2023 | POD-Galerkin reduced order models and physics-informed neural networks for solving inverse problems for the Navier-Stokes equations | S. Hijazi, M.A. Freitag, N. LandwehrZeitschrift: Advanced Modeling and Simulation in Engineering SciencesSeiten: 38 pagesBand: 10Link zur Publikation
,
Link zum Preprint
POD-Galerkin reduced order models and physics-informed neural networks for solving inverse problems for the Navier-Stokes equations
Autoren: S. Hijazi, M.A. Freitag, N. Landwehr
(2023)
We present a Reduced Order Model (ROM) which exploits recent developments in Physics Informed Neural Networks (PINNs) for solving inverse problems for the Navier--Stokes equations (NSE). In the proposed approach, the presence of simulated data for the fluid dynamics fields is assumed. A POD-Galerkin ROM is then constructed by applying POD on the snapshots matrices of the fluid fields and performing a Galerkin projection of the NSE (or the modified equations in case of turbulence modeling) onto the POD reduced basis. A POD-Galerkin PINN ROM is then derived by introducing deep neural networks which approximate the reduced outputs with the input being time and/or parameters of the model. The neural networks incorporate the physical equations (the POD-Galerkin reduced equations) into their structure as part of the loss function. Using this approach, the reduced model is able to approximate unknown parameters such as physical constants or the boundary conditions. A demonstration of the applicability of the proposed ROM is illustrated by two cases which are the steady flow around a backward step and the unsteady turbulent flow around a surface mounted cubic obstacle.
Zeitschrift:
Advanced Modeling and Simulation in Engineering Sciences
2023 | Solving the Parametric Eigenvalue Problem by Taylor Series and Chebyshev Expansion | T. Mach, M.A. FreitagZeitschrift: arXivBand: 2302.03661Link zum Preprint
Solving the Parametric Eigenvalue Problem by Taylor Series and Chebyshev Expansion
Autoren: T. Mach, M.A. Freitag
(2023)
We discuss two approaches to solving the parametric (or stochastic) eigenvalue problem A(μ)λ(μ)=A(μ)v(μ). One of them uses a Taylor expansion and the other a Chebyshev expansion. The parametric eigenvalue problem assumes that the matrix A depends on a parameter μ, where μ might be a random variable. Consequently, the eigenvalues and eigenvectors are also functions of μ. We compute a Taylor approximation of these functions about μ₀ by iteratively computing the Taylor coefficients. The complexity of this approach is O(n³) for all eigenpairs, if the derivatives of A(μ) at μ₀ are given. The Chebyshev expansion works similarly. We first find an initial approximation iteratively which we then refine with Newton's method. This second method is more expensive but provides a good approximation over the whole interval of the expansion instead around a single point.
We present numerical experiments confirming the complexity and demonstrating that the approaches are capable of tracking eigenvalues at intersection points. Further experiments shed light on the limitations of the Taylor expansion approach with respect to the distance from the expansion point μ₀.
2022 | Time-limited Balanced Truncation for Data Assimilation Problems | J. König, M.A. FreitagZeitschrift: arXivBand: 2212.07719Link zum Preprint
Time-limited Balanced Truncation for Data Assimilation Problems
Autoren: J. König, M.A. Freitag
(2022)
Balanced truncation is a well-established model order reduction method in system theory that has been applied to a variety of problems. Recently, a connection between linear Gaussian Bayesian inference problems and the system theoretic concept of balanced truncation was drawn for the first time. Although this connection is new, the application of balanced truncation to data assimilation is not a novel concept: It has already been used in four-dimensional variational data assimilation (4D-Var) in its discrete formulation. In this paper, the link between system theory and data assimilation is further strengthened by discussing the application of balanced truncation to standard linear Gaussian Bayesian inference, and, in particular, the 4D-Var method. similarities between both data assimilation problems allow a discussion of established methods as well as a generalisation of the state-of-the-art approach to arbitrary prior covariances as reachability Gramians. Furthermore, we propose an enhanced approach using time-limited balanced truncation that allows to balance Bayesian inference for unstable systems and in addition mproves the numerical results for short observation periods.
