<!-- BaNnErBlUrFlE-BoDy-start -->
<!-- Preheader Text : BEGIN -->
<div style="display:none !important;display:none;visibility:hidden;mso-hide:all;font-size:1px;color:#ffffff;line-height:1px;height:0px;max-height:0px;opacity:0;overflow:hidden;">
 PETSc provided solvers do not directly use threads. The BLAS used by LAPACK and PETSc may use threads depending on what BLAS is being used and how it was configured. Some of the vector operations in GMRES in PETSc use BLAS that can use threads,
</div>
<!-- Preheader Text : END -->

<!-- Email Banner : BEGIN -->
<div style="display:none !important;display:none;visibility:hidden;mso-hide:all;font-size:1px;color:#ffffff;line-height:1px;height:0px;max-height:0px;opacity:0;overflow:hidden;">ZjQcmQRYFpfptBannerStart</div>

<!--[if ((ie)|(mso))]>
  <table border="0" cellspacing="0" cellpadding="0" width="100%" style="padding: 16px 0px 16px 0px; direction: ltr" ><tr><td>
    <table border="0" cellspacing="0" cellpadding="0" style="padding: 0px 10px 5px 6px; width: 100%; border-radius:4px; border-top:4px solid #90a4ae;background-color:#D0D8DC;"><tr><td valign="top">
      <table align="left" border="0" cellspacing="0" cellpadding="0" style="padding: 4px 8px 4px 8px">
        <tr><td style="color:#000000; font-family: 'Arial', sans-serif; font-weight:bold; font-size:14px; direction: ltr">
          This Message Is From an External Sender
        </td></tr>
        <tr><td style="color:#000000; font-weight:normal; font-family: 'Arial', sans-serif; font-size:12px; direction: ltr">
          This message came from outside your organization.
        </td></tr>

      </table>

    </td></tr></table>
  </td></tr></table>
<![endif]-->

<![if !((ie)|(mso))]>
  <div dir="ltr"  id="pfptBannerd1x4y2c" style="all: revert !important; display:block !important; text-align: left !important; margin:16px 0px 16px 0px !important; padding:8px 16px 8px 16px !important; border-radius: 4px !important; min-width: 200px !important; background-color: #D0D8DC !important; background-color: #D0D8DC; border-top: 4px solid #90a4ae !important; border-top: 4px solid #90a4ae;">
    <div id="pfptBannerd1x4y2c" style="all: unset !important; float:left !important; display:block !important; margin: 0px 0px 1px 0px !important; max-width: 600px !important;">
      <div id="pfptBannerd1x4y2c" style="all: unset !important; display:block !important; visibility: visible !important; background-color: #D0D8DC !important; color:#000000 !important; color:#000000; font-family: 'Arial', sans-serif !important; font-family: 'Arial', sans-serif; font-weight:bold !important; font-weight:bold; font-size:14px !important; line-height:18px !important; line-height:18px">
        This Message Is From an External Sender
      </div>
      <div id="pfptBannerd1x4y2c" style="all: unset !important; display:block !important; visibility: visible !important; background-color: #D0D8DC !important; color:#000000 !important; color:#000000; font-weight:normal; font-family: 'Arial', sans-serif !important; font-family: 'Arial', sans-serif; font-size:12px !important; line-height:18px !important; line-height:18px; margin-top:2px !important;">
This message came from outside your organization.
      </div>

    </div>

    <div style="clear: both !important; display: block !important; visibility: hidden !important; line-height: 0 !important; font-size: 0.01px !important; height: 0px"> </div>
  </div>
<![endif]>

<div style="display:none !important;display:none;visibility:hidden;mso-hide:all;font-size:1px;color:#ffffff;line-height:1px;height:0px;max-height:0px;opacity:0;overflow:hidden;">ZjQcmQRYFpfptBannerEnd</div>
<!-- Email Banner : END -->

<!-- BaNnErBlUrFlE-BoDy-end -->
<html><head><!-- BaNnErBlUrFlE-HeAdEr-start -->
<style>
  #pfptBannerd1x4y2c { all: revert !important; display: block !important; 
    visibility: visible !important; opacity: 1 !important; 
    background-color: #D0D8DC !important; 
    max-width: none !important; max-height: none !important }
  .pfptPrimaryButtond1x4y2c:hover, .pfptPrimaryButtond1x4y2c:focus {
    background-color: #b4c1c7 !important; }
  .pfptPrimaryButtond1x4y2c:active {
    background-color: #90a4ae !important; }
</style>

<!-- BaNnErBlUrFlE-HeAdEr-end -->
<meta http-equiv="content-type" content="text/html; charset=utf-8"></head><body style="overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;"><div><br></div>   PETSc provided solvers do not directly use threads. <div><br></div><div>   The BLAS used by LAPACK and PETSc may use threads depending on what BLAS is being used and how it was configured. </div><div><br></div><div>   Some of the vector operations in GMRES in PETSc use BLAS that can use threads, including axpy, dot, etc. For sufficiently large problems, the use of threaded BLAS can help with these routines, but not significantly for the solver. </div><div><br></div><div>   Dense matrix-vector products MatMult() and dense matrix direct solvers PCLU use BLAS and thus can benefit from threading. The benefit can be significant for large enough problems with good hardware, especially with PCLU. </div><div><br></div><div>   If you run with -blas_view  PETSc tries to indicate information about the threading of BLAS. You can also use -blas_num_threads <n> to set the number of threads, equivalent to setting the environmental variable.  For dense solvers you can vary the number of threads and run with -log_view to see what it helps to improve and what it does not effect.</div><div><br></div><div><br></div><div><div><br><blockquote type="cite"><div>On Apr 22, 2024, at 4:06 PM, Yongzhong Li <yongzhong.li@mail.utoronto.ca> wrote:</div><br class="Apple-interchange-newline"><div><meta charset="UTF-8"><div dir="ltr" id="pfptBannerbeaibqa" style="color-scheme: revert !important; writing-mode: revert !important; -webkit-ruby-position: revert !important; color: revert !important; font-family: revert !important; font-feature-settings: revert !important; font-kerning: revert !important; font-optical-sizing: revert !important; font-palette: revert !important; font-size: revert !important; font-size-adjust: revert !important; font-stretch: revert !important; font-style: revert !important; font-synthesis-small-caps: revert !important; font-synthesis-style: revert !important; font-synthesis-weight: revert !important; font-variant-alternates: revert !important; font-variant-caps: revert !important; font-variant-east-asian: revert !important; font-variant-emoji: revert !important; font-variant-ligatures: revert !important; font-variant-numeric: revert !important; font-variant-position: revert !important; font-variation-settings: revert !important; font-weight: revert !important; text-orientation: revert !important; text-rendering: revert !important; zoom: revert !important; -webkit-font-smoothing: revert !important; -webkit-locale: revert !important; -webkit-text-size-adjust: revert !important; -webkit-text-zoom: revert !important; letter-spacing: revert !important; text-autospace: revert !important; text-spacing-trim: revert !important; accent-color: revert !important; align-content: revert !important; align-items: revert !important; align-self: revert !important; alignment-baseline: revert !important; alt: revert !important; animation-composition: revert !important; animation: revert !important; animation-timeline: revert !important; appearance: revert !important; aspect-ratio: revert !important; backface-visibility: revert !important; background-attachment: revert !important; background-blend-mode: revert !important; background-image: revert !important; background-position: revert !important; background-repeat: revert !important; background-size: revert !important; baseline-shift: revert !important; block-step-insert: revert !important; block-step-size: revert !important; border-collapse: revert !important; border-image: revert !important; box-sizing: revert !important; break-after: revert !important; break-before: revert !important; break-inside: revert !important; buffered-rendering: revert !important; caption-side: revert !important; caret-color: revert !important; clear: revert !important; clip: revert !important; clip-path: revert !important; clip-rule: revert !important; color-interpolation: revert !important; color-interpolation-filters: revert !important; column-count: revert !important; column-fill: revert !important; column-gap: revert !important; column-rule-color: revert !important; column-rule-style: revert !important; column-rule-width: revert !important; column-span: revert !important; column-width: revert !important; contain: revert !important; container-name: revert !important; container-type: revert !important; content: revert !important; content-visibility: revert !important; counter-increment: revert !important; counter-reset: revert !important; counter-set: revert !important; cursor: revert !important; cx: revert !important; cy: revert !important; dominant-baseline: revert !important; empty-cells: revert !important; fill: revert !important; fill-opacity: revert !important; fill-rule: revert !important; filter: revert !important; flex: revert !important; flex-flow: revert !important; float: revert !important; flood-color: revert !important; flood-opacity: revert !important; glyph-orientation-horizontal: revert !important; glyph-orientation-vertical: revert !important; grid: revert !important; grid-column-end: revert !important; grid-column-start: revert !important; grid-row-end: revert !important; grid-row-start: revert !important; hanging-punctuation: revert !important; hyphenate-character: revert !important; hyphens: revert !important; image-orientation: revert !important; image-rendering: revert !important; input-security: revert !important; isolation: revert !important; justify-content: revert !important; justify-items: revert !important; justify-self: revert !important; kerning: revert !important; lighting-color: revert !important; line-break: revert !important; line-height: revert !important; list-style: revert !important; margin-trim: revert !important; marker: revert !important; mask: revert !important; mask-size: revert !important; mask-type: revert !important; masonry-auto-flow: revert !important; math-style: revert !important; mix-blend-mode: revert !important; object-fit: revert !important; object-position: revert !important; offset-anchor: revert !important; offset-distance: revert !important; offset-path: revert !important; offset-position: revert !important; offset-rotate: revert !important; order: revert !important; orphans: revert !important; outline: revert !important; outline-offset: revert !important; overflow-anchor: revert !important; overflow-wrap: revert !important; overflow: revert !important; page: revert !important; paint-order: revert !important; perspective: revert !important; perspective-origin: revert !important; pointer-events: revert !important; position: revert !important; print-color-adjust: revert !important; quotes: revert !important; r: revert !important; resize: revert !important; rotate: revert !important; row-gap: revert !important; rx: revert !important; ry: revert !important; scale: revert !important; scroll-behavior: revert !important; scroll-snap-align: revert !important; scroll-snap-stop: revert !important; scroll-snap-type: revert !important; scroll-timeline: revert !important; scrollbar-color: revert !important; scrollbar-gutter: revert !important; scrollbar-width: revert !important; shape-image-threshold: revert !important; shape-margin: revert !important; shape-outside: revert !important; shape-rendering: revert !important; size: revert !important; speak-as: revert !important; stop-color: revert !important; stop-opacity: revert !important; stroke: revert !important; stroke-color: revert !important; stroke-dasharray: revert !important; stroke-dashoffset: revert !important; stroke-linecap: revert !important; stroke-linejoin: revert !important; stroke-miterlimit: revert !important; stroke-opacity: revert !important; stroke-width: revert !important; tab-size: revert !important; table-layout: revert !important; text-align: left !important; text-align-last: revert !important; text-anchor: revert !important; text-box-edge: revert !important; text-box-trim: revert !important; text-decoration-color: revert !important; text-decoration: revert !important; text-decoration-skip-ink: revert !important; text-decoration-style: revert !important; text-decoration-thickness: revert !important; text-emphasis-color: revert !important; text-emphasis-position: revert !important; text-emphasis-style: revert !important; text-group-align: revert !important; text-indent: revert !important; text-justify: revert !important; text-overflow: revert !important; text-shadow: revert !important; text-transform: revert !important; text-underline-offset: revert !important; text-underline-position: revert !important; text-wrap: revert !important; touch-action: revert !important; transform: revert !important; transform-box: revert !important; transform-origin: revert !important; transform-style: revert !important; transition: revert !important; translate: revert !important; vector-effect: revert !important; vertical-align: revert !important; view-timeline: revert !important; view-transition-name: revert !important; white-space: revert !important; widows: revert !important; will-change: revert !important; word-break: revert !important; x: revert !important; y: revert !important; z-index: revert !important; -apple-color-filter: revert !important; -apple-pay-button-style: revert !important; -apple-pay-button-type: revert !important; -internal-text-autosizing-status: revert !important; border-spacing: revert !important; -webkit-box-align: revert !important; -webkit-box-decoration-break: revert !important; -webkit-box-direction: revert !important; -webkit-box-flex: revert !important; -webkit-box-flex-group: revert !important; -webkit-box-lines: revert !important; -webkit-box-ordinal-group: revert !important; -webkit-box-orient: revert !important; -webkit-box-pack: revert !important; -webkit-box-reflect: revert !important; -webkit-column-axis: revert !important; -webkit-column-progression: revert !important; -webkit-cursor-visibility: revert !important; -webkit-font-size-delta: revert !important; -webkit-hyphenate-limit-after: revert !important; -webkit-hyphenate-limit-before: revert !important; -webkit-hyphenate-limit-lines: revert !important; -webkit-initial-letter: revert !important; -webkit-line-align: revert !important; -webkit-line-box-contain: revert !important; -webkit-line-clamp: revert !important; -webkit-line-grid: revert !important; -webkit-line-snap: revert !important; -webkit-marquee-direction: revert !important; -webkit-marquee-increment: revert !important; -webkit-marquee-repetition: revert !important; -webkit-marquee-speed: revert !important; -webkit-marquee-style: revert !important; -webkit-nbsp-mode: revert !important; -webkit-rtl-ordering: revert !important; -webkit-text-decorations-in-effect: revert !important; -webkit-text-fill-color: revert !important; -webkit-text-security: revert !important; -webkit-text-stroke-color: revert !important; -webkit-text-stroke-width: revert !important; -webkit-user-drag: revert !important; -webkit-user-modify: revert !important; -webkit-user-select: revert !important; word-spacing: revert !important; backdrop-filter: revert !important; background-clip: revert !important; background-origin: revert !important; box-shadow: revert !important; mask-composite: revert !important; mask-mode: revert !important; text-combine-upright: revert !important; -webkit-backdrop-filter: revert !important; -webkit-background-clip: revert !important; -webkit-background-origin: revert !important; -webkit-box-shadow: revert !important; -webkit-mask: revert !important; -webkit-mask-composite: revert !important; -webkit-mask-source-type: revert !important; -webkit-text-combine: revert !important; display: block !important; visibility: revert !important; opacity: revert !important; background-color: rgb(208, 216, 220) !important; block-size: revert !important; border-block: revert !important; border-bottom-color: revert !important; border-bottom-style: revert !important; border-bottom-width: revert !important; border-end-end-radius: revert !important; border-end-start-radius: revert !important; border-inline: revert !important; border-left-color: revert !important; border-left-style: revert !important; border-left-width: revert !important; border-right-color: