http://popups.lib.uliege.be/1373-5411/index.php?id=2200 Publications of Auteurs Gerhard-Wilhelm Weber fr 0 http://popups.lib.uliege.be/1373-5411/index.php?id=3272 Stochastic differential equations are widely used to model noise-affected phenomena in nature, technology and economy (Kloeden et al., 1994). As these equations are usually hard to represent by a computer and hard to resolve we express them in simplified manner. We introduce an approximation by discretization and additive models based on splines. Then, we construct a penalized residual sum of squares (PRSS) for this model. We show when the related minimization program can be written as a Tikhonov regularization problem (ridge regression), and we treat it using continuous optimization techniques. In particular, we apply the elegant framework of conic quadratic programming. Convex optimization problems are very well-structured, resembling linear programs and permit the use of interior point methods (Nesterov & Nemirovskii, 1993). Fri, 13 Sep 2024 13:53:07 +0200 Fri, 13 Sep 2024 13:53:16 +0200 http://popups.lib.uliege.be/1373-5411/index.php?id=3272 http://popups.lib.uliege.be/1373-5411/index.php?id=2195 Optimization theory is a key technology for inverse problems of reconstruction with applications in science, technology and economy. Discrete tomography is a modern research field which deals with finite objects from VLSI chip design or medical imaging. This paper focuses on the utilization of modern optimization methods to approximately resolve the NP-hard reconstruction problem of discrete tomography. Our new approaches and introductions are based on modeling and algorithms from coding theory and optimal experimental design. Here, we combine continuous and discrete optimization with exploiting geometrical symmetries, or more generally, equivariances, in a framework of statistical learning. Tue, 30 Jul 2024 13:22:50 +0200 Tue, 30 Jul 2024 13:23:02 +0200 http://popups.lib.uliege.be/1373-5411/index.php?id=2195 http://popups.lib.uliege.be/1373-5411/index.php?id=1883 This article is a small survey and pioneering as a starting point for a longer research project : to utilize generalized semi-infinite optimization for purposes of prediction. Firstly, it reflects tbe analytical and inverse (intrinsic) behaviour of generalized semi-infinite optimization problems P(f,h,g,u,v) and presents interpretations of them from the viewpoint of anticipatory systems. These differentiable problems admit an infinite set Y(x) of inequality constraints y, which depends on the state x. Under suitable assumptions, we present global stability properties of the feasible set and corresponding structural stability properties of the entire optimization problem (Weber, 2002 ; Weber, 2003). The achieved results are a basis of algorithm design. In the course of explanation, the perturbational approach gives rise to reconstructions. By studying three applications of generalized semi-infinite optimization, secondly, we interpret these aspects of inverse problems in the sense of prediction. The three anticipatory systems are : (i) Reverse Chebycchev approximation, where we describe a given system by a neighbouring easier one as long as possible under some error tolerance. We begin by a motivating problem from chemical engineering and turn then to time-dependent systems. (ii) Time-minimal or -maximal optimization problems, where we want to pull or push the time-horizon of some process to present time or into the future. We mention global warming and turn to further kinds of biosystems. (iii) Computational biology, where we are concerned with prediction and stability of DNA microarray gene-expression patterns. Wed, 17 Jul 2024 12:56:01 +0200 Thu, 10 Oct 2024 16:49:39 +0200 http://popups.lib.uliege.be/1373-5411/index.php?id=1883