Achieving High Accuracy in Neural Network Training for PDEs with Energy Natural Gradient Descent.
Dienstag, 18.4.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We will talk about energy natural gradient descent, a natural gradient method with respect to a Hessian-induced Riemannian metric as an optimization algorithm for physics-informed neural networks (PINNs) and the deep Ritz method. As a main motivation we show that the update direction in function space resulting from the energy natural gradient corresponds to the Newton direction modulo an orthogonal projection onto the model's tangent space. We present numerical results illustrating that energy natural gradient descent yields highly accurate solutions with errors several orders of magnitude smaller than what is obtained when training PINNs with standard optimizers like gradient descent, Adam or BFGS, even when those are allowed significantly more computation time.
The fundamental gap conjecture
Dienstag, 25.4.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
At the end of his study [J. Statist. Phys., 83] on thermodynamic functions\nof a free boson gas, van den Berg conjectured that the difference between the\ntwo smallest eigenvalues\n\nΓ\nV\n(Ω) := λ\nV\n2\n(Ω) − λ\nV\n1\n(Ω);\n\nof the Schr¨odinger operator −∆ + V on a convex domain Ω in R\nd\n, d ≥ 1,\nequipped with homogeneous Dirichlet boundary conditions satisfies\n\nΓ\nV\n(Ω) ≥ Γ (ID) = 3π\n2\nD2\n, (1)\n\nwhere ID is the interval (−D/2, D/2) of length D = diameter(Ω) . The term\nΓ\nV\n(Ω) is called the fundamental gap and describes an important physical quantity: for example, in statistical mechanics, ΓV\n(Ω) measures the energy needed to\njump from the ground state to the next excited eigenstate, or computationally,\nit can control the rate of convergence of numerical methods to compute large\neigenvalue problems [SIAM, 2011]. Thus, one is interested in (optimal) lower\nbounds on ΓV\n(Ω). Since the late 80s, the fundamental conjecture (1) attracts\nconsistently the attention of many researcher including M. S. Ashbaugh & R.\nBenguria [Proc. Amer. Math. Soc., 89], R. Schoen and S.-T. Yau Camb. Press,\n94., B. Andrews and J. Clutterbuck [J. Amer. Math.\nSoc., 11].\n\nIn this talk, I present new results on the fundamental gap conjecture (1)\nfor the Schr¨odinger operator −∆ + V on a convex domain Ω equipped with\nRobin boundary conditions. In particular, we present a proof of this conjecture\nin dimension one, and mention results for the p-Laplacian.\n\nThe talk is based on the joint works [1, 2] with B. Andrews and J. Clutterbuck.\nReferences\n[1] Ben Andrews, Julie Clutterbuck, and Daniel Hauer. Non-concavity of the\nRobin ground state. Camb. J. Math., 8(2):243–310, 2020.\n[2] Ben Andrews, Julie Clutterbuck, and Daniel Hauer. The fundamental gap\nfor a one-dimensional Schr¨odinger operator with Robin boundary conditions.\nProc. Amer. Math. Soc., 149(4): 1481–1493, 2021.
Paths towards Open World Generalization
Montag, 8.5.23, 14:00-15:00, Raum 226, Hermann-Herder-Str. 10
Acceleration of quantum mechanical systems by exploiting similarity
Dienstag, 9.5.23, 09:00-10:00, Raum 226, Hermann-Herder-Str. 10
Total (generalized) variation for images and shapes
Dienstag, 9.5.23, 10:00-11:00, Raum 226, Hermann-Herder-Str. 10
Nonlinear bending-torsion theory for inextensible tapered rods by \(\bGamma\)-convergence
Dienstag, 16.5.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
The upcoming presentation will focus on a modified version of the main theorem initially stated and proven by Mora and Mueller in their\nwork titled "Derivation of the nonlinear bending-torsion theory for inextensible rods by Γ-Convergence", published in 2003. Specifically, we will direct our attention towards studying a rod with a non-constant cross-section, known as a taper, and answer the question of how the \(\bGamma\)-limit of the elastic energy changes in this case. Additionally, we will combine this new result with previous findings, using banana plants (Musa sp.) as an example.\n
Numerical computations and thermodynamically complete models for inelastic behaviour in solids
Dienstag, 6.6.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
In this talk, I will introduce some aspects of the mathematical modelling of inelastic solids, placing particular emphasis on models that are compatible with the second law of thermodynamics. In particular, I will describe a recent model from [Cichra, Pr?ša; 2020] and discuss its numerical approximation via the finite element method. One of the advantages of the approach considered here is that it is not necessary to introduce additional concepts, such as the plastic strain. Moreover, as a consequence of the thermodynamically consistent derivation, one is able to compute the evolution of the temperature field without additional complication. I will also showcase an application of this modelling approach to the Mullins effect, for which up to date there had been no simple yet fully coupled thermo-mechanical model.