2022 | Conditioning analysis for discrete Helmholtz problems | A. Kaya, M.A. FreitagZeitschrift: Computers & Mathematics with ApplicationsVerlag: ElsevierSeiten: 171‒182Band: 118Link zur Publikation
Conditioning analysis for discrete Helmholtz problems
Autoren: A. Kaya, M.A. Freitag
(2022)
In this paper, we examine conditioning of the discretization of the Helmholtz problem. Although the discrete Helmholtz problem has been studied from different perspectives, to the best of our knowledge, there is no conditioning analysis for it. We aim to fill this gap in the literature. We propose a novel method in 1D to observe the near-zero eigenvalues of a symmetric indefinite matrix. Standard classification of ill-conditioning based on the matrix condition number is not true for the discrete Helmholtz problem. We relate the ill-conditioning of the discretization of the Helmholtz problem with the condition number of the matrix. We carry out analytical conditioning analysis in 1D and extend our observations to 2D with numerical observations. We examine several discretizations. We find different regions in which the condition number of the problem shows different characteristics. We also explain the general behavior of the solutions in these regions.
Zeitschrift:
Computers & Mathematics with Applications
2022 | Datenassimilation: Die nahtlose Verschmelzung von Daten und Modellen | M.A. Freitag, S. ReichZeitschrift: Mitteilungen der Deutschen Mathematiker-VereinigungVerlag: De GruyterSeiten: 108‒112Band: 30Link zur Publikation
Datenassimilation: Die nahtlose Verschmelzung von Daten und Modellen
Autoren: M.A. Freitag, S. Reich
(2022)
Der Sonderforschungsbereich SFB1294 „Datenassimilation: Die nahtlose Verschmelzung von Daten und Modellen“ wird seit 2017 von der Deutschen Forschungsgemeinschaft an der Universität of Potsdam gefördert und befindet sich gegenwärtig am Beginn der zweiten Förderphase. Gemeinsam mit den Projektpartnern der Humboldt Universität zu Berlin, der Technischen Universität Berlin, dem Weierstraß-Institut für Angewandte Analysis und Stochastik sowie dem Helmholtz-Zentrum Potsdam: Deutsches GeoForschungsZentrum GFZ, werden innerhalb des Verbundes sowohl die statistischen und mathematischen Grundlagen also auch mannigfaltige Anwendungen zeitabhängiger inverser Problemeuntersucht. In diesem Beitrag werden in kurzen Zügen sowohl die grundlegende mathematische Fragestellung also auch die aktuellen wissenschaftlichen Projekte des SFBs vorgestellt
Zeitschrift:
Mitteilungen der Deutschen Mathematiker-Vereinigung
2021 | Autonomous Exploration and Identification of High Performing Adsorbents using Active Learning | G. Donval, C. Hand, J. Hook, E. Dupont, M. S. Landman, M.A. Freitag, M. Lennox, T. DürenVerlag: submittedLink zum Preprint
Autonomous Exploration and Identification of High Performing Adsorbents using Active Learning
Autoren: G. Donval, C. Hand, J. Hook, E. Dupont, M. S. Landman, M.A. Freitag, M. Lennox, T. Düren
(2021)
MOFs and COFs are porous materials with a large variety of applications including gas storage and separation. Synthesised in a modular fashion from distinct building blocks, a near infinite number of structures can be constructed and the properties of the material can be tailored for a specific application. While this modularity is a very attractive feature it also poses a challenge. Attempting to identify the best performing material(s) for a given application is experimentally intractable. Current research efforts combine molecular simulations and machine learning techniques to evaluate the simulated performance of hundreds of thousands of materials to identify top performing MOFs and COFs for a given application. These approaches typically rely on moderated brute-force screening which is still resource-intensive as typically between 70‒100 % of the hundreds of thousands of materials must be simulated to create a training set for the machine learning models used, restricting screening to relatively simple molecules. In this work we demonstrate our novel Bayesian mining approach to materials screening which allows 62‒92 % of the top 100 porous materials for a range of applications to be readily identified from large materials databases after only assessing less than one percent of all materials. This is a stark contrast to the 0‒1 % achieved by conventional brute-force screening where porous materials are just chosen at random during a high throughput screening. Through this accelerated virtual screening process, the identification of high performing materials can be used to more rapidly inform experimental efforts and hence lead to an acceleration of the entire research and development pipeline of porous materials.