revert !important; border-right-style: revert !important; border-right-width: revert !important; border-start-end-radius: revert !important; border-start-start-radius: revert !important; inset: revert !important; contain-intrinsic-block-size: revert !important; contain-intrinsic-size: revert !important; contain-intrinsic-inline-size: revert !important; height: revert !important; inline-size: revert !important; inset-block: revert !important; inset-inline: revert !important; margin-block: revert !important; margin-inline: revert !important; max-block-size: revert !important; max-height: revert !important; max-inline-size: revert !important; max-width: revert !important; min-block-size: revert !important; min-height: revert !important; min-inline-size: revert !important; overscroll-behavior-block: revert !important; overscroll-behavior-inline: revert !important; overscroll-behavior: revert !important; padding-block: revert !important; padding-inline: revert !important; scroll-margin-block: revert !important; scroll-margin: revert !important; scroll-margin-inline: revert !important; scroll-padding-block: revert !important; scroll-padding: revert !important; scroll-padding-inline: revert !important; width: revert !important; margin: 16px 0px !important; padding: 8px 16px !important; border-radius: 4px !important; min-width: 200px !important; border-top-width: 4px !important; border-top-style: solid !important; border-top-color: rgb(144, 164, 174) !important;"><div id="pfptBannerbeaibqa" style="color-scheme: unset !important; writing-mode: unset !important; -webkit-ruby-position: unset !important; color: unset !important; font-family: unset !important; font-feature-settings: unset !important; font-kerning: unset !important; font-optical-sizing: unset !important; font-palette: unset !important; font-size: unset !important; font-size-adjust: unset !important; font-stretch: unset !important; font-style: unset !important; font-synthesis-small-caps: unset !important; font-synthesis-style: unset !important; font-synthesis-weight: unset !important; font-variant-alternates: unset !important; font-variant-caps: unset !important; font-variant-east-asian: unset !important; font-variant-emoji: unset !important; font-variant-ligatures: unset !important; font-variant-numeric: unset !important; font-variant-position: unset !important; font-variation-settings: unset !important; font-weight: unset !important; text-orientation: unset !important; text-rendering: unset !important; zoom: unset !important; -webkit-font-smoothing: unset !important; -webkit-locale: unset !important; -webkit-text-size-adjust: unset !important; -webkit-text-zoom: unset !important; letter-spacing: unset !important; text-autospace: unset !important; text-spacing-trim: unset !important; accent-color: unset !important; align-content: unset !important; align-items: unset !important; align-self: unset !important; alignment-baseline: unset !important; alt: unset !important; animation-composition: unset !important; animation: unset !important; animation-timeline: unset !important; appearance: unset !important; aspect-ratio: unset !important; backface-visibility: unset !important; background: unset !important; background-blend-mode: unset !important; baseline-shift: unset !important; block-step-insert: unset !important; block-step-size: unset !important; border-collapse: unset !important; border: unset !important; box-sizing: unset !important; break-after: unset !important; break-before: unset !important; break-inside: unset !important; buffered-rendering: unset !important; caption-side: unset !important; caret-color: unset !important; clear: unset !important; clip: unset !important; clip-path: unset !important; clip-rule: unset !important; color-interpolation: unset !important; color-interpolation-filters: unset !important; column-count: unset !important; column-fill: unset !important; column-gap: unset !important; column-rule-color: unset !important; column-rule-style: unset !important; column-rule-width: unset !important; column-span: unset !important; column-width: unset !important; contain: unset !important; container-name: unset !important; container-type: unset !important; content: unset !important; content-visibility: unset !important; counter-increment: unset !important; counter-reset: unset !important; counter-set: unset !important; cursor: unset !important; cx: unset !important; cy: unset !important; dominant-baseline: unset !important; empty-cells: unset !important; fill: unset !important; fill-opacity: unset !important; fill-rule: unset !important; filter: unset !important; flex: unset !important; flex-flow: unset !important; float: left !important; flood-color: unset !important; flood-opacity: unset !important; glyph-orientation-horizontal: unset !important; glyph-orientation-vertical: unset !important; grid: unset !important; grid-column-end: unset !important; grid-column-start: unset !important; grid-row-end: unset !important; grid-row-start: unset !important; hanging-punctuation: unset !important; hyphenate-character: unset !important; hyphens: unset !important; image-orientation: unset !important; image-rendering: unset !important; input-security: unset !important; isolation: unset !important; justify-content: unset !important; justify-items: unset !important; justify-self: unset !important; kerning: unset !important; lighting-color: unset !important; line-break: unset !important; line-height: unset !important; list-style: unset !important; margin-trim: unset !important; marker: unset !important; mask: unset !important; mask-size: unset !important; mask-type: unset !important; masonry-auto-flow: unset !important; math-style: unset !important; mix-blend-mode: unset !important; object-fit: unset !important; object-position: unset !important; offset-anchor: unset !important; offset-distance: unset !important; offset-path: unset !important; offset-position: unset !important; offset-rotate: unset !important; order: unset !important; orphans: unset !important; outline: unset !important; outline-offset: unset !important; overflow-anchor: unset !important; overflow-wrap: unset !important; overflow: unset !important; page: unset !important; paint-order: unset !important; perspective: unset !important; perspective-origin: unset !important; pointer-events: unset !important; position: unset !important; print-color-adjust: unset !important; quotes: unset !important; r: unset !important; resize: unset !important; rotate: unset !important; row-gap: unset !important; rx: unset !important; ry: unset !important; scale: unset !important; scroll-behavior: unset !important; scroll-snap-align: unset !important; scroll-snap-stop: unset !important; scroll-snap-type: unset !important; scroll-timeline: unset !important; scrollbar-color: unset !important; scrollbar-gutter: unset !important; scrollbar-width: unset !important; shape-image-threshold: unset !important; shape-margin: unset !important; shape-outside: unset !important; shape-rendering: unset !important; size: unset !important; speak-as: unset !important; stop-color: unset !important; stop-opacity: unset !important; stroke: unset !important; stroke-color: unset !important; stroke-dasharray: unset !important; stroke-dashoffset: unset !important; stroke-linecap: unset !important; stroke-linejoin: unset !important; stroke-miterlimit: unset !important; stroke-opacity: unset !important; stroke-width: unset !important; tab-size: unset !important; table-layout: unset !important; text-align: unset !important; text-align-last: unset !important; text-anchor: unset !important; text-box-edge: unset !important; text-box-trim: unset !important; text-decoration-color: unset !important; text-decoration: unset !important; text-decoration-skip-ink: unset !important; text-decoration-style: unset !important; text-decoration-thickness: unset !important; text-emphasis-color: unset !important; text-emphasis-position: unset !important; text-emphasis-style: unset !important; text-group-align: unset !important; text-indent: unset !important; text-justify: unset !important; text-overflow: unset !important; text-shadow: unset !important; text-transform: unset !important; text-underline-offset: unset !important; text-underline-position: