Algorithms and basis functions in tomographic reconstruction of ionospheric electron density

Computerized ionospheric tomography (CIT) is a method to investigate ionosphere electron density in two or three dimensions. This method provides a flexible tool for studying ionosphere. Earth based receivers record signals transmitted from the GPS satellites and the computed pseudorange and phase values are used to calculate Total Electron Content (TEC). Computed TEC data and the tomographic reconstruction algorithms are used together to obtain tomographic images of electron density. In this study, a set of basis functions and reconstruction algorithms are used to investigate best fitting two dimensional tomographic images of ionosphere electron density in height and latitude for one satellite and one receiver pair. Results are compared to IRI-95 ionosphere model and both receiver and model errors are neglected

Matrix Multiplication Verification Using Coding Theory

We study the Matrix Multiplication Verification Problem (MMV) where the goal is, given three $n \times n$ matrices $A$, $B$, and $C$ as input, to decide whether $AB = C$. A classic randomized algorithm by Freivalds (MFCS, 1979) solves MMV in $\widetilde{O}(n^2)$ time, and a longstanding challenge is to (partially) derandomize it while still running in faster than matrix multiplication time (i.e., in $o(n^{\omega})$ time). To that end, we give two algorithms for MMV in the case where $AB - C$ is sparse. Specifically, when $AB - C$ has at most $O(n^{\delta})$ non-zero entries for a constant $0 \leq \delta < 2$, we give (1) a deterministic $O(n^{\omega - \varepsilon})$-time algorithm for constant $\varepsilon = \varepsilon(\delta) > 0$, and (2) a randomized $\widetilde{O}(n^2)$-time algorithm using $\delta/2 \cdot \log_2 n + O(1)$ random bits. The former algorithm is faster than the deterministic algorithm of K\"{u}nnemann (ESA, 2018) when $\delta \geq 1.056$, and the latter algorithm uses fewer random bits than the algorithm of Kimbrel and Sinha (IPL, 1993), which runs in the same time and uses $\log_2 n + O(1)$ random bits (in turn fewer than Freivalds's algorithm). We additionally study the complexity of MMV. We first show that all algorithms in a natural class of deterministic linear algebraic algorithms for MMV (including ours) require $\Omega(n^{\omega})$ time. We also show a barrier to proving a super-quadratic running time lower bound for matrix multiplication (and hence MMV) under the Strong Exponential Time Hypothesis (SETH). Finally, we study relationships between natural variants and special cases of MMV (with respect to deterministic $\widetilde{O}(n^2)$-time reductions)

A Lower Bound for Matrix Multiplication

We prove that computing the product of two n × n matrices over the binary field requires at least 2.5 n 2 - o ( n 2 ) multiplications. Key Words : matrix multiplication, arithmetic complexity, lower bounds, linear codes. 1. INTRODUCTION Let x = ( x 1 , . . . , x n ) T and y = ( y 1 , . . . , y m ) T be column vectors of indeterminates. A straight-line algorithm for computing a set of bilinear forms in x and y is called quadratic ( respectively bilinear ), if all its non-scalar multiplication are of the shape l ( x , y ) . l ( x , y ) , (respectively l ( x ) . l ( y ) ) where l and l are linear forms of the indeterminates. 1 In this paper we establish the new 2.5 n 2 - o ( n 2 ) lower bound on the multiplicative complexity of quadratic algorithms for multiplying n × n matrices over the binary field Z 2 . Let M F ( n , m , k ) and M ## F ( n , m , k ) denote the number of multiplications required to compute the product of n ×m and m ×k matrices by means of quadratic ..