Coupled 3D-1D solute transport models: Derivation, model error analysis, and numerical approximation
Tuesday, 17.10.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Starting from full-dimensional models of solute transport, we derive and analyze multi-dimensional (3D-1D) coupled models of time-dependent convection, diffusion, and exchange in and around pulsating vascular and perivascular networks. These models are widely applicable for modelling transport in vascularized tissue, brain perivascular spaces, vascular plants and similar environments. The well-posedness of the full and the multi-dimensional equations is established. In the derivation of the 3D-1D model, a 3D inclusion is reduced to its centerline. Thus, we establish a-priori estimates on the associated modelling errors in evolving Bochner spaces in terms of the inclusion's diameter. We consider both continuous and discontinuous Galerkin approximations to the coupled 3D-1D problems, and we discuss the convergence properties of the numerical schemes. Finally, we present numerical simulations in idealized geometries and in a brain mesh with a large network of vessels on its surface and inside the parenchyma. \n
Adapted Wasserstein distance for SDEs with irregular coefficients
Tuesday, 7.11.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We solve an optimal transport problem under probabilistic constraints, where the marginals are laws of solutions of stochastic differential equations with irregular, that is non-globally Lipschitz continuous coefficients. Numerical methods are employed as a theoretical tool\nto bound the adapted Wasserstein distance. This opens the door for\ncomputing the adapted Wasserstein distance in a simple way.\n\n\nJoint work with B. Robinson (University of Vienna).
1D approximation in Wasserstein spaces
Tuesday, 14.11.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Given a Borel probability measure, we seek to approximate it with a measure uniformly\ndistributed over a 1-dimensional set. With this end, we minimize the Wasserstein distance of this fixed measure to all probability measures uniformly distributed to connected 1 dimensional sets and a regularization term given by their length. To show existence of solution to this problem, one cannot easily resort to the direct method in the calculus of variations due to concentration of mass effects. Therefore, we propose a relaxed problem in the space of probability measures which always admits a solution. In the sequel, we show that whenever the initial measureis absolutely continuous w.r.t. the 1-Hausdorff measure (in particular for absolutely continuous measures w.r.t. Lebesgue measure in R^d) then the solution will be a rectiable measure. This allows us to perform a blow-up argument that, in dimension 2, shows that the solution has a uniform density, being therefore a solution to the original problem. Finally, we prove a phase-field approximation for this problem in the form of a Gamma-convergence result of a functional reminiscent of the Ambrosio-Tortorelli approximation for the Mumford-Shah problem, with the additional property of enforcing connectivity of the 1-dimensional sets that emerges from the approximation. This last feature is achieved with the connectivity functional introduced by Dondl and Wojtowytsch.
Resolvent estimates for one-dimensional Schroedinger operators with complex potentials
Tuesday, 28.11.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We study one-dimensional Schroedinger operators with unbounded complex potentials of various growths (from iterated logs to super-exponentials). We derive asymptotic formulas for the norm of the resolvent as the spectral parameter diverges along the imaginary and real axes. In each case, our analysis yields an explicit leading order term as well as an optimal estimate of the remainder. We also discuss several extensions of the main results, their interrelation with the complementary estimates based on non-semiclassical pseudomode construction in [KS-19] and several examples.\n\nThe talk is based on the joint work [AS-23] with A. Arnal.\n\nReferences:\n\n[AS-23] A. Arnal and P. Siegl: Resolvent estimates for one-dimensional Schroedinger operators with complex potentials, 2023, J. Funct. Anal. 284, 109856\n\n[KS-19] D. Krejcirik and P. Siegl: Pseudomodes for Schroedinger operators with complex potentials, 2019, J. Funct. Anal. 276, 2856-2900
Continuum Limit of Nearest Neighbor and Random Long-Range Interactions
Tuesday, 12.12.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
The thesis deals with the limit behavior of discrete energies with different types of interactions between points. On the one hand, only nearest neighbor interactions are considered and on the other hand random long-range interactions. For the latter some assumptions on the conductance have to be made. In a last step, we will try to combine these two types of interactions and investigate whether some assumptions can be dropped in this case.\n\n \n\n
On strong approximation of SDEs with a discontinuous drift coecient
Tuesday, 19.12.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
The classical assumption in the literature on numerical approximation of stochastic differential equations (SDEs) is global Lipschitz continuity of the coecients of the equation.\nHowever, many SDEs arising in applications fail to have globally Lipschitz continuous coecients.\nIn the last decade an intensive study of numerical approximation of SDEs with nonglobally Lipschitz continuous coecients has begun. In particular, strong approximation\nof SDEs with a drift coecient that is discontinuous in space has recently gained a lot of\ninterest. Such SDEs arise e.g. in mathematical finance, insurance and stochastic control\nproblems. Classical techniques of error analysis are not applicable to such SDEs and well\nknown convergence results for standard methods do not carry over in general.\n\nIn this talk I will present recent results on strong approximation of such SDEs.\n\nThe talk is based on joint work with Thomas M¨uller-Gronbach (University of Passau).