unset !important; text-wrap: unset !important; touch-action: unset !important; transform: unset !important; transform-box: unset !important; transform-origin: unset !important; transform-style: unset !important; transition: unset !important; translate: unset !important; vector-effect: unset !important; vertical-align: unset !important; view-timeline: unset !important; view-transition-name: unset !important; white-space: unset !important; widows: unset !important; will-change: unset !important; word-break: unset !important; x: unset !important; y: unset !important; z-index: unset !important; -apple-color-filter: unset !important; -apple-pay-button-style: unset !important; -apple-pay-button-type: unset !important; -internal-text-autosizing-status: unset !important; border-spacing: unset !important; -webkit-box-align: unset !important; -webkit-box-decoration-break: unset !important; -webkit-box-direction: unset !important; -webkit-box-flex: unset !important; -webkit-box-flex-group: unset !important; -webkit-box-lines: unset !important; -webkit-box-ordinal-group: unset !important; -webkit-box-orient: unset !important; -webkit-box-pack: unset !important; -webkit-box-reflect: unset !important; -webkit-column-axis: unset !important; -webkit-column-progression: unset !important; -webkit-cursor-visibility: unset !important; -webkit-font-size-delta: unset !important; -webkit-hyphenate-limit-after: unset !important; -webkit-hyphenate-limit-before: unset !important; -webkit-hyphenate-limit-lines: unset !important; -webkit-initial-letter: unset !important; -webkit-line-align: unset !important; -webkit-line-box-contain: unset !important; -webkit-line-clamp: unset !important; -webkit-line-grid: unset !important; -webkit-line-snap: unset !important; -webkit-marquee-direction: unset !important; -webkit-marquee-increment: unset !important; -webkit-marquee-repetition: unset !important; -webkit-marquee-speed: unset !important; -webkit-marquee-style: unset !important; -webkit-nbsp-mode: unset !important; -webkit-rtl-ordering: unset !important; -webkit-text-decorations-in-effect: unset !important; -webkit-text-fill-color: unset !important; -webkit-text-security: unset !important; -webkit-text-stroke-color: unset !important; -webkit-text-stroke-width: unset !important; -webkit-user-drag: unset !important; -webkit-user-modify: unset !important; -webkit-user-select: unset !important; word-spacing: unset !important; backdrop-filter: unset !important; box-shadow: unset !important; mask-composite: unset !important; mask-mode: unset !important; text-combine-upright: unset !important; -webkit-backdrop-filter: unset !important; -webkit-background-clip: unset !important; -webkit-background-origin: unset !important; -webkit-box-shadow: unset !important; -webkit-mask: unset !important; -webkit-mask-composite: unset !important; -webkit-mask-source-type: unset !important; -webkit-text-combine: unset !important; display: block !important; visibility: unset !important; opacity: unset !important; block-size: unset !important; border-block: unset !important; border-radius: unset !important; border-end-end-radius: unset !important; border-end-start-radius: unset !important; border-inline: unset !important; border-start-end-radius: unset !important; border-start-start-radius: unset !important; inset: unset !important; contain-intrinsic-block-size: unset !important; contain-intrinsic-size: unset !important; contain-intrinsic-inline-size: unset !important; height: unset !important; inline-size: unset !important; inset-block: unset !important; inset-inline: unset !important; margin-block: unset !important; margin-inline: unset !important; max-block-size: unset !important; max-height: unset !important; max-inline-size: unset !important; min-block-size: unset !important; min-height: unset !important; min-inline-size: unset !important; min-width: unset !important; overscroll-behavior-block: unset !important; overscroll-behavior-inline: unset !important; overscroll-behavior: unset !important; padding-block: unset !important; padding: unset !important; padding-inline: unset !important; scroll-margin-block: unset !important; scroll-margin: unset !important; scroll-margin-inline: unset !important; scroll-padding-block: unset !important; scroll-padding: unset !important; scroll-padding-inline: unset !important; width: unset !important; margin: 0px 0px 1px !important; max-width: 600px !important;"><div id="pfptBannerbeaibqa" style="color-scheme: unset !important; writing-mode: unset !important; -webkit-ruby-position: unset !important; font-family: Arial, sans-serif !important; font-feature-settings: unset !important; font-kerning: unset !important; font-optical-sizing: unset !important; font-palette: unset !important; font-size: 14px !important; font-size-adjust: unset !important; font-stretch: unset !important; font-style: unset !important; font-synthesis-small-caps: unset !important; font-synthesis-style: unset !important; font-synthesis-weight: unset !important; font-variant-alternates: unset !important; font-variant-caps: unset !important; font-variant-east-asian: unset !important; font-variant-ligatures: unset !important; font-variant-numeric: unset !important; font-variant-position: unset !important; font-variation-settings: unset !important; font-weight: bold !important; text-orientation: unset !important; text-rendering: unset !important; zoom: unset !important; -webkit-font-smoothing: unset !important; -webkit-locale: unset !important; -webkit-text-zoom: unset !important; letter-spacing: unset !important; accent-color: unset !important; align-content: unset !important; align-items: unset !important; align-self: unset !important; alignment-baseline: unset !important; alt: unset !important; animation-composition: unset !important; animation: unset !important; appearance: unset !important; aspect-ratio: unset !important; backface-visibility: unset !important; background-attachment: unset !important; background-blend-mode: unset !important; background-image: unset !important; background-position: unset !important; background-repeat: unset !important; background-size: unset !important; baseline-shift: unset !important; border-collapse: unset !important; border: unset !important; box-sizing: unset !important; break-after: unset !important; break-before: unset !important; break-inside: unset !important; buffered-rendering: unset !important; caption-side: unset !important; caret-color: unset !important; clear: unset !important; clip: unset !important; clip-path: unset !important; clip-rule: unset !important; color-interpolation: unset !important; color-interpolation-filters: unset !important; column-count: unset !important; column-fill: unset !important; column-gap: unset !important; column-rule-color: unset !important; column-rule-style: unset !important; column-rule-width: unset !important; column-span: unset !important; column-width: unset !important; contain: unset !important; container-name: unset !important; container-type: unset !important; content: unset !important; counter-increment: unset !important; counter-reset: unset !important; counter-set: unset !important; cursor: unset !important; cx: unset !important; cy: unset !important; dominant-baseline: unset !important; empty-cells: unset !important; fill: unset !important; fill-opacity: unset !important; fill-rule: unset !important; filter: unset !important; flex: unset !important; flex-flow: unset !important; float: unset !important; flood-color: unset !important; flood-opacity: unset !important; glyph-orientation-horizontal: unset !important; glyph-orientation-vertical: unset !important; grid: unset !important; grid-column-end: unset !important; grid-column-start: unset !important; grid-row-end: unset !important; grid-row-start: unset !important; hanging-punctuation: unset !important; hyphenate-character: unset !important; hyphens: unset !important; image-orientation: unset !important; image-rendering: unset !important; isolation: unset !important; justify-content: unset !important; justify-items: unset !important; justify-self: unset !important; kerning: unset !important; lighting-color: unset !important; line-break: unset !important; line-height: 18px !important; list-style: unset !important; margin-trim: unset !important; marker: unset !important; mask: unset !important; mask-type: unset !important; math-style: unset !important; mix-blend-mode: unset !important; object-fit: unset !important; object-position: unset !important; offset-anchor: unset !important; offset-distance: unset !important; offset-path: unset !important; offset-position: unset !important; offset-rotate: unset !important; order: unset !important; orphans: unset !important; outline: unset !important; outline-offset: unset !important; overflow-wrap: unset !important; overflow: unset !important; page: unset !important; paint-order: unset !important; perspective: unset !important; perspective-origin: unset !important; pointer-events: unset !important; position: unset !important; print-color-adjust: unset !important; quotes: unset !important; r: unset !important; resize: unset !important; rotate: unset !important; row-gap: unset !important; rx: unset !important; ry: unset !important; scale: unset !important; scroll-behavior: unset !important; scroll-snap-align: unset !important; scroll-snap-stop: unset !important; scroll-snap-type: unset !important; shape-image-threshold: unset !important; shape-margin: unset !important; shape-outside: unset !important; shape-rendering: unset !important; size: unset !important; speak-as: unset !important; stop-color: unset !important; stop-opacity: unset !important; stroke: unset !important; stroke-color: unset !important; stroke-dasharray: unset !important; stroke-dashoffset: unset !important; stroke-linecap: unset !important; stroke-linejoin: unset !important; stroke-miterlimit: unset !important; stroke-opacity: unset !important; stroke-width: unset !important; tab-size: unset !important; table-layout: unset !important; text-align: unset !important; text-align-last: unset !important; text-anchor: unset !important; text-decoration-color: unset !important; text-decoration: unset !important; text-decoration-skip-ink: unset !important; text-decoration-style: unset !important; text-decoration-thickness: unset !important; text-emphasis-color: unset !important; text-emphasis-position: unset !important; text-emphasis-style: unset !important; text-indent: unset !important; text-overflow: unset !important; text-shadow: unset !important; text-transform: unset !important; text-underline-offset: unset !important; text-underline-position: unset !important; text-wrap: unset !important; touch-action: unset !important; transform: unset !important; transform-box: unset !important; transform-origin: unset !important; transform-style: unset !important; transition: unset !important; translate: unset !important; vector-effect: unset !important; vertical-align: unset !important; white-space: unset !important; widows: unset !important; will-change: unset !important; word-break: unset !important; x: unset !important; y: unset !important; z-index: unset !important; -apple-color-filter: unset !important; -apple-pay-button-style: unset !important; -apple-pay-button-type: unset !important; border-spacing: unset !important; -webkit-box-align: unset !important; -webkit-box-decoration-break: unset !important; -webkit-box-direction: unset !important; -webkit-box-flex: unset !important; -webkit-box-flex-group: unset !important; -webkit-box-lines: unset !important; -webkit-box-ordinal-group: unset !important; -webkit-box-orient: unset !important; -webkit-box-pack: unset !important; -webkit-box-reflect: unset !important; -webkit-column-axis: unset !important; -webkit-column-progression: unset !important; -webkit-cursor-visibility: unset !important; -webkit-hyphenate-limit-after: unset !important; -webkit-hyphenate-limit-before: unset !important; -webkit-hyphenate-limit-lines: unset !important; -webkit-initial-letter: unset !important; -webkit-line-align: unset !important; -webkit-line-box-contain: unset !important; -webkit-line-clamp: unset !important; -webkit-line-grid: unset !important; -webkit-line-snap: unset !important; -webkit-nbsp-mode: unset !important; -webkit-rtl-ordering: unset !important; -webkit-text-decorations-in-effect: unset !important; -webkit-text-fill-color: unset !important; -webkit-text-security: unset !important; -webkit-text-stroke-color: unset !important; -webkit-text-stroke-width: unset !important; -webkit-user-drag: unset !important; -webkit-user-modify: unset !important; -webkit-user-select: unset !important; word-spacing: unset !important; background-clip: unset !important; background-origin: unset !important; box-shadow: unset !important; mask-composite: unset !important; mask-mode: unset !important; text-combine-upright: unset !important; -webkit-backdrop-filter: unset !important; -webkit-background-clip: unset !important; -webkit-background-origin: unset !important; -webkit-box-shadow: unset !important; mask-size: unset !important; -webkit-mask: unset !important; -webkit-mask-composite: unset !important; -webkit-mask-source-type: unset !important; -webkit-text-combine: unset !important; display: block !important; visibility: visible !important; opacity: unset !important; background-color: rgb(208, 216, 220) !important; block-size: unset !important; border-block: unset !important; border-radius: unset !important; border-end-end-radius: unset !important; border-end-start-radius: unset !important; border-inline: unset !important; border-start-end-radius: unset !important; border-start-start-radius: unset !important; inset: unset !important; contain-intrinsic-block-size: unset !important; contain-intrinsic-size: unset !important; contain-intrinsic-inline-size: unset !important; height: unset !important; inline-size: unset !important; inset-block: unset !important; inset-inline: unset !important; margin-block: unset !important; margin: unset !important; margin-inline: unset !important; max-block-size: unset !important; max-height: unset !important; max-inline-size: unset !important; max-width: unset !important; min-block-size: unset !important; min-height: unset !important; min-inline-size: unset !important; min-width: unset !important; overscroll-behavior-block: unset !important; overscroll-behavior-inline: unset !important; overscroll-behavior: unset !important; padding-block: unset !important; padding: unset !important; padding-inline: unset !important; scroll-margin-block: unset !important; scroll-margin: unset !important; scroll-margin-inline: unset !important; scroll-padding-block: unset !important; scroll-padding: unset !important; scroll-padding-inline: unset !important; width: unset !important;">This Message Is From an External Sender</div><div id="pfptBannerbeaibqa" style="color-scheme: unset !important; writing-mode: unset !important; -webkit-ruby-position: unset !important; font-family: Arial, sans-serif !important; font-feature-settings: unset !important; font-kerning: unset !important; font-optical-sizing: unset !important; font-palette: unset !important; font-size: 12px !important; font-size-adjust: unset !important; font-stretch: unset !important; font-style: unset !important; font-synthesis-small-caps: unset !important; font-synthesis-style: unset !important; font-synthesis-weight: unset !important; font-variant-alternates: unset !important; font-variant-caps: unset !important; font-variant-east-asian: unset !important; font-variant-ligatures: unset !important; font-variant-numeric: unset !important; font-variant-position: unset !important; font-variation-settings: unset !important; font-weight: unset !important; text-orientation: unset !important; text-rendering: unset !important; zoom: unset !important; -webkit-font-smoothing: unset !important; -webkit-locale: unset !important; -webkit-text-zoom: unset !important; letter-spacing: unset !important; accent-color: unset !important; align-content: unset !important; align-items: unset !important; align-self: unset !important; alignment-baseline: unset !important; alt: unset !important; animation-composition: unset !important; animation: unset !important; appearance: unset !important; aspect-ratio: unset !important; backface-visibility: unset !important; background-attachment: unset !important; background-blend-mode: unset !important; background-image: unset !important; background-position: unset !important; background-repeat: