Using distributions as statistical evidence in well-structured and ill-structured problems

Research has suggested that understanding in well-structured settings often does not transfer to the everyday, less-structured problems encountered outside of school. Little is known, beyond anecdotal evidence, about how teachers' consideration of distributions as evidence in well-structured settings compares with their use in ill-structured problem contexts. A qualitative study of preservice secondary teachers examined their use of distributions as evidence in four tasks of varying complexity and ill-structuredness. Results suggest that teachers' incorporation of distributions in well-structured settings does not imply that they will be incorporated in less structured problems (and vice-versa). Implications for research and teaching are discussed.

Probability distribution modelling to improve stability in nonlinear MIMO control

We consider the direct adaptive inverse control of nonlinear multivariable systems with different delays between every input-output pair. In direct adaptive inverse control, the inverse mapping is learned from examples of input-output pairs. This makes the obtained controller sub optimal, since the network may have to learn the response of the plant over a larger operational range than necessary. Moreover, in certain applications, the control problem can be redundant, implying that the inverse problem is ill posed. In this paper we propose a new algorithm which allows estimating and exploiting uncertainty in nonlinear multivariable control systems. This approach allows us to model strongly non-Gaussian distribution of control signals as well as processes with hysteresis. The proposed algorithm circumvents the dynamic programming problem by using the predicted neural network uncertainty to localise the possible control solutions to consider.

A procedure for determining a spacewise dependent heat source and the initial temperature

The inverse problem of determining a spacewise dependent heat source, together with the initial temperature for the parabolic heat equation, using the usual conditions of the direct problem and information from two supplementary temperature measurements at different instants of time is studied. These spacewise dependent temperature measurements ensure that this inverse problem has a unique solution, despite the solution being unstable, hence the problem is ill-posed. We propose an iterative algorithm for the stable reconstruction of both the initial data and the source based on a sequence of well-posed direct problems for the parabolic heat equation, which are solved at each iteration step using the boundary element method. The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for a typical benchmark test example, which has the input measured data perturbed by increasing amounts of random noise. The numerical results show that the proposed procedure gives accurate numerical approximations in relatively few iterations.

Boundary integral equations for acoustical inverse sound-soft scattering

The shape of a plane acoustical sound-soft obstacle is detected from knowledge of the far field pattern for one time-harmonic incident field. Two methods based on solving a system of integral equations for the incoming wave and the far field pattern are investigated. Properties of the integral operators required in order to apply regularization, i.e. injectivity and denseness of the range, are proved.

A variational method for identifying a spacewise-dependent heat source

The inverse problem of determining a spacewise-dependent heat source for the parabolic heat equation using the usual conditions of the direct problem and information from one supplementary temperature measurement at a given instant of time is studied. This spacewise-dependent temperature measurement ensures that this inverse problem has a unique solution, but the solution is unstable and hence the problem is ill-posed. We propose a variational conjugate gradient-type iterative algorithm for the stable reconstruction of the heat source based on a sequence of well-posed direct problems for the parabolic heat equation which are solved at each iteration step using the boundary element method. The instability is overcome by stopping the iterative procedure at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented which have the input measured data perturbed by increasing amounts of random noise. The numerical results show that the proposed procedure yields stable and accurate numerical approximations after only a few iterations.

Determination of a spacewise dependent heat source

This paper investigates the inverse problem of determining a spacewise dependent heat source in the parabolic heat equation using the usual conditions of the direct problem and information from a supplementary temperature measurement at a given single instant of time. The spacewise dependent temperature measurement ensures that the inverse problem has a unique solution, but this solution is unstable, hence the problem is ill-posed. For this inverse problem, we propose an iterative algorithm based on a sequence of well-posed direct problems which are solved at each iteration step using the boundary element method (BEM). The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for various typical benchmark test examples which have the input measured data perturbed by increasing amounts of random noise.

Using cognitive load theory to interpret student difficulties with a problem-based learning approach to engineering education:a case study

This article reports on an investigationwith first year undergraduate ProductDesign and Management students within a School of Engineering and Applied Science. The students at the time of this investigation had studied fundamental engineering science and mathematics for one semester. The students were given an open ended, ill-formed problem which involved designing a simple bridge to cross a river.They were given a talk on problemsolving and given a rubric to follow, if they chose to do so.They were not given any formulae or procedures needed in order to resolve the problem. In theory, they possessed the knowledge to ask the right questions in order tomake assumptions but, in practice, it turned out they were unable to link their a priori knowledge to resolve this problem. They were able to solve simple beam problems when given closed questions. The results show they were unable to visualize a simple bridge as an augmented beam problem and ask pertinent questions and hence formulate appropriate assumptions in order to offer resolutions.