2021 | Optimization based model order reduction for stochastic systems | M. Redmann, M.A. FreitagZeitschrift: Appl. Math. Comput., 398Link zur Publikation
Optimization based model order reduction for stochastic systems
Autoren: M. Redmann, M.A. Freitag
(2021)
Zeitschrift:
Appl. Math. Comput., 398
2020 | Inexact methods for the low-rank solution to large-scale Lyapunov equations | P. Kürschner, M.A. FreitagZeitschrift: BIT, 60:1221-1259Link zur Publikation
,
Link zum Preprint
Inexact methods for the low-rank solution to large-scale Lyapunov equations
Autoren: P. Kürschner, M.A. Freitag
(2020)
Zeitschrift:
BIT, 60:1221-1259
2020 | Numerical linear algebra in data assimilation | M.A. FreitagZeitschrift: GAMM MitteilungenLink zur Publikation
Numerical linear algebra in data assimilation
Autoren: M.A. Freitag
(2020)
Zeitschrift:
GAMM Mitteilungen
2019 | Teaching of Computing to Mathematics Students: Programming and Discrete Mathematics | J. Betteridge, J.H. Davenport, M.A. Freitag, W. Heijtljes, S. Kynaston, G. Sankaran, G. TraustasonZeitschrift: Proceedings of the 3rd Conference on Computing Education Practice, 12, 1-4Link zur Publikation
Teaching of Computing to Mathematics Students: Programming and Discrete Mathematics
Autoren: J. Betteridge, J.H. Davenport, M.A. Freitag, W. Heijtljes, S. Kynaston, G. Sankaran, G. Traustason
(2019)
Zeitschrift:
Proceedings of the 3rd Conference on Computing Education Practice, 12, 1-4
2019 | Projection methods for weak constraint variational data assimilation | M.A. Freitag, D.L.H. Green
Projection methods for weak constraint variational data assimilation
Autoren: M.A. Freitag, D.L.H. Green
(2019)
2018 | GMRES convergence bounds for eigenvalue problems | M.A. Freitag, P. Kürschner, J. PestanaZeitschrift: Comput. Methods Appl. Math.Seiten: 203-222Band: 18(2)Link zur Publikation
GMRES convergence bounds for eigenvalue problems
Autoren: M.A. Freitag, P. Kürschner, J. Pestana
(2018)
Zeitschrift:
Comput. Methods Appl. Math.
2018 | Model reduction and approximation: theory and algorithms | M. A. FreitagZeitschrift: SIAM Rev.Seiten: 763–767Band: 60(3)
(review of book)
Model reduction and approximation: theory and algorithms
Autoren: M. A. Freitag
(2018)
2018 | Balanced model order reduction for linear random dynamical systems driven by Lévy noise | M. Redmann, M.A. FreitagZeitschrift: To appear in J. Comput. Dyn.Link zum Preprint
Balanced model order reduction for linear random dynamical systems driven by Lévy noise
Autoren: M. Redmann, M.A. Freitag
(2018)
Zeitschrift:
To appear in J. Comput. Dyn.
