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Öğe ADAPTIVE DISCONTINUOUS GALERKIN FINITE ELEMENTS FOR ADVECTIVE ALLEN-CAHN EQUATION(Amer Inst Mathematical Sciences-Aims, 2021) Uzunca, Murat; Sariaydin-Filibelioglu, AyseWe apply a space adaptive interior penalty discontinuous Galerkin method for solving advective Allen-Cahn equation with expanding and contracting velocity fields. The advective Allen-Cahn equation is first discretized in time and the resulting semi-linear elliptic PDE is solved by an adaptive algorithm using a residual-based a posteriori error estimator. The a posteriori error estimator contains additional terms due to the non-divergence-free velocity field. Numerical examples demonstrate the effectiveness and accuracy of the adaptive approach by resolving the sharp layers accurately.Öğe Energy preserving model order reduction of the nonlinear Schrodinger equation(Springer, 2018) Karasozen, Bulent; Uzunca, MuratAn energy preserving reduced order model is developed for two dimensional nonlinear Schrodinger equation (NLSE) with plane wave solutions and with an external potential. The NLSE is discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of Hamiltonian ordinary differential equations are integrated in time by the energy preserving average vector field (AVF) method. The mass and energy preserving reduced order model (ROM) is constructed by proper orthogonal decomposition (POD) Galerkin projection. The nonlinearities are computed for the ROM efficiently by discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD). Preservation of the semi-discrete energy and mass are shown for the full order model (FOM) and for the ROM which ensures the long term stability of the solutions. Numerical simulations illustrate the preservation of the energy and mass in the reduced order model for the two dimensional NLSE with and without the external potential. The POD-DMD makes a remarkable improvement in computational speed-up over the POD-DEIM. Both methods approximate accurately the FOM, whereas POD-DEIM is more accurate than the POD-DMD.Öğe Global energy preserving model reduction for multi-symplectic PDEs(Elsevier Science Inc, 2023) Uzunca, Murat; Karasozen, Bulent; Aydin, AyhanMany Hamiltonian systems can be recast in multi-symplectic form. We develop a reduced -order model (ROM) for multi-symplectic Hamiltonian partial differential equations (PDEs) that preserves the global energy. The full-order solutions are obtained by finite difference discretization in space and the global energy preserving average vector field (AVF) method. The ROM is constructed in the same way as the full-order model (FOM) applying proper orthogonal decomposition (POD) with the Galerkin projection. The reduced-order system has the same structure as the FOM, and preserves the discrete reduced global energy. Ap-plying the discrete empirical interpolation method (DEIM), the reduced-order solutions are computed efficiently in the online stage. A priori error bound is derived for the DEIM ap-proximation to the nonlinear Hamiltonian. The accuracy and computational efficiency of the ROMs are demonstrated for the Korteweg de Vries (KdV) equation, Zakharov-Kuznetzov (ZK) equation, and nonlinear Schrodinger (NLS) equation in multi-symplectic form. Preser-vation of the reduced energies shows that the reduced-order solutions ensure the long-term stability of the solutions.(c) 2022 Elsevier Inc. All rights reserved.Öğe Intrusive and data-driven reduced order modelling of the rotating thermal shallow water equation(Elsevier Science Inc, 2022) Karasozen, Bulent; Yildiz, Suleyman; Uzunca, MuratIn this paper, we investigate projection-based intrusive and data-driven model order reduction in numerical simulation of rotating thermal shallow water equation (RTSWE) in parametric and non-parametric form. Discretization of the RTSWE in space with centered finite differences leads to Hamiltonian system of ordinary differential equations with linear and quadratic terms. The full-order model (FOM) is obtained by applying linearly implicit Kahan's method in time. Applying proper orthogonal decomposition with Galerkin projection (POD-G), we construct the intrusive reduced-order model (ROM). We apply operator inference (OpInf) with re-projection as data-driven ROM. In the parametric case, we make use of the parameter dependency at the level of the PDE without interpolating between the reduced operators. The least-squares problem of the OpInf is regularized with the minimum norm solution. Both ROMs behave similarly and are able to accurately predict the in the test and training data and capture system behaviour in the prediction phase with several orders of magnitude in computational speed-up over the FOM. The preservation of system physics such as the conserved quantities of the RTSWE by both ROMs enable that the models fit better to data and stable solutions are obtained in long-term predictions which are robust to parameter changes. (C) 2022 Elsevier Inc. All rights reserved.