Applications of interval computations to earthquake-resistant engineering: How to compute derivatives of interval functions fast

One of the main sources of destruction during earthquake is resonance. Therefore, the following idea has been proposed. We design special control linkages between floors that are normally unattached to the building but can be attached if necessary. They are so designed that adding them changes the building’s characteristic frequency. We continuously monitor displacements within the structure, and when they exceed specified limits, the linkages are engaged in a way to control structural motion. This idea can also be applied to avoid vibrational destruction of large aerospace structures. To check for a resonance, one must know not only the displacements x(t), but also the rates ẋ(t) with which they change. So, we must estimate the velocity from the approximately known function values. Since we consider a small time interval, the function x(t) can be well approximated by its first Taylor expansion terms. Therefore, we consider two cases: when x(t) is linear, and when it is quadratic. In mathematical terms, for some n, we know n + 1 real numbers t0, t1, ..., tn (called moments of time), n numbers x1, ..., xn (called measurement results), and n numbers εi (called precisions). The unknown function x(t) = a + bt + ct (or x(t) = a + bt) satisfies the condition that x(ti) ∈ [xi − εi, xi + εi] for i = 1, 2, ..., n. From that, we must find the possible values of the derivative ẋ(t0). This problem can be reformulated as a particular case of linear programming (LP), so in principle, we can apply LP techniques. We prove that for our problems, one can use faster algorithms. Similar algorithms are described for the case, when the measurement error is of statistical nature. BRIEF INTRODUCTION TO AN ENGINEERING PROBLEM The problem of resonance destruction. One of the main sources of destruction during earthquake is resonance. The majority of the earthquake strikes are not powerful enough to destroy a building in one blow, but the earthquake usually contains a wide spectrum of vibrations with different frequencies. When the characteristic frequencies of the building lie inside this spectrum area, resonance and eventually destruction can occur. The vibrations can also occur, when a bridge, an oil platform, or an aerospace construction encounters periodic waves. How can we diminish destruction? Before we enumerate different methods, let us describe (briefly) the corresponding mathematical problem.