## F. Theory of Computation

### Refine

#### Keywords

- Ableitungsfreie Optimierung (1)
- Beschränkte Krümmung (1)
- Bildsegmentierung (1)
- Effizienter Algorithmus (1)
- Gamma-Konvergenz (1)
- Hadamard manifold (1)
- Hadamard space (1)
- Hadamard-Mannigfaltigkeit (1)
- Hadamard-Raum (1)
- Hyperspektraler Sensor (1)

#### Faculty / Organisational entity

Shared memory concurrency is the pervasive programming model for multicore architectures
such as x86, Power, and ARM. Depending on the memory organization, each architecture follows
a somewhat different shared memory model. All these models, however, have one common
feature: they allow certain outcomes for concurrent programs that cannot be explained
by interleaving execution. In addition to the complexity due to architectures, compilers like
GCC and LLVM perform various program transformations, which also affect the outcomes of
concurrent programs.
To be able to program these systems correctly and effectively, it is important to define a
formal language-level concurrency model. For efficiency, it is important that the model is
weak enough to allow various compiler optimizations on shared memory accesses as well
as efficient mappings to the architectures. For programmability, the model should be strong
enough to disallow bogus “out-of-thin-air” executions and provide strong guarantees for well-synchronized
programs. Because of these conflicting requirements, defining such a formal
model is very difficult. This is why, despite years of research, major programming languages
such as C/C++ and Java do not yet have completely adequate formal models defining their
concurrency semantics.
In this thesis, we address this challenge and develop a formal concurrency model that is very
good both in terms of compilation efficiency and of programmability. Unlike most previous
approaches, which were defined either operationally or axiomatically on single executions,
our formal model is based on event structures, which represents multiple program executions,
and thus gives us more structure to define the semantics of concurrency.
In more detail, our formalization has two variants: the weaker version, WEAKEST, and the
stronger version, WEAKESTMO. The WEAKEST model simulates the promising semantics proposed
by Kang et al., while WEAKESTMO is incomparable to the promising semantics. Moreover,
WEAKESTMO discards certain questionable behaviors allowed by the promising semantics.
We show that the proposed WEAKESTMO model resolve out-of-thin-air problem, provide
standard data-race-freedom (DRF) guarantees, allow the desirable optimizations, and can be
mapped to the architectures like x86, PowerPC, and ARMv7. Additionally, our models are
flexible enough to leverage existing results from the literature to establish data-race-freedom
(DRF) guarantees and correctness of compilation.
In addition, in order to ensure the correctness of compilation by a major compiler, we developed
a translation validator targeting LLVM’s “opt” transformations of concurrent C/C++
programs. Using the validator, we identified a few subtle compilation bugs, which were reported
and were fixed. Additionally, we observe that LLVM concurrency semantics differs
from that of C11; there are transformations which are justified in C11 but not in LLVM and
vice versa. Considering the subtle aspects of LLVM concurrency, we formalized a fragment
of LLVM’s concurrency semantics and integrated it into our WEAKESTMO model.

This thesis brings together convex analysis and hyperspectral image processing.
Convex analysis is the study of convex functions and their properties.
Convex functions are important because they admit minimization by efficient algorithms
and the solution of many optimization problems can be formulated as
minimization of a convex objective function, extending much beyond
the classical image restoration problems of denoising, deblurring and inpainting.
\(\hspace{1mm}\)
At the heart of convex analysis is the duality mapping induced within the
class of convex functions by the Fenchel transform.
In the last decades efficient optimization algorithms have been developed based
on the Fenchel transform and the concept of infimal convolution.
\(\hspace{1mm}\)
The infimal convolution is of similar importance in convex analysis as the
convolution in classical analysis. In particular, the infimal convolution with
scaled parabolas gives rise to the one parameter family of Moreau-Yosida envelopes,
which approximate a given function from below while preserving its minimum
value and minimizers.
The closely related proximal mapping replaces the gradient step
in a recently developed class of efficient first-order iterative minimization algorithms
for non-differentiable functions. For a finite convex function,
the proximal mapping coincides with a gradient step of its Moreau-Yosida envelope.
Efficient algorithms are needed in hyperspectral image processing,
where several hundred intensity values measured in each spatial point
give rise to large data volumes.
\(\hspace{1mm}\)
In the \(\textbf{first part}\) of this thesis, we are concerned with
models and algorithms for hyperspectral unmixing.
As part of this thesis a hyperspectral imaging system was taken into operation
at the Fraunhofer ITWM Kaiserslautern to evaluate the developed algorithms on real data.
Motivated by missing-pixel defects common in current hyperspectral imaging systems,
we propose a
total variation regularized unmixing model for incomplete and noisy data
for the case when pure spectra are given.
We minimize the proposed model by a primal-dual algorithm based on the
proximum mapping and the Fenchel transform.
To solve the unmixing problem when only a library of pure spectra is provided,
we study a modification which includes a sparsity regularizer into model.
\(\hspace{1mm}\)
We end the first part with the convergence analysis for a multiplicative
algorithm derived by optimization transfer.
The proposed algorithm extends well-known multiplicative update rules
for minimizing the Kullback-Leibler divergence,
to solve a hyperspectral unmixing model in the case
when no prior knowledge of pure spectra is given.
\(\hspace{1mm}\)
In the \(\textbf{second part}\) of this thesis, we study the properties of Moreau-Yosida envelopes,
first for functions defined on Hadamard manifolds, which are (possibly) infinite-dimensional
Riemannian manifolds with negative curvature,
and then for functions defined on Hadamard spaces.
\(\hspace{1mm}\)
In particular we extend to infinite-dimensional Riemannian manifolds an expression
for the gradient of the Moreau-Yosida envelope in terms of the proximal mapping.
With the help of this expression we show that a sequence of functions
converges to a given limit function in the sense of Mosco
if the corresponding Moreau-Yosida envelopes converge pointwise at all scales.
\(\hspace{1mm}\)
Finally we extend this result to the more general setting of Hadamard spaces.
As the reverse implication is already known, this unites two definitions of Mosco convergence
on Hadamard spaces, which have both been used in the literature,
and whose equivalence has not yet been known.

Optimal Multilevel Monte Carlo Algorithms for Parametric Integration and Initial Value Problems
(2015)

We intend to find optimal deterministic and randomized algorithms for three related problems: multivariate integration, parametric multivariate integration, and parametric initial value problems. The main interest is concentrated on the question, in how far randomization affects the precision of an approximation. We want to understand when and to which extent randomized algorithms are superior to deterministic ones.
All problems are studied for Banach space valued input functions. The analysis of Banach space valued problems is motivated by the investigation of scalar parametric problems; these can be understood as particular cases of Banach space valued problems. The gain achieved by randomization depends on the underlying Banach space.
For each problem, we introduce deterministic and randomized algorithms and provide the corresponding convergence analysis.
Moreover, we also provide lower bounds for the general Banach space valued settings, and thus, determine the complexity of the problems. It turns out that the obtained algorithms are order optimal in the deterministic setting. In the randomized setting, they are order optimal for certain classes of Banach spaces, which includes the L_p spaces and any finite dimensional Banach space. For general Banach spaces, they are optimal up to an arbitrarily small gap in the order of convergence.