Öğe Kernel Principal Component Analysis for Allen-Cahn Equations(Mdpi, 2024) Cakir, Yusuf; Uzunca, MuratDifferent researchers have analyzed effective computational methods that maintain the precision of Allen-Cahn (AC) equations and their constant security. This article presents a method known as the reduced-order model technique by utilizing kernel principle component analysis (KPCA), a nonlinear variation of traditional principal component analysis (PCA). KPCA is utilized on the data matrix created using discrete solution vectors of the AC equation. In order to achieve discrete solutions, small variations are applied for dividing up extraterrestrial elements, while Kahan's method is used for temporal calculations. Handling the process of backmapping from small-scale space involves utilizing a non-iterative formula rooted in the concept of the multidimensional scaling (MDS) method. Using KPCA, we show that simplified sorting methods preserve the dissipation of the energy structure. The effectiveness of simplified solutions from linear PCA and KPCA, the retention of invariants, and computational speeds are shown through one-, two-, and three-dimensional AC equations.Öğe Linearly implicit methods for Allen-Cahn equation(Elsevier Science Inc, 2023) Uzunca, Murat; Karasozen, BuelentIt is well known that the Allen-Cahn equation satisfies a nonlinear stability property, i.e., the free-energy functional decreases in time. Linearly implicit integrators have been de-veloped for energy-preserving methods for conservative systems with polynomial Hamil-tonians, which are based on the concept of polarization. In this paper, we construct lin-early implicit methods for gradient flows preserving the energy dissipation by polarizing the free-energy functional. Two-step linearly implicit methods are derived for the Allen -Cahn equation inheriting energy dissipation law. Numerical experiments for one-, two-, and three-dimensional Allen-Cahn equations demonstrate the energy dissipation and the accuracy of the linearly implicit methods.(c) 2023 Elsevier Inc. All rights reserved.Öğe Linearly implicit methods for the nonlinear Klein-Gordon equation(Elsevier, 2025) Uzunca, Murat; Karasozen, BulentWe present energy-preserving linearly implicit integrators for the nonlinear Klein-Gordon equation, based on the polarization of the polynomial functions. They are symmetric, second- order accurate in time and space, and unconditionally stable. Instead of solving a nonlinear algebraic equation at every time step, the linearly implicit integrators only require solving a linear system, which reduces the computational cost. We propose three types of linearly implicit integrators for the nonlinear Klein-Gordon equation, that preserve the modified, polarized invariants, ensuring the stability of the solutions in long-time integration. Numerical results confirm the theoretical convergence orders and preservation of the Hamiltonians that guarantee the stability of the solutions in long-time simulation.Öğe Nonintrusive model order reduction for cross-diffusion systems(Elsevier, 2022) Karasozen, Bulent; Mulayim, Gulden; Uzunca, MuratIn this paper, we investigate tensor based nonintrusive reduced-order models (ROMs) for parametric cross-diffusion equations. The full-order model (FOM) consists of ordinary differential equations (ODEs) in matrix or tensor form resulting from finite difference discretization of the differential operators by taking the advantage of Kronecker structure. The matrix/tensor differential equations are integrated in time with the implicit-explicit (IMEX) Euler method. The reduced bases, relying on a finite sample set of parameter values, are constructed in form of a two-level approach by applying higher-order singular value decomposition (HOSVD) to the space-time snapshots in tensor form, which leads to a large amount of computational and memory savings. The nonintrusive reduced approximation for an arbitrary parameter value is obtained through tensor product of the reduced basis by the parameter dependent core tensor that contains the reduced coefficients. The reduced coefficients for new parameter values are computed with the radial basis functions. The efficiency of the proposed method is illustrated through numerical experiments for two-dimensional Schnakenberg and three-dimensional Brusselator cross-diffusion equations. The spatiotemporal patterns are accurately predicted by the reduced-order models with speedup factors of orders two and three over the full-order models. (c) 2022 Elsevier B.V. All rights reserved.Öğe Nonlinear Reduced Order Modelling for Korteweg-de Vries Equation(International Society of Academicians, 2024) Çakir, Yusuf; Uzunca, MuratEfficient computational techniques that maintain the accuracy and invariant preservation property of the Korteweg-de Vries (KdV) equations have been studied by a wide range of researchers. In this paper, we introduce a reduced order model technique utilizing kernel principle component analysis (KPCA), a nonlinear version of the classical principle component analysis, in a non-intrusive way. The KPCA is applied to the data matrix, which is formed by the discrete solution vectors of KdV equation. In order to obtain the discrete solutions, the finite differences are used for spatial discretization, and linearly implicit Kahan's method for the temporal one. The back-mapping from the reduced dimensional space, is handled by a non-iterative formula based on the idea of multidimensional scaling (MDS) method. Through KPCA, we illustrate that the reduced order approximations conserve the invariants, i.e., Hamiltonian, momentum and mass structure of the KdV equation. The accuracy of reduced solutions, conservation of invariants, and computational speed enhancements facilitated by classical (linear) PCA and KPCA are exemplified through one-dimensional KdV equation.