unset !important; background-size: unset !important; baseline-shift: unset !important; border-collapse: unset !important; border: unset !important; box-sizing: unset !important; break-after: unset !important; break-before: unset !important; break-inside: unset !important; buffered-rendering: unset !important; caption-side: unset !important; caret-color: unset !important; clear: unset !important; clip: unset !important; clip-path: unset !important; clip-rule: unset !important; color-interpolation: unset !important; color-interpolation-filters: unset !important; column-count: unset !important; column-fill: unset !important; column-gap: unset !important; column-rule-color: unset !important; column-rule-style: unset !important; column-rule-width: unset !important; column-span: unset !important; column-width: unset !important; contain: unset !important; container-name: unset !important; container-type: unset !important; content: unset !important; counter-increment: unset !important; counter-reset: unset !important; counter-set: unset !important; cursor: unset !important; cx: unset !important; cy: unset !important; dominant-baseline: unset !important; empty-cells: unset !important; fill: unset !important; fill-opacity: unset !important; fill-rule: unset !important; filter: unset !important; flex: unset !important; flex-flow: unset !important; float: unset !important; flood-color: unset !important; flood-opacity: unset !important; glyph-orientation-horizontal: unset !important; glyph-orientation-vertical: unset !important; grid: unset !important; grid-column-end: unset !important; grid-column-start: unset !important; grid-row-end: unset !important; grid-row-start: unset !important; hanging-punctuation: unset !important; hyphenate-character: unset !important; hyphens: unset !important; image-orientation: unset !important; image-rendering: unset !important; isolation: unset !important; justify-content: unset !important; justify-items: unset !important; justify-self: unset !important; kerning: unset !important; lighting-color: unset !important; line-break: unset !important; line-height: 18px !important; list-style: unset !important; margin-trim: unset !important; marker: unset !important; mask: unset !important; mask-type: unset !important; math-style: unset !important; mix-blend-mode: unset !important; object-fit: unset !important; object-position: unset !important; offset-anchor: unset !important; offset-distance: unset !important; offset-path: unset !important; offset-position: unset !important; offset-rotate: unset !important; order: unset !important; orphans: unset !important; outline: unset !important; outline-offset: unset !important; overflow-wrap: unset !important; overflow: unset !important; page: unset !important; paint-order: unset !important; perspective: unset !important; perspective-origin: unset !important; pointer-events: unset !important; position: unset !important; print-color-adjust: unset !important; quotes: unset !important; r: unset !important; resize: unset !important; rotate: unset !important; row-gap: unset !important; rx: unset !important; ry: unset !important; scale: unset !important; scroll-behavior: unset !important; scroll-snap-align: unset !important; scroll-snap-stop: unset !important; scroll-snap-type: unset !important; shape-image-threshold: unset !important; shape-margin: unset !important; shape-outside: unset !important; shape-rendering: unset !important; size: unset !important; speak-as: unset !important; stop-color: unset !important; stop-opacity: unset !important; stroke: unset !important; stroke-color: unset !important; stroke-dasharray: unset !important; stroke-dashoffset: unset !important; stroke-linecap: unset !important; stroke-linejoin: unset !important; stroke-miterlimit: unset !important; stroke-opacity: unset !important; stroke-width: unset !important; tab-size: unset !important; table-layout: unset !important; text-align: unset !important; text-align-last: unset !important; text-anchor: unset !important; text-decoration-color: unset !important; text-decoration: unset !important; text-decoration-skip-ink: unset !important; text-decoration-style: unset !important; text-decoration-thickness: unset !important; text-emphasis-color: unset !important; text-emphasis-position: unset !important; text-emphasis-style: unset !important; text-indent: unset !important; text-overflow: unset !important; text-shadow: unset !important; text-transform: unset !important; text-underline-offset: unset !important; text-underline-position: unset !important; text-wrap: unset !important; touch-action: unset !important; transform: unset !important; transform-box: unset !important; transform-origin: unset !important; transform-style: unset !important; transition: unset !important; translate: unset !important; vector-effect: unset !important; vertical-align: unset !important; white-space: unset !important; widows: unset !important; will-change: unset !important; word-break: unset !important; x: unset !important; y: unset !important; z-index: unset !important; -apple-color-filter: unset !important; -apple-pay-button-style: unset !important; -apple-pay-button-type: unset !important; border-spacing: unset !important; -webkit-box-align: unset !important; -webkit-box-decoration-break: unset !important; -webkit-box-direction: unset !important; -webkit-box-flex: unset !important; -webkit-box-flex-group: unset !important; -webkit-box-lines: unset !important; -webkit-box-ordinal-group: unset !important; -webkit-box-orient: unset !important; -webkit-box-pack: unset !important; -webkit-box-reflect: unset !important; -webkit-column-axis: unset !important; -webkit-column-progression: unset !important; -webkit-cursor-visibility: unset !important; -webkit-hyphenate-limit-after: unset !important; -webkit-hyphenate-limit-before: unset !important; -webkit-hyphenate-limit-lines: unset !important; -webkit-initial-letter: unset !important; -webkit-line-align: unset !important; -webkit-line-box-contain: unset !important; -webkit-line-clamp: unset !important; -webkit-line-grid: unset !important; -webkit-line-snap: unset !important; -webkit-nbsp-mode: unset !important; -webkit-rtl-ordering: unset !important; -webkit-text-decorations-in-effect: unset !important; -webkit-text-fill-color: unset !important; -webkit-text-security: unset !important; -webkit-text-stroke-color: unset !important; -webkit-text-stroke-width: unset !important; -webkit-user-drag: unset !important; -webkit-user-modify: unset !important; -webkit-user-select: unset !important; word-spacing: unset !important; background-clip: unset !important; background-origin: unset !important; box-shadow: unset !important; mask-composite: unset !important; mask-mode: unset !important; text-combine-upright: unset !important; -webkit-backdrop-filter: unset !important; -webkit-background-clip: unset !important; -webkit-background-origin: unset !important; -webkit-box-shadow: unset !important; mask-size: unset !important; -webkit-mask: unset !important; -webkit-mask-composite: unset !important; -webkit-mask-source-type: unset !important; -webkit-text-combine: unset !important; display: block !important; visibility: visible !important; opacity: unset !important; background-color: rgb(208, 216, 220) !important; block-size: unset !important; border-block: unset !important; border-radius: unset !important; border-end-end-radius: unset !important; border-end-start-radius: unset !important; border-inline: unset !important; border-start-end-radius: unset !important; border-start-start-radius: unset !important; inset: unset !important; contain-intrinsic-block-size: unset !important; contain-intrinsic-size: unset !important; contain-intrinsic-inline-size: unset !important; height: unset !important; inline-size: unset !important; inset-block: unset !important; inset-inline: unset !important; margin-block: unset !important; margin-bottom: unset !important; margin-inline: unset !important; margin-left: unset !important; margin-right: unset !important; max-block-size: unset !important; max-height: unset !important; max-inline-size: unset !important; max-width: unset !important; min-block-size: unset !important; min-height: unset !important; min-inline-size: unset !important; min-width: unset !important; overscroll-behavior-block: unset !important; overscroll-behavior-inline: unset !important; overscroll-behavior: unset !important; padding-block: unset !important; padding: unset !important; padding-inline: unset !important; scroll-margin-block: unset !important; scroll-margin: unset !important; scroll-margin-inline: unset !important; scroll-padding-block: unset !important; scroll-padding: unset !important; scroll-padding-inline: unset !important; width: unset !important; margin-top: 2px !important;">This message came from outside your organization.