Seismic Imaging with the Elliptic Radon Transform in 3D: Analytical and Numerical Aspects

A well-established method to investigate subsurface material parameters is to generate pressure waves on the surface and measure their reflections returning there at different points. In this thesis, we consider a scanning geometry with constant distance from source to receiver in three space dimensions. After linearisation this situation is modelled by the elliptic Radon transform which integrates over ellipsoids. As an inversion formula of this transform is unknown, we propose a certain imaging operator appropriate to apply the method of the approximate inverse and develop a migration scheme to reconstruct singularities in the speed of sound. Further, we calculate the top order symbol of the imaging operator as a pseudodifferential operator and analyse its behaviour in different situations. We use the obtained results to achieve reconstructions of the subsurface, which are relatively independent of the distance to the surface and the offset. Last, we present numerical experiments and test our implementation with data generated by solving the wave equation numerically

The Guarded Lambda-Calculus: Programming and Reasoning with Guarded Recursion for Coinductive Types

We present the guarded lambda-calculus, an extension of the simply typed lambda-calculus with guarded recursive and coinductive types. The use of guarded recursive types ensures the productivity of well-typed programs. Guarded recursive types may be transformed into coinductive types by a type-former inspired by modal logic and Atkey-McBride clock quantification, allowing the typing of acausal functions. We give a call-by-name operational semantics for the calculus, and define adequate denotational semantics in the topos of trees. The adequacy proof entails that the evaluation of a program always terminates. We introduce a program logic with L\"ob induction for reasoning about the contextual equivalence of programs. We demonstrate the expressiveness of the calculus by showing the definability of solutions to Rutten's behavioural differential equations.Comment: Accepted to Logical Methods in Computer Science special issue on the 18th International Conference on Foundations of Software Science and Computation Structures (FoSSaCS 2015

Guarded Cubical Type Theory: Path Equality for Guarded Recursion

This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of program logics, and for programming and reasoning with coinductive types. We wish to implement GDTT with decidable type-checking, while still supporting non-trivial equality proofs that reason about the extensions of guarded recursive constructions. CTT is a variation of Martin-L\"of type theory in which the identity type is replaced by abstract paths between terms. CTT provides a computational interpretation of functional extensionality, is conjectured to have decidable type checking, and has an implemented type-checker. Our new type theory, called guarded cubical type theory, provides a computational interpretation of extensionality for guarded recursive types. This further expands the foundations of CTT as a basis for formalisation in mathematics and computer science. We present examples to demonstrate the expressivity of our type theory, all of which have been checked using a prototype type-checker implementation, and present semantics in a presheaf category.Comment: 17 pages, to be published in proceedings of CSL 201

Denoising Diffusion Samplers

Denoising diffusion models are a popular class of generative models providing state-of-the-art results in many domains. One adds gradually noise to data using a diffusion to transform the data distribution into a Gaussian distribution. Samples from the generative model are then obtained by simulating an approximation of the time-reversal of this diffusion initialized by Gaussian samples. Practically, the intractable score terms appearing in the time-reversed process are approximated using score matching techniques. We explore here a similar idea to sample approximately from unnormalized probability density functions and estimate their normalizing constants. We consider a process where the target density diffuses towards a Gaussian. Denoising Diffusion Samplers (DDS) are obtained by approximating the corresponding time-reversal. While score matching is not applicable in this context, we can leverage many of the ideas introduced in generative modeling for Monte Carlo sampling. Existing theoretical results from denoising diffusion models also provide theoretical guarantees for DDS. We discuss the connections between DDS, optimal control and Schr\"odinger bridges and finally demonstrate DDS experimentally on a variety of challenging sampling tasks.Comment: In The Eleventh International Conference on Learning Representations, 202

An Empirical Investigation in to the Factors Influencing the Economic Incentive to Retain Ownership of Weaned Steer Calves

Marketing and production data collected from weaned calves (628 head) in a university sponsored retained ownership demonstration program are analyzed to identify factors affecting the annualized rate of return when retaining ownership versus selling the calves at weaning. Data were collected on the following characteristics associated with the calves: 1) ranch-of-origin production management practices; 2) feedlot performance; 3) carcass merit; 4) health history; and 5) market prices. Retained ownership until slaughter was more profitable, on average, when compared to selling calves at weaning. The calculated annualized rate of return to retained ownership versus selling calves at weaning averaged 11.5% per head. Regression analyses indicate that market prices paid for weaned calves and fed cattle have the greatest influence on the rate of return to retained ownership. The other five categories (ranch-of-origin, production management practices, feedlot performance, carcass merit, health history) also contributed to explaining the variability in the rate of return per head. Marketing and production risks were not incorporated into the regression model. However, summary statistics indicate that coefficient of variation associated with per-head retained ownership revenue is 50% higher than the estimated per-head revenue for weaned calves

Risk and the Economic Incentive to Retain Ownership of Steer Calves

Retained ownership of steer calves is an investment decision for cow/calf producers. Data collected over a three-year period on 845 steer calves reveals that retaining ownership of steer calves is, on average, profitable. Systematic and unsystematic risks associated with retaining ownership of steer calves are identified. Empirical results indicate that unsystematic risk account for 67% of the variability in the rate of return to retained ownership. Empirical evidence also suggests that retaining ownership is a riskier investment decision than assumed in the earlier literature. This suggests that the lack of enthusiasm for retaining ownership by cow/calf producers is the result of the level of risk associated with retaining ownership rather than producers being too risk averse