2018 | A low-rank approach to the solution of weak constraint variational data assimilation problems | M.A. Freitag, D.L.H. GreenZeitschrift: J. Comput. Phys.Seiten: 263-281Band: 357Link zur Publikation
A low-rank approach to the solution of weak constraint variational data assimilation problems
Autoren: M.A. Freitag, D.L.H. Green
(2018)
Zeitschrift:
J. Comput. Phys.
2015 | Tuned preconditioners for inexact two-sided inverse and Rayleigh quotient iteration | M.A. Freitag, P. KürschnerZeitschrift: Numer. Linear Algebra Appl.Seiten: 175–196Band: 22(1)Link zur Publikation
Tuned preconditioners for inexact two-sided inverse and Rayleigh quotient iteration
Autoren: M.A. Freitag, P. Kürschner
(2015)
Zeitschrift:
Numer. Linear Algebra Appl.
2015 | The effect of numerical model error on data assimilation | S.E. Jenkins, C.J. Budd, M.A. Freitag, N.D. SmithZeitschrift: J. Comput. Appl. Math.Seiten: 567 – 588Band: 290Link zur Publikation
The effect of numerical model error on data assimilation
Autoren: S.E. Jenkins, C.J. Budd, M.A. Freitag, N.D. Smith
(2015)
Zeitschrift:
J. Comput. Appl. Math.
2014 | A new approach for calculating the real stability radius | M.A. Freitag, A. SpenceZeitschrift: BITVerlag: SpringerSeiten: 381-400Band: 54(2)Link zur Publikation
A new approach for calculating the real stability radius
Autoren: M.A. Freitag, A. Spence
(2014)
2014 | Calculating the $H_\infty$-norm using the implicit determinant method | M.A. Freitag, A Spence, P. Van DoorenZeitschrift: SIAM J. Matrix Anal. Appl.Seiten: 619-635Band: 35(2)Link zur Publikation
Calculating the $H_\infty$-norm using the implicit determinant method
Autoren: M.A. Freitag, A Spence, P. Van Dooren
(2014)
Zeitschrift:
SIAM J. Matrix Anal. Appl.
2014 | The computation of Jordan blocks in parameter-dependent matrices | R.O. Akinola, M.A. Freitag, A. SpenceZeitschrift: IMA J. Numer. Anal.Verlag: Oxford University PressSeiten: 955-976Band: 34(3)Link zur Publikation
The computation of Jordan blocks in parameter-dependent matrices
Autoren: R.O. Akinola, M.A. Freitag, A. Spence
(2014)
Zeitschrift:
IMA J. Numer. Anal.
Verlag:
Oxford University Press
2014 | The calculation of the distance to a nearby defective matrix | R.O. Akinola, M.A. Freitag, A. SpenceZeitschrift: Numer. Linear Algebra Appl.Seiten: 403-414Band: 21(3)Link zur Publikation
The calculation of the distance to a nearby defective matrix
Autoren: R.O. Akinola, M.A. Freitag, A. Spence
(2014)
Zeitschrift:
Numer. Linear Algebra Appl.
2013 | Large Scale Inverse Problems: Computational Methods and Applications in the Earth Sciences | M. Cullen, M.A. Freitag, S. Kindermann and R. Scheichl (Editors)Reihe: Radon Series on Computational and Applied Mathematics, Vol. 13Verlag: De Gruyter, BerlinLink zur Publikation
Large Scale Inverse Problems: Computational Methods and Applications in the Earth Sciences
Autoren: M. Cullen, M.A. Freitag, S. Kindermann and R. Scheichl (Editors)
(2013)
Reihe:
Radon Series on Computational and Applied Mathematics, Vol. 13
Verlag:
De Gruyter, Berlin
2013 | Synergy of inverse problems and data assimilation techniques. Large Scale Inverse Problems | M.A. Freitag, R.W.E PotthastZeitschrift: Radon Ser. Comput. Appl. Math.Verlag: De GruyterBand: 13
(Preprint)
Synergy of inverse problems and data assimilation techniques. Large Scale Inverse Problems
Autoren: M.A. Freitag, R.W.E Potthast
(2013)
Zeitschrift:
Radon Ser. Comput. Appl. Math.
