Unbiased Black-Box Complexities of Jump Functions

We analyze the unbiased black-box complexity of jump functions with small, medium, and large sizes of the fitness plateau surrounding the optimal solution. Among other results, we show that when the jump size is $(1/2 - \varepsilon)n$, that is, only a small constant fraction of the fitness values is visible, then the unbiased black-box complexities for arities $3$ and higher are of the same order as those for the simple \textsc{OneMax} function. Even for the extreme jump function, in which all but the two fitness values $n/2$ and $n$ are blanked out, polynomial-time mutation-based (i.e., unary unbiased) black-box optimization algorithms exist. This is quite surprising given that for the extreme jump function almost the whole search space (all but a $\Theta(n^{-1/2})$ fraction) is a plateau of constant fitness. To prove these results, we introduce new tools for the analysis of unbiased black-box complexities, for example, selecting the new parent individual not by comparing the fitnesses of the competing search points, but also by taking into account the (empirical) expected fitnesses of their offspring.Comment: This paper is based on results presented in the conference versions [GECCO 2011] and [GECCO 2014

Complexity plots

In this paper, we present a novel visualization technique for assisting in observation and analysis of algorithmic\ud complexity. In comparison with conventional line graphs, this new technique is not sensitive to the units of\ud measurement, allowing multivariate data series of different physical qualities (e.g., time, space and energy) to be juxtaposed together conveniently and consistently. It supports multivariate visualization as well as uncertainty visualization. It enables users to focus on algorithm categorization by complexity classes, while reducing visual impact caused by constants and algorithmic components that are insignificant to complexity analysis. It provides an effective means for observing the algorithmic complexity of programs with a mixture of algorithms and blackbox software through visualization. Through two case studies, we demonstrate the effectiveness of complexity plots in complexity analysis in research, education and application

Laplace deconvolution on the basis of time domain data and its application to Dynamic Contrast Enhanced imaging

In the present paper we consider the problem of Laplace deconvolution with noisy discrete non-equally spaced observations on a finite time interval. We propose a new method for Laplace deconvolution which is based on expansions of the convolution kernel, the unknown function and the observed signal over Laguerre functions basis (which acts as a surrogate eigenfunction basis of the Laplace convolution operator) using regression setting. The expansion results in a small system of linear equations with the matrix of the system being triangular and Toeplitz. Due to this triangular structure, there is a common number $m$ of terms in the function expansions to control, which is realized via complexity penalty. The advantage of this methodology is that it leads to very fast computations, produces no boundary effects due to extension at zero and cut-off at $T$ and provides an estimator with the risk within a logarithmic factor of the oracle risk. We emphasize that, in the present paper, we consider the true observational model with possibly nonequispaced observations which are available on a finite interval of length $T$ which appears in many different contexts, and account for the bias associated with this model (which is not present when $T\rightarrow\infty$). The study is motivated by perfusion imaging using a short injection of contrast agent, a procedure which is applied for medical assessment of micro-circulation within tissues such as cancerous tumors. Presence of a tuning parameter $a$ allows to choose the most advantageous time units, so that both the kernel and the unknown right hand side of the equation are well represented for the deconvolution. The methodology is illustrated by an extensive simulation study and a real data example which confirms that the proposed technique is fast, efficient, accurate, usable from a practical point of view and very competitive.Comment: 36 pages, 9 figures. arXiv admin note: substantial text overlap with arXiv:1207.223

Adjoint methods for computing sensitivities in local volatility surfaces

In this paper we present the adjoint method of computing sensitivities of option prices with respect to nodes in the local volatility surface. We first introduce the concept of algorithmic differentiation and how it relates to\ud path-wise sensitivity computations within a Monte Carlo framework. We explain the two approaches available: forward mode and adjoint mode. We illustrate these concepts on the simple example of a model with a geometric Brownian motion driving the underlying price process, for which\ud we compute the Delta and Vega in forward and adjoint mode. We then go on to explain in full detail how to apply these ideas to a model where the underlying has a volatility term defined by a local volatility surface. We provide source codes for both the simple and the more complex case and\ud analyze numerical results to show the strengths of the adjoint approach