The energy technique for BDF methods
Dienstag, 20.6.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Abstract The application of the energy technique to numerical methods with very good stability properties for parabolic equations, such as algebraically stable Runge–Kutta methods or\nA-stable multistep methods, is straightforward. The extension to high order multistep methods requires some effort; the main difficulty concerns suitable choices of test functions. We\ndiscuss the energy technique for all six backward difference formula (BDF) methods. In the\ncases of the A-stable one- and two-step BDF methods, the application is trivial. The energy\ntechnique is applicable also to the three-, four- and five-step BDF methods via Nevanlinna–\nOdeh multipliers. The main new results are: i) No Nevanlinna–Odeh multipliers exist for the\nsix-step BDF method. ii) The energy technique is applicable under a relaxed condition on\nthe multipliers. iii) We present multipliers that make the energy technique applicable also to\nthe six-step BDF method. Besides its simplicity, the energy technique for BDF methods is\npowerful, it leads to several stability estimates, and flexible, it can be easily combined with\nother stability techniques.
Stability properties of the \(L^2\)-projection mapping to finite element spaces on adaptively generated meshes
Dienstag, 11.7.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
\n\nThe \(L^2\)-projection mapping to Lagrange finite element spaces is a crucial tool in numerical analysis. Its Sobolev stability is known to be the key to discrete stability and quasi-optimality estimates for parabolic problems. For adaptively generated meshes the proof of Sobolev stability is challenging and requires conditions on how strongly the mesh size varies.\n\nHence, for the newest vertex bisection and its generalisation to higher dimensions by Maubach and Traxler we present optimal estimates on the mesh grading. Previously, grading estimates have been available only for 2D mesh refinement strategies. For such adaptively generated meshes we discuss Sobolev stability of the \(L^2\)-projection mapping to Lagrange finite element spaces under certain conditions on the polynomial degree and on the space dimension. In particular, the \(L^2\)-projection is \(W^{1,2}\)-stable for any polynomial degree, for any space dimension smaller than \(7\).\n\nThis is joint work with Lars Diening and Johannes Storn (Bielefeld University).\n
On the role of discrete Green’s operator preconditioning in FFT-based computational homogenization methods
Dienstag, 25.7.23, 15:00-16:00, Raum 226, Hermann-Herder-Str. 10
Solving computational homogenization problems on fine grids leads to systems of linear equations with millions to billions of unknowns, which favour iterative solvers over direct solvers.\nHowever, the number of iterations of iterative solvers can grow with increasing system size. To\novercome this issue, the so-called FFT-based solvers use the discrete Green’s operator preconditioning, which makes the condition number of the resulting linear system independent of the\nsystem/grid size [1, 2]. We studied the discrete Green’s operator preconditioning from a linear\nalgebra viewpoint and showed that all individual eigenvalues of such preconditioned systems can\nbe bounded purely from the knowledge of the material data of the problems, both original and\nreference. We developed a simple algorithm to compute these bounds [3, 4]. In my talk, I will\ndiscuss the theoretical aspects of these results and practical applications of the discrete Green’s\noperator preconditioning to periodic homogenisation problems discretised on regular grids [5].\n\n\nReferences\n\n[1] Moulinec, H. and Suquet, P., A fast numerical method for computing the linear and nonlinear\nmechanical properties of composites, Comptes Rendus de l’Acad´emie des sciences. S´erie II.\nM´ecanique, physique, chimie, astronomie, 318 (1994) 1417–1423\n\n[2] Schneider, M., A review of nonlinear FFT-based computational homogenization methods,\nActa Mechanica, 29 (2021) DOI 10.1007/s00707-021-02962-1,\n\n[3] Ladeck´y, M. and Pultarov´a, I. and Zeman, J., Guaranteed two-sided bounds on all eigenvalues of preconditioned diffusion and elasticity problems solved by the finite element method,\nApplications of Mathematics, 66, 21–42 (2020) DOI 10.21136/AM.2020.0217-19\n\n[4] Pultarov´a, I., Ladeck´y, M., Two-sided guaranteed bounds to individual eigenvalues of preconditioned finite element and finite difference problems, Numerical Linear Algebra Applications\n28 (2021) e2382. DOI 10.1002/nla.2382.\n\n[5] Ladeck´y, M., Leute, J.R., Falsafi, A., Pultarov´a, I., Pastewka, L., Junge, T., and Zeman,\nJ. An Optimal Preconditioned FFT-accelerated Finite Element Solver for Homogenization.\nApplied Mathematics and Computation 446 (2023) 127835 DOI 10.1016/j.amc.2023.127835