</div></div><div style="height: 0px; clear: both !important; display: block !important; visibility: hidden !important; line-height: 0 !important; font-size: 0.01px !important;"></div></div><div class="WordSection1" style="page: WordSection1; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 18px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;"><div style="margin: 0cm; font-size: 10pt; font-family: "Helvetica Neue";">Hello all,<o:p></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: "Helvetica Neue";"><o:p> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: "Helvetica Neue";">I am writing to ask if PETSc’s KSPSolver makes use of OpenMP/multithreading, specifically when performing iterative solutions with the GMRES algorithm.<o:p></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: "Helvetica Neue";"><o:p> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: "Helvetica Neue";">The questions appeared when I was running a large numerical program based on boundary element method. I used the PETSc's GMRES algorithm in KSPSolve to solve a shell matrix system iteratively. I observed that threads were being utilized, controlled by the<span class="Apple-converted-space"> </span><b>OPENBLAS_NUM_THREADS</b><span class="Apple-converted-space"> </span>environment variable. However, I noticed no significant performance difference between running the solver with multiple threads versus a single thread.<br><br>Could you please<span class="Apple-converted-space"> </span><b>confirm if GMRES in KSPSolve leverages multithreading, and also whether it is influenced by the multithreadings of the low-level math libraries such as BLAS and LAPACK?</b><span class="Apple-converted-space"> </span><b>If so</b>, how can I enable multithreading effectively to see noticeable improvements in solution times when using GMRES?<span class="Apple-converted-space"> </span><b>If not</b>, why do I observe that threads are being used during the GMERS solutions?<o:p></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: "Helvetica Neue";"><o:p> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: "Helvetica Neue";"><b>For reference, I am using PETSc version 3.16.0, configured in CMakelists as follows:</b><o:p></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: "Helvetica Neue";"><br>./configure PETSC_ARCH=config-release --with-scalar-type=complex --with-fortran-kernels=1 --with-debugging=0 COPTFLAGS=-O3 -march=native CXXOPTFLAGS=-O3 -march=native FOPTFLAGS=-O3 -march=native --with-cxx=g++ --download-openmpi --download-superlu --download-opencascade --with-openblas-include=${OPENBLAS_INC} --with-openblas-lib=${OPENBLAS_LIB} --with-threadsafety --with-log=0 --with-openmp<br><br>To simplify the diagnosis of potential issues, I have also written a small example program using GMRES to solve a sparse matrix system derived from a 2D Poisson problem using the finite difference method. I found similar issues on this piece of codes. The code is as follows:<br><br>#include <petscksp.h><br><br>/* Monitor function to print iteration number and residual norm */<br>PetscErrorCode MyKSPMonitor(KSP ksp, PetscInt n, PetscReal rnorm, void *ctx) {<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>PetscErrorCode ierr;<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = PetscPrintf(PETSC_COMM_WORLD, "Iteration %D, Residual norm %g\n", n, (double)rnorm);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>CHKERRQ(ierr);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>return 0;<br>}<br><br>int main(int argc, char **args) {<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>Vec x, b, x_true, e;<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>Mat A;<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>KSP ksp;<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>PetscErrorCode ierr;<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>PetscInt i, j, Ii, J, n = 500; // Size of the grid n x n<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>PetscInt Istart, Iend, ncols;<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>PetscScalar v;<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>PetscMPIInt rank;<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>PetscInitialize(&argc, &args, NULL, NULL);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>PetscLogDouble t1, t2;<span class="Apple-converted-space"> </span><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>// Variables for timing<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>MPI_Comm_rank(PETSC_COMM_WORLD, &rank);<br><br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>// Create vectors and matrix<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = VecCreateMPI(PETSC_COMM_WORLD, PETSC_DECIDE, n*n, &x); CHKERRQ(ierr);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = VecDuplicate(x, &b); CHKERRQ(ierr);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = VecDuplicate(x, &x_true); CHKERRQ(ierr);<br><br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>// Set true solution as all ones<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = VecSet(x_true, 1.0); CHKERRQ(ierr);<br><br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>// Create and assemble matrix A for the 2D Laplacian using 5-point stencil<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = MatCreate(PETSC_COMM_WORLD, &A); CHKERRQ(ierr);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = MatSetSizes(A, PETSC_DECIDE, PETSC_DECIDE, n*n, n*n); CHKERRQ(ierr);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = MatSetFromOptions(A); CHKERRQ(ierr);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = MatSetUp(A); CHKERRQ(ierr);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = MatGetOwnershipRange(A, &Istart, &Iend); CHKERRQ(ierr);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>for (Ii = Istart; Ii < Iend; Ii++) {<br><span class="apple-converted-space">       <span class="Apple-converted-space"> </span></span>i = Ii / n; // Row index<br><span class="apple-converted-space">       <span class="Apple-converted-space"> </span></span>j = Ii % n; // Column index<br><span class="apple-converted-space">       <span class="Apple-converted-space"> </span></span>v = -4.0;<br><span class="apple-converted-space">       <span class="Apple-converted-space"> </span></span>ierr = MatSetValues(A, 1, &Ii, 1, &Ii, &v, INSERT_VALUES); CHKERRQ(ierr);<br><span class="apple-converted-space">       <span class="Apple-converted-space"> </span></span>if (i > 0) { // South<br><span class="apple-converted-space">           <span class="Apple-converted-space"> </span></span>J = Ii - n;<br><span class="apple-converted-space">           <span class="Apple-converted-space"> </span></span>v = 1.0;<br><span class="apple-converted-space">           <span class="Apple-converted-space"> </span></span>ierr = MatSetValues(A, 1, &Ii, 1, &J, &v, INSERT_VALUES); CHKERRQ(ierr);<br><span class="apple-converted-space">       <span class="Apple-converted-space"> </span></span>}<br><span class="apple-converted-space">       <span class="Apple-converted-space"> </span></span>if (i < n - 1) { // North<br><span class="apple-converted-space">           <span class="Apple-converted-space"> </span></span>J = Ii + n;<br><span class="apple-converted-space">           <span class="Apple-converted-space"> </span></span>v = 1.0;<br><span class="apple-converted-space">           <span class="Apple-converted-space"> </span></span>ierr = MatSetValues(A, 1, &Ii, 1, &J, &v, INSERT_VALUES); CHKERRQ(ierr);<br><span class="apple-converted-space">       <span class="Apple-converted-space"> </span></span>}<br><span class="apple-converted-space">       <span class="Apple-converted-space"> </span></span>if (j > 0) { // West<br><span class="apple-converted-space">           <span class="Apple-converted-space"> </span></span>J = Ii - 1;<br><span class="apple-converted-space">           <span class="Apple-converted-space"> </span></span>v = 1.0;<br><span class="apple-converted-space">           <span class="Apple-converted-space"> </span></span>ierr = MatSetValues(A, 1, &Ii, 1, &J, &v, INSERT_VALUES); CHKERRQ(ierr);<br><span class="apple-converted-space">       <span class="Apple-converted-space"> </span></span>}<br><span class="apple-converted-space">       <span class="Apple-converted-space"> </span></span>if (j < n - 1) { // East<br><span class="apple-converted-space">           <span class="Apple-converted-space"> </span></span>J = Ii + 1;<br><span class="apple-converted-space">           <span class="Apple-converted-space"> </span></span>v = 1.0;<br><span class="apple-converted-space">           <span class="Apple-converted-space"> </span></span>ierr = MatSetValues(A, 1, &Ii, 1, &J, &v, INSERT_VALUES); CHKERRQ(ierr);<br><span class="apple-converted-space">       <span class="Apple-converted-space"> </span></span>}<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>}<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY); CHKERRQ(ierr);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY); CHKERRQ(ierr);<br><br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>// Compute the RHS corresponding to the true solution<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = MatMult(A, x_true, b); CHKERRQ(ierr);<br><br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>// Set up and solve the linear system<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = KSPCreate(PETSC_COMM_WORLD, &ksp); CHKERRQ(ierr);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = KSPSetOperators(ksp, A, A); CHKERRQ(ierr);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = KSPSetType(ksp, KSPGMRES); CHKERRQ(ierr);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = KSPSetTolerances(ksp, 1e-5, PETSC_DEFAULT, PETSC_DEFAULT, PETSC_DEFAULT); CHKERRQ(ierr);<br><br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>/* Set up the monitor */<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = KSPMonitorSet(ksp, MyKSPMonitor, NULL, NULL); CHKERRQ(ierr);<br><br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>// Start timing<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>PetscTime(&t1);<br><br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = KSPSolve(ksp, b, x); CHKERRQ(ierr);<br><br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>// Stop timing<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>PetscTime(&t2);<br><br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>// Compute error<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = VecDuplicate(x, &e); CHKERRQ(ierr);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = VecWAXPY(e, -1.0, x_true, x); CHKERRQ(ierr);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>PetscReal norm_error, norm_true;<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = VecNorm(e, NORM_2, &norm_error); CHKERRQ(ierr);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = VecNorm(x_true, NORM_2, &norm_true); CHKERRQ(ierr);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>PetscReal relative_error = norm_error / norm_true;<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>if (rank == 0) { // Print only from the first MPI process<br><span class="apple-converted-space">       <span class="Apple-converted-space"> </span></span>PetscPrintf(PETSC_COMM_WORLD, "Relative error ||x - x_true||_2 / ||x_true||_2: %g\n", (double)relative_error);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>}<br><br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>// Output the wall time taken for MatMult<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>PetscPrintf(PETSC_COMM_WORLD, "Time taken for KSPSolve: %f seconds\n", t2 - t1);<br><br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>// Cleanup<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = VecDestroy(&x); CHKERRQ(ierr);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = VecDestroy(&b); CHKERRQ(ierr);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = VecDestroy(&x_true); CHKERRQ(ierr);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = VecDestroy(&e); CHKERRQ(ierr);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = MatDestroy(&A); CHKERRQ(ierr);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>ierr = KSPDestroy(&ksp); CHKERRQ(ierr);<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>PetscFinalize();<br><span class="apple-converted-space">   <span class="Apple-converted-space"> </span></span>return 0;<br>}<br><br>Here are some profiling results for GMERS solution.<br><br>OPENBLAS_NUM_THREADS = 1, iteration steps<span class="apple-converted-space"> <span class="Apple-converted-space"> </span></span>= 859, solution time = 16.1<br>OPENBLAS_NUM_THREADS = 2, iteration steps<span class="apple-converted-space"> <span class="Apple-converted-space"> </span></span>= 859, solution time = 16.3<o:p></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: "Helvetica Neue";">OPENBLAS_NUM_THREADS = 4, iteration steps<span class="apple-converted-space"> <span class="Apple-converted-space"> </span></span>= 859, solution time = 16.7<o:p></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: "Helvetica Neue";">OPENBLAS_NUM_THREADS = 8, iteration steps<span class="apple-converted-space"> <span class="Apple-converted-space"> </span></span>= 859, solution time = 16.8<o:p></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: "Helvetica Neue";">OPENBLAS_NUM_THREADS = 16, iteration steps<span class="apple-converted-space"> <span class="Apple-converted-space"> </span></span>= 859, solution time = 17.8<o:p></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: "Helvetica Neue";"><br><b>I am using one workstation with Intel® Core™ i9-11900K Processor, 8 cores, 16 threads. Note that I am not using multiple MPI processes, such as mpirun/mpiexec, the default number of MPI processes should be 1, correct if I am wrong.</b><o:p></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: "Helvetica Neue";"><o:p> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: "Helvetica Neue";">Thank you in advance!<br><br>Sincerely,<o:p></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: "Helvetica Neue";">Yongzhong<o:p></o:p></div><div style="margin: 0cm; font-size: 12pt; font-family: Aptos, sans-serif;"><span style="font-size: 11pt;"><o:p> </o:p></span></div><div style="margin: 0cm; font-size: 12pt; font-family: Aptos, sans-serif;"><span lang="EN-US" style="font-size: 11pt; font-family: Calibri, sans-serif;">-----------------------------------------------------------<o:p></o:p></span></div><div style="margin: 0cm; font-size: 12pt; font-family: Aptos, sans-serif;"><b><span lang="EN-US" style="font-size: 11pt; font-family: Calibri, sans-serif;">Yongzhong Li</span></b><span lang="EN-US" style="font-size: 11pt; font-family: Calibri, sans-serif;"><o:p></o:p></span></div><div style="margin: 0cm; font-size: 12pt; font-family: Aptos, sans-serif;"><span lang="EN-US" style="font-size: 11pt; font-family: Calibri, sans-serif;">PhD student | Electromagnetics Group<o:p></o:p></span></div><div style="margin: 0cm; font-size: 12pt; font-family: Aptos, sans-serif;"><span lang="EN-US" style="font-size: 11pt; font-family: Calibri, sans-serif;">Department of Electrical & Computer Engineering<o:p></o:p></span></div><div style="margin: 0cm; font-size: 12pt; font-family: Aptos, sans-serif;"><span lang="EN-US" style="font-size: 11pt; font-family: Calibri, sans-serif;">University of Toronto<o:p></o:p></span></div><div style="margin: 0cm; font-size: 12pt; font-family: Aptos, sans-serif;"><span lang="EN-US" style="font-size: 11pt; font-family: Calibri, sans-serif;"><a href="https://urldefense.us/v3/__http://www.modelics.org__;!!G_uCfscf7eWS!efVv_hPkRBEhyquXer2c8sFeGrjOtTjEGicYg2niCyfT9swzjLFyf6k4XrhKElaF-cX_Q02y9KSTRNFHPlKhXMtuzaTekCWcXgw$"><span style="color: rgb(5, 99, 193);">http://www.modelics.org</span></a></span></div></div></div></blockquote></div><br></div></body></html>