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We propose a uniform block diagonal preconditioner for condensed H(div)-conforming HDG schemes for parameter-dependent saddle point problems including the generalized Stokes problem and the linear elasticity. An optimal preconditioner is obtained for the stiffness matrix on the global velocity/displacement space via the auxiliary space preconditioning (ASP) technique [49]. A robust preconditioner spectrally equivalent to the Schur complement on the element-wise constant pressure space is also constructed. Finally, numerical results of the generalized Stokes and the steady linear elasticity equations verify the robustness of our proposed preconditioner with respect to model parameters and mesh size.

Most physical processes posses structural properties such as constant energies, volumes, and other invariants over time. When learning models of such dynamical systems, it is critical to respect these invariants to ensure accurate predictions and physically meaningful behavior. Strikingly, state-of-the-art methods in Gaussian process (GP) dynamics model learning are not addressing this issue. On the other hand, classical numerical integrators are specifically designed to preserve these crucial properties through time. We propose to combine the advantages of GPs as function approximators with structure preserving numerical integrators for dynamical systems, such as Runge-Kutta methods. These integrators assume access to the ground truth dynamics and require evaluations of intermediate and future time steps that are unknown in a learning-based scenario. This makes direct inference of the GP dynamics, with embedded numerical scheme, intractable. Our key technical contribution is the evaluation of the implicitly defined Runge-Kutta transition probability. In a nutshell, we introduce an implicit layer for GP regression, which is embedded into a variational inference-based model learning scheme.

We derive novel error estimates for Hybrid High-Order (HHO) discretizations of Leray-Lions problems set in W^(1,p) with p in (1,2]. Specifically, we prove that, depending on the degeneracy of the problem, the convergence rate may vary between (k+1)(p-1) and (k+1), with k denoting the degree of the HHO approximation. These regime-dependent error estimates are illustrated by a complete panel of numerical experiments.

An energy stable finite element scheme within arbitrary Lagrangian Eulerian (ALE) framework is derived for simulating the dynamics of millimetric droplets in contact with solid surfaces. Supporting surfaces considered may exhibit non--homogeneous properties which are incorporated into system through generalized Navier boundary conditions (GNBC). Numerical scheme is constructed such that the counterpart of (continuous) energy balance holds on the discrete level. This ensures that no spurious energy is introduced into the discrete system, i.e. the discrete formulation is stable in the energy norm. The newly proposed scheme is numerically validated to confirm the theoretical predictions. Of a particular interest is the case of droplet on a non-homogeneous inclined surface. This case shows the capabilities of the scheme to capture the complex droplet dynamics (sliding and rolling) while maintaining stability during the long time simulation.

We introduce a new hybridized discontinuous Galerkin method for the incompressible magnetohydrodynamics equations. If particular velocity, pressure, magnetic field, and magnetic pressure spaces are employed for both element and trace solution fields, we arrive at an energy stable method which returns pointwise divergence-free velocity fields and magnetic fields and properly balances linear momentum. We discretize in time using a second-order-in-time generalized-$\alpha$ method, and we present a block iterative method for solving the resulting nonlinear system of equations at each time step. We numerically examine the effectiveness of our method using a manufactured solution and observe our method yields optimal convergence rates in the $L_2$ norm for the velocity field, pressure field, magnetic field, and magnetic pressure field. We further find our method is pressure robust. We then apply our method to a selection of benchmark problems and numerically confirm our method is energy stable.

For the general class of residual distribution (RD) schemes, including many finite element (such as continuous/discontinuous Galerkin) and flux reconstruction methods, an approach to construct entropy conservative/ dissipative semidiscretizations by adding suitable correction terms has been proposed by Abgrall (J.~Comp.~Phys. 372: pp. 640--666, 2018). In this work, the correction terms are characterized as solutions of certain optimization problems and are adapted to the SBP-SAT framework, focusing on discontinuous Galerkin methods. Novel generalizations to entropy inequalities, multiple constraints, and kinetic energy preservation for the Euler equations are developed and tested in numerical experiments. For all of these optimization problems, explicit solutions are provided. Additionally, the correction approach is applied for the first time to obtain a fully discrete entropy conservative/dissipative RD scheme. Here, the application of the deferred correction (DeC) method for the time integration is essential. This paper can be seen as describing a systematic method to construct structure preserving discretization, at least for the considered example.

In this paper we discuss a reduced basis method for linear evolution PDEs, which is based on the application of the Laplace transform. The main advantage of this approach consists in the fact that, differently from time stepping methods, like Runge-Kutta integrators, the Laplace transform allows to compute the solution directly at a given instant, which can be done by approximating the contour integral associated to the inverse Laplace transform by a suitable quadrature formula. In terms of the reduced basis methodology, this determines a significant improvement in the reduction phase - like the one based on the classical proper orthogonal decomposition (POD) - since the number of vectors to which the decomposition applies is drastically reduced as it does not contain all intermediate solutions generated along an integration grid by a time stepping method. We show the effectiveness of the method by some illustrative parabolic PDEs arising from finance and also provide some evidence that the method we propose, when applied to a simple advection equation, does not suffer the problem of slow decay of singular values which instead affects methods based on time integration of the Cauchy problem arising from space discretization.

The positive definiteness of discrete time-fractional derivatives is fundamental to the numerical stability (in the energy sense) for time-fractional phase-field models. A novel technique is proposed to estimate the minimum eigenvalue of discrete convolution kernels generated by the nonuniform L1, half-grid based L1 and time-averaged L1 formulas of the fractional Caputo's derivative. The main discrete tools are the discrete orthogonal convolution kernels and discrete complementary convolution kernels. Certain variational energy dissipation laws at discrete levels of the variable-step L1-type methods are then established for time-fractional Cahn-Hilliard model.They are shown to be asymptotically compatible, in the fractional order limit $\alpha\rightarrow1$, with the associated energy dissipation law for the classical Cahn-Hilliard equation. Numerical examples together with an adaptive time-stepping procedure are provided to demonstrate the effectiveness of the proposed methods.

This PhD thesis contains several contributions to the field of statistical causal modeling. Statistical causal models are statistical models embedded with causal assumptions that allow for the inference and reasoning about the behavior of stochastic systems affected by external manipulation (interventions). This thesis contributes to the research areas concerning the estimation of causal effects, causal structure learning, and distributionally robust (out-of-distribution generalizing) prediction methods. We present novel and consistent linear and non-linear causal effects estimators in instrumental variable settings that employ data-dependent mean squared prediction error regularization. Our proposed estimators show, in certain settings, mean squared error improvements compared to both canonical and state-of-the-art estimators. We show that recent research on distributionally robust prediction methods has connections to well-studied estimators from econometrics. This connection leads us to prove that general K-class estimators possess distributional robustness properties. We, furthermore, propose a general framework for distributional robustness with respect to intervention-induced distributions. In this framework, we derive sufficient conditions for the identifiability of distributionally robust prediction methods and present impossibility results that show the necessity of several of these conditions. We present a new structure learning method applicable in additive noise models with directed trees as causal graphs. We prove consistency in a vanishing identifiability setup and provide a method for testing substructure hypotheses with asymptotic family-wise error control that remains valid post-selection. Finally, we present heuristic ideas for learning summary graphs of nonlinear time-series models.