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7.2. Iterative Regularization Methods

  • Landweber Iteration
  • The alternating iterative method
  • The Conjugate Gradient Method for Ill-Posed Problems

7.3. Bayesian Inference for Inverse Problems

  • Modeling Unknowns as Random Variables
  • Prior, Likelihood, and Posterior Distributions
  • Maximum a Posteriori (MAP) Estimation

                7.4. Applications 

  • Medical Imaging: X-Ray CT, MRI, and Optical Tomography
  • Geophysics: Seismic Inversion and Ground-Penetrating Radar
  • Remote Sensing and Image Deblurring
  • Material Science and Non-Destructive Evaluation

 

          For Registration, please click here