2013 | The Origin of Power-Law Emergent Scaling in Large Binary Networks | D.P. Almond, C.J. Budd, M.A. Freitag, G.W. Hunt, N.J. McCullen, N.D. SmithZeitschrift: Phys. ASeiten: 1004-1027Band: 392(4)Link zur Publikation
,
Link zum Preprint
The Origin of Power-Law Emergent Scaling in Large Binary Networks
Autoren: D.P. Almond, C.J. Budd, M.A. Freitag, G.W. Hunt, N.J. McCullen, N.D. Smith
(2013)
2013 | Resolution of sharp fronts in the presence of model error in variational data assimilation | M.A. Freitag, N.K. Nichols, C.J. BuddZeitschrift: Q. J. R. Meteorol. Soc.Seiten: 742-757Band: 139Link zur Publikation
Resolution of sharp fronts in the presence of model error in variational data assimilation
Autoren: M.A. Freitag, N.K. Nichols, C.J. Budd
(2013)
Zeitschrift:
Q. J. R. Meteorol. Soc.
2011 | A Newton-based method for the calculation of the distance to instability | M.A. Freitag, A. SpenceZeitschrift: Linear Alg. Appl.Seiten: 3189-3205Band: 435(12)Link zur Publikation
A Newton-based method for the calculation of the distance to instability
Autoren: M.A. Freitag, A. Spence
(2011)
Zeitschrift:
Linear Alg. Appl.
2011 | Regularization techniques for ill-posed inverse problems in data assimilation | M.A. Freitag, N.K. Nichols, C.J. BuddZeitschrift: Comput. & FluidsSeiten: 168-173Band: 46(1)Link zur Publikation
Regularization techniques for ill-posed inverse problems in data assimilation
Autoren: M.A. Freitag, N.K. Nichols, C.J. Budd
(2011)
Zeitschrift:
Comput. & Fluids
2010 | L1-regularisation for ill-posed problems in variational data assimilation | M.A. Freitag, N.K. Nichols, C.J. BuddZeitschrift: PAMMSeiten: 665-668Band: 10(1)Link zur Publikation
L1-regularisation for ill-posed problems in variational data assimilation
Autoren: M.A. Freitag, N.K. Nichols, C.J. Budd
(2010)
2009 | Shift-invert Arnoldi's method with preconditioned iterative solves | M.A. Freitag, A. SpenceZeitschrift: SIAM J. Matrix Anal. Appl.Seiten: 942-969Band: 31(3)Link zur Publikation
Shift-invert Arnoldi's method with preconditioned iterative solves
Autoren: M.A. Freitag, A. Spence
(2009)
Zeitschrift:
SIAM J. Matrix Anal. Appl.
2009 | The calculation of the distance to instability by the computation of a Jordan block | M. Freitag (joint work with A. Spence)
Oberwolfach Report No. 37/2009. (Report)
The calculation of the distance to instability by the computation of a Jordan block
Autoren: M. Freitag (joint work with A. Spence)
(2009)
Oberwolfach Report No. 37/2009. (
Report)
2008 | Rayleigh quotient iteration and simplified Jacobi-Davidson with preconditioned iterative solves for generalised eigenvalue problems | M.A. Freitag, A. Spence, E. Vainikko
(Gzipped PostScript), (PDF)
Rayleigh quotient iteration and simplified Jacobi-Davidson with preconditioned iterative solves for generalised eigenvalue problems
Autoren: M.A. Freitag, A. Spence, E. Vainikko
(2008)
2008 | The Impact of Multiplication Operators on the Ill-Posedness of Inverse Problems | Melina Freitag
168 pages, VDM Verlag Dr. Müller Aktiengesellschaft & Co. KG (2008). (Available online here)
The Impact of Multiplication Operators on the Ill-Posedness of Inverse Problems
Autoren: Melina Freitag
(2008)
168 pages, VDM Verlag Dr. Müller Aktiengesellschaft & Co. KG (2008). (Available online
here)
2008 | Rayleigh Quotient iteration and simplified Jacobi-Davidson method with preconditioned iterative solves | M.A. Freitag, A. SpenceZeitschrift: Linear Alg. Appl.Verlag: ElsevierSeiten: 2049-2060Band: 428(8-9)Link zur Publikation
Rayleigh Quotient iteration and simplified Jacobi-Davidson method with preconditioned iterative solves
Autoren: M.A. Freitag, A. Spence
(2008)
Zeitschrift:
Linear Alg. Appl.
