Solving a large nonlinear system of equations is very computationally complex due to several numerical issues, such as high linear-algebra cost and large memory requirements. Model-Order Reduction (MOR) has been proposed as a way to overcome the issues associated with large dimensions, the most used approach for doing so being Proper Orthogonal Decomposition (POD). The key idea of POD is to reduce a large number of interdependent variables (snapshots) of the system to a much smaller number of uncorrelated variables while retaining as much as possible of the variation in the original variables.

In this work, we show how intervals and constraint solving techniques can be used to compute all the snapshots at once (I-POD). This new process gives us two advantages over the traditional POD method: 1. handling uncertainty in some parameters or inputs; 2. reducing the snapshots computational cost.

]]>l^p-regularization leads to good results, but it is not used as widely as should be, because it lacks a convincing theoretical explanation -- and thus, practitioners are often reluctant to use it, especially in critical situations. In this paper, we show that fuzzy techniques provide a theoretical explanation for the l^p-regularization.

Fuzzy techniques also enables us to come up with natural next approximations to be used when the accuracy of the l^p-based de-noising and de-blurring is not sufficient.

]]>We also discuss which expert-motivated nonlinear models should be used to get a more accurate description of economic and financial phenomena.

]]>Before we concentrate our efforts on designing such algorithms, it is important to make sure that such an algorithm is possible in the first place, i.e., that the corresponding problem is algorithmically computable. In this paper, we analyze the computability of such uncertainty-related problems. It turns out that in a naive (straightforward) formulation, many such problems are not computable, but they become computable if we reformulate them in appropriate practice-related terms.

]]>In other situations, e.g., when buying a house to live in or selecting a movie to watch, the result of the decision is the decision maker's own satisfaction. In such situations, a more adequate approach is to use utilities - a quantitative way of describing user's preferences. In this talk, after a brief introduction describing what are utilities, how to evaluate them, and how to make decisions based on utilities, we explain how to make decisions in situations with user uncertainty - a realistic situation when a decision maker cannot always decide which alternative is better for him or her.

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