Alik Ismail-Zadeh is a Chief Scientist / Research Professor of the Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences (RusAS) at Moscow, Russia, and a Senior Scientist at the Institute of Applied Geosciences, Karlsruhe Institute of Technology, Germany. He graduated from the Baku State and Lomonossov Moscow State Universities (mathematics and physics) and received his PhD and Doctor of Science degrees (in geophysics) from RusAS. He has been lecturing on computational geodynamics at the University of Karlsruhe, Abdus Salam International Center for Theoretical Physics in Trieste, and Moscow State University of Oil and Gas, and working as visiting scholar/professor in the universities of California (UCLA), Cambridge, Paris (IPGP), Stockholm (KTH), Tokyo, Trieste, and Uppsala. His scientific interests cover studies of the crust and mantle dynamics, basin evolution, salt tectonics, and seismic hazard through theoretical analysis and numerical modelling. He is an author of over 100 peer-reviewed papers and four books. Alik Ismail-Zadeh is an honorary fellow of the Royal Astronomical Society and has been awarded the Academia Europaea Medal, AGU International Award, and several prestigious fellowships including Alexander von Humboldt, Royal Society of London, Royal Swedish Academy of Sciences, and Russian President.Alexander Korotkii is Head of the Department of Applied Problems of the Institute of Mathematics and Mechanics, Russian Academy of Sciences at Yekaterinburg, Russia. He graduated from Ural State University (mathematics and mechanics) and received his PhD and Doctor of Science degrees (in mathematics) from RusAS. He is also Professor in Mathematics at the Chair of Numerical Mathematics of the Ural Federal University and teach on computational methods and mathematical modeling. His scientific interests cover studies of partial differential equations, optimal control theory, systems of distributed parameters with uncertainties and conflicts, direct and inverse problems in fluid dynamics. He is an author and co-author of 80 research papers.Igor Tsepelev is a Senior Scientist of the Department of Applied Problems of the Institute of Mathematics and Mechanics, RusAS at Yekaterinburg, Russia. He graduated from the Ural State University (mathematics and mechanics) and received his PhD in mathematics (differential equations and inverse problems) from RusAS. His scientific interests cover studies of inverse ill-posed problems, development of numerical methods for solving systems of partial differential equations and algorithms for parallel computing, and mathematical and numerical modeling in geophysics. He is an author and co-author of more than 50 research papers.

• Preface.- Introduction.- Backward advection method and application to salt tectonics.- Variational method and its application to mantle plume evolution.- Application of the VAR method to volcanic lava dynamics.- Quasi-reversibility method and its applications.- Application of the QRV method to reconstruction of plate subduction.- Comparison of data assimilation methods for mantle convection models.

Alik Ismail-Zadeh is a Chief Scientist / Research Professor of the Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences (RusAS) at Moscow, Russia, and a Senior Scientist at the Institute of Applied Geosciences, Karlsruhe Institute of Technology, Germany. He graduated from the Baku State and Lomonossov Moscow State Universities (mathematics and physics) and received his PhD and Doctor of Science degrees (in geophysics) from RusAS. He has been lecturing on computational geodynamics at the University of Karlsruhe, Abdus Salam International Center for Theoretical Physics in Trieste, and Moscow State University of Oil and Gas, and working as visiting scholar/professor in the universities of California (UCLA), Cambridge, Paris (IPGP), Stockholm (KTH), Tokyo, Trieste, and Uppsala. His scientific interests cover studies of the crust and mantle dynamics, basin evolution, salt tectonics, and seismic hazard through theoretical analysis and numerical modelling. He is an author of over 100 peer-reviewed papers and four books. Alik Ismail-Zadeh is an honorary fellow of the Royal Astronomical Society and has been awarded the Academia Europaea Medal, AGU International Award, and several prestigious fellowships including Alexander von Humboldt, Royal Society of London, Royal Swedish Academy of Sciences, and Russian President.Alexander Korotkii is Head of the Department of Applied Problems of the Institute of Mathematics and Mechanics, Russian Academy of Sciences at Yekaterinburg, Russia. He graduated from Ural State University (mathematics and mechanics) and received his PhD and Doctor of Science degrees (in mathematics) from RusAS. He is also Professor in Mathematics at the Chair of Numerical Mathematics of the Ural Federal University and teach on computational methods and mathematical modeling. His scientific interests cover studies of partial differential equations, optimal control theory, systems of distributed parameters with uncertainties and conflicts, direct and inverse problems in fluid dynamics. He is an author and co-author of 80 research papers.Igor Tsepelev is a Senior Scientist of the Department of Applied Problems of the Institute of Mathematics and Mechanics, RusAS at Yekaterinburg, Russia. He graduated from the Ural State University (mathematics and mechanics) and received his PhD in mathematics (differential equations and inverse problems) from RusAS. His scientific interests cover studies of inverse ill-posed problems, development of numerical methods for solving systems of partial differential equations and algorithms for parallel computing, and mathematical and numerical modeling in geophysics. He is an author and co-author of more than 50 research papers.