2008 | A tuned preconditioner for inexact inverse iteration applied to Hermitian eigenvalue problems | M.A. Freitag, A. SpenceZeitschrift: IMA J. Numer. Anal.Verlag: Oxford University PressSeiten: 522-551Band: 28(3)Link zur Publikation
A tuned preconditioner for inexact inverse iteration applied to Hermitian eigenvalue problems
Autoren: M.A. Freitag, A. Spence
(2008)
Zeitschrift:
IMA J. Numer. Anal.
Verlag:
Oxford University Press
2007 | Convergence theory for inexact inverse iteration applied to the generalised nonsymmetric eigenproblem | M.A. Freitag, A. SpenceZeitschrift: Electron. Trans. Numer. Anal.Seiten: 40-64Band: 28Link zum Preprint
Convergence theory for inexact inverse iteration applied to the generalised nonsymmetric eigenproblem
Autoren: M.A. Freitag, A. Spence
(2007)
Zeitschrift:
Electron. Trans. Numer. Anal.
2007 | Inner-outer Iterative Methods for Eigenvalue Problems - Convergence and Preconditioning | M. A. Freitag
PhD Thesis, Department of Mathematical Sciences, University of Bath (2007). (Gzipped PostScript), (PDF)
Inner-outer Iterative Methods for Eigenvalue Problems - Convergence and Preconditioning
Autoren: M. A. Freitag
(2007)
2007 | The Preissmann box scheme and its modification for transcritical flows | M.A. Freitag, K. W. MortonZeitschrift: Internat. J. Numer. Methods Engrg.Verlag: WileySeiten: 791-811Band: 70(7)Link zur Publikation
The Preissmann box scheme and its modification for transcritical flows
Autoren: M.A. Freitag, K. W. Morton
(2007)
Zeitschrift:
Internat. J. Numer. Methods Engrg.
2007 | Convergence rates for inexact inverse iteration with application to preconditioned iterative solves | M.A. Freitag, A. SpenceZeitschrift: BITVerlag: SpringerSeiten: 27-44Band: 47(1)Link zur Publikation
Convergence rates for inexact inverse iteration with application to preconditioned iterative solves
Autoren: M.A. Freitag, A. Spence
(2007)
2005 | Analytical and numerical studies on the influence of multiplication operators for the ill-posedness of inverse problems | M. Freitag, B. HofmannZeitschrift: J. Inverse Ill-Posed Probl.Seiten: 123-148Band: 13(2)Link zur Publikation
Analytical and numerical studies on the influence of multiplication operators for the ill-posedness of inverse problems
Autoren: M. Freitag, B. Hofmann
(2005)
Zeitschrift:
J. Inverse Ill-Posed Probl.
2004 | On the Influence of Multiplication Operators on the Ill-posedness of Inverse Problems | M. Freitag
Diplomarbeit, Fakultät für Mathematik, Technische Universität Chemnitz (2004). (Available online here)
On the Influence of Multiplication Operators on the Ill-posedness of Inverse Problems
Autoren: M. Freitag
(2004)
Diplomarbeit, Fakultät für Mathematik, Technische Universität Chemnitz (2004). (Available online
here)
2003 | Transcritical flow modelling with the Box Scheme | M. Freitag
MSc Thesis, Department of Mathematical Sciences, University of Bath (2003). (GZipped PostScript), (PDF)
Transcritical flow modelling with the Box Scheme
Autoren: M. Freitag
(2003)
