https://quantfilab.github.io/pmarupanthorn/tags/computational-mathematics/Computational MathematicsHugo Blox Builder (https://hugoblox.com)en-usThu, 20 Mar 2014 00:00:00 +0000https://quantfilab.github.io/pmarupanthorn/media/icon_hu68170e94a17a2a43d6dcb45cf0e8e589_3079_512x512_fill_lanczos_center_3.pnghttps://quantfilab.github.io/pmarupanthorn/tags/computational-mathematics/https://quantfilab.github.io/pmarupanthorn/publication/amm2014/Thu, 20 Mar 2014 00:00:00 +0000https://quantfilab.github.io/pmarupanthorn/publication/amm2014/<div class="research-bilingual" data-research-bilingual> <section id="research-content-AMM2014-en" class="research-language-panel" lang="en"> <figure class="research-concept-map research-concept-map--image"> <img src="https://quantfilab.github.io/pmarupanthorn/pmarupanthorn/publication/amm2014/concept-map-en.png" alt="Conceptual map: combining density-of-states sampling with within-bin averages" loading="eager"> </figure> <h2>The problem</h2> <p>Ordinary Monte Carlo sampling can miss narrow, sharply peaked regions and severely underestimate ill-behaved integrals. The conventional 1/t method samples these regions more effectively, but its fixed representative value within each bin produces bin-width error and eventual error saturation.</p> <p><strong>Who benefits:</strong></p> <ul> <li>Computational scientists evaluating difficult multidimensional integrals</li> <li>Monte Carlo and density-of-states researchers</li> <li>Applied mathematicians studying numerical convergence</li> <li>Physics and engineering analysts working with sharply peaked functions</li> </ul> <h2>Method</h2> <p>The proposed estimator retains the 1/t random walk for estimating the density of states g(y), but replaces the fixed representative value of each y-space bin with the continuously updated mean of the integrand values actually sampled in that bin. The resulting approximation is a sum of g(y) times the within-bin mean. Tests use sharply peaked Gaussian integrals in one, two, and three dimensions with zero means, standard deviations equal to the square root of 0.4, and the domain [-10, 10] in each dimension. Simple sampling, the conventional 1/t algorithm, and the proposed method are compared over four bin widths, 10^10 trials per run, and 50 independent simulations using exact integral values and fractional accuracy.</p> <h2>Results</h2> <p>Simple sampling severely underestimated the sharply localized integrals, while the conventional 1/t estimates remained sensitive to bin width and eventually reached an error plateau. At bin width 0.0005, the proposed estimates for the one-, two-, and three-dimensional cases were 1.585321, 2.513090, and 3.984088, close to the respective exact values 1.585331, 2.513274, and 3.984371. Its error continued to decrease approximately as N^-1/2 without saturation. Because accuracy was no longer governed by bin width, wider bins could reach all states faster and reduce computational effort.</p> </section> </div>