Quasi-Regular Solutions for 3D Equations of Tricomi and Keldish Types

Цветан Д. Христов, Недю Ив. Попиванов, Манфред Шнайдер - Изучени са някои тримерни гранични задачи за уравнения от смесен тип. За уравнения от типа на Трикоми те са формулирани от М. Протер през 1952, като тримерни аналози на задачите на Дарбу или Коши–Гурса в равнината. Добре известно е, че новите задачи са некоректни. Ние формулираме нова гранична задача за уравнения от типа на Келдиш и даваме понятие за квазиругулярно решение на тази задача и на eдна от задачите на Протер. Намерени са достатъчни условия за единственост на такива решения.

Resolução em tomografia de tempo de trânsito poço-a-poço: a dependência da iluminação

The key aspect limiting resolution in crosswell traveltime tomography is illumination, a well known result but not as well exemplified. Resolution in the 2D case is revisited using a simple geometric approach based on the angular aperture distribution and the Radon Transform properties. Analitically it is shown that if an interface has dips contained in the angular aperture limits in all points, it is correctly imaged in the tomogram. By inversion of synthetic data this result is confirmed and it is also evidenced that isolated artifacts might be present when the dip is near the illumination limit. In the inverse sense, however, if an interface is interpretable from a tomogram, even an aproximately horizontal interface, there is no guarantee that it corresponds to a true interface. Similarly, if a body is present in the interwell region it is diffusely imaged in the tomogram, but its interfaces - particularly vertical edges - can not be resolved and additional artifacts might be present. Again, in the inverse sense, there is no guarantee that an isolated anomaly corresponds to a true anomalous body because this anomaly can also be an artifact. Jointly, these results state the dilemma of ill-posed inverse problems: absence of guarantee of correspondence to the true distribution. The limitations due to illumination may not be solved by the use of mathematical constraints. It is shown that crosswell tomograms derived by the use of sparsity constraints, using both Discrete Cosine Transform and Daubechies bases, basically reproduces the same features seen in tomograms obtained with the classic smoothness constraint. Interpretation must be done always taking in consideration the a priori information and the particular limitations due to illumination. An example of interpreting a real data survey in this context is also presented.

Resolução em tomografia de tempo de trânsito poço-a-poço: a dependência da iluminação

The key aspect limiting resolution in crosswell traveltime tomography is illumination, a well known result but not as well exemplified. Resolution in the 2D case is revisited using a simple geometric approach based on the angular aperture distribution and the Radon Transform properties. Analitically it is shown that if an interface has dips contained in the angular aperture limits in all points, it is correctly imaged in the tomogram. By inversion of synthetic data this result is confirmed and it is also evidenced that isolated artifacts might be present when the dip is near the illumination limit. In the inverse sense, however, if an interface is interpretable from a tomogram, even an aproximately horizontal interface, there is no guarantee that it corresponds to a true interface. Similarly, if a body is present in the interwell region it is diffusely imaged in the tomogram, but its interfaces - particularly vertical edges - can not be resolved and additional artifacts might be present. Again, in the inverse sense, there is no guarantee that an isolated anomaly corresponds to a true anomalous body because this anomaly can also be an artifact. Jointly, these results state the dilemma of ill-posed inverse problems: absence of guarantee of correspondence to the true distribution. The limitations due to illumination may not be solved by the use of mathematical constraints. It is shown that crosswell tomograms derived by the use of sparsity constraints, using both Discrete Cosine Transform and Daubechies bases, basically reproduces the same features seen in tomograms obtained with the classic smoothness constraint. Interpretation must be done always taking in consideration the a priori information and the particular limitations due to illumination. An example of interpreting a real data survey in this context is also presented.

How does problem based learning fit with cognitive load theory?

