Related papers: Geometric modeling and regularization of algebraic…
Recent quasi-optimal error estimates for the finite element approximation of total-variation regularized minimization problems require the existence of a Lipschitz continuous dual solution. We discuss the validity of this condition and…
The Lasso and the basis pursuit in compressed sensing and machine learning are convex optimization problems with three parameters: the regularization scalar, the observation vector and the data matrix. Relative to the first two parameters,…
We obtain conditions for the differentiability of weak solutions for a second-order uniformly elliptic equation in divergence form with a homogeneous co-normal boundary condition. The modulus of continuity for the coefficients is assumed to…
In recent years, several numerical methods for solving the unique continuation problem for the wave equation in a homogeneous medium with given data on the lateral boundary of the space-time cylinder have been proposed. This problem enjoys…
This paper studies stability aspects of solutions of parametric mathematical programs and generalized equations, respectively, with disjunctive constraints. We present sufficient conditions that, under some constraint qualifications…
We consider geodesics in both Riemannian and Lorentzian manifolds with metrics of low regularity. We discuss existence of extremal curves for continuous metrics and present several old and new examples that highlight their subtle…
Overdetermined systems of first kind integral equations appear in many applications. When the right-hand side is discretized, the resulting finite-data problem is ill-posed and admits infinitely many solutions. We propose a numerical method…
Optimization problems arising in data science have given rise to a number of new derivative-based optimization methods. Such methods often use standard smoothness assumptions -- namely, global Lipschitz continuity of the gradient function…
In this paper, we obtain general conditions under which the wave equation is well-posed in spacetimes with metrics of Lipschitz regularity. In particular, the results can be applied to spacetimes where there is a loss of regularity on a…
We derive the solvability and regularity of the Dirichlet problem for fully non-linear elliptic equations possibly with degenerate right-hand side on Hermitian manifolds, through establishing a quantitative version of boundary estimate…
In this paper, we study Lipschitz continuity of the solution mappings of regularized least-squares problems for which the convex regularizers have (Fenchel) conjugates that are $\mathcal{C}^2$-cone reducible. Our approach, by using…
We consider a class of abstract quasilinear parabolic problems with lower--order terms exhibiting a prescribed singular structure. We prove well--posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global…
With distributed computing and mobile applications becoming ever more prevalent, synchronizing diverging replicas of the same data is a common problem. Reconciliation -- bringing two replicas of the same data structure as close as possible…
Beside the minimization of the prediction error, two of the most desirable properties of a regression scheme are stability and interpretability. Driven by these principles, we propose continuous-domain formulations for one-dimensional…
Employing two distinct types of regularization terms, we propose two regularized extragradient methods for solving equilibrium problems on Hadamard manifolds. The sequences generated by these extragradient algorithms converge to a solution…
We investigate the effect of explicitly enforcing the Lipschitz continuity of neural networks with respect to their inputs. To this end, we provide a simple technique for computing an upper bound to the Lipschitz constant---for multiple…
Partial differential equations describing compressible fluids are prone to the formation of shock singularities, arising from faster upstream fluid particles catching up to slower, downstream ones. In geometric terms, this causes the…
We provide a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity. The functionals considered here range amongst those with unbalanced…
We consider finite element approximations of unique continuation problems subject to elliptic equations in the case where the normal derivative of the exact solution is known to reside in some finite dimensional space. To give quantitative…
In this paper, we mainly study tilt stability and Lipschitz stability of convex optimization problems. Our characterizations are geometric and fully computable in many important cases. As a result, we apply our theory to the group Lasso…