TBA
Tuesday, 16.1.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
TBA
The p-Laplacien in the setting of multi-valued operators.
Tuesday, 16.1.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
\nAbstract: The thesis provides proofs for the existence of solutions for both the stationary problem\n\[ \blambda u + Au \bni f \bin \bOmega \bsubset R^N;~~ u=0 ~\btext{on}~~ \bpartial \bOmega\] \nand the non-stationary problem \n\[dy(t)/dt + Ay(t) \bni f(t) ~ \btext{for}~ t\bin[0,T];~~ y(0)=y_0 \]for A being a maximal monotone/accretive operator on \(L^2(\bOmega)\). It especially considers such operators A that arise as the sub-differential of some energy-functional and also shows some regularity for them in the non-stationary case. As an example the theory is applied to the p-Laplace operator.\n
A-priori bounds for geometric FE discretizations of a Cosserat rod and simulations for microheterogeneous prestressed rods
Tuesday, 23.1.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
In summary, this thesis focuses on developing an a priori theory for geometric finite\nelement discretizations of a Cosserat rod model, which is derived from incompatible\nelasticity. This theory will be supported by corresponding numerical experiments\nto validate the convergence behavior of the proposed method.\nThe main result describes the qualitative behavior of intrinsic H 1 -errors and\n\nL^2 -errors in terms of the mesh diameter 0 < h ≪ 1 of the approximation scheme:\n\nD 1,2 (u, u h ) ≲ h m , d L 2 (u, u h ) ≲ h m+1 ,\n\nfor a sequence of m-order discrete solutions u h and an exact solution u.
Convergence of computational homogenization methods based on the fast Fourier transform
Tuesday, 6.2.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Since their inception in the mid 1990s, computational methods based on the fast Fourier transform (FFT) have been established as efficient and powerful tools for computing the effective mechanical properties of composite materials. These methods operate on a regular grid, employ periodic boundary conditions for the displacement fluctuation and utilize the FFT to design matrix-free iterative schemes whose iteration count is (most often) bounded independently of the grid spacing.\nIn the talk at hand, we will take a look both at the convergence of the used iterative schemes and the convergence of the underlying spectral discretization. Remarkably, despite the presence of discontinuous coefficients, the spectral discretization enjoys the same convergence rate as a finite-element discretization on a regular grid. Moreover, the convergence behavior of the effective stresses profit from a superconvergence phenomenon apparently inherent to computational homogenization problems.\n---------------------------------------------------------
On strong approximation of SDEs with a discontinuous drift coefficient
Tuesday, 13.2.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
The classical assumption in the literature on numerical approximation of stochastic differential equations (SDEs) is global Lipschitz continuity of the coefficients of the equation.\nHowever, many SDEs arising in applications fail to have globally Lipschitz continuous coefficients.\nIn the last decade an intensive study of numerical approximation of SDEs with nonglobally Lipschitz continuous coefficients has begun. In particular, strong approximation\nof SDEs with a drift coefficient that is discontinuous in space has recently gained a lot of\ninterest. Such SDEs arise e.g. in mathematical finance, insurance and stochastic control\nproblems. Classical techniques of error analysis are not applicable to such SDEs and well\nknown convergence results for standard methods do not carry over in general.\nIn this talk I will present recent results on strong approximation of such SDEs.\nThe talk is based on joint work with Thomas M¨uller-Gronbach (University of Passau)