Abstract : We consider optimization problems that are posed in the space of Radon measures as well as the space of functions of bounded variation. We present recent theoretical advances, in particular with respect to optimization problems that involve integrality conditions on the distributed optimization variables. These include approximation results in function space that are based on Gamma-convergence, optimality conditions, and consistent discretization schemes. Moreover, the presentations also include recent algorithmic convergence analysis and discrete and stochastic algorithms that allow for efficient solutions of the arising subproblems.
[02974] Proximal methods for point source localisation
Format : Talk at Waseda University
Author(s) :
Tuomo Valkonen (Escuela Politécnica Nacional & University of Helsinki)
Abstract : Point source localisation is generally modelled as a Lasso-type problem on measures. However, optimisation methods in such non-Hilbert spaces are not yet very well understood. Most numerical algorithms are based on the Frank-Wolfe conditional gradient method. We develop extensions of proximal-type methods to spaces of measures. This includes forward-backward splitting, its inertial version, and primal-dual proximal splitting. Their convergence proofs follow standard patterns. We demonstrate their numerical efficacy.
[04816] Nonsmooth minimization in Banach spaces meets sparse dictionary learning
Format : Talk at Waseda University
Author(s) :
Daniel Walter (Humboldt Universität zu Berlin)
Abstract : We propose a novel method for problems involving nonsmooth regularization terms over infinite dimensional function spaces . It resembles a dictionary learning algorithm which updates a dictionary $\mathcal{A}_k$ of extremal points of the unit ball of the regularizer and of a sparse represenation of the iterate $u_k$ in its conic hull. Imposing additional assumptions on the dual variables, its asymptotic linear convergence is shown.
[04270] L^q-quasinorm sparse optimal control problems with controls in BV functions
Format : Talk at Waseda University
Author(s) :
Pedro Martín Merino (Escuela Politécnica Nacional)
Abstract : We consider an elliptic optimal control of elliptic linear partial differential equations that involves a nonconvex regularization in terms of the $L^q$ quasi–norm (with $q$ in (0, 1)) together with a TV penalization in the cost function, given by:
$$\mathcal{F}(y,u)=\frac{1}{2}\| y-y_d \|^2_{L^2(\Omega)}+\frac{ \alpha}{2}\|u\|^2_{L^2(\Omega)}+\beta \int_\Omega |u|^{q} dx + \gamma \int_\Omega |Du|, $$
where $u$ is control and $y$ are the control and state variables, respectively.
When $\gamma=0$, the classical direct method fails to argue the existence of solutions due to the lack of lower semicontinuity of $\mathcal{F}$. However, given the topological properties of $BV(\Omega)$ and the continuity of the $L^q$-quasinorm in $L^1(\Omega)$, the existence of solutions can be obtained for controls in $BV(\Omega)$. In this talk, we address the optimality conditions for this problem and a numerical approach for its numerical solution based on primal-dual splitting methods.
[04282] Robust Optimal Experimental Design for Bayesian Inverse Problems
Format : Talk at Waseda University
Author(s) :
Ahmed Attia (Argonne National Laboratory)
Todd Munson (Argonne National Laboratory)
Sven Leyffer (Argonne National Laboratory)
Abstract : An optimal design is defined as the one that maximizes a predefined utility function which is formulated in terms of the elements of an inverse problem. An example being optimal sensor placement for parameter identification. This formulation generally overlooks misspecification of the elements of the inverse problem such as the prior or the measurement uncertainties. In this talk, we present efficient recipes for designing optimal experimental design schemes, for Bayesian inverse problems, such that the optimal design is robust with respect to misspecification of elements of the inverse problem.
[03634] Opial property in Wasserstein spaces and applications
Format : Talk at Waseda University
Author(s) :
Emanuele Naldi (TU Braunschweig)
Abstract : The Opial property is a metric characterization of weak convergence for a suitable class of Banach spaces. It plays an important role in the study of weak convergence of iterates of mappings and of the asymptotic behavior of nets satisfying some metric properties. Since it involves only metric quantities, it is possible to define this property also in metric spaces provided with a suitable notion of weak convergence. This is the case for spaces of probability measures endowed with the Kantorovich-Rubinstein-Wasserstein metric deriving by optimal transport. In particular, in this talk, we present an Opial property in the Wasserstein space of Borel probability measures with finite quadratic moment on a separable Hilbert space. We present applications of this property to convergence of Wasserstein gradient flows of lower semicontinuous and geodesically convex functionals defined on the space of probability measures. We show further application to convergence of sequences generated by the proximal point algorithm and a proximal gradient algorithm when the functional satisfy some additional hypothesis. We conclude with one last application of the property to convergence to a fixed point for iterations of a non-exapansive map defined on a weakly closed set.
