{"id":1412,"date":"2023-11-16T11:30:49","date_gmt":"2023-11-16T19:30:49","guid":{"rendered":"https:\/\/dornsife.usc.edu\/sergey-lototsky\/?page_id=1412"},"modified":"2023-11-16T11:30:50","modified_gmt":"2023-11-16T19:30:50","slug":"gaussian-fields-matlab","status":"publish","type":"page","link":"https:\/\/dornsife.usc.edu\/sergey-lototsky\/gaussian-fields-matlab\/","title":{"rendered":"Gaussian-Fields (MATLAB)"},"content":{"rendered":"\n\n  \n    \n\n\n\n\n\n\n<div\n  class=\"cc--component-container cc--rich-text \"\n\n  \n  \n  \n  \n  \n  \n  >\n  <div class=\"c--component c--rich-text\"\n    \n      >\n\n    \n      \n<div class=\"f--field f--wysiwyg\">\n\n    \n  <pre>clear;\r\nclear\r\nN=200;\r\ndx=1\/N;\r\na=2\/pi;\r\nb=2\/pi^2;\r\nx=[0:dx:1];\r\ny=[0:dx:1];\r\nM=80;\r\nz=zeros(N+1);\r\nksi=randn(N+1);\r\nfor m=1:M\r\n    for k=1:M\r\n        %z=z+a*sin(pi*k*x')*sin(pi*m*y)*ksi(m,k)\/sqrt(m^2+k^2); % GFF\r\n        %z=z+b*sin(pi*k*x')*sin(pi*m*y)*ksi(m,k)\/(m*k); %Tensor of two BB\r\n        %z=z+b*sin(pi*(k-0.5)*x')*sin(pi*m*y)*ksi(m,k)\/(m*(k-0.5)); %Kiefer: Tensor of BM and BB\r\n        z=z+b*sin(pi*(k-0.5)*x')*sin(pi*(m-0.5)*y)*ksi(m,k)\/((m-0.5)*(k-0.5)); %Br. Sheet: Tensor of BM and BM\r\n    end\r\nend\r\nmesh(x,y,z)\r\nhold\r\n%title('GFF on [0,1]x[0,1]; 80^2 terms, 200^2 points'); xlabel('X-axis'); ylabel('Y-axis');\r\n%title('BB-BB on [0,1]x[0,1]; 80^2 terms, 200^2 points'); xlabel('X-axis'); ylabel('Y-axis');\r\n%title('Kiefer Field on [0,1]x[0,1]; 80^2 terms, 200^2 points'); xlabel('BrM'); ylabel('BrBr');\r\ntitle('Brownian Sheet on [0,1]x[0,1]; 80^2 terms, 200^2 points'); xlabel('X-axis'); ylabel('Y-axis');\r\n\r\n\r\n% using the argument about crossing a linear barrier, we concluded that E sup_{t&gt;0} W(t)\/(1+t) = \\sqrt{pi\/8} \\approx 0.63\r\n% Monte Carlo simulations confirm it. Note that if X(t)=W(t)\/(1+t), then dX(t)=-X(t)dt\/(1+t) + dW(t)\/(1+t). \r\n\r\nclear; dt=0.001; ddt=sqrt(dt); M=200; N=10000; ksi=randn(N,M); for n=1:M X(1,n)=0; for k=1:N-1 X(k+1,n)=X(k,n)+(ddt*ksi(k,n)-dt*X(k,n))\/(1+(k+1)*dt); end; end; sum(max(X))\/M<\/pre>\n\n\n\n<\/div>\n\n\n  <\/div><\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":370,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-1412","page","type-page","status-publish","hentry"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.1.1 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Gaussian-Fields (MATLAB) - Sergey Lototsky<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/gaussian-fields-matlab\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Gaussian-Fields (MATLAB) - 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