This paper reports on an investigation with first year undergraduate Product Design and Management students within a School of Engineering. The students at the time of this investigation had studied fundamental engineering science and mathematics for one semester. The students were given an open ended, ill formed problem which involved designing a simple bridge to cross a river. They were given a talk on problem solving and given a rubric to follow, if they chose to do so. They were not given any formulae or procedures needed in order to resolve the problem. In theory, they possessed the knowledge to ask the right questions in order to make assumptions but, in practice, it turned out they were unable to link their a priori knowledge to resolve this problem. They were able to solve simple beam problems when given closed questions. The results show they were unable to visualise a simple bridge as an augmented beam problem and ask pertinent questions and hence formulate appropriate assumptions in order to offer resolutions.

On Provable Algorithms for Determination of Continuous Protein Interdomain Motions from Residual Dipolar Couplings

Dynamics of biomolecules over various spatial and time scales are essential for biological functions such as molecular recognition, catalysis and signaling. However, reconstruction of biomolecular dynamics from experimental observables requires the determination of a conformational probability distribution. Unfortunately, these distributions cannot be fully constrained by the limited information from experiments, making the problem an ill-posed one in the terminology of Hadamard. The ill-posed nature of the problem comes from the fact that it has no unique solution. Multiple or even an infinite number of solutions may exist. To avoid the ill-posed nature, the problem needs to be regularized by making assumptions, which inevitably introduce biases into the result.

Here, I present two continuous probability density function approaches to solve an important inverse problem called the RDC trigonometric moment problem. By focusing on interdomain orientations we reduced the problem to determination of a distribution on the 3D rotational space from residual dipolar couplings (RDCs). We derived an analytical equation that relates alignment tensors of adjacent domains, which serves as the foundation of the two methods. In the first approach, the ill-posed nature of the problem was avoided by introducing a continuous distribution model, which enjoys a smoothness assumption. To find the optimal solution for the distribution, we also designed an efficient branch-and-bound algorithm that exploits the mathematical structure of the analytical solutions. The algorithm is guaranteed to find the distribution that best satisfies the analytical relationship. We observed good performance of the method when tested under various levels of experimental noise and when applied to two protein systems. The second approach avoids the use of any model by employing maximum entropy principles. This 'model-free' approach delivers the least biased result which presents our state of knowledge. In this approach, the solution is an exponential function of Lagrange multipliers. To determine the multipliers, a convex objective function is constructed. Consequently, the maximum entropy solution can be found easily by gradient descent methods. Both algorithms can be applied to biomolecular RDC data in general, including data from RNA and DNA molecules.

An alternative benchmarking approach for electricity utility regulation using maximum entropy

The main purpose of this study is to present an alternative benchmarking approach that can be used by national regulators of utilities. It is widely known that the lack of sizeable data sets limits the choice of the benchmarking method and the specification of the model to set price controls within incentive-based regulation. Ill-posed frontier models are the problem that some national regulators have been facing. Maximum entropy estimators are useful in the estimation of such ill-posed models, in particular in models exhibiting small sample sizes, collinearity and non-normal errors, as well as in models where the number of parameters to be estimated exceeds the number of observations available. The empirical study involves a sample data used by the Portuguese regulator of the electricity sector to set the parameters for the electricity distribution companies in the regulatory period of 2012-2014. DEA and maximum entropy methods are applied and the efficiency results are compared.

Soft computing based parameter identification in pavements and geomechanical systems