Öğe Pricing European and American options under Heston model using discontinuous Galerkin finite elements(Elsevier, 2020) Kozpinar, Sinem; Uzunca, Murat; Karasozen, Bulent[No abstract available]Öğe Reduced order modelling of nonlinear cross-diffusion systems(Elsevier Science Inc, 2021) Karasozen, Bulent; Mulayim, Gulden; Uzunca, Murat; Yildiz, SuleymanIn this work, we present reduced-order models (ROMs) for a nonlinear cross-diffusion problem from population dynamics, the Shigesada-Kawasaki-Teramoto (SKT) equation with Lotka-Volterra kinetics. The formation of the patterns of the SKT equation consists of a fast transient phase and a long stationary phase. Reduced order solutions are computed by separating the time into two time-intervals. In numerical experiments, we show for one- and two-dimensional SKT equations with pattern formation, the reduced-order solutions obtained in the time-windowed form, i.e., principal decomposition framework, are more accurate than the global proper orthogonal decomposition solutions obtained in the whole time interval. The finite-difference discretization of the SKT equation in space results in a system of linear-quadratic ordinary differential equations. The ROMs have the same linear-quadratic structure as the full order model. Using the linear-quadratic structure of the ROMs, the computation of the reduced-order solutions is further accelerated by the use of proper orthogonal decomposition in a tensorial framework so that the computations in the reduced system are independent of the full-order solutions. Furthermore, the prediction capabilities of the ROMs are illustrated for one- and two-dimensional patterns. Finally, we show that the entropy is decreasing by the reduced solutions, which is important for the global existence of solutions to the nonlinear cross-diffusion equations such as the SKT equation. (C) 2021 Elsevier Inc. All rights reserved.Öğe Reduced order optimal control of the convective FitzHugh-Nagumo equations(Pergamon-Elsevier Science Ltd, 2020) Karasozen, Bulent; Uzunca, Murat; Kucukseyhan, TugbaIn this paper, we compare three model order reduction methods: the proper orthogonal decomposition (POD), discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD) for the optimal control of the convective FitzHugh-Nagumo (FHN) equations. The convective FHN equations consist of the semi-linear activator and the linear inhibitor equations, modeling blood coagulation in moving excitable media. The semilinear activator equation leads to a non-convex optimal control problem (OCP). The most commonly used method in reduced optimal control is POD. We use DEIM and DMD to approximate efficiently the nonlinear terms in reduced order models. We compare the accuracy and computational times of three reduced-order optimal control solutions with the full order discontinuous Galerkin finite element solution of the convection dominated FHN equations with terminal controls. Numerical results show that POD is the most accurate whereas POD-DMD is the fastest. (C) 2019 Elsevier Ltd. All rights reserved.Öğe Reduced-order modeling for Ablowitz-Ladik equation(Elsevier, 2023) Uzunca, Murat; Karasoezen, BuelentIn this paper, reduced-order models (ROMs) are constructed for the Ablowitz-Ladik equation (ALE), an integrable semi-discretization of the nonlinear Schrodinger equation (NLSE) with and without damping. Both ALEs are non-canonical conservative and dissipative Hamiltonian systems with the Poisson matrix depending quadratically on the state variables, and with quadratic Hamiltonian. The full-order solutions are obtained with the energy preserving midpoint rule for the conservative ALE and exponential midpoint rule for the dissipative ALE. The reduced-order solutions are constructed intrusively by preserving the skew-symmetric structure of the reduced non-canonical Hamiltonian system by applying proper orthogonal decomposition (POD) with the Galerkin projection. For an efficient offline-online decomposition of the ROMs, the quadratic nonlinear terms of the Poisson matrix are approximated by the discrete empirical interpolation method (DEIM). The computation of the reduced-order solutions is further accelerated by the use of tensor techniques. Preservation of the Hamiltonian and momentum for the conservative ALE, and preservation of dissipation properties of the dissipative ALE, guarantee the long-term stability of soliton solutions. & COPY; 2023 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.Öğe Reduced-Order modeling for Heston stochastic volatility model(Hacettepe Univ, Fac Sci, 2024) Kozpinar, Sinem; Uzunca, Murat; Karasozen, BulentIn this paper, we compare the intrusive proper orthogonal decomposition (POD) with Galerkin projection and the data-driven dynamic mode decomposition (DMD), for Heston's option pricing model. The full order model is obtained by discontinuous Galerkin discretization in space and backward Euler in time. Numerical results for butterfly spread, European and digital call options reveal that in general DMD requires more modes than the POD modes for the same level of accuracy. However, the speed-up factors are much higher for DMD than POD due to the non-intrusive nature of the DMD.