[03671] An Optimal Transport-based approach to Total-Variation regularization for the Diffusion MRI problem
Format : Talk at Waseda University
Author(s) :
Rodolfo Assereto (University of Graz)
Kristian Bredies (University of Graz)
Marion I. Menzel (GE Global Research, Munich)
Emanuele Naldi (TU Braunschweig)
Claudio Mayrink Verdun (TU München)
Abstract : Diffusion Magnetic Resonance Imaging (dMRI) is a non-invasive imaging technique that draws structural information from the interaction between water molecules and biological tissues. Common ways of tackling the derived inverse problem include, among others, Diffusion Tensor Imaging (DTI), High Angular Resolution Diffusion Imaging (HARDI) and Diffusion Spectrum Imaging (DSI). However, these methods are structurally unable to recover the full diffusion distribution, only providing partial information about particle displacement. In our work, we introduce a Total-Variation (TV) regularization defined from an optimal transport perspective using 1-Wasserstein distances. Such a formulation produces a variational problem that can be handled by well-known algorithms enjoying good convergence properties, such as the primal-dual proximal method by Chambolle and Pock. It allows for the reconstruction of the complete diffusion spectrum from measured undersampled k/q space data.
[04791] A minimization problem in the space of bounded deformations arising in visco-plastic fluid flows
Format : Talk at Waseda University
Author(s) :
Lukas Holbach (Johannes Gutenberg University Mainz)
Christian Meyer (TU Dortmund)
Georg Stadler (New York University)
Abstract : Plasticity and material failure play an important role in Earth's plate motion. These phenomena are commonly modeled by incompressible Stokes flows with visco-plastic rheologies. Weak solutions of these nonlinear equations can be characterized as minimizers of a convex energy functional. While the solution is unique and lies in $H^1$ if a lower-bound regularization on the viscosity is employed, the problem becomes singular without regularization and the solution must be sought in the space of bounded deformations (BD).
We present an existence result in BD and show that the regularized solutions converge to a solution of the singular problem with respect to a suitable topology when the regularization parameter tends to zero.
[03554] Solving Discrete Subproblems of a Trust-Region Algorithm for MIOCP
Format : Talk at Waseda University
Author(s) :
Marvin Severitt (TU Dortmund University)
Paul Manns (TU Dortmund University)
Abstract : We consider a trust-region algorithm for the solution of control problems, where the control input is an integer-valued function and is regularized with a total variation term in the objective. A class of integer linear programs arises as discretizations of the trust-region subproblems. We discuss how, in the one-dimensional case, the discretized subproblems can be solved with a graph-based approach and how the information obtained can be used for the two-dimensional case.
[03555] Regularization and outer approximation for optimal control problems in BV
Format : Talk at Waseda University
Author(s) :
Annika Müller (TU Dortmund University)
Christian Meyer (TU Dortmund University)
Abstract : We consider optimal control problems with a constraint on the TV-seminorm of the control. We replace the TV-seminorm by a regularized version and solve the resulting optimization problems with an outer approximation algorithm. We prove convergence of the algorithm to the globally optimal solutions, which in turn converge to the optimal solution to the original problem as the regularization parameter vanishes.
[03256] On integer optimal control problems with total variation regularization
Format : Talk at Waseda University
Author(s) :
Jonas Marko (BTU Cottbus-Senftenberg)
Gerd Wachsmuth (BTU Cottbus-Senftenberg)
Abstract : We investigate integer optimal control problems of the form
$$ \text{Minimize}\quad F(u) + \beta \text{TV}(u)\quad \text{s.t.}\quad u(t)\in\{\nu_1,\dots,\nu_d\}\subset\mathbb{Z}\text{ for a.a. }t\in(0,T)$$
with $\beta>0$. The contribution $F$ is assumed to be differentiable and could e.g. realize the tracking of the state given by an ODE or PDE dependent on $u$.
We show local optimality conditions of first and second order as well as non-local optimality conditions. Also, we will calculate numerical solutions exemplary on two specific control problems.
[04006] Non-uniform Grid Refinement for the Combinatorial Integral Approximation
Format : Talk at Waseda University
Author(s) :
Christoph Hansknecht (TU Clausthal)
Paul Manns (TU Dortmund University)
Abstract : We examine mixed-integer optimal control problems (MIOCP) with a discrete-valued control variable distributed on a two-dimensional domain.
We compute integral controls by rounding fractional ones according to the combinatorial integral approximation (CIA) framework with switching costs modeling total variation. The rounding problem becomes computationally challenging in two dimensions, leading us to examine both heuristic and exact solution approaches, reducing the number of problem variables based on non-uniform grid refinements.