• Preface.- Introduction.- Backward advection method and application to salt tectonics.- Variational method and its application to mantle plume evolution.- Application of the VAR method to volcanic lava dynamics.- Quasi-reversibility method and its applications.- Application of the QRV method to reconstruction of plate subduction.- Comparison of data assimilation methods for mantle convection models.

Alik Ismail-Zadeh is a Chief Scientist / Research Professor of the Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences (RusAS) at Moscow, Russia, and a Senior Scientist at the Institute of Applied Geosciences, Karlsruhe Institute of Technology, Germany. He graduated from the Baku State and Lomonossov Moscow State Universities (mathematics and physics) and received his PhD and Doctor of Science degrees (in geophysics) from RusAS. He has been lecturing on computational geodynamics at the University of Karlsruhe, Abdus Salam International Center for Theoretical Physics in Trieste, and Moscow State University of Oil and Gas, and working as visiting scholar/professor in the universities of California (UCLA), Cambridge, Paris (IPGP), Stockholm (KTH), Tokyo, Trieste, and Uppsala. His scientific interests cover studies of the crust and mantle dynamics, basin evolution, salt tectonics, and seismic hazard through theoretical analysis and numerical modelling. He is an author of over 100 peer-reviewed papers and four books. Alik Ismail-Zadeh is an honorary fellow of the Royal Astronomical Society and has been awarded the Academia Europaea Medal, AGU International Award, and several prestigious fellowships including Alexander von Humboldt, Royal Society of London, Royal Swedish Academy of Sciences, and Russian President.Alexander Korotkii is Head of the Department of Applied Problems of the Institute of Mathematics and Mechanics, Russian Academy of Sciences at Yekaterinburg, Russia. He graduated from Ural State University (mathematics and mechanics) and received his PhD and Doctor of Science degrees (in mathematics) from RusAS. He is also Professor in Mathematics at the Chair of Numerical Mathematics of the Ural Federal University and teach on computational methods and mathematical modeling. His scientific interests cover studies of partial differential equations, optimal control theory, systems of distributed parameters with uncertainties and conflicts, direct and inverse problems in fluid dynamics. He is an author and co-author of 80 research papers.Igor Tsepelev is a Senior Scientist of the Department of Applied Problems of the Institute of Mathematics and Mechanics, RusAS at Yekaterinburg, Russia. He graduated from the Ural State University (mathematics and mechanics) and received his PhD in mathematics (differential equations and inverse problems) from RusAS. His scientific interests cover studies of inverse ill-posed problems, development of numerical methods for solving systems of partial differential equations and algorithms for parallel computing, and mathematical and numerical modeling in geophysics. He is an author and co-author of more than 50 research papers.

• Preface.- Introduction.- Backward advection method and application to salt tectonics.- Variational method and its application to mantle plume evolution.- Application of the VAR method to volcanic lava dynamics.- Quasi-reversibility method and its applications.- Application of the QRV method to reconstruction of plate subduction.- Comparison of data assimilation methods for mantle convection models.