Öğe Structure preserving model order reduction of shallow water equations(Wiley, 2021) Karasozen, Bulent; Yildiz, Suleyman; Uzunca, MuratIn this paper, we present two different approaches for constructing reduced-order models (ROMs) for the two-dimensional shallow water equation (SWE). The first one is based on the noncanonical Hamiltonian/Poisson form of the SWE. After integration in time by the fully implicit average vector field method, ROMs are constructed with proper orthogonal decomposition(POD)/discrete empirical interpolation method that preserves the Hamiltonian structure. In the second approach, the SWE as a partial differential equation with quadratic nonlinearity is integrated in time by the linearly implicit Kahan's method, and ROMs are constructed with the tensorial POD that preserves the linear-quadratic structure of the SWE. We show that in both approaches, the invariants of the SWE such as the energy, enstrophy, mass and circulation are preserved over a long period of time, leading to stable solutions. We conclude by demonstrating the accuracy and the computational efficiency of the reduced solutions by a numerical test problem.Öğe Structure preserving reduced order modeling for gradient systems(Elsevier Science Inc, 2019) Akman Yildiz, Tugba; Uzunca, Murat; Karasozen, BulentMinimization of energy in gradient systems leads to formation of oscillatory and Turing patterns in reaction-diffusion systems. These patterns should be accurately computed using fine space and time meshes over long time horizons to reach the spatially inhomogeneous steady state. In this paper, a reduced order model (ROM) is developed which preserves the gradient dissipative structure. The coupled system of reaction-diffusion equations are discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of ordinary differential equations (ODEs) are integrated in time by the average vector field (AVF) method, which preserves the energy dissipation of the gradient systems. The ROMs are constructed by the proper orthogonal decomposition (POD) with Galerkin projection. The nonlinear reaction terms are computed efficiently by discrete empirical interpolation method (DEIM). Preservation of the discrete energy of the FOMs and ROMs with POD-DEIM ensures the long term stability of the steady state solutions. Numerical simulations are performed for the gradient dissipative systems with two specific equations; real Ginzburg-Landau equation and Swift-Hohenberg equation. Numerical results demonstrate that the POD-DEIM reduced order solutions preserve well the energy dissipation over time and at the steady state. (C) 2018 Elsevier Inc. All rights reserved.Öğe Structure-Preserving Reduced- Order Modeling of Non-Traditional Shallow Water Equation(Springer Science and Business Media Deutschland GmbH, 2021) Yildiz, Süleyman; Uzunca, Murat; Karasözen, BülentAn energy- preserving reduced -order model (ROM) is developed for the non-traditional shallow water equation (NTSWE) with full Coriolis force. The NTSWE in the noncanonical Hamiltonian/Poisson form is discretized in space by finite differences. The resulting system of ordinary differential equations is integrated in time by the energy preserving average vector field (AVF) method. The Poisson structure of the discretized NTSWE exhibits a skew-symmetric matrix depending on the state variables. An energy- preserving, computationally efficient reduced order model (ROM) is constructed by proper orthogonal decomposition with Galerkin projection. The nonlinearities are computed for the ROM efficiently by discrete empirical interpolation method. Preservation of the discrete energy and the discrete enstrophy are shown for the full- order model, and for the ROM which ensures the long- term stability of the solutions. The accuracy and computational efficiency of the ROMs are shown by two numerical test problems. © 2021, Springer Nature Switzerland AG.Öğe Structure-preserving reduced-order modeling of Korteweg-de Vries equation(Elsevier, 2021) Uzunca, Murat; Karasozen, Bulent; Yildiz, SuleymanComputationally efficient, structure-preserving reduced-order methods are developed for the Korteweg-de Vries (KdV) equations in Hamiltonian form. The semi-discretization in space by finite differences is based on the Hamiltonian structure. The resulting skew-gradient system of ordinary differential equations (ODES) is integrated with the linearly implicit Kahan's method, which preserves the Hamiltonian approximately. We have shown, using proper orthogonal decomposition (POD), the Hamiltonian structure of the full-order model (EOM) is preserved by the reduced-order model (ROM). The reduced model has the same linear-quadratic structure as the FOM. The quadratic nonlinear terms of the KdV equations are evaluated efficiently by the use of tensorial framework, clearly separating the offline-online cost of the FOMs and ROMs. The accuracy of the reduced solutions, preservation of the conserved quantities, and computational speed-up gained by ROMs are demonstrated for the one-dimensional single and coupled KdV equations, and two-dimensional Zakharov-Kuznetsov equation with soliton solutions. (C) 2021 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.
