Numerical methods for linear least squaresNumerical methods for linear least squares entails the numerical analysis of linear least squares problems. A general approach to the least squares problem can be described as follows. Suppose that we can find an n by m matrix S such that XS is an orthogonal projection onto the image of X. Then a solution to our minimization problem is given by simply because is exactly a sought for orthogonal projection of onto an image of X (see the picture below and note that as explained in the next section the image of X is just a subspace generated by column vectors of X).
Numerical stabilityIn the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues.
Computer simulationComputer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be determined by comparing their results to the real-world outcomes they aim to predict. Computer simulations have become a useful tool for the mathematical modeling of many natural systems in physics (computational physics), astrophysics, climatology, chemistry, biology and manufacturing, as well as human systems in economics, psychology, social science, health care and engineering.
Soil consolidationSoil consolidation refers to the mechanical process by which soil changes volume gradually in response to a change in pressure. This happens because soil is a two-phase material, comprising soil grains and pore fluid, usually groundwater. When soil saturated with water is subjected to an increase in pressure, the high volumetric stiffness of water compared to the soil matrix means that the water initially absorbs all the change in pressure without changing volume, creating excess pore water pressure.
Types of volcanic eruptionsSeveral types of volcanic eruptions—during which lava, tephra (ash, lapilli, volcanic bombs, and volcanic blocks), and assorted gases are expelled from a volcanic vent or fissure—have been distinguished by volcanologists. These are often named after famous volcanoes where that type of behavior has been observed. Some volcanoes may exhibit only one characteristic type of eruption during a period of activity, while others may display an entire sequence of types all in one eruptive series.
Volcanic rockVolcanic rock (often shortened to volcanics in scientific contexts) is a rock formed from lava erupted from a volcano. Like all rock types, the concept of volcanic rock is artificial, and in nature volcanic rocks grade into hypabyssal and metamorphic rocks and constitute an important element of some sediments and sedimentary rocks. For these reasons, in geology, volcanics and shallow hypabyssal rocks are not always treated as distinct. In the context of Precambrian shield geology, the term "volcanic" is often applied to what are strictly metavolcanic rocks.
Failure causeFailure causes are defects in design, process, quality, or part application, which are the underlying cause of a failure or which initiate a process which leads to failure. Where failure depends on the user of the product or process, then human error must be considered. A part failure mode is the way in which a component failed "functionally" on the component level. Often a part has only a few failure modes. For example, a relay may fail to open or close contacts on demand.
Numerical weather predictionNumerical weather prediction (NWP) uses mathematical models of the atmosphere and oceans to predict the weather based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of computer simulation in the 1950s that numerical weather predictions produced realistic results. A number of global and regional forecast models are run in different countries worldwide, using current weather observations relayed from radiosondes, weather satellites and other observing systems as inputs.
Volcanic glassVolcanic glass is the amorphous (uncrystallized) product of rapidly cooling magma. Like all types of glass, it is a state of matter intermediate between the closely packed, highly ordered array of a crystal and the highly disordered array of liquid. Volcanic glass may refer to the interstitial material, or matrix, in an aphanitic (fine-grained) volcanic rock, or to any of several types of vitreous igneous rocks. Volcanic glass is formed when magma is rapidly cooled.
Shear strength (soil)Shear strength is a term used in soil mechanics to describe the magnitude of the shear stress that a soil can sustain. The shear resistance of soil is a result of friction and interlocking of particles, and possibly cementation or bonding of particle contacts. Due to interlocking, particulate material may expand or contract in volume as it is subject to shear strains. If soil expands its volume, the density of particles will decrease and the strength will decrease; in this case, the peak strength would be followed by a reduction of shear stress.
