Luisa D’Amore

Regularization of a Fourier series method for the laplace transform inversion with real data

We propose a numerical method for computing a function, given its Laplace transform function on the real axis. The inversion algorithm is based on the Fourier series expansion of the unknown function and the Fourier coefficients are approximated using a Tikhonov regularization method. The key point of this approach is the use of the regularization scheme in order to improve the conditioning of the discrete problem: the value of the regularization parameter is that giving a tradeoff between the discretization error, including the regularization error, and the conditioning of the discrete problem.

Monitoring and Migration of a PETSc-based Parallel Application for Medical Imaging in a Grid computing PSE

In last decades, imaging techniques became central to the diagnostic process providing the medical community with a fast growing amounts of information held in images. This implies developing computational tools which allow a reliable, robust and efficient processing of data and enhanced analysis. Moreover, clinicians may have the need to explore collaborative approaches and to exchange diagnostic information from available data. A medical experiment often involves not a single approach but a set of processings that should be sometimes executed concurrently.

A smoothing spline that approximates Laplace transform functions only known on measurements on the real axis

The scientific and application-oriented interest in the Laplace transform and its inversion is testified by more than 1000 publications in the last century. Most of the inversion algorithms available in the literature assume that the Laplace transform function is available everywhere. Unfortunately, such an assumption is not fulfilled in the applications of the Laplace transform. Very often, one only has a finite set of data and one wants to recover an estimate of the inverse Laplace function from that. We propose a fitting model of data. More precisely, given a finite set of measurements on the real axis, arising from an unknown Laplace transform function, we construct a dth degree generalized polynomial smoothing spline, where d = 2m − 1, such that internally to the data interval it is a dth degree polynomial complete smoothing spline minimizing a regularization functional, and outside the data interval, it mimics the Laplace transform asymptotic behavior, i.e. it is a rational or an exponential function (the end behavior model), and at the boundaries of the data set it joins with regularity up to order m − 1, with the end behavior model. We analyze in detail the generalized polynomial smoothing spline of degree d = 3. This choice was motivated by the (ill)conditioning of the numerical computation which strongly depends on the degree of the complete spline. We prove existence and uniqueness of this spline. We derive the approximation error and give a priori and computable bounds of it on the whole real axis. In such a way, the generalized polynomial smoothing spline may be used in any real inversion algorithm to compute an approximation of the inverse Laplace function. Experimental results concerning Laplace transform approximation, numerical inversion of the generalized polynomial smoothing spline and comparisons with the exponential smoothing spline conclude the work.

A Scalable Numerical Algorithm for Solving Tikhonov Regularization Problems

We present a numerical algorithm for solving large scale Tikhonov Regularization problems. The approach we consider introduces a splitting of the regularization functional which uses a domain decomposition, a partitioning of the solution and modified regularization functionals on each sub domain. We perform a feasibility analysis in terms of the algorithm and software scalability, to this end we use the scale-up factor which measures the performance gain in terms of time complexity reduction. We verify the reliability of the approach on a consistent test case (the Data Assimilation problem for oceanographic models).

DD-OceanVar: A Domain Decomposition Fully Parallel Data Assimilation Software for the Mediterranean Forecasting System☆

Abstract OceanVar is a Data Assimilation (DA) software which is being used in Italy within the Mediterranean Forecasting System (MFS) to combine observational data (Sea level anomaly, sea-surface temperatures, etc.) with backgrounds produced by computational models of ocean currents for the Mediterranean Sea (namely, the NEMO framework). OceanVAR is based on a three-dimensional variational approach. We describe computational efforts aimed to design a fully parallel OceanVar software, based on Domain Decomposition (DD), which involves modification of the variational scheme on each sub domain. Our approach aims to face to the ever greater multi-level parallelism and scalability of the current and of the next generation of leadership computing facility systems, while fulfilling the specific requirements of OceanVar within the MFS.

A Parallel Three‐dimensional Variational Data Assimilation Scheme

Data Assimilation (DA) refers to the methods for merging observed (generally sparse and noisy) information into the numerical model. Good assimilations make the modeled state more consistent with the observations. Effective data assimilation systems tend to make forecasts more accurate within the ability of the model. In this work we discuss some computational efforts towards the development of parallel three dimensional data assimilation scheme, based on the oceanographic 3D‐VAR assimilation scheme, named OCEANVAR.

A Decomposition of the Tikhonov Regularization Functional Oriented to Exploit Hybrid Multilevel Parallelism

We introduce a decomposition of the Tikhonov Regularization (TR) functional which split this operator into several TR functionals, suitably modified in order to enforce the matching of their solutions. As a consequence, instead of solving one problem we can solve several problems reproducing the initial one at smaller dimensions. Such approach leads to a reduction of the time complexity of the resulting algorithm. Since the subproblems are solved in parallel, this decomposition also leads to a reduction of the overall execution time. Main outcome of the decomposition is that the parallel algorithm is oriented to exploit the highest performance of parallel architectures where concurrency is implemented both at the coarsest and finest levels of granularity. Performance analysis is discussed in terms of the algorithm and software scalability. Validation is performed on a reference parallel architecture made of a distributed memory multiprocessor and a Graphic Processing Unit. Results are presented on the Data Assimilation problem, for oceanographic models.

On the Numerical Approximation of the Laplace Transform Function from Real Samples and Its Inversion

Many applications are tackled using the Laplace Transform (LT) known on a countable number of real values [J. Electroanal. Chem. 608, 37–46 (2007), Int. J. solid Struct. 41, 3653–3674 (2004), Imaging 26, 1183–1196 (2008), J. Magn. Reson. 156, 213–221 (2002)]. The usual approach to solve the LT inverse problem relies on a regularization technique combined with information a priori both on the LT function and on its inverse (see for instance [http://s-provencher.com/pages/ contin.shtml]). We propose a fitting model enjoying LT properties: we define a generalized spline that interpolates the LT function values and mimics the asymptotic behavior of LT functions. Then, we prove existence and uniqueness of this model and, through a suitable error analysis, we give a priori approximation error bounds to confirm the reliability of this approach. Numerical results are presented.

A Modification of Weeks’ Method for Numerical Inversion of the Laplace Transform in the Real Case Based on Automatic Differentiation

Numerical inversion of the Laplace transform on the real axis is an inverse and ill-posed problem. We describe a powerful modification of Weeks’ Method, based on automatic differentiation, to be used in the real inversion. We show that the automatic differentiation technique assures accurate and efficient numerical computation of the inverse Laplace function.

Using GPGPU Accelerated Interpolation Algorithms for Marine Bathymetry Processing with On-Premises and Cloud Based Computational Resources

Data crowdsourcing is one of most remarkable results of pervasive and internet connected low-power devices making diverse and different “things” as a world wide distributed system. This paper is focused on a vertical application of GPGPU virtualization software exploitation targeted on high performance geographical data interpolation. We present an innovative implementation of the Inverse Distance Weight (IDW) interpolation algorithm leveraging on CUDA GPGPUs. We perform tests in both physical and virtualized environments in order to demonstrate the potential scalability in production. We present an use case related to high resolution bathymetry interpolation in a crowdsource data context.