Accurate estimation of road pavement geometry and layer material properties through the use of proper nondestructive testing and sensor technologies is essential for evaluating pavement’s structural condition and determining options for maintenance and rehabilitation. For these purposes, pavement deflection basins produced by the nondestructive Falling Weight Deflectometer (FWD) test data are commonly used. The nondestructive FWD test drops weights on the pavement to simulate traffic loads and measures the created pavement deflection basins. Backcalculation of pavement geometry and layer properties using FWD deflections is a difficult inverse problem, and the solution with conventional mathematical methods is often challenging due to the ill-posed nature of the problem. In this dissertation, a hybrid algorithm was developed to seek robust and fast solutions to this inverse problem. The algorithm is based on soft computing techniques, mainly Artificial Neural Networks (ANNs) and Genetic Algorithms (GAs) as well as the use of numerical analysis techniques to properly simulate the geomechanical system. A widely used pavement layered analysis program ILLI-PAVE was employed in the analyses of flexible pavements of various pavement types; including full-depth asphalt and conventional flexible pavements, were built on either lime stabilized soils or untreated subgrade. Nonlinear properties of the subgrade soil and the base course aggregate as transportation geomaterials were also considered. A computer program, Soft Computing Based System Identifier or SOFTSYS, was developed. In SOFTSYS, ANNs were used as surrogate models to provide faster solutions of the nonlinear finite element program ILLI-PAVE. The deflections obtained from FWD tests in the field were matched with the predictions obtained from the numerical simulations to develop SOFTSYS models. The solution to the inverse problem for multi-layered pavements is computationally hard to achieve and is often not feasible due to field variability and quality of the collected data. The primary difficulty in the analysis arises from the substantial increase in the degree of non-uniqueness of the mapping from the pavement layer parameters to the FWD deflections. The insensitivity of some layer properties lowered SOFTSYS model performances. Still, SOFTSYS models were shown to work effectively with the synthetic data obtained from ILLI-PAVE finite element solutions. In general, SOFTSYS solutions very closely matched the ILLI-PAVE mechanistic pavement analysis results. For SOFTSYS validation, field collected FWD data were successfully used to predict pavement layer thicknesses and layer moduli of in-service flexible pavements. Some of the very promising SOFTSYS results indicated average absolute errors on the order of 2%, 7%, and 4% for the Hot Mix Asphalt (HMA) thickness estimation of full-depth asphalt pavements, full-depth pavements on lime stabilized soils and conventional flexible pavements, respectively. The field validations of SOFTSYS data also produced meaningful results. The thickness data obtained from Ground Penetrating Radar testing matched reasonably well with predictions from SOFTSYS models. The differences observed in the HMA and lime stabilized soil layer thicknesses observed were attributed to deflection data variability from FWD tests. The backcalculated asphalt concrete layer thickness results matched better in the case of full-depth asphalt flexible pavements built on lime stabilized soils compared to conventional flexible pavements. Overall, SOFTSYS was capable of producing reliable thickness estimates despite the variability of field constructed asphalt layer thicknesses.

A novel methodology for simulating contact-line behavior in capillary-driven flows

Despite the wide swath of applications where multiphase fluid contact lines exist, there is still no consensus on an accurate and general simulation methodology. Most prior numerical work has imposed one of the many dynamic contact-angle theories at solid walls. Such approaches are inherently limited by the theory accuracy. In fact, when inertial effects are important, the contact angle may be history dependent and, thus, any single mathematical function is inappropriate. Given these limitations, the present work has two primary goals: 1) create a numerical framework that allows the contact angle to evolve naturally with appropriate contact-line physics and 2) develop equations and numerical methods such that contact-line simulations may be performed on coarse computational meshes.

Fluid flows affected by contact lines are dominated by capillary stresses and require accurate curvature calculations. The level set method was chosen to track the fluid interfaces because it is easy to calculate interface curvature accurately. Unfortunately, the level set reinitialization suffers from an ill-posed mathematical problem at contact lines: a ``blind spot'' exists. Standard techniques to handle this deficiency are shown to introduce parasitic velocity currents that artificially deform freely floating (non-prescribed) contact angles. As an alternative, a new relaxation equation reinitialization is proposed to remove these spurious velocity currents and its concept is further explored with level-set extension velocities.

To capture contact-line physics, two classical boundary conditions, the Navier-slip velocity boundary condition and a fixed contact angle, are implemented in direct numerical simulations (DNS). DNS are found to converge only if the slip length is well resolved by the computational mesh. Unfortunately, since the slip length is often very small compared to fluid structures, these simulations are not computationally feasible for large systems. To address the second goal, a new methodology is proposed which relies on the volumetric-filtered Navier-Stokes equations. Two unclosed terms, an average curvature and a viscous shear VS, are proposed to represent the missing microscale physics on a coarse mesh.

All of these components are then combined into a single framework and tested for a water droplet impacting a partially-wetting substrate. Very good agreement is found for the evolution of the contact diameter in time between the experimental measurements and the numerical simulation. Such comparison would not be possible with prior methods, since the Reynolds number Re and capillary number Ca are large. Furthermore, the experimentally approximated slip length ratio is well outside of the range currently achievable by DNS. This framework is a promising first step towards simulating complex physics in capillary-dominated flows at a reasonable computational expense.