Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Institut für Mathematik

Linear partial differential equations (PDEs) are an important, widely applied class of mechanistic models, describing physical processes such as heat transfer, electromagnetism, and wave propagation. In practice, specialized numerical methods based on discretization are used to solve PDEs. They generally use an estimate of the unknown model parameters and, if available, physical measurements for initialization. Such solvers are often embedded into larger scientific models with a downstream application and thus error quantification plays a key role. However, by ignoring parameter and measurement uncertainty, classical PDE solvers may fail to produce consistent estimates of their inherent approximation error. In this work, we approach this problem in a principled fashion by interpreting solving linear PDEs as physics-informed Gaussian process (GP) regression. Our framework is based on a key generalization of the Gaussian process inference theorem to observations made via an arbitrary bounded linear operator. Crucially, this probabilistic viewpoint allows to (1) quantify the inherent discretization error; (2) propagate uncertainty about the model parameters to the solution; and (3) condition on noisy measurements. Demonstrating the strength of this formulation, we prove that it strictly generalizes methods of weighted residuals, a central class of PDE solvers including collocation, finite volume, pseudospectral, and (generalized) Galerkin methods such as finite element and spectral methods. This class can thus be directly equipped with a structured error estimate. In summary, our results enable the seamless integration of mechanistic models as modular building blocks into probabilistic models by blurring the boundaries between numerical analysis and Bayesian inference.

This talk focuses on constrained optimization problems that can be efficiently solved using first-order methods, particularly Frank-Wolfe methods (also known as Conditional Gradients). These algorithms have emerged as a crucial class of methods for minimizing smooth convex functions over polytopes, and their applicability extends beyond this domain. Recently, they have garnered significant attention due to their ability to facilitate structured optimization, a key aspect in some machine learning applications. I will provide a broad overview of these methods, highlighting their applications and presenting recent advances in both traditional optimization and machine learning. Time permitting, I will also discuss an extension of these methods to the mixed-integer convex optimization setting.

For directed graphs with arc capacities, Nagamochi and Ibaraki (1989) showed that if the cut condition guarantees the existence of a fractional multiflow, and this implication holds in a certain hereditary way, then the cut condition also guarantees the existence of an integer multiflow. Motivated by our recent results with Mohammed Majthoub Almoghrabi and Philipp Warode on integer and unsplittable multiflows in series-parallel digraphs, we discuss a hierarchy of cut conditions and a somewhat refined version of the Nagamochi-Ibaraki Theorem. 

In this talk, we provide an overview of Gurobi's algorithmic
components for solving nonlinear optimization problems to global optimality. In essence, we extend our existing mixed-integer programming (MIP) framework to handle such problems. This includes our presolve algorithms, an extension of the branch-and-bound method utilizing spatial relaxations, and an interior point algorithm for nonlinear problems, which serves as a primal heuristic to find high-quality solutions. As a result, we can compute solutions to nonlinear optimization  problems along with certificates for global optimality. Finally, we have extended gurobipy to facilitate the easy formulation of expression-based nonlinear optimization problems in Python.
Gurobi's nonlinear solver applies to explicit expression-based
constraints and does not require the supply of derivative data.

The Abs-Smooth Frank-Wolfe algorithm (ASFW) is a non-smooth variant of the popular Frank-Wolfe algorithms. In this talk we sketch and analyze the "vanilla" and the "heavy-ball" variants of the ASFW algorithm. We provide stronger and more general primal-dual convergence results for ASFW when applied in the convex setting. We derive a convergence rate for our algorithm which is identical to the smooth case. So far, there is limited understanding of accelerated convergence regimes in the context of ASFW. We also provide some answers in this context by looking into some special cases.

The theory of energetic rate-independent systems is an elegant way to describe nonlinear systems in mechanics and other fields. One particular advantage is that it yields a natural time discretization that consists of a sequence of minimization problems. Unfortunately, in many interesting cases the objective functional is only given implicitly as the solution of a second minimization problem for a
curve length in the state space. Therefore, its evaluation and obtaining derivatives can be very costly. Instead, we present a transformation based on the Finsler exponential map that turns the second minimization problem into an initial-value-problem for a second-order ODE. Solutions of this can be found much cheaper numerically, or may even be available in closed form. We show examples of this construction, and how to use it to obtain fast and robust Proximal Newton solvers for finite-strain elastoplasticity. 

tLean is an interactive theorem prover and programming language. It allows its users to write definitions and proofs from mathematics or computer science in a formal language, so that they may be verified. We'll introduce formalization in Lean with a simple study of linked lists. Then, we'll proceed with a discussion of what it means to verify algorithms in Lean, and how it differs from algorithms that produce proofs. Finally, we'll implement a variant of subgradient descent in Lean and prove a convergence theorem for it.